The Theory of Ontic Modalities 9783110326895, 9783110326574

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Uwe Meixner The Theory of Ontic Modalities

Philosophische Analyse Philosophical Analysis Herausgegeben von / Edited by Herbert Hochberg • Rafael Hüntelmann • Christian Kanzian Richard Schantz • Erwin Tegtmeier Band 13 / Volume 13

Uwe Meixner

The Theory of Ontic Modalities

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Preface The basic idea of this book has been around for some time, David Hume being, arguably, one of its first explicit proponents (Ludwig Wittgenstein, in the Tractatus, following in his footsteps). The idea is this: (so-called) logical (conceptual, analytic) necessity is the only alethic necessity. But, contrary to Hume, this is not to say that statements of nomological, metaphysical, or other non-logical alethic necessity – for example, the everyday necessity that is exhibited in the statement “it is impossible to open this door” – have no objective meaning, or if they had an objective meaning would never be true. I take it that such statements of necessity do not only purport to have an objective meaning but have it, and are often true in that very meaning. It is just that the non-logical alethic necessities are reducible in a certain sense (which I will expound and justify) reducible to logical necessity. This necessity I do not take to be a verbal or conventional, or epistemic matter. I do not treat logical necessity in the manner that is suggested by the logical empiricists’ treatment of analytic truth. Logical necessity, like all other alethic necessities, I take to be an ontic necessity, not an epistemic one.1 My designating logical necessity with the help of the word “logical,” which does suggest a close connection of logical necessity to words and opinions, is merely a concession to a widely accepted way of speaking about this kind of necessity. People intend a certain ontic necessity, and they (still) quite frequently call it “logical necessity.” Often the “logical” in “logical necessity” is modified by a “broadly.” This is indeed a helpful modification, because it removes the misunderstanding that logical necessity is related to some specific logical system (for example, the calculus of elementary predicate logic). A better, but less frequently used designation for this ontic necessity – a designation which I shall also use – is “conceptual necessity.” This designation, however, still suggests a connection to words and conceptions, with which concepts are, after all, usually thought to be very closely bound up. Moreover, also entities that are not designable as concepts seem to help in producing logical necessities. (That I am identical with something is a 1

There is bound to be some controversy about this, and it will be addressed in the body of the book.

Preface

logical necessity whose being a logical necessity seems to have much to do with myself; but I am certainly not a concept.) The best way, as far as descriptive adequacy is concerned, to designate the ontic necessity which is often called “logical necessity” is to call it “intrinsic” or “inner necessity.” Because it is unfamiliar, I will not usually employ this designation. But by using it here the basic idea of this book can be expressed in the following way: all outer ontic necessity is reducible to inner necessity. In books on modality (that is, on necessity, possibility, etc.), modeltheoretic semantics for modal logics often take up considerable space. The method of treating modality which is applied in this book is entirely different. In this book, a theory for all ontic modalities, non-relational or relational, is constructed in the manner that is also used in constructing, most notably, set theory (but, note, the constructed theory is not an extension of set theory): by stating ontological laws, within the framework of first-order predicate logic and without providing model-theoretic semantics for the constructed theory (which, accordingly, will be frequently characterized by the adjective “onto-nomological”). Analyses – namely, explicit definitions – of modal operators are provided directly within the theory itself, not within a meta-linguistic meta-theory that is built on top of it. On the basis of these definitions, the logics for the defined modal operators prove to be logically derivable within the theory, and this is all the justification these logics will get, and all they require. The intended theory, which is expounded – and confronted with the views of other philosophers – in the Chapters 3 and 4, and 7 and 8 of this book (while Chapters 5 and 6 are dedicated to an important application of the theory, and Chapters 1 and 2 to the exposition of the ontological and epistemological problem of ontic modalities),2 goes some way to being a general theory of coarse-grained (i.e., coarsely individuated) intensional entities. It is an ontological theory of states of affairs and properties (and hence also of individuals), capable of encompassing set theory as a special case. Although non-actual possible worlds play no role at all in my basic 2

Concerning the overall structure of the book: the odd-numbered chapters focus on the presentation of my own thoughts, whereas each even-numbered chapter focuses on the critical discussion of the thoughts of other philosophers regarding the issues raised by the immediately preceding odd-numbered chapter. I found this separation of presentation and discussion necessary in order to preserve at least some perspicuity in my handling of a large amount of complex material. I hope that it will also prove helpful to the reader.

2

Preface

analyses of modal operators, as a believer in this intensionalistic theory of states of affairs and properties, I suppose I must be called a modal realist, since believing in (a non-trivial form of) the theory requires believing in non-actual possible states of affairs and properties that are not abstract constructions. I also believe that there are non-actual but possible concrete individuals (conceived of in a way completely different from David Lewis’s), though this is not an integral part of the theory. I hope to have provided good arguments for all these non-actualist realist positions. In fact, the only real problem I take to be connected with modal realism is the purely epistemological problem of how non-actual entities can be objects of human cognition. Though largely about ontological matters, this book also addresses the epistemological issues connected with ontic modalities. I believe it addresses them with more attention than most other books on the subject of modality. I also believe that the epistemological problem of ontic modality – part of which Hume so forcefully brought home to us – is, to the extent that it is solvable at all, solved by the theory of modality that is developed in this book. For the sake of completeness, I would have liked to include a ninth and tenth chapter on modal properties: on potentialities and dispositions. But my results in this regard must be reserved for a separate publication. This book is already long enough as it stands. Though not complete in its exposition of my views on modality, the reader will find the central issues of the ontology and epistemology of modality treated in it. Work on this book was funded during an 18-months-period by the German Research Foundation (DFG), for which I am grateful. Last but not least, I would like to thank Hans Rott, Professor of Theoretical Philosophy at the University of Regensburg, for his crucial support in all matters that made the writing of this book financially possible – and for his friendly encouragement along the way. Uwe Meixner

3

To every possible person who is necessary if actual

Contents 1

The Problem of Ontic Modalities .......................................... 9

1.1 1.2 1.3 1.4 1.5

Three Problems, Two Requirements ........................................................ 9 The Seriousness of the Epistemological Problem ...................................11 An Argument against Knowledge of a Mere Possibility.........................14 Existence and Non-Existence ..................................................................16 Actualisms ...............................................................................................23

2

Actualism, Modal Realism, and Modal Epistemology....... 27

2.1 2.2 2.3 2.4

Alvin Plantinga on Existence, Actuality, and Actualism ........................27 What Is Modal Realism, and What, Really, Is It Good For? ..................35 Conceivability and (Ontic) Possibility ....................................................40 On Peter van Inwagen’s Modal Skepticism ............................................52

3

An Onto-Nomological Theory of Modality......................... 63

3.1 3.2 3.3

What Statements of Possibility and Necessity Mean ..............................63 The Basic Mereology of States of Affairs...............................................66 The Actuality of States of Affairs in the Mereology of States of Affairs.......................................................................................74 Further Principles for “That”...................................................................79 The Number of States of Affairs .............................................................82 Bases of Necessity and Modal Principles in the Mereology of States of Affairs...................................................................................83 The Classical Modal Principles and the Bases-Theory of Necessity..................................................................................................89 Necessity and Possible Worlds................................................................94 Three Final Observations: Another “That”-Principle, the Trickiness of “w*,” and the Consistency of the Mereology of States of Affairs ................................................................96 Another Apparent Paradox ....................................................................100

3.4 3.5 3.6 3.7 3.8 3.9

3.9.1

Contents

4

States of Affairs, Modality, and the Bases-Theory of Necessity..................................... 105

4.1 4.2 4.3 4.4 4.5

States of Affairs, Propositions, and Sets of Possible Worlds ................105 Possible Worlds as States of Affairs......................................................112 Supercontingent States of Affairs..........................................................119 Armstrong’s States of Affairs................................................................133 Van Fraassen’s Objection against the Basic Idea of a Bases-Theory of Necessity .............................................................147 Metaphysical Necessity? Nomological Necessity? ...............................152 An Assertion of Plantinga’s: Every State of Affairs Is Actual in Itself....................................................................................156 On Chihara’s Finding of a Contradiction in Plantinga’s Ontological Theory.............................................................157

4.6 4.7 4.8

5

The Theory of Conditionals in the Onto-NomologicalTheory of Modality .............................. 159

5.1 5.2 5.2.1 5.2.2 5.3 5.3.1

Relational and Non-Relational Necessity..............................................161 Conditionals as Strict Implications with Variable Bases.......................163 Intrinsic Vagueness of Basis and Conditions for Bases ........................169 Determining a Basis...............................................................................174 The Logic of Conditionals .....................................................................177 Further Logical Difficulties for the Bases-Theory of Conditionals and Their Solutions ......................................................185 Might-Conditionals, Explanatory Conditionals, and Causation............190 The Solution to the Epistemological Problem of Ontic Modalities.......194 But There Is a Puzzle Concerning Logical Modalities..........................199

5.4 5.5 5.5.1

6

Other Theories of Conditionals versus the Bases-Theory of Conditionals ...................................... 207

6.1 6.2 6.3

Are Indicative Conditionals (Always) Material Implications?..............207 The Assertability of Indicative Conditionals.........................................210 The Metalinguistic Theory of Conditionals and the Bases-Theory of Conditionals .........................................................215 The Argument of David Lewis against the Idea of a Bases-Theory of Conditionals ........................................................221 Comparative Similarity (Closeness) of Worlds, Lewis’s and Stalnaker’s Interpretations of Counterfactual Conditionals, and the Bases-Theory of Conditionals ............................226 Is Logical Necessity Verbal Necessity? ................................................242

6.4 6.5

6.6

6

Contents

7

The Onto-Nomological Theory of Modality Extended: Adding Properties and Individuals..................249

7.1 7.2 7.3 7.4

7.5.1 7.6 7.7

Individuals, Properties, and Exemplification ........................................251 The Identity of Properties, Essential Properties, and Sets.....................254 Essences – and Parthood for Properties ................................................259 Regarding the Question What Properties There Are – and More on Essence and Essentiality .........................................................262 The (Transworld) Identity of Individuals and Logical I-Essences – and Regarding the Question What Individuals There Are ..................................................................283 Notiones Completae and L(eibniz)-Individuals ....................................290 The Actuality of Properties ...................................................................294 The Actuality of Individuals..................................................................298

8

Properties, Essences, and Actualism (Again) ....................309

8.1 8.2 8.3 8.4 8.5

David Lewis on Possible Worlds, Individuals, and I-Properties...........309 Alvin Plantinga and David Lewis on Essence and Essentiality ............329 Negative, Disjunctive, and Empty Properties .......................................341 Haecceitism? Anti-Haecceitism?...........................................................345 Graeme Forbes and Alvin Plantinga on Property Actualism and Quantifier Actualism ......................................................................348

7.5

Bibliography ...........................................................................................361 Index........................................................................................................365

7

1

The Problem of Ontic Modalities

Ontic (or alethic) modalities are the expressions “it is possible that __,” “it is necessary that __” and “if __, then __” if taken to have a non-doxastic (hence also non-epistemic) and non-deontic sense. (In order to simplify formulation, I will from now on restrict the expressions “it is possible that __,” “it is necessary that __” and “if __, then __” in this book to having an ontic sense only.) Philosophers widely appreciate that there is a philosophical problem connected with ontic modalities, and this chapter is dedicated to elucidating its nature. 1.1

Three Problems, Two Requirements

In fact, there are three philosophical problems connected with ontic modalities: a semantical problem, an epistemological problem, and an ontological one. The three problems are closely connected, as will become apparent immediately. But let me begin by stating the three problems: The Semantical Problem What are the (necessary and sufficient) truth conditions of sentences of the forms “It is possible that A,” “It is necessary that A,” and “If A, then B”? The answer to this question is not clear. The Epistemological Problem How can we know, in principle, that a true / false sentence of the form “It is possible that A,” or the form “It is necessary that A,” or the form “If A, then B” is true / is false? The answer to this question is not clear. The Ontological Problem Which items must we assume in our ontology for making true sentences of the forms “It is possible that A,” “It is necessary that A,” and “If A, then B” true, and false sentences of these forms false? The answer to this question is not clear.

1 The Problem of Ontic Modalities

Obviously, any solution of the Epistemological and the Ontological Problem of Ontic Modalities presupposes a solution of the Semantical Problem, and every solution of the Semantical Problem of Ontic Modalities should make a solution of the Epistemological Problem and the Ontological Problem possible: First Requirement for a Theory of Ontic Modalities Every solution of the Semantical Problem of Ontic Modalities must enable (though it need not uniquely determine) a solution of the Epistemological and the Ontological Problem of Ontic Modalities. In addition it is required that every solution to each one of the three Problems of Ontic Modalities conform to the following: Second Requirement for a Theory of Ontic Modalities Every solution of the Semantical Problem, or the Epistemological Problem, or the Ontological Problem of Ontic Modalities must be compatible with the truth and the falsity of those sentences of the forms “it is possible that A,” “it is necessary that A,” and “if A, then B” that we uncontroversially know to be true, respectively false. I will from now on call sentences that have the forms just mentioned “modal sentences.” The Second Requirement would be empty if there were no modal sentences the truth or falsity of which is uncontroversially known. But, in fact, there are such sentences, many of them. This might lead one to the opinion that the Epistemological Problem of Ontic Modalities is not really serious (since we know so many modal sentences to be true, respectively false). But this would be a misconception. The Epistemological Problem is central because the fact of modal knowledge is the presupposition of the Second Requirement for a Theory of Ontic Modalities. This fact must therefore be taken into account – not only as a whole, but also in detail – by all attempted solutions to the three Problems of Ontic Modalities. Given the fact of modal knowledge, the Epistemological Problem poses the question how modal knowledge is possible. And an answer to that question is not at all obvious. Let me explain.

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1.2

The Seriousness of the Epistemological Problem

Because “A” logically implies “it is possible that A,” because “it is not the case that A” logically implies “it is not necessary that A,” and because “A, and it is not the case that B” logically implies “ it is not the case that if A, then B” (where arbitrary – but true or false – sentences can be substituted for the capital letters in the expressions in quotes), there are – already for this reason – many modal sentences that are known to be true, respectively false. This is the easy part of the Epistemological Problem of Ontic Modalities: Verifying a sentence of the form “it is possible that A” is, in many cases, just a matter of verifying the sentence “A.” If that sentence is devoid of modal expressions, then there is no new epistemological problem at all here. In turn, falsifying a sentence of the form “it is necessary that A” or of the form “if A, then B” is, in many cases, just a matter of falsifying the sentence “A” or the sentence “A ⊃ B” (i.e., the corresponding material implication). Again, if these latter sentences are devoid of modal expressions, then there is no new epistemological problem at all. The hard part of the Epistemological Problem of Ontic Modalities is constituted by the following questions: How could we verify “it is possible that A,” given that we have falsified “A”? How could we falsify “it is necessary that A,” given that we have verified “A”? How could we falsify “if A, then B,” given that we have verified “A ⊃ B”? Only if these three questions have epistemologically satisfactory answers, a solution to the Semantical Problem of Ontic Modalities is warranted which does not trivialize ontic modalities, that is, which does not logically equate “it is possible that A” and “it is necessary that A” with “A,” and “if A, then B” with “A ⊃ B.” It is not at all obvious that the above three questions have epistemologically satisfactory answers. The seriousness of the issue can be brought out by dramatizing it a bit. Consider the following (entirely fictitious) story. The philosopher Ernst Antipopper (“Sir Ernst”) claims that every general statement can be matched by a similar general statement

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that is unfalsifiable.1 The logical form of a general statement is ∀x(F[x] ⊃ G[x]), and the matching general statement envisaged by Antipopper has the logical form ∀x(F[x] ∧ ◊G[x] ⊃ G[x]). Antipopper calls a statement of the latter form a “secure” general statement. A secure general statement is generated from an arbitrary general statement by inserting into it what Antipopper calls “a modal lightning rod.” Clearly, every general statement has exactly one secure general statement corresponding to it, and different general statements have different secure general statements corresponding to them. Antipopper then suggests that if a theory is threatened by the falsification of one of its general statements, it should shift into “the secure mode,” that is, replace the problematic general statement by the secure general statement corresponding to it. For although the verification of F[a] ∧ ¬G[a] would falsify ∀x(F[x] ⊃ G[x]), the verification of F[a] ∧ ¬G[a] would, of course, not yet falsify ∀x(F[x] ∧ ◊G[x] ⊃ G[x]). For falsifying the latter, one would have to verify in addition ◊G[a], and this verification, thus Antipopper, can, in the presence of ¬G[a], be simply blocked by claiming that ¬G[a] is true. To the objection that this is an ad hoc procedure, Antipopper replies that there can be nothing ad hoc about a procedure which is always applicable. To the objection that ◊G[a] is “obviously” true and ¬G[a] “obviously” false in all interesting cases, Antipopper replies with the challenge to make this alleged obviousness obvious to someone for whom the points in question are not obvious at all. Antipopper concedes that ◊G[a] might indeed be obviously true in all interesting cases if the possibility in question were logical possibility (although even under this interpretation of “it is possible that” he has doubts about it2). But who says that the possibility in question has to be logical possibility? Antipopper recommends choosing a possibility that is different from logical possibility when securing a theory by installing modal lightning rods in its general statements. This story highlights one central epistemological difficulty regarding ontic modalities: the verification of ◊A in the face of having falsified A, or in other words, the falsification of ¬A in the face of having verified ¬A. One may call this difficulty “the Difficulty of Ascertaining Mere Possibility.” Another central epistemological difficulty regarding ontic 1

See The Logic of Scientific Non-Discovery, pp. 213-220. “Consider: it was once unanimously thought a logical possibility that a person may exist without a body; most philosophers nowadays tend to deny that there is such a logical possibility.” (The Logic of Scientific Non-Discovery, p. 219.) 2

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modalities – let us stick for the time being, in order to simplify discussion, to monadic ontic modalities – is the falsification of A in the face of having verified A, or in other words, the verification of ◊¬A in the face of having falsified ¬A. It is clear that the two difficulties described are really one and the same difficulty, and that they can be both described as “the difficulty of ascertaining mere possibility.” Besides the Difficulty of Ascertaining Mere Possibility there are two further epistemological difficulties regarding modalities: the difficulty of verifying A, and the difficulty of falsifying ◊A. Again, the two difficulties are really one and the same. For the difficulty of verifying A is – expressed in other words – the difficulty of falsifying ◊¬A, and the difficulty of falsifying ◊A is – expressed in other words – the difficulty of verifying ¬A. The two difficulties, which really are one and the same, can both be subsumed under the heading “the Difficulty of Ascertaining Necessity.” For appreciating this difficulty, note, first, that A cannot be verified without verifying A. But even if A is verified, there usually remains a large epistemic gap between the truth of A and the truth of A; this gap has been pointed out – in the context of the discussion of causation and of laws of nature – by empiricist thinkers (from David Hume to Bas van Fraassen); they have also asserted that the gap cannot be bridged. Whether this second claim is true, remains to be seen. The two epistemological difficulties that have now been described – the Difficulty of Ascertaining Mere Possibility and the Difficulty of Ascertaining Necessity – form the utterly recalcitrant parts of the Epistemological Problem of Ontic Modalities. The source of these difficulties is likely to be found in the meaning that is assigned to the ontic modalities, and one might hope to make the Epistemological Problem of Ontic Modalities more tractable by a solution to the Semantical Problem that fits – is just right for – the purpose of obtaining a solution to the Epistemological Problem of Ontic Modalities. But there might not be enough space for such a conceptual maneuver, since the ordinary meanings of “it is possible that” and “it is necessary that,” though not sacrosanct, have to be by and large respected. In any case, the two epistemological difficulties seem to be able to assume more or less vexing forms. Let us now look at an especially vexing form – due to a semantical indecision – of the Difficulty of Ascertaining Mere Possibility.

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1.3

An Argument against Knowledge of a Mere Possibility

The Difficulty of Ascertaining Mere Possibility arises when we know that A is not the case, but we still want to assert that is possible that A (or when we know that A is the case, but we still want to assert that it is not necessary that A). How can we know that it is possible that A (alternatively: possible that ¬A), even though we know that it is not the case that A (alternatively: not the case that ¬A)? Our prospects of finding a philosophically defensible way of knowing this are dark indeed if the following argument is correct: If a state of affairs is not the case, then it is non-existent. But there is no knowledge of the non-existent. Hence, for any state of affairs that is not the case, we do not know that it is, nevertheless, possible. Corollary: If it is not the case that A, then we do not know that it is possible that A. (In other words: we know that it is possible that A only if A.) Set out in formal terms, the argument looks surprisingly complex: 1. 2. 3. 4.

∀x(S(x) ∧ ¬O(x) ⊃ ¬E(x)) ∀x(¬E(x) ⊃ ¬∃y(S(y) ∧ A(y, x) ∧ ∃zK(z, y))) ∀x(S(x) ∧ ¬O(x) ⊃ ¬K(b, that ◊(x))) ¬O(that A) ⊃ ¬K(b, that ◊(that A))

Premise 1 Premise 2 (1., 2.) (3.)

The following formalizations are employed: S(x) : x is a state of affairs O(x) : x is the case E(x) : x exists A(y, x) : y is a state of affairs about x K(z, y) : z knows y ◊(x) : x is possible. “b” is designating one of us, one arbitrarily chosen. Besides the usual logical constants, there is an operator, “that,” which forms singular terms out of sentences (or functional expressions out of open sentences). Note that the sentence-connectives of knowledge and

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possibility can be defined with the help of “that” on the basis of the predicates of knowledge and possibility: KzA =Def K(z, that A) ◊A =Def ◊(that A). The third line of the argument is obtained from the two lines preceding it in the following manner: Assume S(x) ∧ ¬O(x). Hence by 1.: ¬E(x). Hence by 2.: ¬∃y(S(y) ∧ A(y, x) ∧ ∃zK(z, y)). But: S(that ◊(x)) ∧ A(that ◊(x), x). Hence: ¬∃zK(z, that ◊(x)). Hence: ¬K(b, that ◊(x)).

The corollary in the fourth line of the argument is obtained from the argument’s third line in the following manner: Assume ¬O(that A). But: S(that A). Hence by 3.: ¬K(b, that ◊(that A)).

Given the above definitions of the sentence-connectives of knowledge and possibility, and considering that “O(that A)” is logically equivalent to “A,” the corollary can also be expressed in the following way: ¬A ⊃ ¬Kb◊A, or: Kb◊A ⊃ A. A second corollary is this: ¬Kb(¬A ∧ ◊A). For suppose Kb(¬A ∧ ◊A). Hence by elementary epistemic logic: Kb¬A ∧ Kb◊A. Hence, again by elementary epistemic logic: Kb◊A ∧ ¬A – contradicting Kb◊A ⊃ A. And a third corollary is of course: Kb¬A ⊃ ¬Kb◊A, which is logically equivalent to the second corollary. Now, “knowledge of a mere possibility” can mean either one of four things: (1) knowledge that a state of affairs is possible, while that state of affair is not the case [¬A ∧ Kb◊A]; (2) knowledge of the fact that a state of affairs is not the case but possible [Kb(¬A ∧ ◊A)]; (3) knowledge of the fact that a state of affairs that is known to be not the case is possible [Kb(Kb¬A ∧ ◊A)]; (4) knowledge that a state of affairs is not the case, accompanied by the knowledge that it is possible [Kb¬A ∧ Kb◊A].

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In all four senses (the senses (2), (3) and (4) are logically equivalent according to standard epistemic logic) the above argument, if correct, would prove that there is no knowledge (by anyone of us) of a mere possibility. We have already seen that the logic of the above argument is impeccable. If we want to escape from it, we must criticize its premises. The decisive criticism, I believe, is this: Although there is an adequate interpretation of E(x) (“x exists”) in which Premise 1 is clearly true, and also an adequate interpretation of E(x) in which Premise 2 is clearly true, there is no interpretation of E(x) in which both premises are clearly true. In order to see this, we must get clear about the notion of existence. 1.4

Existence and Non-Existence

The notion of existence seems trivial. But the amount of philosophical literature generated by this notion belies that seeming. The first distinction to be made is that between the quantifier of existence – “∃ex” – and the predicate of existence – “E(x).” Note that I do not identify the quantifier of existence with the expression that is often, negligently, identified with it, i.e., with “∃x”; rather, I distinguish the quantifier of existence from that latter quantifier by the uppercase index “e.” There are two reasons for this carefulness: (1) “∃x” can be understood substitutionally, and then it has nothing to do with (extralinguistic) existence; (2) even if “∃x” is understood in an ontic, non-substitutional sense, “∃x” can be understood in a purely numerical sense (as it will, in fact, be understood by me): “at least one x is such that …,” without any clear reference to existence at all. But the purely numerical quantifier “∃x,” commonly called “quantifier of existence,” and the quantifier of existence properly speaking, “∃ex,” are certainly not unrelated. With the help of the existence predicate, the latter quantifier can be defined on the basis of the former quantifier: ∃exA[x] =Def ∃x(E(x) ∧ A[x]). Thus, if ∀xE(x) turned out to be a logical truth, then the two quantifiers would indeed be logically indistinguishable and express logically equivalent concepts.

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But is ∀xE(x) a logical truth? There are certain phenomena that militate against assuming this: (A) There are true singular statements of non-existence: ¬E(a). If classical predicate logic is assumed, then the falsity of ∀xE(x) must be concluded from these statements, and ∀xE(x) turns out to be not a logical truth. (B) There are true singular statements of the possibility of non-existence: ◊¬E(a). If classical predicate logic is assumed and also the modallogical principle that with A being possible all logical consequences of A are possible, too, then the truth of ◊∃x¬E(x) must be concluded from those statements. Now, if ∀xE(x) is a logical truth, then ∀xE(x) is a logical truth, too – according to an elementary principle of modal logic. But ∀xE(x) cannot be a logical truth because it contradicts ◊∃x¬E(x),3 whose truth has just been concluded on the basis of true premises. ∀xE(x), therefore, is not a logical truth, either. Yet the revulsion against “Meinong’s shocker,” i.e., the revulsion against the truth of ∃x¬E(x),4 against the very logical consistency of it, has been so great that philosophers have sought ways to defuse the arguments (A) and (B), and to uphold the logical truth of ∀xE(x) in the teeth of them. What are those ways? Concerning (A): (i)

One might stick to classical predicate logic, postulate that all singular terms have a referent within the quantificational universe of discourse – i.e., inside the class5 of all entities quantified over6

3

Considering that “◊” means as much as “¬¬,” and “¬∃x¬” as much as “∀x.” It seems to me that the revulsion against the truth of ∃x¬E(x) is in part due to the fact that ∃x¬E(x) is confused with ∃ex¬E(x), the latter statement being indeed an elementary logical falsehood. 5 I here (in this preliminary chapter, but not in the other chapters) use the term “class” as a potentially more general term than “set”: all sets are classes, but perhaps not all classes are sets (namely, those classes that are not elements of other classes would not be sets). 6 The quantificational universe of discourse must be distinguished from the referential universe of discourse: the class of all entities referred to by singular terms. The normal 4

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– and identify the meaning of the word “exist” with the meaning of “is an element of the quantificational universe of discourse.” Drawback: One is flatly denying that there are true singular statements of non-existence. (ii) One might identify the meaning of “exists” with the meaning of “is an element of the quantificational universe of discourse” and admit that there are true singular statements of non-existence, while still maintaining that all singular terms have a referent (although not all of them have a referent within the quantificational universe of discourse). Drawback: One has to give up classical predicate logic; quantification and singular reference become dissociated. (For example, the inferenceschema A[c] ⇒ ∃yA[y] is no longer logically valid.) (iii) One might interpret singular statements of existence in the following manner: “E(a)” : “the singular term ‘a’ has a referent within the quantificational universe of discourse”; “¬E(a)” : “the singular term ‘a’ does not have a referent within the quantificational universe of discourse.”7 In quantified contexts, however, “E(x)” is simply taken to mean “x is an element of the quantificational universe of discourse.” Moreover, one admits that there are true singular statements of non-existence; their truth can be (1) due to the fact that the singular term concerned has a referent, but not in the quantificational universe of discourse, or (2) due to the fact that it does not have any referent at all. Drawback: One has to give up classical predicate logic; quantification and singular reference become even further dissociated. In addition, there is no longer a uniform sense of “exist”: in singular statements, “exist” is a disguised metalinguistic predicate.

– classical – relationship between the two is that the referential universe of discourse is a subclass – usually a proper subclass – of the quantificational universe of discourse. 7 Note that this does not automatically mean the same as “a is not an element of the quantificational universe of discourse.” If the singular term “a” does not have a referent, then the sentence “the singular term ‘a’ does not have a referent within the quantificational universe of discourse” is true, while the sentence “a is not an element of the quantificational universe of discourse” can be considered to be neither true nor false.

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As has just been seen, each of these three ways to uphold the logical truth of ∀xE(x) in the face of apparent counterexamples has its drawback. Things become even more difficult for the defenders of the logical truth of ∀xE(x) when they are confronted with apparently true singular statements of the possibility of non-existence. For some of these statements do not appear to be reasonably deniable.8 The best way to uphold the logical truth of ∀xE(x) while not denying that there are true instances of ◊¬E(a) may be the following: Concerning (B): (iv) With respect to each possible world w the meaning of the word “exist” is identified with the meaning of “is an element of the quantificational universe of discourse of w,” each world having its own quantificational universe of discourse.9 It is admitted that there are true singular statements of the possibility of nonexistence, while for each possible world reference remains within the possible world’s quantificational universe of discourse (hence no sentence of the form ¬E(a) is true at any possible world). This way of treating the matter has interesting consequences: It turns out that whatever possible worlds and corresponding quantificational universes of discourse we are looking at: ∀xE(x) is true at all possible worlds. Hence ∀xE(x) is a logical truth, as is ∀xE(x). But we can also look at possible worlds and corresponding universes of discourse such that ∃x◊¬E(x) is true at some possible world w´. How can this be? Easily: there merely needs to be a possible world w´´ (among the possible worlds considered) to whose quantificational universe of discourse some entity does not belong that belongs to the quantificational universe of discourse of w´ (and nothing in (iv) has forbidden this situation). Moreover, we can even look at possible worlds and corresponding universes of discourse 8

We should nevertheless keep in mind the quite general epistemological difficulty that was pointed out in Section 1.2 and further developed in Section 1.3: the Difficulty of Ascertaining Mere Possibility. How, for example, do I know that I might not have existed (a mere possibility of which I am firmly convinced)? Presumably, there is a satisfactory answer to this question, but we do not know it yet. 9 I leave it open whether the quantificational universe of discourse of a possible world w can be identical with the quantificational universe of discourse of another possible world w´.

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such that ◊¬E(a) is true at some possible world w´ (say, the real world). For this, there merely needs to be a possible world w´´ to whose quantificational universe of discourse a certain entity x does not belong, but which is also such that the singular term “a” has it as its referent with respect to w´.10 In this manner the logical truth of ∀xE(x) and ∀xE(x) can be regarded as compatible with the truth of ◊¬E(a). But, of course, this has a price. The modal-logical principle that with A being possible all logical consequences of A are possible, too, can no longer be generally upheld; nor can the modal-logical principle be generally upheld that with A being necessary all logical consequences of A are necessary, too.11 The loss of these principles is a very grave drawback. Leaving these logical matters aside, there is a common philosophical drawback to all suggested procedures for escaping the arguments (A) and (B): by explicating “exist” with reference to quantificational universes of discourse these procedures, in effect, turn existence and non-existence into a matter of semantics (the semantics of reference). But this is inadequate: existence and non-existence are a matter of ontology, not of semantics. We must, therefore, look for explications of “exist” that do without semantical concepts (such as the concept of a quantificational universe of discourse, or the concept of reference). We must give “exist” a purely ontological sense – in which sense neither E(a) nor ¬E(a) can be true if “a” is a nonreferring singular term. We must, in addition, observe the restraint that “exist” is not allowed to have a meaning in singular statements that is different from the meaning that it has in quantified statements. But once “exist” is given a uniform ontological meaning, is there a way of escaping from the arguments (A) and (B)? Note that requiring “exist” to have a uniform ontological meaning must not be taken to imply that there is only one possible ontological meaning of “exist.” There might be at least two such meanings (as will be seen presently); but if one of these meanings is adopted, then “exist” must 10

For example: “U.M. might not have existed” can be regarded as true in the real world because there is a possible world w´´ to whose quantificational universe of discourse I do not belong, while the singular term “U.M.” certainly has me as its referent with respect to the real world. 11 Within classical predicate logic, ¬∀xE(x) is a logical consequence of ¬E(a), and E(a) is a logical consequence of ∀xE(x). Both logical relationships do not obtain in free logic. But (iv) does not have free logic as its backgound, nor is shifting to free logic as a basis for modal predicate logic an obviously recommendable step to take.

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be accorded that meaning uniformly, in all contexts. Otherwise, we would be proceeding in a philosophically unsatisfactory ad hoc manner. One possible ontological meaning of “x exists” – “E(x)” – is the following: First Ontological Meaning of Existence: E(x) =Def x = x. In this sense (with “=” in the definiendum designating the ontological relation of numerical identity), ∀xE(x) and ∀xE(x) are self-evident logical truth, and in this sense, there simply are no true singular statements of the forms ¬E(a) or ◊¬E(a). True, even in the ontological sense now under scrutiny E(a) can be not true: if, and only if, the singular term “a” does not refer to anything. But it does not follow that ¬E(a) would then be true. Rather, given the ontological sense of E(a) now in consideration, ¬E(a), too, would be not true if E(a) were not true. In fact, given this ontological sense of E(a), ¬E(a) cannot be true – on pain of contradiction; for if ¬E(a) were true, then E(a) (i.e., a = a) would be not true, and hence “a” would not refer to anything (otherwise a = a would certainly be true) – which means that ¬E(a) would be not true (given the ontological – hence non-semantical – sense of E(a) now under consideration). The considerations that apply to ¬E(a) apply, mutatis mutandis, also to ◊¬E(a). Thus, if the First Ontological Meaning of Existence is adopted, one escapes the arguments (A) and (B), because both in the case of (A) and in the case of (B) the main premise is falsified (that is, “There are true singular statements of non-existence” in the case of (A), and “There are true singular statements of the possibility of non-existence” in the case of (B)). But what is to become now of the strong intuition that there are true singular statements of non-existence? And what is to become of the even stronger intuition that there are true singular statements of the possibility of non-existence? Are these intuitions mere illusions? Probably not. But given that we require “exist” to have a uniform ontological meaning, there is only one way to preserve these intuitions: There must be another possible ontological meaning of “x exists,” of “E(x),” besides the possible meaning of it we just considered. According to this other meaning, it turns out that there are true singular statements of non-existence, or at least that there are true singular statements of the possibility of non-existence.

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Second Ontological Meaning of Existence: E(x) =Def A(x). Here “A(x)” stands for “x is actual.” The meaning of this predicate is to a certain extent vague, but not in all its applications to the same degree. Its meaning becomes particularly clear if it is applied to states of affairs: an actual state of affairs is simply an obtaining state of affairs, in other words, a fact. As is rather plausible, there are states of affairs that do not obtain, for example, the state of affairs that U.M. was born in 1964. Hence the state of affairs that U.M. was born in 1964 is not actual, and therefore, according to the Second Ontological Meaning of Existence, the state of affairs that U.M. was born in 1964 does not exist. Therefore, trivially, it is also possible that the state of affairs that U.M. was born in 1964 does not exist. Hence, according to the Second Ontological Meaning of Existence, it is not a logical truth that everything exists – because it is not true that everything is actual, and it is not true that everything is actual because some states of affairs do not obtain.12 Thus, both argument (A) and argument (B) are entirely vindicated, in premise and conclusion, as soon as states of affairs figure as objects of quantification and reference, and there is then not the slightest well-founded motivation to get in some way or other around these arguments. It is useful to call the Second Ontological Meaning of Existence actual existence, and the First Ontological Meaning of Existence numerical existence. From the present vantage point, it can easily be seen that the argument in Section 1.3 – the Argument against Knowledge of a Mere Possibility – equivocates between actual existence and numerical existence. If “to exist” is taken to mean the same as “to be actual,” then Premise 1 is clearly true, but Premise 2 is certainly not clearly true. If, on the other hand, “to exist” is taken to mean the same as “to be selfidentical,” then Premise 2 is clearly true (indeed trivially true), but Premise 1 is clearly not true. Which of the two ontological meanings should be given to “E(x),” actual existence or numerical existence? This is a matter of stipulation. In ordinary language, “exist” is used equivocally: When I say “I might have not existed,” I am certainly using “exist” in a sense that is different from 12

One might think to deny that some states of affairs do not obtain by denying that there are states of affairs: if it were true that there are no states of affairs, then it would be trivially true that all states of affairs obtain. But nihilism regarding states of affairs does not seem to be a plausible position.

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the sense in which I use it when I say “At least one possible world exists besides the real world.” I herewith decide to use “x exists” (or “E(x)”) as expressing actual existence; therefore, “x is actual” (or “A(x)”) is in this book taken to be a synonym for “x exists.” I will (try to) avoid using the word “exists” in the sense of numerical existence; if occasion should arise for employing the concept of numerical existence, I will (if I remember my precepts) express it by the phrase “is a something.” Moreover, instead of the term “numerical existence,” which strongly but wrongly insinuates that existence simpliciter is logically implied by it, I will frequently use the word “being.” The ordinary language use of “exist” equivocates between actuality (actual existence) and (mere) being (i.e., numerical existence); for the purposes of this book, I eliminate this equivocation by reserving the term “exist” for expressing actuality, whereas being is expressed by me in a different way. Where I lapse from my precepts, I trust that the context will make clear which concept of existence I mean by “exist.” 1.5

Actualisms

Non-existence (or non-actuality) is widely regarded as a problematic, even an absurd notion. To some extent, such views are due to a confusion of actual existence with numerical existence, flowing from a failure to realize that there are at least two first-order ontological concepts of existence operative in ordinary language. This confusion, as we have seen, is also responsible for the Argument against Knowledge of a Mere Possibility that was presented in Section 1.3. It is indeed absurd to assert the numerical non-existence – the non-being – of anything. But existence simpliciter – as I have stipulated in the previous section (being free to settle matters in this way) – is not numerical existence; it is actual existence, or actuality, and there is nothing obviously absurd in asserting the non-actuality of something. Thus, the position of general actualism – “Everything exists” / “Everything is actual” – is not obviously correct, and indeed it is false, as we have already seen (assuming, of course, that we accord states of affairs a place in our ontology). Although general actualism is false – because actualism with respect to states of affairs is false – there may be specialized actualisms that are true. Here are two likely candidates:

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Actualism with respect to individuals: All individuals exist. Actualism with respect to properties: All properties exist. But, on reflection, it turns out that actualism with respect to properties is also false. Consider the property of being a unicorn. This property of individuals is actual if, and only if, it is exemplified by at least one actual invidual. But no actual individual exemplifies it (at least, this is very likely the case). Therefore, the property of being a unicorn is not actual, in other words, it is non-existent.13 The only actualism which seems at all plausible is actualism with respect to individuals. But even with regard to an individual it does not seem absurd to assert that it does not exist – once actual existence is distinguished from numerical existence and the interpretation of existence simpliciter as actual existence is firmly adhered to. One may well wonder what can make this or that actualism an attractive position if there cannot be any question of a confusion of actuality with (mere) being and “exist” is firmly interpreted as expressing actuality. The attractions of actualism flow, I believe, from an epistemological source; one can hardly maintain that non-actuality is intrinsically an ontological problem. On the contrary, it is felt to be an ontological problem because, and only to the extent, it is an epistemological problem. What, then, is the epistemological problem that is thought to be connected with non-actuality? It is simply the one that is indicated by the second premise of the Argument against Knowledge of a Mere Possibility: There is no knowledge of the non-existent. As I have said above, this statement is not clearly true if “the non-existent” is interpreted to mean as much as “the non-actual”; but a convincing case remains to be made for the statement’s being clearly false under that interpretation. Else, the Argument against Knowledge of a Mere Possibility remains a threat to modal theory, especially considering that Premise 1 of that argument turns out to be clearly true if “exists” is interpreted to mean 13

Remember that its being (its numerical existence) is not being denied by this; its being – just like the being (the numerical existence) of the state of affairs that U.M. is born in 1964 – remains beyond reasonable doubt, at least for everybody who is not a nominalist.

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as much as “is actual.” But, in fact, in view of the considerations in the previous section and above, in this section, it is already clear that there is at least some knowledge of the non-existent, namely, at least some knowledge about non-existent states of affairs and non-existent properties. Since the question “Is there knowledge of the non-existent?” has been answered by a clear “Yes,” we must question next, “To what extent is there knowledge of the non-existent?” Serious limitations to our knowledge of the non-existent would provide strong motivation for upholding special forms of actualism, in case strong arguments against these actualisms were absent. For example, if we had no strong argument to the conclusion that some individuals are non-existent and if it turned out that, in any case, we have no logically contingent knowledge of nonexistent individuals, then the assumption that all individuals exist – i.e., the assumption of actualism with respect to individuals – would become very attractive indeed. It is one question how much non-actuality is warrantable from the epistemological point of view. It is another question how much nonactuality is needed for modal theory. But clearly, there can be an epistemologically satisfactory theory of modality only if no more nonactuality is needed for modal theory than is warrantable from the epistemological point of view. Modal theory is at least committed to non-actual states of affairs. For suppose every state of affairs were actual; hence the possibility of a state of affairs would imply its actuality, and consequently modal theory would become otiose. Even its logical motivation would vanish: not only B ⊃ ◊B (a central principle of alethic modal logic) but also ◊B ⊃ B would turn out to be true for every (true or false) sentence B. This can easily be demonstrated: Suppose ◊B, hence ◊(that B), hence A(that B) [because ∀x(S(x) ⊃ A(x)), and S(that B)], hence O(that B) [because ∀x(S(x) ⊃ (A(x) ≡ O(x))], hence B [because B ≡ O(that B)]. The demonstration is independent of any particular interpretation of possibility (◊). It will have been noticed that the demonstration also contains the necessary steps for proving that every (true or false) sentence would be true if every state of affairs were actual; it thus provides a perfect reductio ad absurdum of actualism with respect to states of affairs.

Thus, there is no way to avoid the truth that modal theory is committed to states of affairs that are non-actual. But fortunately we have seen above that the assumption of nonactual states of affairs is well supported, and hence warranted from the

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epistemological point of view. It remains to be seen how and to what extent there can be knowledge of non-actual states of affairs over and above the mere knowledge that there are such things and that this or that state of affairs is non-actual. In particular, we must inquire how and to what extent (a) there can be knowledge regarding the – logical or nonlogical – possibility of non-actual states of affairs if they are possible, and (b) knowledge regarding their impossibility if they are impossible. Question (a) regards the Difficulty of Ascertaining Mere Possibility, and question (b), in fact, regards the Difficulty of Ascertaining Necessity. Since the impossibility of non-actual states of affairs is nothing else than the necessity of the negations of these states of affairs, which are the necessarily actual states of affairs, it is clear that the Difficulty of Ascertaining Necessity (of the actual) can be reformulated as the Difficulty of Ascertaining Impossibility (of the non-actual). Later, the question needs to be addressed whether the assumption of non-actual individuals, too, is warranted from the epistemological point of view, and if so, how and to what extent there can be knowledge of nonactual individuals over and above the mere knowledge that there are such things. An answer must also be found to the question to what extent nonactual individuals are required by modal theory. And hopefully the requirements of modal theory do not lie outside the epistemological warranty regarding non-actual individuals.

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2

Actualism, Modal Realism, and Modal Epistemology

This chapter, preceding my presentation of the basics of a theory of ontic modality in Chapter 3, is dedicated to discussing in some more detail the issues that have been raised or adumbrated in Chapter 1.

2.1

Alvin Plantinga on Existence, Actuality, and Actualism

According to Alvin Plantinga, one can discern ... a dominant conception of possible worlds, properties and propositions. ... [T]his conception presupposes that there are or could have been things that do not exist; it presupposes that in addition to all the things that exist, there are some more – “possible objects” – that do not exist but could have. This presupposition seems to me resoundingly false. (“Self-Profile,” p. 91.)

Readers (if they are like me) may wonder how the conception Plantinga refers to could be the “dominant conception” it in fact is if its presupposition really were “resoundingly false.” It is, after all, somewhat unlikely that most of the extant philosophical intelligence in modal matters is concentrated in Alvin Plantinga. Readers may feel inclined to suspect that the supposition of non-existent, merely possible objects can be interpreted in such a way as to make it not false, or at least: not resoundingly false, this way of interpreting it being the way in which the supposition of merely possible objects is in fact predominantly understood (though not by Plantinga). The distinction between two possible meanings of existence in Section 1.4 is helpful here. If “x exists” has the First Ontological Meaning of Existence – according to which “x exists” means as much as “x is selfidentical” –, then, of course, there neither are nor could have been things that do not exist, nor are there, a fortiori, non-existent things that could have existed. The simple reason for this is that it is necessary in the strongest sense that everything is self-identical. Thus, if Plantinga has the First Ontological Meaning of Existence in mind when he uses the word

2 Actualism, Modal Realism, and Modal Epistemology

“exist” – and I suspect he has –, then he is quite right about the resounding falsity of supposing that some things do not exist but could have. But one may well question whether Plantinga has the relevant ontological meaning of existence in mind. Is anybody, who assumes that some things do not exist but could have, assuming that some things are not self-identical but could have been? Is anybody assuming this? It is unlikely, to put it mildly. But if the champions of merely possible objects are not dunces in that way, perhaps they are dunces in another? Here is a ready suggestion in this direction, one explicitly offered by Plantinga: I am now inclined to think that ‘there are’ and ‘there exist’ are ordinarily and in most contexts no more than stylistic variants – in which case ‘There are some things that do not exist’ expresses the same proposition as ‘There exist some things that do not exist’. But then, clearly enough, it is logically false that there are some things that do not exist. (“Self-Profile,” p. 67; see also “Replies,” p. 315.)

But, in fact, “there are” and “there exist” are not stylistic variants of each other. “There are some things that do not exist” unequivocally expresses the same proposition as “Some things do not exist.” But is this also true of “There exist some things that do not exist”? Does also this latter sentence unequivocally express the same proposition as “Some things do not exist,” as it must if “there exist” is to be a stylistic variant of “there are”? No. Plantinga, for one, takes “There exist some things that do not exist” to express the same proposition as “Some existent things do not exist,” and the logically false proposition that this second sentence in fact expresses is not clearly – and most people would add: is clearly not – the same proposition as the proposition that some things do not exist. Thus, “there are” and “there exist” are not stylistic variants of each other, or evident synonyms – and certainly not in the present context: the context of discussing the question whether there are some things that do not exist. Plantinga, too, has certainly been touched by doubts regarding these matters, for he writes: We should note … that philosophers of great intellectual power have asserted that there are things that do not exist; and they haven’t taken themselves to be making … the absurd claim that there exist things that do not exist … [H]ow, then, can we avoid saddling Castañeda, Meinong and Russell with the absurd view that there exist things that do not exist, and indeed, things that not only do not but could not have existed? (“Replies,” p. 315.)

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The following quotation presents the central part of what Plantinga has to offer in his laudable effort to show more respect to the mentioned philosophers and to others who have proposed that there are some things that do not exist: The possibilist [or, more accurately speaking, the nonactualist, as Plantinga himself points out] believes that there is (and exists) a property that does not entail existence, but is entailed by every property. (Ibid., p. 315.)

The property of being self-identical does certainly seem to be a property which is just as Plantinga’s non-actualist requires: it is entailed – logically implied – by every property, and it does not appear to entail – logically imply – existence. And, indeed, it does not entail existence if existence, contra Plantinga (see below), is equated with (the property of) actuality. If, however, existence is equated with the property of being self-identical, pro Plantinga (as it seems to me), then, obviously, self-identity cannot be a property such as Plantinga’s non-actualist requires (since it certainly entails itself and therefore also existence, with which it is equated). In fact, if existence is equated with self-identity, then there is no property that meets the requirements of Plantinga’s non-actualist (since, obviously, there is no property that does not entail self-identity). Thus, the decision regarding the issue at hand proves to be dependent on the interpretation of existence. Plantinga insists that existence must not be equated with actuality (see the quotations below). Others are less dogmatic about this, and, I believe, legitimately so. I propose that people who hold that there are some things that do not exist accord to “x exists” the Second Ontological Meaning of Existence (see Section 1.4); for then, and only then, does their assertion have a chance of being true (as I am entirely confident they are intelligent enough to realize). Thus, by holding that there are some things that do not exist, they are holding that some things are not actual, or in other words: that some things are not real. I also propose that to accord to the word “exist” the Second Ontological Meaning of Existence is no less legitimate than to accord to it the First. What is of prime importance, of course, is to recognize which of the two Ontological Meanings of Existence is employed when the word “exist” is used on a given occasion. Plantinga is unable to recognize actuality as a legitimate ontological interpretation of existence. Instead, he separates actuality from existence with extreme emphasis:

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[A]ctuality must not be confused with existence. (“Self-Profile,” p. 88.) I said above that actuality and existence must not be confused. This is worth repeating; actuality and existence must not be confused. (Ibid., p. 90.) Actuality must be distinguished from existence. (Ibid., p. 90.)

Well, one can be of the opinion that people should not say “x exists” and mean x is actual – for the sake of avoiding the misunderstandings that are likely to arise due to the fact that, by saying “x exists,” one can also mean x is something, or in other words: x is self-identical, or again in other words: x is identical with something. Nevertheless, people do say “x exists,” and mean x is actual. And one’s normative convictions on how words should be used should not make one blind to the way in which they are in fact used – should not mislead one, above all, into according the thesis that there are some things that do not exist an interpretation which is both uncharitable and irrelevant. Plantinga’s blind spot regarding existence and actuality makes him see inadequacies where there are none: The truth of the matter is, then, that there neither are nor could have been objects that do not exist. This view is sometimes called ‘actualism’; I call it that myself. Nevertheless, this is unfortunate terminology, for it tends to perpetuate a confusion between actuality and existence. (“Self-Profile,” pp. 91-92.)

The truth of the matter is (we may count on it) that anybody who is numerically different from Plantinga and a few others and who calls the view that there neither are nor could have been objects that do not exist “actualism” is using “x exists” in the sense of “x is actual.”1 Since such a 1

One should be careful with the claim that there could not have been objects that do not exist, since that claim is even less plausible than the claim that there are no objects that do not exist. Since it is uncontroversial that you and I might not have existed (that is, that I and you might not have been actual), it must be equally uncontroversial that there are some objects that might not have existed. And doesn’t this entail that there could have been objects that do not exist (namely, you and I)? Of course, one can, and many people do, block this inference by the ad hoc device of a fitting modal logic: a modal logic that does not endorse the inference of ◊∃x¬E(x) from ∃x◊¬E(x). Consider, however, an analogous case. For the time being, everybody, I expect, will agree that there is nothing wrong with inferring ◊∃x(x is not physical) from ∃x◊(x is not physical). But perhaps, in a few more years, this inference will be generally considered invalid: because it produces a conflict with necessary physicalism, the

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use of “x exists” is no less legitimate than any other use of “x exists,” there is no “unfortunate terminology” involved, and certainly no “confusion between actuality and existence.” The alleged confusion between actuality and existence is fostered [according to Plantinga] by the ill-advised habit of speaking of “unactualized possibles”, where the alleged reference is not to states of affairs that are not actual but might have been, but to objects that do not exist but might have. (“Self-Profile,” p. 90.)

But when most people speak of “unactualized possibles,” then (we may count on it) they mean precisely this: entities that are not actual but might have been. This is their meaning in every case, and not only in case the intended “unactualized possibles” are states of affairs. If entities which are not states of affairs are called “unactualized possibles” by them – which entities, they say, do not exist but might have –, then they are not indulging an “ill-advised habit of speaking,” but are exercising their perfect right to use “x exists” in the sense of “x is actual.” While acknowledging possible but unactual states of affairs (see “Self-Profile,” p. 92), Plantinga, it should be noted, is curiously silent on possible individuals that are not actual (or, for that matter, on possible properties that are not actual). Does he or does he not believe that there are such things? While Plantinga calls himself an “actualist” (cf. the above quotation), it must be remembered that ‘actualism’ is for him the misnamed doctrine that there neither are nor could have been objects that do not exist,2 where “x exists” means something else than “x is actual.” Like everything else, all individuals exist for Plantinga, but that does not mean that all individuals are actual for him. Regarding Plantinga’s views on existence and actuality, it is instructive to compare the earlier The Nature of Necessity with the later “Self-Profile.” By holding that all unactual possible worlds exist – “they exist all right, but they just are not actual” (The Nature of Necessity, p. 131) – Plantinga is separating existence and

negation of necessary physicalism – the thesis that there could have been objects that are not physical – having finally come to be generally considered “resoundingly false.” It should be obvious that the described manner of proceeding – hand-tailoring logic – is as question-begging regarding the truth of necessary actualism as it is questionbegging regarding the truth of necessary physicalism. 2 Plantinga suggests (“Self-Profile,” p. 92) that the doctrine should really be called “existentialism.”

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actuality. The separation he intends is emphasized when he regards Socrates being a carpenter as an existent but unactual state of affairs. But then, inconsistently, Plantinga seems to be nevertheless identifying the thesis that there are possible but unactual objects with the thesis that there are possible but non-existent objects, rejecting the former thesis together with the latter. See the heading of chapter VIII of The Nature of Necessity, p. 149: “Possible But Unactual Objects: On What There Isn’t,” the very first sentence of that chapter referring to the contention that Plantinga denies: “the venerable contention that there are or could be possible objects that do not exist.” If we proceed on the assumption that what Plantinga really wants to do is to separate existence and actuality (in “Self-Profile,” he endorses that separation, as we have seen), then he does essentially the same thing that Meinong can be seen to be doing in the quotation Plantinga adduces (see The Nature of Necessity, p. 133): Meinong, clearly enough, distinguishes two notions of being: one according to which everything is, and one according to which some things are, and others are not. The latter notion of being – being2 – is actuality (or reality). The former notion of being – being1 – is also described by Moore and Russell in two further quotations Plantinga adduces (ibid., pp. 133-134); it might be called “somethingness.” Now, the following account seems to me very likely true: Meinong distinguishes somethingness and actuality, and calls actuality “existence”; Plantinga also distinguishes somethingness and actuality, but calls somethingness “existence.”3 Both hold that some entities are not actual, and that all entities are something. But Meinong maintains the former doctrine by asserting “something does not exist,” while Plantinga maintains the latter doctrine by asserting “everything exists.” Clearly, their disagreement is entirely verbal, in substance they do not disagree. But is there not also some substantial disagreement between Meinong and Plantinga in these matters? Yes, there is. But what that substantial disagreement consists in is not easily determinable. Both Meinong and Plantinga believe that the sentence “Pegasus does not exist” is true, but they accord different interpretations to this sentence. For Meinong, “Pegasus does not exist” means as much as “Pegasus is not actual”; for Plantinga, in contrast, “Pegasus does not exist” means as much as “Pegasus is not something.” So far, no obviously substantial disagreement between Meinong and Plantinga has become apparent, but merely a difference in the interpretation of a certain sentence. 3

Curiously, Plantinga remarks: “[T]his implausible notion of being or thereisness is uncalled for” (The Nature of Necessity, p. 152; my emphasis). The remark is curious, for if thereisness – or somethingness – is interpreted as the property of being identical with something (= the property of being self-identical), then the notion of thereisness certainly seems plausible enough. And, in fact, the indicated interpretation of thereisness is its only plausible interpretation; it is the same interpretation as the one that Plantinga employs with regard to existence (and the only plausible interpretation of existence if existence is to be both a non-world-indexical property and different from actuality).

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Meinong and Plantinga even agree, I suppose, on the truth of the sentence “Pegasus is not actual.” But Meinong, contradicting Plantinga, holds that Pegasus, though non-actual, is at least something, his simple reason for this being that everything is something. The position that everything is something is shared by Plantinga (see above); but Plantinga, in contrast to Meinong, denies the validity of inferring “Pegasus is something” from “everything is something,” his reason for this being that the name “Pegasus” does not refer to anything. Thus, the substantial disagreement between Meinong and Plantinga comes down to this: Meinong takes “Pegasus” to be a referring name and sticks to classical predicate logic; Plantinga takes “Pegasus” to be a non-referring name and gives up classical predicate logic.4

According to the terminological decisions made in this book, general actualism – or actualism simpliciter – is the doctrine that everything exists, where “x exists” is stipulated to mean as much as “x is actual.” Plantinga is not an actualist in this sense (though still an actualist in his sense) because he recognizes non-actual states of affairs. By the same token, he is not an actualist with respect to states of affairs, according to the terminology employed in this book. But Plantinga might still be an actualist with respect to individuals, according to the terminology of this book, that is, he might still believe that all individuals are actual. I suspect that this is precisely his position: that all individuals are (not only existent, but also) actual. If he envisaged individuals that are (existent but) not actual – as might be expected, since he has emphasized so strongly that actuality is not existence (and why, one is inclined to ask, should the behavior of individuals regarding actuality be different from the behavior of states of affairs?) –, then he would certainly make mention of this. But he doesn’t. 4

In The Nature of Necessity, Plantinga wrote as if he had never heard of free logic. But his lengthy analyses of singular negative existentials (i.e., singular negative existence-statements) do culminate in the position that the names occurring in true singular negative existentials are non-referring (hence, if not paraphrased away, necessitate a modification of classical predicate logic), although he does not put matters in exactly that way: “But now we see the error of our ways: although some singular negative existentials are possibly true, none of these are predicative. ... In worlds where he does not exist, Socrates [or Pegasus] has no properties at all, not even that of non-existence.” (The Nature of Necessity, p. 152.) If Pegasus, who, unlike Socrates, does not exist in the actual world, has no properties at all, then Pegasus does also not have the property of being identical with something. But how can “Pegasus does not have the property of being identical with something” be true unless “Pegasus” does not refer to anything (and logic is modified accordingly, departing from classical logic)?

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Summing up, I take Plantinga’s position to be this: Everything exists. Everything is actual, except some states of affairs and ... . The dots “...” can be replaced by further “some”-phrases in case the need to do so should happen to arise. Except for not recognizing actuality as a legitimate interpretation of existence – which, however, is absolutely essential for doing justice to those non-actualists who traditionally express their views in terms of nonexistence – Plantinga seems clear enough on actuality: [Actuality] must … be distinguished from actuality in α, a property had only by α, from actuality in itself, a property had by every states of affairs, and from actuality in some possible world or other, a property had by every possible state of affairs. (“Self-Profile,” p. 90.)

As is urged by this quotation, one must distinguish between actuality simpliciter, actuality in a state of affairs x, actuality in α (“α” being a rigid designator for the actual world), and actuality in a possible world w. A unified picture arises if one takes into account that possible worlds, and a fortiori the actual world, are for Plantinga maximal-consistent states of affairs (see Section 4.2). By employing notions intuitively that will be comprehensively introduced, described and examined in Chapter 3, we can here define: y is actual in the state of affairs x =Def the state of affairs that y is actual is an intensional part of the state of affairs x. [The concept of actuality employed in the definiens is actuality simpliciter.] y is actual in itself =Def y is actual in the state of affairs y. y is actual in the possible world x = y is actual in the state of affairs x, and x is a possible world [possible worlds being taken to be maximal-consistent states of affairs]. y is actual in α =Def y is actual in the possible world α [“α” being a rigid designator for the maximal-consistent state of affairs which is the actual world].

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On the basis of these definitions, two of Plantinga’s assertions are provable within the theory that will be developed in Chapter 3: every state of affairs is actual in itself,5 every possible state of affairs is actual in some possible world. Plantinga also asserts: α is the only entity that is actual in α. This is false, since, for example, the state of affairs that Plantinga is a philosopher in the year 2001 – a state of affairs which is obviously different from α – is actual in α (whereas in some other possible world that state of affairs is not actual). It must be added that Plantinga’s distinctions regarding actuality are not all the distinctions that are metaphysically relevant regarding actuality. For further perspectives on actuality, see Section 4.3.

2.2

What Is Modal Realism, and What, Really, Is It Good For?

The first discussion of modal realism can be found in the philosophical correspondence of Gottfried Wilhelm Leibniz and Antoine Arnauld. In the letter of July 14, 1686, Leibniz writes (and I translate from the original French): All that is actual can be conceived as possible, and if the actual Adam will have in the course of time such and such posterity, then one cannot deny that predicate of this same Adam, conceived as possible, and even more so considering that you have conceded that God envisages all these predicates in him when he determines to create him. They therefore apply to him. And I do not see that what you have said of the reality of the possibles [la réalité des possibles] is contrary to this. For calling something possible, it is, in my eyes, sufficient that one can form a notion of it, even if it were only in the divine understanding, which is, so to speak, the realm of possible realities [le pays des réalités possibles]. Thus, when one speaks of possibles, I am content that one can form true sentences about them, as, for example, one can judge that a perfect square does not imply a contradiction, even if there were no perfect square in the world. And if one wanted to absolutely reject the pure possibles [les purs possibles], one would destroy contingency, and freedom; because if 5

To give an impression of what is involved in deducing this assertion: According to definition, “Every state of affairs is actual in itself” means as much as “For every state of affairs y: the state of affairs that y is actual is an intensional part of y.” This follows on the basis of two principles: every state of affairs is an intensional part of itself (compare P2 in Chapter 3), every state of affairs y is identical with the state of affairs that y is actual (mainly responsible for this is P24 in Chapter 3; for further details see Section 4.7).

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there were nothing possible except what God effectively creates, then what God creates would be necessary, and God, wanting to create something [some X], could only create just this [this X], without having the freedom of choice. (Der Briefwechsel mit Antoine Arnauld, pp. 146, 148.)

From Leibniz’s brief defense of modal realism, we can also gather what modal realism is. It is, put in the briefest manner, the doctrine that there are possible realities that are merely possible, or in other words: the doctrine that there are possible beings which are (ontologically) independent of the human mind (that is, “possible realities”) and which are not actual. That Leibniz stands by this doctrine is very clear from the above citation. A modern subscriber to modal realism in the defined sense is David Lewis. I myself am also a modal realist in the defined sense, though my modal realism is fitting the definition in a manner that is very different from Lewis’s or Leibniz’s modal realism (as will become amply clear in this book). Charles Chihara defines modal realism in a way that is somewhat different from mine: Modal Realism, as I use the term, is the doctrine that, besides the actual world we live in, there exist other possible worlds. (The Worlds of Possibility, p. 76.)

Thus described, modal realism does not exclude modal constructivism; for nothing in Chihara’s definition forbids that these other possible worlds besides the actual world are abstract constructions, depending, in the last analysis, on the human mind. But Chihara immediately distinguishes between modal realism and modal realism “in the full-blown sense”: Modal Realism, in the full-blown sense of the term, maintains that these other possible worlds are not mere abstractions or descriptions – they contain things that are just as real and concrete as things like tables, chairs, people, and trees in the actual world. (Ibid.)

Chihara’s “Modal Realism, in the full-blown sense of the term,” obviously comes closer than his “Modal Realism, as I [Chihara] use the term,” to modal realism as I myself define it; for modal realism, as I define it, can

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very well be taken to be entailed by Chihara’s “Modal Realism in the fullblown sense of the term.” The reverse entailment, however, is not so clear. In any case, Chihara is opposed to modal realism, be it in his sense or in mine, for he states: A thing that does not exist, but is a possible being, has no place in my account of modal logic. (Ibid., p. 219.)6

Though this statement refers only to his account of modal logic, we can take it to indicate (and the tenor of Chihara’s book, The Worlds of Possibility, confirms this) that a thing that is not an actual but a possible being has no place in Chihara’s account of being. For him, there simply are no non-actual possible beings, and this contradicts modal realism as Chihara defines it, whether in the “full-blown sense” or in the less than full-blown one – since the other possible worlds besides the actual world that are mentioned in both of his definitions of modal realism must certainly be taken to be possible beings that are not actual (or in other words, “possible things that do not exist”7). And, of course, Chihara’s actualism is also in opposition to modal realism as I define it (see above). Because the denial of actualism is so clearly a part of modal realism, it comes as a surprise to read of “Plantinga’s Actualistic Modal Realism” in Chihara’s The Worlds of Possibility (p. 112). The solution to the puzzle is that the word “actualistic” in the cited phrase refers merely to actualism with regard to individual objects, not to actualism in a completely general sense (i.e., with regard to all entities). Chihara “find[s] it curious that Plantinga wants to assert the existence [i.e., the there-being, I take it] of merely possible worlds (possible worlds that are not actual), but not merely possible individual objects” (ibid., p. 114). Plantinga can make this differentiation, since possible worlds are for him states of affairs, and not individual objects. But the asymmetry in his views regarding the actuality of individual objects and the actuality states of affairs is indeed somewhat curious. In any case, Plantinga is a modal realist, 6

Compare the occurrence of “exist” in the first quotation from Chihara with the occurrence of “exist” in this third quotation. Do the two occurrences have the same meaning? No, the first occurrence of “exist” (reading “there exist other possible worlds” as “there are other possible worlds that exist”) has the First Ontological Meaning of Existence, and the second occurrence of “exist” the Second Ontological Meaning of Existence. (For these meanings, see Section 1.4.) 7 Note that taking “to exist” to mean as much as “to be actual” (that is, employing the Second Ontological Meaning of Existence) is the only way to make reasonable sense of the phrase “possible thing that does not exist.” The notion of a possible thing that does not exist in the sense of not being identical with itself (employing the First Ontological Meaning of Existence) is a blatantly self-contradictory notion.

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according to Chihara’s original definition of modal realism (presented above), although he – unlike David Lewis – is not a modal realist “in the full-blown sense of the term” (presented above), since “Plantinga’s possible worlds are abstract” (ibid., p. 113). I myself find it curious that Chihara feels constrained to assert that “it is ... clear that Plantinga is not a Modal Realist in my [Chihara’s] sense. For Plantinga does not assert the existence of possible worlds that are as real and concrete as the actual world” (ibid.). Apparently, Chihara has forgotten his own original distinction between “Modal Realism, as I use the term,” and “Modal Realism, in the full-blown sense of the term.”

But let us continue. Chihara asks a question and immediately provides his own answer to it: How is it that these philosophers [“the many outstanding philosophers who, in varying degrees, believe in possible worlds of some kind or other,” says Chihara] have come to believe in possible worlds? Although many different explanations can be given, it is clear that the striking success of a kind of semantic theory known as “possible worlds semantics”, especially in the area of modal logic, has played a major role in making the postulation of possible worlds attractive. (The Worlds of Possibility, pp. 1-2.)

I am inclined to believe that this empirical “played-de-facto-a-major-role” explanation of a certain psychological phenomenon among present-day outstanding philosophers is, as far as it goes, correct. But the striking success of possible-worlds-semantics was certainly no motive for Leibniz – the most outstanding believer in non-actual possibilia, including possible worlds – to believe in such things. He had rather different motives (see the above quotation), which I shall immediately address and which I consider to be of much greater philosophical weight and fundamentality than the mere usefulness of possible worlds for providing model-theoretic semantics for modal logic. We may rest assured that possible worlds are not needed for the semantics of modal logic.8 The deeper philosophical reasons for modal realism lie elsewhere. For example, in the considerations that Leibniz offers in the above quotation from one of his letters to Arnauld. Stripped of their theological 8

My own efforts to show that the semantics of modal logic can do without possible worlds can be found in my 1995 paper “Ontologically Minimal Logical Semantics.” There, in stating truth-rules for a modal language, I employ modal operators (just as, in standard model-theoretic semantics for a truth-functional language, truth-functional operators are invariably employed). My approach in that paper is, therefore, consistent with modalism (the doctrine that modal concepts are to be taken as primitive concepts), but, of course, it does not entail modalism.

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trappings and with some of their implications filled in, we can reformulate these considerations as follows: (1) Notions like chance, freedom, contingency are indispensable modal notions, and their full content – the way we actually mean them – and the extent we in fact assume that there are instances of them can only be maintained within modal realism: the doctrine that there are possible beings which are (ontologically) independent of the human mind and which are not actual. (2) And, in fact, we can speak of non-actual possible beings and form true predications about them. For we can certainly speak of actual beings and form true predications about them, and these very same beings might have turned out to be non-actual, while still retaining at least some of the properties we now truthfully attribute to them. Here, (2) is a supporting, defensive argument for modal realism, addressing certain (rather modern) worries regarding non-actual possibles. Leibniz’s main reason for modal realism, however, is contained in (1). According to (1), denying modal realism while retaining the full content of the notions chance, freedom, contingency will lead to the result that there are no such things: no chance, no freedom, no contingency; and eschewing the implication of modal realism when explicating the notions of chance, freedom, contingency will leave us with emasculated ersatz for these notions, with mere shadows of their true selves, so to speak. Thus, what Leibniz can be taken to argue is this: modal realism is implicit in modal common-sense assumptions employing modal common-sense meanings; if modal realism is dislodged, then these common-sensical elements will be dislodged as well; and this is just too high a price to pay. It seems to me that neither Charles Chihara nor the hosts of others that have opposed modal realism have done enough to seriously endanger the cogency of these elementary Leibnizian considerations. Modal realism just isn’t a mere figment of sophisticated modal logicians, who are so blinded by the dazzling efficiency of their conceptual tools that they ontologize them. It is already implicit in everyday discourse (for example, if somebody says that of the several possible alternatives he indeed had, he put into effect the one that fitted best his well-considered intentions). It remains to make a few comments about (2), which may seem rather puzzling. An extended metaphor will be helpful for understanding the thrust of the Leibnizian

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argument. Suppose actuality is like a light falling arbitrarily (perhaps due to the action of a lighting-technician) on certain, but not on all, objects in a room that, if left alone, is entirely dark.9 Clearly, the objects in that room have most of their properties independently of whether or not the light (of actuality) happens to fall upon them, and, consequently, we can speak about the objects in the dark and form true predications about them to an extent that is in principle not much less than the extent we can form true predications about the objects in the light. It must be admitted that error is much more likely to occur with regard to the objects in the dark than with regard to the objects in the light. Nevertheless, there is no reason to assume that we are always in error with regard to the objects in the dark, or that we have no clues for the predications we form about them (so that the truth of these predications, if they turn out true, can only be utterly accidental). All the objects in the room are variants of each other, and therefore we know, for example, that some objects in the dark are similar, respectively dissimilar, in certain nameable respects to certain objects in the light. (A certain merely possible Adam – to use a favorite example of Leibniz’s – is similar to the actual Adam in, say, being male, but different from him in not having sinned.)

2.3

Conceivability and (Ontic) Possibility

The relationship between conceivability and possibility is the subject of a disproportionally large portion of the literature on modality,10 one reason for this being surely the important role that conceivability arguments play in the philosophy of mind. In this section, I am going to make my own contribution to this portion of the literature by saying what seems necessary to me to say with regard to the relationship between conceivability and possibility. I will also touch on relevant thought by David Chalmers, Robert Stalnaker, and George Bealer. (1) Is conceivability necessary for ontic possibility? The answer to this question seems simple and uncontroversial: “No, it is not. Something can be ontically possible and totally inconceivable (for us human beings).” But there are second thoughts. Has anybody ever presented an example of an ontic possibility that is inconceivable? It seems 9

The non-metaphorical conception of actuality behind this metaphor is expounded in my book Ereignis und Substanz. Essentially, it is also the conception of actuality Leibniz had. 10 The voluminous collection Conceivability and Possibility, which appeared in 2002, is merely the peak of the iceberg.

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(ontically) impossible to present an example of an ontic possibility that is inconceivable. This seems just as impossible as presenting an example (in the full sense of the phrase) of, say, a rose that is never an object of consciousness. Nevertheless, it can hardly be doubted that there are many roses that are never objects of consciousness. By analogy, it would seem that, even if it is impossible to present an example of an ontic possibility that is inconceivable, it can hardly be doubted that there many ontic possibilities that are inconceivable. But is it really impossible to present an example of an ontic possibility that is inconceivable? This depends on the precise meaning that is accorded to the word “inconceivable” (and hence, of course, also to the word “conceivable”). It may mean: (a) not understandable, (b) not imaginable, (c) not mindable (i.e., not capable of becoming an object of the – human – mind). What, in turn, is the possibility indicated by “-able” in “conceivable” (“understandable,” “imaginable,” “mindable”)? Though circularity or infinite regress are not automatic consequences, it had better not be the same concept of possibility as the concept of possibility whose application to a certain state of affairs is to be established in a conceivability argument. The simplest plan is to take “conceivable” to be just short for “conceived by somebody at some time” (and analogously for “understandable,” “imaginable,” “mindable”). Would – should – anybody call anything “conceivable” if it is never conceived by anybody? I do not think so. (The treatment of “conceivable for x” is of course analogous: it is just short for “conceived by x at some time.”)

Now, one can obviously present examples of ontic possibilities that are not imaginable: for example, the possibility that space has more than three (spatial) dimensions. The difficulty here lies as much in the three-dimensional spatial constitution of perception as in the disjunctive nature of the state of affairs to be imagined. Even if we could somehow manage to imagine space having, say, four spatial dimensions, we would not thereby have managed to imagine space having more than three dimensions: we would only have managed to imagine a state of affairs that has much more content than – and logically includes – this latter state of affairs. The principle if one imagines a state of affairs, then one imagines every state of affairs logically contained in it just isn’t true.

It is rather more difficult to present to a forum of philosophers convincing examples of ontic possibilities that are not understandable.

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What makes this difficult is the lack of modesty philosophers generally display regarding their capability of understanding possibilities. But here is a suggestion. Either it is possible that all conscious experiences are physical, or it is possible that some conscious experience is not physical. The history of philosophical thought rather forcefully suggests that, whichever of these two alternatives is true (and at least one of the two must be true), there will be at least one ontic possibility that is not understandable. What is unequivocally impossible, however, is to present an example of an ontic possibility that is not mindable. The reason is that by presenting a state of affairs one is already “minding” it, and hence it must be mindable. Thus, by the very act of presenting an example of an ontic possibility – that is, an example of a state of affairs that is ontically possible – one necessarily displays it as mindable. The three ways pointed out above of understanding “inconceivable” (and therefore “conceivable”) will have to be kept in mind. And, in fact, further distinctions of interpretation will become necessary. (2) Is conceivability sufficient for ontic possibility? This is precisely the question that is discussed in the many papers that have grown around the conceivability arguments in the philosophy of mind, the reason for this being that all these arguments proceed on the premise that conceivability is in some significant sense indicative of ontic possibility. The answer to question (2) depends, of course, on what is meant by “conceivable.” And note: if the answer to question (2) is yes, for some particular meaning of the word “conceivable,”11 then there is a further question: is it really helpful for the intended purposes if “conceivable” is accorded this particular meaning? If “conceivable” is taken to mean as much as “mindable,” then it is not true that conceivability is sufficient for ontic possibility. The state of affairs that 1 is a larger number than 2 is capable of becoming an object of the mind: it is mindable, but it is nevertheless not ontically possible. If “conceivable” is taken to mean as much as “understandable,” then the situation is more complicated. There is a minimal sense of 11

Already Leibniz gave the affirmative answer to question (2), for one particular meaning of the word “conceivable.” For he says: “For calling something possible, it is, in my eyes, sufficient that one can form a notion of it” (in the quotation at the beginning of the previous section).

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“understandable” according to which a state of affairs is understandable if, and only if, some sentence is understandable which expresses that state of affairs. If “conceivable” is taken to mean as much as “understandable” in that minimal sense, then, once again, it is not true that conceivability is sufficient for ontic possibility. The state of affairs that 1 is a larger number than 2 is understandable in the sense of being expressed by an understandable sentence, but it is nevertheless not ontically possible. The minimal sense of “understandable” was, however, not intended when understandability was brought up as a possible interpretation of conceivability in the discussion of question (1) above. Rather, “understandable” was there intended to mean as much as “really graspable as an ontic possibility.” Presupposing precisely this interpretation, it was suggested above that neither the state of affairs that all conscious experiences are physical is understandable (i.e., really graspable as an ontic possibility), nor the state of affairs that some conscious experience is not physical. If this is indeed the case, then there is bound to be at least one ontic possibility that is not understandable, and hence not conceivable if conceivability is identified with understandability. However matters may stand in this regard, it seems undeniable that every state of affairs that is really graspable as an ontic possibility is ontically possible. Thus, if “conceivable” is taken to mean as much as “really graspable as an ontic possibility,” then, indeed, conceivability is sufficient for ontic possibility. But now we must ask the further question: is it really helpful for the intended purposes – mainly, the success of the conceivability arguments – if “conceivable” is accorded the meaning of “really graspable as an ontic possibility”? Unfortunately, it is not helpful. For when the premise that a certain state of affairs is conceivable is used in an argument, then it is usually intended as epistemically establishing or at least supporting the – initially doubtful – conclusion that the state of affairs in question is in fact ontically possible. But the premise that a certain state of affairs is really graspable as an ontic possibility cannot epistemically establish or support the conclusion that the state of affairs in question is ontically possible, since adducing that premise with the intention of establishing or supporting this conclusion is a move that is blatantly question-begging. It is comparable to the procedure of someone who, wishing to establish that the state of affairs p obtains, adduces for establishing this conclusion the premise that p is known to obtain. If, in turn, “conceivable” is taken to mean as much as “imaginable,” then the situation is bifurcated, as it was in the previous case (of taking

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“conceivable” to mean as much as “understandable”). There is a minimal sense of the word “imaginable” according to which a state of affairs is imaginable if, and only if, one can form a (not necessarily detailed) mental image of it. If “conceivable” is taken to mean as much as “imaginable” in this minimal sense, then, once again, it is not true that conceivability is sufficient for ontic possibility. For we can form mental images of certain arrangements of geometrical objects (that is, of certain geometrical states of affairs), which arrangements, however, are not ontically possible.12 If, on the other hand, “conceivable” is taken to mean as much as “imaginable” in the sense of “consistently imaginable,” then, indeed, conceivability is sufficient for ontic possibility; for then conceivability is sufficient for logical (or conceptual) possibility, and logical possibility is certainly sufficient for ontic possibility. Or so say I. While it is uncontroversial that everything that is ontically possible is logically possible, it is often denied that everything that is logically possible is ontically possible. Typically, this denial is used in criticizing conceivability arguments. It is conceded that a certain conceivability argument or other shows the logical possibility of X, meaning: the possibility of X in a broad, a priori sense. But, it is said, logical possibility is only epistemic possibility; it does not entail ontic – or “metaphysical”13– possibility, which, however, is (we may suppose) just the relevant possibility: the possibility needed for the conclusion of the argument. The failure of entailment invoked in such criticism is usually claimed to be made plausible by some example like the following. It is ontically (“metaphysically”) impossible that water is not H2O, but it is nevertheless logically possible (and conceivable) that water is not H2O. Hence the logical possibility that water is not H2O must be a mere epistemic possibility. As David Chalmers puts it: 12

For a simple example, see Oscar Reutersvärd’s “Opus 2B,” reproduced in Das verzauberte Auge, p. 70. 13 “Metaphysical” in association with “possibility” and “necessity” can be used either in a broad or in a narrow sense. In the broad sense, metaphysical possibility and necessity is just ontic (or alethic) possibility; this is the sense used in the present context. In the narrow sense, metaphysical possibility is broader than nomological possibility, but narrower than logical possibility, and metaphysical necessity, in turn, narrower than nomological necessity, but broader than logical necessity (all six modalities just mentioned being ontic, alethic modalities). For details, see Section 4.6. In that section, it is also shown that the ambiguity (just pointed out) of the expression “metaphysically possible/necessary” does, as a matter of fact, generate confusion.

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It is often said that it is conceivable that Hesperus is not Phosphorus, or that water is not H2O, or that heat is not the motion of molecules, but none of these states of affairs is in fact possible. In these cases, we have a posteriori necessities and impossibilities, out of reach of a priori methods. … [T]here remains a sense in which a world with XYZ in the oceans can be seen as satisfying the statement ‘water is not H2O’. … [T]here is clearly a broad sense in which it is epistemically possible that water is not H2O, in that the hypothesis is not ruled out a priori. Intuitively, there are ways our world could turn out such that, if they turn out that way [“such that, if our world turns out in one of these ways”?], it will turn out that water is not H2O. (“Does Conceivability Entail Possibility?”, p. 161, p. 162.)

Analogously, it is claimed that the conceded logical possibility that pain is not a brain-state is nothing for a physicalist to worry about. For even if the state of affairs that pain is not a brain-state is logically possible, this means only that it is epistemically possible; the state of affairs may yet be ontically impossible (to the physicalist’s complete satisfaction), just like the state of affairs that water is not H2O. These thoughts – loaded to the helm with unclarity (What is it, precisely, that makes logical possibility epistemic? Is logical possibility really something that is intrinsically related to us? If there had not been any human beings, would there also not have been logical possibility?) – are rendered irrelevant if one keeps in mind that logical possibility is in this book simply the most comprehensive (or, in the intensional idiom, the weakest) ontic possibility: the ontic possibility entailed by all ontic possibilities, and that a state of affairs is ontically possible if, and only if, there is some ontic possibility that applies to it. It follows immediately that a state of affairs p is ontically possible if, and only if, it is logically possible.14 Thus, when Robert Stalnaker writes that [s]ome have questioned the move from the premise that zombies are conceivable, or conceptually possible, to the conclusion that they are metaphysically possible, and I will consider whether there might be a gap between conceptual and metaphysical possibility, but for the moment I will assume that metaphysical possibility is possibility in the widest sense (“What Is It Like to Be a Zombie?,” p. 386),

14

If p is logically possible, then the most comprehensive ontic possibility applies to it, and hence p is ontically possible. Conversely, if p is ontically possible, then some ontic possibility applies to it, and hence also the most comprehensive ontic possibility applies to it, which means that p is logically possible.

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then he has in my eyes (at least “for the moment”) already asserted that there is no gap between ontic (metaphysical) possibility and logical (conceptual) possibility; for, according to him, ontic (metaphysical) possibility is possibility in the widest sense, and – agreeing with him – I merely add that logical (conceptual) possibility is also just this: possibility in the widest sense. I believe that I am entirely within my rights if I understand logical possibility in this way. The identity of logical possibility and ontic possibility is also endorsed by George Bealer: What about ‘metaphysical possibility’? This is a technical term which Kripke stipulatively introduced, solely for heuristic purposes, as a synonym of his term ‘logically possible’ – that is, of ‘possible’ [in the widest sense; U.M.]. Thus, according to this standard philosophical usage, p is possible iff p is logically possible iff p is metaphysically possible (“Modal Epistemology and the Rationalist Renaissance,” p. 78).

But what, then, about the intuition that is shared by so many: that the state of affairs that water is not H2O is logically, “epistemically” possible, but not ontically, “metaphysically” possible? – This intuition is based on confusing two different states of affairs that can each be connected with the sentence “water is not H2O.” For “water” can be taken to designate, in virtue of its meaning, either (I) water as it actually is, or (II) a substance that is phenomenally indistinguishable from water as it actually is. Here “actual” has the – usual – sense, in which it makes the reference of an expression in each linguistic context be identical with its reference at the actual world.

If the first designatum of “water” is effective, then it is neither ontically nor logically possible that water is not H2O; for water as it actually is cannot (logically or ontically) be anything but H2O.15 If, however, the second designatum of “water” is effective, then it is both ontically and logically possible that water is not H2O; for a substance that is phenomenally indistinguishable from water as it actually is can (logically and ontically) be something else than H2O.16 But according to both alternatives for the designatum of the word “water” in the sentence “water is not H2O,” logical and ontic possibility are in unison: either they both 15

According to possible-worlds-semantics, the intension of “water is not H2O,” with “water” taken to have the first designatum, is the empty set. 16 According to possible-worlds-semantics, the intension of “water is not H2O,” with “water” taken to have the second designatum, is not the empty set.

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apply, or they both do not apply, to the state of affairs that corresponds to that sentence according to the effective alternative in point; according to neither alternative do we have a state of affairs before us that is logically possible but ontically impossible. The contrary impression can only be generated by mixing up the two alternative designata, or rather: by vacillating between them. Although Bealer endorses the identification of logical and ontic possibility (see the above quotation), he, unfortunately, is misled by his righteous zeal against the two-dimensionalism promulgated by Chalmers and others (see below) into offering the following objection, which strikes out against the analysis in the previous paragraph as much as against twodimensionalism, which objection, however, does not do justice to the psychological fact that so many people believe that they can distinguish logical possibility as “epistemic” from ontic, “metaphysical” possibility: Kripke and Putnam taught us that, for every possible situation, the English word ‘water’ denotes H2O in that situation. This just reflects the standard use of the English expressions ‘water’ and ‘denote’. It is simply false that, for some possible situation, the English word ‘water’ denotes XYZ in that situation! (“Modal Epistemology and the Rationalist Renaissance,” p. 89.)

Is that so? I do not think so. And I do not believe that Kripke and Putnam taught us what Bealer says they taught us. I believe that, by almost everybody outside of philosophy, “water” (or the corresponding word in other languages) is taken to mean as much as is meant by “the colorless, transparent liquid occurring on earth as rivers, lakes, oceans, etc., and falling from the clouds as rain” (see Webster’s New World Dictionary, Second College Edition of 1976), which means that though “water” – for philosophers and other people alike – denotes, in virtue of its meaning, H2O in the actual situation, it does certainly no do so, for most people, in every possible situation. No, for most people, “water” – in virtue of its meaning – denotes, in some possible situation, XYZ: a substance that is phenomenally indistinguishable from water as it actually is, but which is not water as it actually is. This, and not what is offered by Bealer, truly describes the standard use of the English expression “water.” And according to this standard use, it indeed turns out to be an empirical, an a posteriori truth that water is H2O, but – contrary to Chalmers (see the above quotation) – this truth is not an a posteriori necessity, because it is not a necessity at all (since in

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some worlds it is simply false that water is H2O, given the standard use of the English term “water”).17 But, fortunately, there is also the philosophical use of the term – the use that Kripke and Putnam taught us –, according to which “water,” in virtue of its meaning, can also be taken to denote, in every possible situation (including the actual one), water as it actually is. And since water is in actual fact H2O and “H2O” cannot but denote in every possible situation H2O, the philosophical use – not the standard use – of the English term “water” does indeed result in its being (ontically) necessary that water is H2O. Now, do we here have an a posteriori necessity after all? – It is a necessity, but calling it “a posteriori” is just like calling it an “a posteriori necessity” that the number of planets is 9, given that “the number of planets” is taken to designate, in virtue of its (modified) meaning, the actual number of the planets (and not whatever number the number of planets turns out to be). And proudly announcing that science, by verifying that water(-as-it-actually-is) is H2O, has empirically discovered the essence of water is just like saying that science, by verifying that the number of planets(-as-it-actually-is) is 9, has empirically discovered the essence of the number of the planets. The above-described semantical complexities are reflected in socalled two-dimensional semantics, but they are reflected there in a distorted and therefore misleading manner. Chalmers (like Bealer) starts out with presupposing erroneously that “water” can only be properly understood in the philosophical Kripke-way, and not also in the non-philosophical Webster’s-New-World-Dictionary-way. This presupposition, of course, must lead to certain difficulties, which Chalmers describes as follows: When we apparently conceive of a world in which water is not H2O, we conceive of a situation in which some other substance (XYZ, say) is the clear liquid surrounding us in the oceans and lakes, and so on. And this situation is indeed metaphysically possible – so our act of conceiving has indeed yielded access to a possible world. It is just that in a certain sense we have misdescribed it in calling it a world where water is not H2O, or a world in which water is XYZ. If Kripke is right, it is in fact a world in which XYZ is watery stuff, but not water, and a world in which the only water that exists is H2O. (“Does Conceivability Entail Possibility?”, p. 162.)

17

Exactly analogous observations as to “water is H2O” apply to “Hesperus is Venus,” “the number of planets is 9,” “the president of the United States in the year 2004 is George W. Bush,” etc.

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Nevertheless, Chalmers concedes (ibid.) that “there remains a sense in which a world with XYZ in the oceans can be seen as satisfying the statement ‘water is not H2O’.” Indeed. And in order to combine this sense with his deeply entrenched Kripkeanism, Chalmers offers, on the pages of his paper that follow the above quotation (and in other places), a body of ideas whose bare skeleton is the following: The sentence “water is not H2O” (for example) has two intensions, a primary (or “epistemic”) intension and a secondary (or “subjunctive”) intension. Corresponding to these two intensions, there are two possibilities regarding “water is not H2O,” the primary (or “epistemic” [or logical]) possibility and the secondary possibility. It is primarily possible [that water is not H2O]1, it is secondarily impossible [that water is not H2O]2, where [that water is not H2O]1 is the primary, and [that water is not H2O]2 is the secondary intension of “water is not H2O.” This is the way Chalmers sees the situation, which, however, is much better described – I believe – in the following way: Every sentence has, in a given context, at most one intension: the state of affairs (others would prefer: the proposition) expressed by it in that context. And there is just one global possibility: logical possibility, which is an ontic possibility – the broadest ontic possibility – and not intrinsically related to epistemic matters at all. But some sentences, as for example “water is not H2O,” have different intensions in different contexts. In most contexts, “water is not H2O” expresses an intension that is logically possible (namely, the state of affairs that a substance that is phenomenally indistinguishable from water as it actually is is not H2O). In some (philosophical) contexts, however, “water is not H2O” expresses an intension that is not logically possible (namely, the state of affairs that water as it actually is is not H2O). Returning after this (relevant) excursion regarding epistemic, ontic and logical possibility to the focus of this section on conceivability and possibility, we must, when identifying conceivability and consistent imaginability (as was envisaged above), once again ask the crucial, further question: is it really helpful for the intended purposes – mainly, the success

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of the conceivability arguments – if “conceivable” is accorded the meaning of “consistently imaginable”? No question in this direction is asked by David Chalmers in his longish paper “Does Conceivability Entail Possibility?” Chalmers does arrive at the conclusion that, indeed, there is a sense of “conceivable” in which it entails “possible” (see ibid., p. 194). But this is no great surprise. What is crucial is whether the entailment is (at least sometimes) useful for establishing a possibility-thesis that stands in doubt. For certainly not every entailment is useful for establishing a thesis that stands in doubt, not even every non-trivial entailment. Consider: knowledge non-trivially entails truth;18 but one can certainly not establish a doubtful thesis by claiming that one knows it to be true (not even if it is true that one knows it to be true).

And the answer to the question asked just before the above note must once more be this: it is not helpful. In justification, we can repeat with regard to consistently imaginable what has been said above regarding really graspable as an ontic possibility. When the premise that a certain state of affairs is conceivable is used in an argument, then it is usually intended as epistemically establishing or (at least) supporting the, initially doubtful, conclusion that the state of affairs in question is in fact ontically possible. But the premise that a certain state of affairs is consistently imaginable cannot epistemically establish or support the conclusion that the state of affairs in question is ontically possible, since adducing that premise with the intention of establishing or supporting this conclusion is a question-begging move – though, in this case, a move not quite as blatantly question-begging as the analogous move in the case of really graspable as an ontic possibility. The situation is once again different if “conceivable” is taken to mean as much as “apparently consistently imaginable.” True, conceivability in this sense is not sufficient for ontic possibility. But normally in a conceivability argument, the true premise that a state of affairs can apparently be consistently imagined does epistemically support (though not establish) the conclusion that that state of affairs is in fact ontically possible; this time, there is no circularity involved. Normally, even the mere true claim that a state of affairs can be imagined epistemically supports (to a lesser extent) the conclusion that the state of 18

If it were a trivial matter that knowledge entails truth, then it would be utterly unreasonable to draw this entailment into doubt. But it is drawn into doubt, for example, by Jay Rosenberg, and not utterly unreasonably, in his paper “Was epistemische Externalisten vergessen [What Epistemic Externalists Forget].”

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affairs in question is ontically possible. It should be noted that the imaginability-claims stand to ontic possibility as the visual-experienceclaims stand to factuality. Hence: in the absence of any evidence to the contrary, the imaginability of a state of affairs p, and even more so the apparently consistent imaginability of p, does render epistemic support to the claim that p is ontically possible. This indicates who has the burden of proof (meant to include also the burden of disproof) regarding the crucial step in a conceivability argument in which “conceivable” is understood to mean apparently consistently imaginable. Sincerity being taken for granted, it is not the proponent of the argument who has the burden of proof regarding this step; it is the opponent who has this burden. If, for example, David Chalmers claims that he can conceive of a zombie world (that is, a world which is in all physical respects exactly like the real world, but devoid of all consciousness) – meaning: that, apparently, he can consistently imagine such a world – and that, therefore, such a world is ontically possible,19 then the burden of proof lies with his opponents: it is incumbent upon them to show that this particular reasoning is implausible. As long as they have not done so, Chalmers’ claim of conceivability renders strong epistemic support to the conclusion that a zombie world is, indeed, ontically possible (which implies, in turn, that consciousness does not supervene logically upon the physical), especially in view of the fact that he is not alone in making this claim and in drawing the inference. The conceivability claim – the claim of being apparently able to imagine p consistently – does not logically imply the possibility claim, the claim that p is ontically possible: p, though apparently consistently imaginable, might nevertheless be ontically impossible. But as long as this impossibility has not been demonstrated or at least made sufficiently credible, the possibility claim is strongly supported – strongly made credible – by the conceivability claim. This – no more and no less – is the justification that can be given for conceivability arguments, which all involve, as a crucial step, the inference of ontic possibility from conceivability. If conceivability is interpreted as apparently consistent imaginability, then one is epistemically justified in moving from the true claim that p is conceivable to the claim that p is ontically possible, as long as there is no considerable evidence against doing so. The range of this verdict is much extended, and appropriately so, if one understands “to imagine” – and hence “to conceive,” that is: “to imagine apparently in a consistent way” – not in the narrow, perceptual 19

See The Conscious Mind.

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sense, but extends its meaning to apply also to mental states that are not tied to mental images, in other words: if imagination is seen to be an intellectual capacity as much as – perhaps even more than – a perceptual one. The subject of our epistemic grasp on ontic modalities will be taken up again in Section 5.5.

2.4

On Peter van Inwagen’s Modal Skepticism

Skepticism – of more or less great extent – is usually expressed as a negative thesis about our human capabilities regarding the attainment of some sort of knowledge. This kind of skepticism is epistemological skepticism. But besides epistemological skepticism, there is also conceptual skepticism. Conceptual skepticism simply consists in the denial that there is a certain concept, hence in the denial that a certain word has indeed the concept connected to it that people believe they intend by it. As is well known, David Hume was both an epistemological and a conceptual skeptic, and so is, in fact, Peter van Inwagen, although, of course, to a much lesser extent than Hume. In this book on modality, I am specifically interested in van Inwagen’s epistemological skepticism regarding a certain type of modal knowledge, and in his conceptual skepticism regarding a certain modal concept. I begin with discussing the latter skepticism. Peter van Inwagen believes that a certain modal concept does not exist (i.e., that there is no such concept): [I]t is often supposed that there is a species of possibility that goes by that name [“logical possibility”] and that one can determine a priori whether a concept or state of affairs is logically possible. But there is no such thing as logical possibility – not, at least, if it is really supposed to be a species of possibility. (“Modal epistemology,” p. 247.)

How does van Inwagen arrive at this view?20 He begins with the concept of logical impossibility, which is real enough. Logical impossibility is an epistemological category: the logically impossible is that which can be seen to be impossible on the basis of

20

Note that van Inwagen himself does not regard the view that there is no such thing as logical possibility as a kind of skepticism.

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logical considerations alone – or, to be liberal, logical and semantical considerations alone. (Ibid.)

His denial of logical possibility is then based on the fact that one cannot infer from something’s not being logically impossible (in the sense just described) that it is “logically possible”: What I dispute is the contention that if a concept or state of affairs is not logically impossible, then it is “logically possible.” It hardly follows that, because a certain thing cannot be proved to be impossible by a certain method, it is therefore possible in any sense of “possible” whatever. (Ibid., p. 248.)

Given the interpretation van Inwagen accords to “logically impossible,” this cannot be disputed. What can be disputed, however, is the interpretation van Inwagen accords to “logically impossible.” Van Inwagen misconstrues logical impossibility; perhaps he even confuses logical impossibility with (humanly) knowable logical impossibility. That a state of affairs is logically impossible does not mean that it “can be seen to be impossible on the basis of logical [and semantical] considerations alone.” Rather, that a state of affairs is logically impossible means that it is impossible on logical grounds alone, no matter to what extent these grounds can be known or not, no matter to what extent human beings can manage to see what these grounds imply. Logical impossibility thus interpreted – correctly, I believe; at least I am no less within my rights to interpret it that way than van Inwagen takes himself to be when interpreting logical impossibility his way – is not an epistemological category: it is an ontological one. And, of course, logical possibility is no longer a non-concept under my interpretation of logical impossibility: a state of affairs is logically possible if, and only if, it is not logically impossible, that is: if, and only if, it is not impossible on logical grounds alone, that is: if, and only if, it is possible as far as purely logical grounds go. Suppose no one (who is human) could ever have seen that the negation of Fermat’s Last Theorem is impossible on the basis of (broadly) logical considerations alone. Suppose that we human beings had just not been intelligent enough for this. With the notable exception of Peter van Inwagen and perhaps a few others, everyone agrees that the negation of Fermat’s Last Theorem would have been logically impossible nonetheless. This indicates that van Inwagen has not managed to capture the standard (non-epistemic) concept of logical impossibility, of which logical possibility, as a species of possibility, is simply the negation. Nor has van Inwagen

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offered any considerations that might be able to cast the shadow of a doubt on this standard concept of logical impossibility.

But my virtual dispute with van Inwagen is largely verbal, because van Inwagen acknowledges a concept of possibility which I, for one, have no scruples to call “logical”: A proposition is physically possible if its conjunction with the laws of nature is ... well, possible. Possible tout court. Possible simpliciter. Possible period. Explanations come to an end somewhere. I can say only that by possibility I mean possibility without qualification. (“Modal epistemology,” p. 248.)

And in a footnote he adds: What I have called possibility without qualification, some have called “absolute” or “intrinsic” or “ontological” or “metaphysical” possibility. The first two seem good enough names. I don’t find “ontological” or “metaphysical” particularly appropriate tags, however. I don’t think that unqualified or absolute or intrinsic possibility is in any clear sense an ontological or metaphysical concept. (Ibid., p. 249.)

People who call the possibility van Inwagen has in mind “ontological” or “metaphysical” merely want to indicate that this possibility is not intrinsically epistemic, that it is not intrinsically relative to us, as subjects of belief; this is the rationale behind their naming-practice, which nevertheless – I am inclined to agree with van Inwagen – is somewhat infelicitous, especially if the word “metaphysical” is used. I myself have suggested the names “inner” or “intrinsic” for the possibility van Inwagen has in mind (see the Preface of this book), but usually I call it “logical” or “conceptual” possibility. It can be shown to coincide with the concept of ontic (or alethic) possibility, that is, with the concept of being possible in some ontic (alethic) sense or other (see the previous section). The very same concept that van Inwagen calls “possibility without qualification” is called “broadly logical possibility” by Alvin Plantinga and many others, and “metaphysical possibility” by David Chalmers and many others; quite common is also the designation “possibility in principle” for it. Whatever we call it – “without qualification,” “in principle,” “logical,” or “metaphysical” – we all intend, I trust, the very same objective, nonepistemic concept: ontic possibility in the widest sense. And there can be

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no doubt that there is such a concept (and hence also the corresponding concept of ontic necessity in the narrowest sense).21 So much for van Inwagen’s conceptual skepticism regarding modality, which he does not himself avow under the name of “skepticism.” I now turn to discussing his epistemological skepticism regarding modality, which he does avow under the name of “modal skepticism.” Like his conceptual skepticism, his epistemological skepticism regarding modality is local. According to van Inwagen, we do have modal knowledge, but in some areas we don’t. According to him, we do know that it is possible that the legs and top of this table might never have been joined together, but we do not and cannot know, say, whether it is possible that I exist and nothing material exist, or whether it is possible for there to be a perfect being (see “Modal epistemology,” pp. 243-245; the examples are van Inwagen’s). To many people the preceding paragraph will suggest that van Inwagen’s beliefs regarding the knowability of modal propositions are better described as the simple denial of modal omniscience (for human beings) than as “skepticism.” Yet the two there-mentioned propositions (or states of affairs22) about the possibility-status of which, van Inwagen tells us, we must be in the dark are indeed philosophically important propositions, because they figure in philosophically important arguments: the argument for the immateriality of the (kernel of the) human person (originally conceived by Descartes, and recently defended by the author of the present book23), and the ontological proof of the existence of God (originally conceived by Anselm of Canterbury, endlessly attacked, 21

My agreement with van Inwagen in subject matter, not words, is underscored by the following statement of his: “If there were no such thing as modality without qualification, there could be no qualified modalities like physical and biological possibility and necessity. If we understand “qualified” modal statements (of any sort), we must understand “unqualified” modal statements.” (“Modal epistemology,” pp. 248-249.) The present book can in large part be read as intending to show that it is not only true that the understanding of “qualified” modal statements presupposes the understanding of “unqualified” modal statements (that is, of statements of logical possibility and necessity), but also that, in a certain (reductive) sense, the understanding of “unqualified” modal statements is sufficient for understanding “qualified” ones. 22 In my eyes, there is a difference between propositions and states of affairs (see Section 4.1) and states of affairs, not propositions, are the primary bearers of modalities. Nevertheless, both a proposition and the corresponding state of affairs can be referred to by the very same “that”-phrase, and modal predicates can, accordingly, be applied to both via one and the same sentence. 23 See chapter 3 of my The Two Sides of Being.

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endlessly defended). These arguments belong to a group of arguments which van Inwagen calls “possibility arguments” (ibid., p. 244). Possibility arguments have a common feature: using a fitting auxiliary premise, they validly (as far as logic is concerned) draw a – for some people – devastating factual conclusion (of the greatest import) from a possibilitypremise that, in the right way, corresponds to this conclusion. For example (cf. ibid.): It is possible [without qualification, or logically] that I exist and nothing material exist. Necessarily, whatever is material is necessarily material. Hence: I am not material. But although this argument – and all other possibility arguments – are momentous and controversial, and always will be, one may well ask whether this fact alone can be justification enough for even a local “modal skepticism,” as van Inwagen calls his position (ibid., p. 245), namely, an epistemological skepticism regarding the propositions expressed by the possibility-premises of possibility arguments. I, for one, believe that I know that it is possible (without qualification, or logically) that I exist and nothing material exist. Others believe that they know that it is impossible that they exist and nothing material exist. Others, again, are agnostics about the matter. But this pattern fits very many propositions p: some believe that they know that p is true, others believe that they know that p is not true, others again believe that they neither know that p is true nor that p is not true. Practically every non-logical proposition discussed in philosophy is of this kind. But hardly any philosopher feels constrained to become even a local (epistemological) skeptic on account of this situation. Most philosophers refuse to infer that we (that is, we human beings) do not and cannot know whether p is true from the mere fact that other philosophers do not concur with their opinions regarding the truth of p, or have no opinion whatever about it. I believe that these philosophers are quite justified in this refusal. The philosophical importance of the propositions expressed by the possibility-premises of possibility arguments and the controversial nature of these premises cannot, in itself, justify even a local epistemological skepticism regarding modality. But van Inwagen has more to offer. According to him, we do not know and cannot know whether it is possible

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(without qualification) that iron be transparent (“Modal epistemology,” p. 247). This time, I agree that we do not know; perhaps we even cannot know, and perhaps I will come to firmly believe that we cannot. Then, if someone were to call me a “local modal skeptic” on this account, I would not deny the charge, although I still would find it more appropriate if I were merely called a denier of modal omniscience, which is a very reasonable epistemological position, not a whiff of extremism about it. But van Inwagen still has more to offer, and this time it is something like a principled argument for modal skepticism (that is, for epistemological skepticism regarding modality), and for a bit more than just strictly local modal skepticism. Inspired by a paper by Stephen Yablo (“Is Conceivability a Guide to Possibility?”), van Inwagen asserts the following: To assert that p is possible, after all, is to commit oneself, willy-nilly, to the thesis that there is a whole, coherent reality – a possible world – in which p is true, of which the truth of p is an integral part. ... Although it is in a sense trivial that to assert the possibility of p is to commit oneself to the possibility of a whole, coherent reality of which the truth of p is an integral part, examination of the attempts of philosophers to justify their modal convictions shows that this triviality is rarely if ever an operative factor in these attempts. (“Modal epistemology,” p. 254.)

Accordingly, van Inwagen believes (or so I interpret him) that, for many propositions p, one cannot know that p is possible without knowing of a very comprehensive (conjunctive) totality of propositions which contains p (sufficiently representative of a world in which p is true) that it is possible; and for certain of these propositions p – not a few – attaining (via imagination in a broad, not merely imagistic sense) the latter, comprehensive knowledge is, according to van Inwagen, bound to be quite beyond the scope of human capacities (see ibid., pp. 253-258). He concludes: If the only way to determine whether a proposition … is possible is by attempting to imagine a world we take to verify this proposition [and there are such cases], then we should be modal skeptics: while we shall certainly know some propositions of this type to be possible, we shall not be able to know whether the premises of our illustrative possibility arguments are true; and neither shall we able to know whether it is possible for there to be transparent iron or naturally purple cows. (“Modal epistemology,” p. 258.)

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But even if it were indeed true that there are many propositions that cannot be known to be possible (nor known to be impossible), I do not think that epistemological holism regarding possibility (as I designate the Yablo-van-Inwagen-position) could in any of these cases be the reason of the unknowability; nor do I think that this kind of epistemological holism can serve as a basis for modal skepticism that is more than just strictly local. For I do not think that epistemological holism regarding possibility is true to any considerable extent. On the contrary, epistemological holism regarding possibility seems to me, for most propositions we human beings have to deal with, even more untrue than epistemological holism regarding truth, which is the doctrine that one cannot know that p is true without knowing of a very comprehensive totality of propositions which contains p that it is true. My argument against epistemological holism regarding the possibility of normal propositions is the following. Like every proposition (and one could as well speak about states of affairs instead of propositions), a normal proposition has a certain content. Normal propositions are just those propositions whose content is not too big, in comparison to human capacities of grasping content. The content of a proposition is of course not separable from anything that is contained in it; but it is separable from everything that is not contained in it. And what is separable from the content of a proposition (which is exactly that which is not contained in it) cannot have anything (which cannot in principle be neglected) to do with whether or not the proposition is possible – possibility being possibility “without qualification,” “intrinsic,” or “absolute” possibility (or “logical” possibility, as I prefer to call it). Thus, whether or not p is possible is determined by the content of p, or one may simply say: by what p in itself is. A way (the canonical way) of knowing the possibility or impossibility of p is, therefore, knowing p (i.e., what p in itself is) well. Thus, if we know a proposition well (and, aside from inconsistent propositions, only normal propositions – propositions with normal sizes of content – can be known well by us), then we will know whether it is possible or not; if we do not know whether it is possible or not, then we just do not know it well. The latter situation obtains, for example, with regard to the proposition of there being transparent iron: we do not know this proposition well. We do not know it well, since we have not delved deeply enough into its content. Since we do not know it well, we do not see whether there is an inner incoherence to it or not. And this, the not knowing it well, is the real reason why we do not know whether it

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is possible, and not, as van Inwagen would have it, our failure to embed it imaginatively in some specific large totality of propositions compossible with it (sufficiently representative of a world in which it is true). Perhaps we even are, qua human beings, unable to delve deeply enough into its content to be able to see whether there is an inner incoherence to it or not. But we cannot be reasonably sure of this skeptical verdict unless we have made a serious effort of coming to know the proposition of there being transparent iron better; this is done by carefully unfolding its content. In doing so, we may yet hit the point where we know it well. It should be noted that the content of a normal proposition about X will often contain large amounts of information about entities Y that are different from X. This is may give rise to the idea that there is something to epistemological holism regarding possibility after all. There isn’t.

However, it cannot be said once and for all how well is well. The relevant knowing-it-well is certainly different for different propositions. It should be noted that perfectly understanding a statement does not mean that one knows the proposition expressed by the statement well. Mathematicians perfectly understand the statement “Every positive even integer is either a prime or the sum of two primes”; but it is not known, to date, whether it is possible that every even number is either a prime or the sum of two primes. (Note that this proposition belongs to those that are possible only if they are true.) On the other hand, if one does not perfectly understand any statement that expresses proposition p, then it seems unlikely that one knows that proposition well. – Indeed, it seems unlikely. But it may sometimes be the case nonetheless: although some animals do not understand (not even imperfectly) any statement whatever, they nevertheless make – or so it is plausible to assume – accurate nonverbalized judgments concerning the possibility of certain propositions, which suggests, though it does not entail, that they know these propositions well enough, perhaps well. A starved mouse, absolutely needing to feed, has before it two exits that lead out of its hiding place: A and B. In front of A, a cat is sitting; in front of B, no cat is sitting. Both exits are, however, rather close to each other. All of these states of affairs are perceived by the mouse. The mouse, though starving, will hesitate to leave its hiding place, since it judges that it is impossible for it, in the situation, to leave the hiding place and escape the cat. Then the situation is changed. The cat is still sitting in front of B. But now there is a third exit, C, fairly far away from B. No cat is sitting in front of C. Again, all of these states of affairs are perceived by the mouse. And now

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the mouse will attempt to leave its hiding place, since it judges that it is possible for it, in the situation, and therefore also in principle, to leave the hiding place and escape the cat. (If mice are not sufficiently intelligent for this experiment, substitute other animals that are. The cat may need to be replaced as well.)

It should also be noted that if we believe to know a proposition well, it does not follow that we know it well in fact. I, for example, believe that I know well the proposition of my existing without anything material existing; but perhaps I am demonstrably mistaken about this. If it were so, this would of course put in serious doubt my claim to know that it is (logically) possible that I exist and nothing material exist. A famous example of a person who did not know a certain proposition well, but believed that he did, is Gottlob Frege. Consider the proposition that is expressed by the conjunction of the axioms of Frege’s logical reconstruction of arithmetic. Frege, for a fairly long time, believed that he knew this proposition well. And in another linguistic guise he would certainly have known it well in fact; but in the guise it had put on in his system, he did not even recognize its essential nature – for a fairly long time, until Bertrand Russell revealed the proposition expressed by the axioms of the Fregean system for what it is: contradictio horribilis, and wrote a letter to Frege. This example shows that knowing or not knowing a proposition is relative to a mode of presentation of it. Even if, in certain contexts, we can do with just two propositions – the true proposition (“tautology”) and the false proposition (“contradiction”) –, the number of the modes of presentation for propositions remains nonetheless infinite: infinitely many modes of presentation for each proposition. Hence the number of potential ways of knowing any single proposition is infinite; and so must be the number of potential ways of not knowing it, let alone the number of potential ways of not knowing it well. Correspondingly, the number of potential ways of being mistaken or just ignorant about the possibilitystatus of any single proposition is infinite, too (even if we deal with only two propositions). But it should be clear that this is no reason for epistemological skepticism regarding modality. Due to the infinity of modes of presentation for each proposition, the number of potential ways of being mistaken or just ignorant about the truth-status of any single proposition is also infinite. Is this a reason for epistemological skepticism regarding truth and falsity? Certainly not.

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It remains to be mentioned that Peter van Inwagen very clearly formulates the two central questions of modal epistemology: First, how can we know that (or find out whether) a proposition is possible when we do not know that it is true – that is, when we either know that it is false or do not know whether it is true or false? Secondly, how can we know that (or find out whether) a proposition that we know to be true is also necessary? (“Modal epistemology,” p. 251.)24

But I do not believe that van Inwagen has offered anything that might serve to answer these questions in a constructive way.

24

Compare: in Section 1.2, the Difficulty of Ascertaining Mere Possibility and the Difficulty of Ascertaining Necessity were described as the utterly recalcitrant part of the Epistemological Problem of Ontic Modalities.

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This chapter is dedicated to the formulation of a basic theory of modality. As will be seen, notwithstanding its restrictions, the theory has a pleasing richness to it, as well as formal rigor and intuitive satisfactoriness.

3.1

What Statements of Possibility and Necessity Mean

In Section 1.1, I described the semantical problem for ontic modalities in the following way: What are the (necessary and sufficient) truth conditions of sentences of the forms “it is possible that A,” “it is necessary that A,” and “if A, then B”? Here is an answer to this question in the form of a series of explicit definitions: ◊n(x) =Def S(x) ∧ ¬P(neg(x), bn) n(x) =Def P(x, bn) n

→(x, y) =Def P(y, conj(bn, x))

◊nA =Def ◊n(that A) nA =Def n(that A) A n→ B =Def n→(that A, that B). At the moment, this entire series must be quite incomprehensible. So let me explain.

3 An Onto-Nomological Theory of Modality

The predicate P(x, y) says that the state of affairs x is an intensional part of the state of affairs y. Hence ∀x∀y(P(x, y) ⊃ S(x) ∧ S(y)) is analytically true. What is meant by a state of affairs x being an intensional part of a state of affairs y can be effectively illustrated by an example: the state of affairs that this object has a surface is an intensional part of the state of affairs that this object is colored. The notion of intensional parthood is readily comprehensible with respect to states of affairs (and also with respect to properties). No antecedent comprehension of the term “possible world” is necessary for grasping that notion, and no checking of possible worlds is required in applying that notion either affirmatively or negatively. The concept of intensional parthood is more basic than the concept of possible world. For the sake of brevity, I will usually omit the modifier “intensional” and simply speak of a state of affairs being a part of another state of affairs (or of itself). “bn” denotes the Nth basis of necessity (hence “b1” denotes the first basis of necessity, “b2” the second basis of necessity, etc.). A basis of necessity is always some state of affairs or other. Hence S(bn) is analytically true. The functor “conj(x, y)” denotes the conjunction of x and y. For states of affairs x and y, their conjunction, conj(x, y), is identical with ιz[S(z) ∧ P(x, z) ∧ P(y, z) ∧ ∀u(P(x, u) ∧ P(y, u) ⊃ P(z, u))] (“the unique state of affairs of which x and y are parts and which itself is a part of every state of affairs of which x and y are parts”). There is of course a theoretical background to this identity: there are principles which, given that x and y are states of affairs, guarantee its truth (and the unique fulfillment of the describing complex predicate in the above definite description). This theoretical background is stated later in this chapter. The functor “neg(x)” denotes the negation of x, and is defined as follows: neg(x) =Def CONJy(QA(y) ∧ ¬P(y, x)) (“the conjunction of all quasi-atomic states of affairs that are not parts of x”). This definition will be elucidated below, when the theoretical background of it has been stated. We are now in a position to grasp the content of the above series of definitions. The first three definitions define modal predicates, the other three definitions define the corresponding modal sentence-connectives. Based on the first and fourth of the above definitions, “it is possible that A” in the Nth sense is true if, and only if, the negation of the state of affairs that A is not a part of the Nth basis of necessity. In other words: ◊nA

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≡ ¬P(neg(that A), bn) is a logical truth (on the basis of the first and fourth of the above definitions). Based on the second and fifth of the above definitions, “it is necessary that A” in the Nth sense is true if, and only if, the state of affairs that A is part of the Nth basis of necessity. In other words: nA ≡ P(that A, bn) is a logical truth (on the basis of the second and fifth of the above definitions). Based on the third and sixth of the above definitions, “if A, then B” in the Nth sense is true if, and only if, the state of affairs that B is part of the conjunction of the Nth basis of necessity with the state of affairs that A. In other words: A n→ B ≡ P(that B, conj(bn, that A)) is a logical truth (on the basis of the third and sixth of the above definitions). Clearly, the above definitions do not fully specify modal concepts, in one word: modalities. For the full specification of a modality, one needs to specify the relevant basis of necessity. If we intend to speak about an ontic modality, then the corresponding basis of necessity needs to be specified ontically. But there is certainly more than one ontically specifiable basis of necessity. For the time being, let me indicate two salient ontic bases of necessity: b1: the basis of logical (or conceptual) necessity – this is simply the minimal (or “tautological”) state of affairs (for further clarification, see the next section). b2: the basis of nomological (or natural) necessity – this is the state of affairs which is the conjunction of all states of affairs that are laws of nature. Given the above definitions, it is now clear that ◊1A means as much as “it is logically possible that A,” and ◊2A as much as “it is nomologically possible that A” (and the corresponding senses of 1A and 2A are likewise clear). Even so, as long as the general theory of states of affairs that provides the background for the above definitions is not specified, the content of an assertion of ◊1A remains vague. The content of an assertion of ◊2A remains vague even after that specification (which is provided in the next section). For giving precise content to ◊2A, one has to specify, in addition to the general theory of states of affairs, the concept of law of nature, and one has to specify which states of affairs are laws of nature in the sense of the specified concept of law of nature. These tasks constitute a

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considerable philosophical (and scientific) problem: the problem of natural (or nomological) necessity. It indicates a much more general problem: Can just any state of affairs be a basis of necessity, and if not, what distinguishes a state of affairs that can be a basis of necessity from a state of affairs that cannot be a basis of necessity? This question will be tackled in due course. Even here I note that the bases of necessity implicitly invoked in making assertions of the form “if A, then B” are singularly unstable, even if those assertions are meant in an ontic sense, and vary from context to context much more so than the bases of monadic modalities do. In fact, they may even vary in the same context (of utterance). This generates the illusion that inferences that are prima facie logically valid for “if _, then _” – for example, the transitivity-inference: If A, then B. If B, then C. Therefore: If A, then C – are, more closely considered, not logically valid for it. (The transitivity-inference is logically valid for “if _, then _” if the same basis of necessity is employed for all three conditionals involved; but the transitivity-inference can easily fail to be valid if the basis of necessity is allowed to vary.) For more on this, see Chapter 5.

3.2

The Basic Mereology of States of Affairs

The semantics of modality outlined above – without the application of model theory, simply by presenting explicit definitions of modal terms – is utterly incomplete without stating the ontological background theory that is connected with it. Here is this background theory: P0

∀x∀y(P(x, y) ⊃ S(x) ∧ S(y)).

P1

∀x∀y∀z(P(x, y) ∧ P(y, z) ⊃ P(x, z)).

P2

∀x(S(x) ⊃ P(x, x)).

P3

∀x∀y(P(x, y) ∧ P(y, x) ⊃ x = y).

P4

∃z[S(z) ∧ ∀x(S(x) ∧ A[x] ⊃ P(x, z)) ∧ ∀y(S(y) ∧ ∀x(S(x) ∧ A[x] ⊃ P(x, y)) ⊃ P(z, y))].

P5

∀z∀z´(S(z) ∧ S(z´) ∧ ∀x(QA(x) ∧ P(x, z) ⊃ P(x, z´)) ⊃ P(z, z´)).

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P6

∀x[P(x, CONJyA[y]) ∧ ¬M(x) ⊃ ∃k´(P(k´, x) ∧ ¬M(k´) ∧ ∃z(P(k´, z) ∧ A[z]))].

P7

w* ≠ k*.

P8

QC(w*).

P9

w* ≠ t*.

P10 A ≡ O(that A). These eleven statements or schemata of statements are the basic principles of a mereology of states of affairs; they contain some defined terms which I will explain as I go through them. The first four principles describe the basic properties of P as the relation of (proper or improper) intensional parthood between states of affairs. Note that P3 formulates an identity criterion for states of affairs. P4 states that for the states of affairs satisfying an arbitrary description A[x] there is a smallest state of affairs that comprises them all. P3, moreover, requires that that state of affairs is unique. In other words, for any description A[x] there is the unique state of affairs which is the sum or conjunction of all states of affairs that satisfy A[x]. Accordingly, we can formulate the following definition: D1

CONJxA[x] =Def ιz[S(z) ∧ ∀x(S(x) ∧ A[x] ⊃ P(x, z)) ∧ ∀y(S(y) ∧ ∀x(S(x) ∧ A[x] ⊃ P(x, y)) ⊃ P(z, y))].

The fulfillment of the so-called existence-condition1 for this definition by definite description (ι is the operator of definite description) is guaranteed for all predicates A[x] by P4. The fulfillment of the uniqueness-condition2 for it is guaranteed for all predicates A[x] by P3. Here are two prominent states of affairs that can be defined as conjunctions: D2

t* =Def CONJx¬S(x).

D3

k* =Def CONJxS(x).

1 2

One should rather call it “the at-least-one-condition.” One could also call it “the at-most-one-condition.”

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One can easily prove that CONJx¬S(x) is identical with ιz[S(z) ∧ ∀y(S(y) ⊃P(z, y))] (“the state of affairs that is a part of every state of affairs”), and that CONJxS(x) is identical with ιz[S(z) ∧ ∀x(S(x) ⊃ P(x, z))] (“the state of affairs of which every state of affairs is a part”). Thus, for obvious reasons, k* can be called “the maximal [or total] state of affairs.” Because intensional parthood is the ontological counterpart of logical implication, k* can also be called, metaphorically, “the (self-)contradictory state of affairs.” In turn, for obvious reasons, t* can be called “the minimal state of affairs,” and because intensional parthood is the ontological counterpart of logical implication, t* can also be called, metaphorically, “the tautological state of affairs.” It is useful to have “z is a minimal state of affairs” and “z is a total state of affairs” as defined predicates, although they each apply (provably) only to one state of affairs (t*, respectively k*): D4

M(z) =Def S(z) ∧ ∀y(S(y) ⊃P(z, y)).

D5

T(z) =Def S(z) ∧ ∀x(S(x) ⊃ P(x, z)).

The fact (provable on the basis of P3) that there is only one “tautological” state of affairs, and only one “contradictory” one, reveals that the conception of states affairs here employed is coarse-grained. A coarsegrained conception of states of affairs is not adequate if the aim is to employ states of affairs as meanings of sentences that are either true or false; for if states of affairs are to be the meanings of such sentences, then, under the coarse-grained conception of states of affairs, all logically true sentences turn out to have the same meaning (i.e., t*), and all logically false sentences turn out to have the same meaning, too (i.e., k*) – what does not seem to be at all desirable. Nevertheless, although coarse-grained states of affairs cannot serve adequately as the meanings of true or false sentences, they can serve in a respectable semantical function, namely, as the intensions of true or false sentences, where intensions are taken to be rough approximations to meanings.3 For the logic of ontic modalities, and quite generally for the purposes of a philosophical theory of ontic modalities, coarse-grained states of affairs are entirely sufficient; finegrained states of affairs need not be taken into account for these purposes. 3

Identity of meaning implies identity of intension, but not vice versa. Identity of intension implies identity of truth-value, but not vice versa.

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The following predicate is useful to the point of being indispensable: D6

O(x) =Def S(x) ∧ A(x).

In other words, to obtain (to be the case, to be a fact) is nothing else than to be an actual state of affairs. Hence it has become clear what is meant by P10, and hence it will also be clear which other prominent state of affairs is defined by the following definition: D7

w* =Def CONJxO(x).

The defined state of affairs is the world in Wittgenstein’s sense: the sum of all obtaining states of affairs. The import of principles P7 and P9 is now clear; according to them, the world is neither the minimal (or “tautological”) nor the maximal (or “contradictory”) state of affairs. Note, furthermore, that the “small” conjunction of states of affairs can be defined on the basis of the “big” conjunction: D8

conj(x, y) =Def CONJv(v = x ∨ v = y).

As is easily seen, CONJv(v = x ∨ v = y) is for all states of affairs x and y identical with ιz[S(z) ∧ P(x, z) ∧ P(y, z) ∧ ∀u(P(x, u) ∧ P(y, u) ⊃ P(z, u))], in other words: the conjunction of states of affairs x and y is the smallest state of affairs of which both x and y are parts. Moving on to P5: this principle turns out to be an atomistic principle once the predicate QA(x) is defined: D9

QA(x) =Def S(x) ∧ ∀y(P(y, x) ⊃ y = x ∨ M(y)).

According to D9, a quasi-atomic state of affairs is a state of affairs whose only proper part, if it has a proper part, is the minimal state of affairs (in consideration of the fact that ∀y(M(y) ≡ y = t*)). One does well to distinguish quasi-atomic states of affairs, atomic states of affairs – states of affairs that have no proper (intensional) part – and elemental states of affairs – states of affairs that have exactly one proper part. It is easily seen that all atomic or elemental states of affairs are quasi-atomic, and vice versa: that all quasi-atomic states of affairs are either atomic or elemental. In the mereology of states of affairs, atomicity coincides with minimality;

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one can prove: ∀z(S(z) ⊃ [¬∃x(x≠z ∧ P(x, z)) ≡ ∀x(S(x) ⊃ P(z, x))]).4 For this reason, I do not here introduce a formally defined predicate for expressing atomicity of states of affairs. But I do formally introduce a predicate for expressing elementalness of states of affairs: D10 EL(x) =Def QA(x) ∧ ¬M(x). While there is exactly one atomic state of affairs – a consequence of the theorem proven in footnote 4 and of the provable fact that there is exactly one minimal state of affairs – the above eleven principles do not determine the exact number of elemental states of affairs. Some mereologists have argued that entities that have the same proper parts must be identical – which, if correct, would imply that all elemental states of affairs are identical to each other, since they all have exactly one proper part and the very same proper part, namely t*. But the invoked identity-principle is false for non-material mereologies, as, for example, the mereology of states of affairs, or the mereology of sets.5 Thus there can easily be more than one elemental state of affairs, and in fact the eleven principles already require that there be at least two elemental states of affairs. P5 states that states of affairs are, in a manner, exhausted by the quasi-atomic states of affairs that are parts of them: if all the quasi-atomic states of affairs that are parts of one state of affairs are also parts of the other, then this is already sufficient for concluding that the former state of affairs is itself a part of the latter. Equivalently, one can also say that states of affairs are exhausted by the elemental states of affairs that are parts of them; for the quasi-atomic states of affairs nearly coincide with the elemental states of affairs, and the one state of affairs that blocks their Suppose (i) S(z), ∀x(S(x) ⊃ P(z, x)), and P(x, z); hence also P(z, x) [using P0]; hence x=z [using P3]. Suppose (ii) S(z), ¬∃x(x≠z ∧ P(x, z)); hence P(t*, z) [using the theorem ∀x(S(x) ⊃ P(t*, x))]; hence t*=z; hence ∀x(S(x) ⊃ P(z, x)) [using ∀x(S(x) ⊃ P(t*, x))]. 5 The validity of the proposed identity-principle is also dubious for material mereologies. Since material atoms have no proper parts (in the relevant material sense), they all have the same proper parts, and hence, according to the identityprinciple at issue, all material atoms are identical to each other – a consequence which is simply absurd (unless there just are no material atoms). If, on the other hand, material atoms are excepted from the identity-principle at issue, then the obvious question is this: Why not make more exceptions? If material atoms can differ from each other although they all have no proper parts, why may not elemental states of affairs differ from each other although they all have the same proper part? 4

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coinciding – that is, t* – is a part of every state of affair, and therefore does not make a difference regarding the exhaustion of states of affairs by states of affairs that are parts of them. Moving on to P6: this principle is the very important principle that regulates how the parts of a conjunction of states of affairs stand to the description (or the property) used in specifying that conjunction. P6 states that every non-minimal part of a conjunction of states of affairs has a nonminimal overlap with some state of affairs that satisfies the description used for specifying that conjunction. Thus stated, P6 is a perfectly evident principle for a mereology of states of affairs. Given the principles P0 – P6, one can prove: ∀x(S(x) ⊃ x = CONJy(QA(y) ∧ P(y, x)), and ∀x(S(x) ⊃ x = CONJy(EL(y) ∧ P(y, x)). In fact, as is suggested by the second theorem, the number of elemental states of affairs determines the total number of states of affairs according to the simple equation: card(S) = 2card(EL) (“The cardinal number of states of affairs is 2 put to the power of the cardinal number of elemental states of affairs”). But for proving this, the system needs to be set-theoretically embedded. Furthermore, given the principles P0 – P6, it can be shown that if the functor of negation is defined as follows: D11 neg(x) =Def CONJy(QA(y) ∧ ¬P(y, x)), that then the principles that one would expect to hold true of a negation of states of affairs all become provable – for example, ∀x(S(x) ⊃ neg(neg(x)) = x), neg(t*) = k*, ∀x∀y(S(x) ∧ S(y) ∧ neg(x) = neg(y) ⊃ x = y). It should not go unmentioned – as a further demonstration of the great definitional power of the proposed mereology of states of affairs – that the “big” disjunction can be defined on the basis of the “big” conjunction as follows: D12 DISJxA[x] =Def CONJy∀x(A[x] ⊃ P(y, x)). The “big” disjunction, in turn, can be used for defining the “small” disjunction: D13 disj(x, y) =Def DISJv(v = x ∨ v = y). As can easily be seen, DISJv(v = x ∨ v = y), or disj(x, y), is for all states of affairs x and y identical with ιz[S(z) ∧ P(z, x) ∧ P(z, y) ∧ ∀u(P(u, x) ∧ P(u,

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y) ⊃ P(u, z))], in other words: the disjunction of states of affairs x and y is the largest state of affair that is a part both of x and of y. Given the above definitions of the three functors neg(x), conj(x, y), and disj(x, y), all the well-known and less well-known Boolean principles that form the stock of truth-functional propositional logic are provable on the basis of P0 – P6.6 The one principle that has not yet been touched upon is P8, which contains one more as yet undefined predicate: QC(x). Here is its definition: D14 QC(x) =Def S(x) ∧ ∀y(P(x, y) ⊃ x = y ∨ T(y)). According to D14, a quasi-complete state of affairs is a state of affairs which is a proper part, if it is a proper part of anything, only of the total (or maximal) state of affairs (in consideration of the fact that ∀y(T(y) ≡ y = k*)). The predicate QC(x) is the counterpart of the predicate QA(x). As the quasi-atomic states of affairs are divided into the one minimal state of affairs, t*, and all the other quasi-atomic states of affairs: the elemental state of affairs, so the quasi-complete states of affairs are divided into the one maximal state of affairs, k*, and all the other quasi-complete states of affairs: the maximal-consistent states of affairs, the definitional description of which is given by D15 MC(x) =Def QC(x) ∧ ¬T(x). It can be shown that the maximal-consistent states of affairs are precisely the negations of the elemental states of affairs (and therefore the elemental states of affairs precisely the negations of the maximal-consistent states of affairs), and it is easily verified on the basis of P8 and P7 that w* – the world – is a maximal-consistent state of affairs. Because of this fact, it is justified to call all and only the states of affairs that are maximal-consistent “possible worlds.” The following two remarkable theorems are provable on the basis of P0 – P6: ∀x∀y[MC(x) ∧ S(y) ⊃ (P(y, x) ≡ ¬P(neg(y), x))] To be precise: every logically true formula φ[p1, …, pn] of truth-functional propositional logic can be translated (in the obvious way) into a functional term φ*[x1, …, xn] of the mereology of states of affairs such that ∀x1…∀xn(S(x1) ∧ … ∧ S(xn) ⊃ φ*[x1, …, xn] = t*) is provable on the basis of P0 – P6. 6

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∀x∀y∀z[MC(x) ∧ S(y) ∧ S(z) ⊃ (P(disj(y, z), x) ≡ P(y, x) ∨ P(z, x))]. This concludes my brief survey of the basic mereology of states of affairs. (It is explored in much greater detail in my book Axiomatic Formal Ontology.) Some philosophers would certainly prefer that the present theory of states of affairs be not called “a mereology.” There may be historical reasons for restricting the use of the term “mereology” to theoretical systems that deal with individuals and their part-relations, but there are no compelling systematical reasons for such a restriction. The present theory of states of affairs is indeed a powerful Boolean algebra, but at the same time it is a mereology: a non-material mereology, and a mereology that has a minimal element, indeed, exactly one minimal element, which one might call “its center.” This centered or Boolean mereology of coarse-grained states of affairs requires universes of states of affairs that have a spindle-shaped structure, with k* at one end (the top) and t* at the other (the bottom). If, for example, we put the number of elemental states of affairs at three,7 and call the elemental states of affairs that we are considering “a,” “b” and “c,” then we obtain the universe of states of affairs U3: [k*:]abc ab ac [w*:]bc a b c t* If S(x) is interpreted to be true of exactly the eight elements in U3 and if P(x, y) is interpreted to hold true of exactly those ordered pairs of elements in U3 such that x is (proper or improper) part of y (in the obvious sense suggested by the diagram), then this universe of states of affairs8 7

This number is wildly unrealistic if we are talking about reality; but it may be entirely appropriate if we are talking about some very restricted “virtual reality,” for example the reality of some very simple game. 8 I omit in the diagram, as obvious, any arrow that indicates that the state of affairs from which the arrow originates is a part of the state of affairs at which the arrow ends (with its head). Inserting such arrows would, by the way, also be misleading: because it would suggest that there are more states of affairs in the universe U3 than are

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fulfills the principles P0 – P9. (See Section 3.4. on the way to make U3 also fulfill P10 and further principles.) There are universes of states of affairs smaller than U3 that also fulfill P0 – P9. In fact, three such universes are contained in U3. One of them is the universe U2: [k*:]ac a [w*:]c t* There is no universe of states of affairs smaller than U2 that fulfills the principles P0 – P9. But the universe of states of affairs U1: k* [w*:]t* still fulfills P0 – P8. And the universe of states of affairs U0: [w*:]t*[:k*] still fulfills P0 – P6.

3.3

The Actuality of States of Affairs in the Mereology of States of Affairs

The mereology of states of affairs stated so far is the basic mereology of states of affairs, but it is far from being a complete mereology of states of affairs. First of all, two principles regarding actuality have to be added. They are obviously true, but nevertheless not provable on the basis of the principles already stated: P11 ∀x∀y(S(x) ∧ A(x) ∧ P(y, x) ⊃ A(y)). designated by the eight basic designators used in its diagram. Note that the universe of states of affairs U3 can be a small part of a much larger universe of discourse.

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P12 A(w*). P11 states that every intensional part of an actual state of affairs is itself actual (namely, an actual state of affairs, according to P0). P12 states that the world is actual. Given the mereology of states of affairs as it now stands, there is a sober truth about the non-actuality of states of affairs. It is this: if the number of elemental states of affairs is N (and the number of elemental state of affairs is ≥ 2 according to the stated principles), then the number of non-actual (or non-obtaining) states of affairs is this: 2N – 2(N –1) (= 2(N –1) = 2N/2). And this implies that actualism about states of affairs – the doctrine that there are no non-actual (or non-existent) states of affairs, that all states of affairs are facts – is not merely false, it is also incoherent. But how is the equation on which this result is based, card(S ∧ ¬A) = card(EL) 2 – 2(card(EL) – 1), obtained? Central to obtaining it is the following theorem: The Actuality Principle for States of Affairs ∀x[S(x) ⊃ (A(x) ≡ P(x, w*))] – “A state of affairs is actual if, and only if, it is a part of the world.” Proof: It is easily seen – given P3, P4, the definition of w* and the definition of O(x) – that ∀x[S(x) ⊃ (A(x) ⊃ P(x, w*))] is true. It remains to be seen that ∀x[S(x) ⊃ (P(x, w*) ⊃ A(x))] is also true. Suppose, therefore, P(x, w*) (the additional supposition S(x) is not needed, since S(x) follows from the already stated supposition by P0). From this, A(x) can be inferred on the basis of P11 and P12, considering that we can prove S(w*) (on the basis of P3, P4, D7).

The number of non-actual states of affairs is obtained by subtracting from the number of all states of affairs – that is, from 2card(EL) – the number of all actual states of affairs. According to the theorem just proven, the number of all actual states of affairs is the number of all states of affairs that are parts of w*. How many states of affairs are there that are parts of w*? To find out, one needs to determine the number of the elemental states of affairs that are parts of w*. This number is equal to card(EL) − 1 (since it can be proven that there is exactly one elemental state of affairs that is not a part of w*, namely, neg(w*)). Since every subset of the set of elemental states of affairs that are parts of w* determines a different part of w*, and

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vice versa, the number of parts of w* is seen to be 2card(EL) − 1. Hence one obtains: card(S ∧ ¬A) = 2card(EL) – 2(card(EL) – 1). And further: 2card(EL) – 2(card(EL) – 1) = 2card(EL)/2 = card(S)/2. Thus it turns out that, provably, there are just as many non-actual states of affairs as there are actual ones. It should not go without mention in this connection that, since the elemental states of affairs can be mapped one-toone upon the maximal-consistent states of affairs (the mapping function is simply negation), we have: card(MC) = card(EL). Therefore, the number of non-actual possible worlds is card(EL) – 1, since the possible worlds have here been identified with the maximal-consistent states of affairs and since there is exactly one possible world that is actual. The latter assertion can be proven in the mereology of states of affairs: According to P7 and P8, w* is a maximal-consistent state of affairs, a possible world, and w* is actual, according to P12. Suppose w´ is another possible world that is actual, another actual state of affairs that is maximal-consistent. Hence P(w´, w*), according to the Actuality Principle for States of Affairs, and because of QC(w´) and P(w´, w*) we have: w´ = w* ∨ T(w*). Hence – contradicting the assumption – w´ = w*, because of ¬T(w*) (P7). There is, therefore, no actual possible world besides w*.

When considering the five principles that govern the actuality of states of affairs: P7, P8, P9, P11, and P12, one may well wonder which of them are true for conceptual reasons only, and which of them, though true, are not true for conceptual reasons only. Incidentally, there can be no reasonable doubt about the purely conceptual nature of the truth of the principles P0 – P6 (or, for that matter, about the conceptual nature of the truth of principle P10). If this be nevertheless found doubtful, one should consider that the truth of P0 – P6 is compatible with there being only one state of affairs. Thus, the prodigious definitional power of the subsystem P0 – P6 is combined with its utter ontological weakness; not even t* ≠ k* can be deduced in it. The ontological weakness of P0 – P6 should do much to remove doubts about the purely conceptual nature of these principles. True, one can deduce ∃xS(x) from them. But ∃xS(x) seems to be a truth that is safely conceptual.

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In my view, number-statements of any kind are not excluded from being conceptually true. In my view, “card(S) ≥ N” and “card(S) = N´” are conceptually true for whatever numbers N and N´ they are true at all. Therefore, since “card(S) ≥ 1” is true – being deducible from true principles – it is conceptually true; and since “card(S) ≥ 2” is true as well – being also deducible from true principles – it is conceptually true as well. The idea that true number-statements cannot be conceptually true is rooted in the positivistic prejudice that conceptual truths cannot say anything informative about the universe of discourse (sometimes also called “the world,” which designation, however, is here reserved for something else), and hence cannot say, for example, how many entities of a certain kind are – at least or exactly – in it. But back to P7, P8, P9, P11, and P12. The only principle out of these five whose purely conceptual truth is, I believe, evident is P11. The purely conceptual truth of the other four is not evident. Perhaps some of them, though true, are not conceptually true after all? But it seems clear (1) that k* is a non-actual state of affairs purely for conceptual reasons, and (2) that t* is an actual state of affairs purely for conceptual reasons.9 (Accepting (1) and (2) is, by the way, another way of establishing that it is a conceptual truth that there are at least two states of affairs.) Moreover, one can plausibly hold (3) that the conjunction of all actual states of affairs is itself actual purely for conceptual reasons.

9

The propositions (1) and (2) can be shown to be true; their truth is, however, not immediately forced upon us. Concerning (2): The statement A(t*) is, on the basis of the analytic principles P0 – P6 and P11, analytically equivalent to ∃x(S(x) ∧ A(x)). Is ∃x(S(x) ∧ A(x)) indubitably a conceptual truth? Is it indubitably a conceptual absurdity to suppose that no state of affairs is actual? Yes, indeed, it is: Suppose ¬∃x(S(x) ∧ A(x)); hence on the basis of the conceptually true principle P10: O(that ¬∃x(S(x) ∧ A(x))); hence by applying D6: S(that ¬∃x(S(x) ∧ A(x))) ∧ A(that ¬∃x(S(x) ∧ A(x))); hence: ∃x(S(x) ∧ A(x)) – contradicting the assumption. Concerning (1): The statement ¬A(k*) is, on the basis of the analytic principles P0 – P6 and P11, analytically equivalent to ∃x(S(x) ∧ ¬A(x)). Is ∃x(S(x) ∧ ¬A(x)) indubitably a conceptual truth? Is it indubitably a conceptual absurdity to suppose that all states of affairs are actual? Yes, indeed, it is: Suppose ∀x(S(x) ⊃ A(x)); S(that ∃x(S(x) ∧ ¬A(x))) is a conceptual truth (it is an instance of a conceptually true general principle stated in the next Section: P13); hence, according to the assumption made: A(that ∃x(S(x) ∧ ¬A(x))); hence by applying D6: O(∃x(S(x) ∧ ¬A(x))); hence on the basis of the conceptually true principle P10: ∃x(S(x) ∧ ¬A(x)) – contradicting the assumption.

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Now, (3) has the consequence that P12 is a conceptual truth (in view of definitions D6 and D7, and P0 – P6); (1), in turn, has the consequence that P7 is a conceptual truth. Unlike P12, P7 is, as a conceptual truth, not an independent principle with regard to the given axiomatization, since ¬A(k*), and therefore ¬P(k*, w*) (in view of the Actuality Principle for States of Affairs), and consequently w* ≠ k* (in view of P2) are conceptually true theorems, because they are logically deducible from conceptually true axiomatic principles (not including P7; see the above proof of the Actuality Principle and the proof of ¬A(k*) in footnote 9). But we still have no hint why P8 and P9 should also be conceptual truths. We can, however, make a case for P8 ⊃ P9 – i.e., QC(w*) ⊃ w* ≠ t* – being a conceptual truth. I have said above that “card(S) ≥ N” is a conceptual truth if it is a truth at all. But “card(S) ≥ 4” is certainly true; it is, therefore, conceptually true. Now, from this conceptual truth – “card(S) ≥ 4,” or in other words: “∃≥4xS(x)” – it follows on the basis of the conceptually true principles P0 – P6 that QC(w*) ⊃ w* ≠ t* is true.10 Therefore, QC(w*) ⊃ w* ≠ t* is not only true but also conceptually true. But this leaves it open whether (a) w* ≠ t* is a conceptual truth, and QC(w*) is not, or whether (b) both sentences are conceptual truths, or whether (c) neither sentence is a conceptual truth. How one decides in this matter depends on one’s theory of actuality, in particular, on one’s theory of the actuality of states of affairs. One might, for example, hold that a certain quasi-complete state of affairs is actual purely for conceptual reasons; from this it would follow that QC(w*) is a conceptual truth, and therefore also w* ≠ t*. But, personally, I do not believe that this position is very plausible; in my view, not even ∃y(S(y) ∧ y ≠ t* ∧ A(y)) is a conceptual truth. But then w* ≠ t* (in view of the fact that S(w*) and A(w*) are conceptual truths) cannot be a conceptual truth, and therefore QC(w*) cannot be one either.

10

Consider that the QC-level of a universe of states of affairs is its top plus the level immediately below the top, and that t* is always at the universe’s bottom (cf. the diagrams of U3 and U2 in the previous section). If a universe of states of affairs has a certain size – i.e., card(EL) ≥ 2, or in other words: card(S) ≥ 4 – its QC-level and its bottom level become separated. Therefore, if w* is at the QC-level, it must be different from t* if there are at least four states of affairs (two elemental states of affairs).

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3.4

Further Principles for “That”

In the previous section, I have added to the basic principles of the mereology of states of affairs, i.e., P0 – P10, two further actuality principles: P11 and P12. Thus, there are now in addition to the seven principles of intensional parthood, P0 – P6, five principles of actuality: P7, P8, P9, P11, and P12, of which P7 proved to be redundant. The only principle for “that” is so far P10; but more principles for “that” than P10 are needed: P13 S(that A). P14 that ¬A = neg(that A). P15 that (A ∧ B) = conj(that A, that B). P16 that ∀xA[x] = CONJy∃x(y = that A[x]). Using the definition A ∨ B =Def ¬(¬A ∧ ¬B), one can prove on the basis of P14 and P15: that (A ∨ B) = neg(conj(neg (that A), neg(that B))) = disj(that A, that B). Using the definition A ⊃ B =Def ¬A ∨ B, one can prove in addition: that (A ⊃ B) = disj(neg(that A), that B). Using the definition ∃xA[x] =Def ¬∀x¬A[x], one can prove on the basis of P14 and P16: that ∃xA[x] = neg(CONJy∃x(y = neg(that A[x]))) = DISJy∃x(y = that A[x]).

The twelve stated principles that do not concern “that” – i.e., P0 – P9, P11, and P12 – form a consistent set of statements, since they have a model that verifies them all: see above, at the end of Section 3.2, the universe of states of affairs U2; in order to accommodate also the principles P11 and P12, imagine circles – symbolizing the property of actuality – drawn around “c” and “t*” in the diagram, but not around “ac” and “a”. And the consistency of all stated principles (P0 – P16) follows, because it is easily seen that the principles P10 and P13 – P16 can be proven on the basis of the principles P0 – P9, P11 and P12 if the following definition of “that A” is adopted: D*

that A =Def CONJy(y = y ∧ ¬A).

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Thus, the universe of states of affairs U2 (and of course also U3) is not only a verifying model for P0 – P9, P11, and P12, it is also a verifying model for the “whole story”: for P0 – P16. This particular series of principles, or in other words: this theory of states of affairs, has herewith been proven consistent. (Of that theory, P13 – P16 are conceptual truths, just like all the other principles of it, with the likely exception of P8 and P9.) The above definition of “that A” allows to deduce this pleasing result. But of course it is not a definition that captures the intended meaning of “that A,” because, according to it, “that A” always denotes either t* or k*, whatever true or false sentence A we are looking at. Interesting enough, though, what is sometimes thought to initiate all by itself an inflation of intensional entities – namely, the introduction of “that” as a name-forming operator applicable to sentences – does not necessarily do any such thing. But the equation “that A = CONJy(y = y ∧ ¬A)” and the extreme ontological restrictions to which this equation gives rise – restrictions regarding the states of affairs that serve as intensions of sentences: according to it, all true sentences have the same intension, t*, and all false sentences also the same intension, k*11 – are not provable if “that A” serves as a basic (and not as a defined) operator, although at first sight one might think it to be provable by an argument that is reminiscent of an argument of some notoriety (called “the Slingshot”): Suppose A is a true sentence; hence (provably) ID1: ιx(x = b) = ιx(A ∧ x = b), and (provably) ID2: CONJy(y = y ∧ ¬A) = t*. Furthermore: (1) (2) (3) (4) 11

A ≡ ιx(x = b) = ιx(A ∧ x = b) that A = that (ιx(x = b) = ιx(A ∧ x = b)) that A = that (ιx(x = b) = ιx(x = b)) that (ιx(x = b) = ιx(x = b)) = t*

a (provable) logical truth12 from (1) by EQU from (2) and ID1 a plausible assumption

Note that the intension of a statement that is designated by a singular term “A” (for A, substitute the statement itself) is designated by the singular term “that A.” (Thus the intension of “snow is white” is designated by “that snow is white.”) Note also that outside of the present context the singular term “that A” is an ambiguous designator and might also be taken to designate at least two other things besides an intension: (quite) naturally, the meaning of the statement designated by “A,” and (somewhat) artificially, the truth-value of that statement. 12 (1) is a provable logical truth if the singular term b is appropriately chosen: the entity denoted by it must be provably different from c*, i.e., from the artificial referent of all definite descriptions ιxA[x] for which ∃=1xA[x] is not true.

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(5)

that A = CONJy(y = y ∧ ¬A)

from (3), (4), and ID2

In this argument, the principle designated by “EQU,” which is used in line (2), is the principle that logically equivalent sentences have the same intensions. Suppose now that A is a false sentence; hence ¬A is a true sentence, and therefore according to the above argument: that ¬A = CONJy(y = y ∧ ¬¬A). But, according to P14, neg(that A) = that ¬A; and we also have: CONJy(y = y ∧ ¬¬A) = t* = neg(k*) = neg(CONJy(y = y ∧ ¬A)) [because CONJy(y = y ∧ ¬A) = k*, since ¬A, according to assumption, is true]. Hence: neg(that A) = neg(CONJy(y = y ∧ ¬A)). And therefore we obtain the very same result as above: that A = CONJy(y = y ∧ ¬A). This entire deduction of the equation corresponding to definition D* is almost impeccable. EQU, indeed, is not deducible in the system P0 – P16. But one can add the provability-rule EQU* to the provability-rules of the system: If A ≡ B is logically provable, then that A = that B is also logically provable.13 This provability-rule, though weaker than EQU, serves the same purpose as EQU in the above deduction, and there is surely nothing wrong with it. There is also nothing wrong with the plausible assumption in line (4); it is also not deducible in the system P0 – P16, but it can be added to that system without scruples. Well, what is it, then, that is wrong with the above deduction? – The false step occurs in moving from (2) to (3) on the basis of ID1. But there is nothing whatever wrong with ID1; the problem is the implicitly applied inference-rule, the substitution of identicals. This rule is not universally valid; there are occasions when its application leads from a true sentence to a false one, and above we have such an occasion: (2) and ID1 are true: (2) purely logically, and ID1 on the basis of the initial assumption; but (3) is certainly not true for any sentence A that is not logically true. The rule of substitution of identicals is safe as long as identicals are not substituted into contexts that are ruled by an occurrence of “that”; for the name-forming operator “that” creates intensional contexts – contexts in which the topical (or contextual) referential function of some singular terms is sometimes different from their factual referential function – a functional difference that had better not be ignored when making an

13

On the concept of (broadly) logical provability, see footnote 17.

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inference via substitution of identicals.14 (Singular terms for which the topical referential function in some contexts is different from the factual referential function are called “referentially unstable”; for more on this matter, see Section 3.9 below.) P13 – P16 and P10 are still not all the principles that are needed for “that.” Yet more “that”-principles will follow in Sections 3.7 and 3.9 below.

3.5

The Number of States of Affairs

The ontological weakness – that is, weakness in the postulation of entities – of the mereology of states of affairs has been preserved up to this point. No principle demanding a dramatically high number of states of affairs has so far been added to the axiomatic crew. The principles so far accepted merely require that there be at least four states of affairs. As far as they are concerned, there need not be more states of affairs than four. From one point of view, this is as it should be, since the mereology of states of affairs should be applicable to finite artificial universes of states of affairs, with more or less restricted numbers of states of affairs (with 4, or 8, or 16, or 32, or 64 states of affairs, for example). If the mereology of states of affairs is to remain applicable as a multipurpose abstract machine, then it must remain open to having any principle of finite number consistent with it added to it that the occasion of application asks for, i.e., any principle that looks like this: ∃=NxEL(x) – “There are exactly N elemental states of affairs,” where N is some natural number ≥ 2. However, from the realistic point of view, i.e., if the mereology of states of affairs is to describe the real universe of states of affairs, then there are only two envisageable principles of the number of states of affairs that are not entirely arbitrary: (1) There is exactly one elemental state of affairs, in other words: there are exactly two states of affairs, namely, t* and k*. (2) There are infinitely many elemental states of affairs, in other words: there are infinitely many states of affairs. On the basis of the principles already stated as axioms (P0 – P16), (1) has already been ruled out. This leaves us with (2); every other envisageable principle of the The factual referential function of „ιx(A ∧ x = b)” is to refer to b (since A is supposed to be true). But the contextual referential function of “ιx(A ∧ x = b)” in “that (ιx(x = b) = ιx(A ∧ x = b))” is not the same as its factual referential function if A is not a logically true sentence. 14

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number of states of affairs would be entirely arbitrary as an assertion about the real universe of states of affairs (since (1) has been already ruled out). With the means of a first-order language, (2) can be formulated in the following way: P17 ∃≥NxEL(x) ⊃ ∃≥N+1xEL(x), for every natural number N ≥ 2.15

3.6

Bases of Necessity and Modal Principles in the Mereology of States of Affairs

In Section 3.1 above, two bases of necessity were indicated: b1 – the basis of logical necessity – and b2 – the basis of nomological necessity. For characterizing the basis of logical necessity completely, we merely need to add the principle P18 S(bn) ∧ b1 = t*.16 For characterizing the basis of nomological necessity completely, one would have to indicate exhaustively which states of affairs are laws of nature – which, to date, is something nobody can do. But the following principles are known to be true for the basis of nomological necessity even independently of knowing exactly which states of affairs are laws of nature: P19 P(b2, w*). P20 b2 ≠ t*. P21 b2 ≠ w*.

The initial assertion ∃≥2xEL(x) is already provable. Note that the reference to natural numbers is not necessary but can be eliminated, since ∃≥NxEL(x) is definable without reference to natural numbers: ∃≥1xEL(x) =Def ∃xEL(x); ∃≥2xEL(x) =Def ∃x∃x´(EL(x) ∧EL(x´) ∧ x ≠ x´), ∃≥3xEL(x) =Def ∃x∃x´∃´´(EL(x) ∧EL(x´) ∧ EL(x´´) ∧ x ≠ x´ ∧ x ≠ x´´ ∧ x´ ≠ x´´); etc. 16 Strictly speaking, S(bn) does nothing to characterize b1 beyond what is already asserted by b1 = t* (since one can already prove S(t*)). But S(bn) also asserts – in as good as place as any – a general truth that will be needed frequently. 15

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In general we have: If P(bn, w*) is true of the basis of necessity bn, then the (corresponding) necessity n is faithful to truth, and conversely; if P(bn, w*) is false of the basis bn, then the necessity n is unfaithful to truth, and conversely. For necessities n that are faithful to truth one can prove in the mereology of states of affairs: nB ⊃ B. Proof: Assume nB, where n is a necessity faithful to truth. Hence P(that B, bn), according to the definitions at the beginning of Section 3.1. Hence P(that B, w*), because of P(bn, w*) and P1. Hence A(that B), because of ∀x[S(x) ⊃ (A(x) ≡ P(x, w*))] and P13. Hence O(that B), because of D6 and P13. Hence B, because of P10.

One can also prove in the mereology of states of affairs: P(bn, w*) ≡ ∀x(n(x) ⊃ O(x)), where ∀x(n(x) ⊃ O(x)) is the obvious predicate-logical correlate of nB ⊃ B, the latter being regarded as a general schema. For the time being, I leave open the question whether one can deduce P(bn, w*) in the mereology of states of affairs from assuming nB ⊃ B as a general schema (but see Section 3.9). Note that the fact that a necessity is faithful to truth does not by itself mean that the necessity is ontic or alethic, for the basis which is an intensional part of w*, making the corresponding necessity faithful to truth, may have been picked out according to criteria which are wholly or partly epistemic (for example). Nor does it seem necessary that every ontic necessity is faithful to truth. A property of necessity that is weaker than faithfulness to truth (in view of the provable statement P(w*, k*), and the principles P3 and P7) is the property of consistency: If bn ≠ k* is true of the basis of necessity bn, then the necessity n is a consistent necessity, and conversely; if bn ≠ k* is false of the basis bn, then n is an inconsistent necessity, and conversely. For necessities n that are consistent one can prove in the mereology of states of affairs: nB ⊃ ¬n¬B. Proof: Assume nB, where n is a consistent necessity. Hence P(that B, bn), according to definition. Assume also n¬B. Hence P(that ¬B, bn), according to definition. Hence P(neg(that B), bn), because of P14. In the mereology of states of affairs it is provable that ∀x∀y(P(x, y) ∧ P(neg(x), y) ≡ y = k*). Hence bn = k* – contradicting the assumed consistency of n. Therefore: ¬n¬B.

One can also prove in the mereology of states of affairs: bn ≠ k* ≡ ∀x[n(x) ⊃ ¬n(neg(x))], where ∀x[n(x) ⊃ ¬n(neg(x))] is the obvious

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predicate-logical correlate of nB ⊃ ¬n¬B, the latter being regarded as a general schema. Another question I leave open for the time being is whether one can deduce bn ≠ k* in the mereology of state of affairs from assuming nB ⊃ ¬n¬B as a general schema (but see again Section 3.9). Further: If bn ≠ t* is true of the basis of necessity bn, then the necessity n is a contingent necessity, and conversely; if bn ≠ t* is false of the basis bn, then n is a non-contingent necessity, and conversely. I use this terminology although the expression “contingent necessity” is unclear, and still remains so when the gloss is added that what is meant by “contingent necessity” is logically contingent necessity. In precisely what sense so-called contingent necessities n can be called “(logically) contingent” becomes apparent when one considers the following theorem that is deducible in the mereology of states of affairs: bn ≠ t* ≡ ∃x(n(x) ∧¬1(x)). Finally: If bn ≠ w* is true of a basis of necessity bn, then the necessity n is a proper necessity, and conversely; if bn ≠ w* is false of the basis bn, then n is an improper necessity, and conversely. In the mereology of states of affairs one can deduce: bn ≠ w* ≡ ∃x(O(x) ∧ ¬n(x)), for every consistent necessity n. Proof: Let n be a consistent necessity. (1) Assume bn = w*, and O(x). Hence P(x, w*), according to D6 and ∀x[S(x) ⊃ (A(x) ≡ P(x, w*))]. Hence P(x, bn). Hence n(x), according to definition. (2) Assume conversely: ∀x(O(x) ⊃ n(x)). Because of S(w*) [a consequence of D7, P3 and P4], P12, D6: O(w*). Hence n(w*). Hence P(w*, bn). According to P8: QC(w*). Hence w* = bn ∨ T(bn), according to D14. Since n is a consistent necessity, we have bn ≠ k*, and therefore ¬T(bn). Hence bn = w*.

Since there are, according to P17, infinitely many states of affairs, there can be infinitely many necessities faithful to truth, just as many necessities faithful to truth as there are bases for them. The extremes are marked, on the one side, by logical necessity, with the basis t* (= b1), which is a proper and non-contingent necessity faithful to truth, and, on the other side, by factuality, with the basis w*, which is an improper and contingent necessity faithful to truth. There can be infinitely many proper and contingent necessities faithful to truth in between, nomological necessity, with the basis b2, being one of them. Whether there are in fact infinitely many proper and contingent necessities faithful to truth in between logical necessity and factuality depends on a decision regarding

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the question whether just any obtaining state of affairs that is different from t* and w* is a basis of necessity. It seems indeed objectively arbitrary to accept one of those states of affairs as a basis of necessity, but to deny that status to any other of them. Thus, from an objective point of view, it seems true that either every obtaining state of affairs that is different from t* and w* is a basis of (proper and contingent) necessity, or none of them is. If this is true, then, since we accept the conjunction of all states of affairs that are laws of nature – i.e., the obtaining state of affairs b2, which is different from both t* and w* (see P19 – P21) – as a basis of necessity, we must accept every obtaining state of affairs that is different from t* and w* as a basis of necessity. Accordingly, we must accept that there are infinitely many necessities faithful to truth. This conclusion can, of course, be rejected. But its rejection comes at a price. One can deny that it is objectively arbitrary to accord the status of basis of necessity to certain obtaining states of affairs that are different from t* and w*, but not to all such states of affairs. One will then have to provide objective criteria for distinguishing, among the obtaining states of affairs that are different from t* and w*, those that are bases of necessity from those that are not. I do not believe that such criteria have been provided by anyone. Consider, as a salient example, the conjunction of all states of affairs that are laws of nature: n*, and an obtaining state of affairs p* of which no law of nature is an intensional part and which is different both from t* and w*. The state of affairs n* is not an intensional part of p*, for a conjunction of laws of nature is itself a law of nature, and, according to assumption, no law of nature is an intensional part of p*. Moreover, the state of affairs p* is not an intensional part of n*; if it were, it would be a law of nature, since n* is a law of nature and every intensional part of a law of nature is itself a law of nature; but p* is not a law of nature (otherwise a law of nature would be an intensional part of p*, contradicting the assumption). The state of affairs n* is such that one would like to call it “a basis of necessity”; the state of affairs p* is such that one would like to deny it that status. But for what objective reason? One such reason could be that n* is somehow objectively necessary while p* is not. But what could be meant by “objectively necessary” in this context, since neither p* nor n* is logically necessary? The intended meaning of “objectively necessary” is utterly mysterious. It is, of course, trivially true that n* is nomologically necessary and p* is not, if n* is allowed as a basis of necessity. But it is also trivially true that p* is, say, anti-nomologically necessary and n* is

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not, if p* is allowed as a basis of necessity. Once they are allowed to be bases of necessity, each of the two states of affairs bestows (prima facie objective) necessity upon itself and denies it to the other state of affairs – in view of the definition n(x) =Def P(x, bn), the principle P2, and the fact that ¬P(p*, n*) and ¬P(n*, p*). But, obviously, these matters are entirely useless for warranting that n* is acceptable as a basis of necessity while p* is not. Indeed, there does not seem to be any objective reason for making a distinction regarding the status of basis of necessity between n* and p*. From the objective point of view, either both states of affairs are bases of necessity, or none of them is. Once it is accepted that it is objectively arbitrary to accord the status of basis of necessity to certain obtaining states of affairs that are different from t* and w*, and not to every other such state of affairs, there is still the option of denying that status to every obtaining state of affairs that is different from t* and w*. This would leave us with only two necessities faithful to truth (logical necessity and factuality), with only one proper necessity faithful to truth (logical necessity) and with no contingent proper necessity faithful to truth. This is not unheard of. Starting with Hume, the restriction of truth-implying ontic necessity that is stronger than truth to logical necessity alone has been the position of empiricist philosophers. The trouble with this austere position is that it does not fit the common practice of modal discourse; like other forms of asceticism, it has no chance of becoming commonly accepted – not even among philosophers (as is amply demonstrated by the history of ideas since Hume). This is not much of a philosophical argument; yet the stated matter of fact has to be respected by philosophers on the grounds of what one might term “intellectual propriety.” If philosophers do not accommodate themselves to the common practice of modal discourse, which requires that there are more truth-implying ontic necessities that are stronger than truth than merely logical necessity, they will violate intellectual propriety, and the price for violating intellectual propriety is to be counted an irrelevant fanatic. Most of the infinitely many necessities faithful to truth we are now ready to countenance do not have a designation. The sequence 1, 2, 3, 4, 5, 6, … – even if prolonged to infinity – does not contain enough terms to give a designation to every proper and contingent necessity faithful to truth. (But there are certainly enough designations in the sequence for every necessity that we could ever want to specifically refer to.) It is a matter of fact that we do not give much thought to most of the infinitely

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many proper and contingent necessities faithful to truth. But this should not mislead us into thinking that, somehow, they are not “real” necessities. They are real enough, considering that they are all based on a real – an obtaining – state of affairs. The strength (or force) of a necessity is inversely proportional to the (intensional) strength of its basis. Logical necessity is the absolutely strongest necessity, since it has the absolutely weakest basis: the minimal state of affairs t*, which has no intensional content. Factuality is the weakest necessity faithful to truth, since it has the strongest basis that is faithful to truth: the maximal-consistent state of affairs w* (just one level below the absolutely strongest basis: the maximal state of affairs k*, which encompasses all intensional content). We have the following correlation: n is a stronger necessity than m if, and only if, bn is a proper intensional part of bm. There are, of course, many pairs of necessities which are such that the basis of each member of them does not contain the basis of the other member of them; in this case, neither one of the paired necessities is stronger than the other. An important fact should be noted. General determinism is sometimes asserted as the thesis that every obtaining state of affairs is necessary, where “is necessary” is thought to express a necessity n that is faithful to truth, hence consistent, and that has more force than factuality. But, as a matter of fact, the intended necessity – whatever it is – cannot have more force than factuality. Those who accept ∀x(O(x) ⊃n(x)) for a certain consistent necessity n must, according to the theorem stated and proven above in this section, also accept bn = w*, and therefore it turns out for them that the necessity n is certainly not stronger than factuality, since it is factuality, which is the weakest necessity faithful to truth. This shows that general determinism is an incoherent position: it aims to spread a proper and consistent necessity over more states of affairs than it can, as a proper and consistent necessity, apply to, namely, over all obaining states of affairs. Leibniz famously asserted, in effect, that logical general determinism is true, i.e., that ∀x(O(x) ⊃ 1(x)) is true. But if ∀x(O(x) ⊃ 1(x)) were true, then – according to the theorem bn ≠ w* ≡ ∃x(O(x) ∧ ¬n(x)) (for consistent necessities n) – b1 = w* would also be true (1 being a consistent necessity), or in other words: w* = t* would be true (because b1 = t*, according to P18), contradicting P9. Logical general determinism is therefore false – provably false in the mereology of states of affairs. Hence, regarding his position on logical general determinism, Leibniz is provably

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mistaken. But it should also be noted that P9 is one of the two principles of the mereology of states of affairs (the other being P8) that, presumably, are not conceptually true (cf. the end of Section 3.3). If w* ≠ t* is indeed not conceptually true (but only true), then Leibniz did not commit a logical mistake in asserting the truth of logical general determinism. It seems that Leibniz would have endorsed MC(w*) – “the world is maximal-consistent” – which is provably equivalent to the conjunction of P7: w* ≠ k*, and P8: QC(w*). The only way to square w* = t*, to which Leibniz is committed by being a logical general determinist, with MC(w*) is to assume (1) that there are less than four states of affairs (see the end of Section 3.3: if there are at least four states of affairs, then QC(w*) in MC(w*) is bound to imply w* ≠ t*), and (2) that there is more than one state of affairs (as is required by w* ≠ k* in MC(w*)). Therefore, the only way to square w* = t* with MC(w*) is to assume that there are exactly two states of affairs, because the mereology of states of affairs excludes that there are exactly three states of affairs: there is no number of elemental states of affairs N such that 3 = 2N. Hence Leibniz, as a logical general determinist and as a “maximal-consistentist,” is committed to the view that there are only two states of affairs, namely, k* and w*, constituting the universe of states of affairs U1 (see the end of Section 3.2). It would be striking if there were any evidence in his works that he did believe this.

3.7

The Classical Modal Principles and the Bases-Theory of Necessity

We have already seen – in the previous section – that for nomological necessity, 2, two classical modal principles are true: B ⊃ B, and B ⊃ ¬¬B. (I leave out the index attached to “” if I refer to some unspecified necessity or other; I also leave out the index attached to “b” if I refer to some unspecified basis of necessity or other.) The same two principles are also true for logical necessity, 1. What about the following other classical principles: (1) (A ⊃ B) ⊃ (A ⊃ B), (2) B ⊃ B, (3) ¬B ⊃ ¬B? Let us see whether we need further principles for proving them, and which principles. It is easily seen that (A ⊃ B) ⊃ (A ⊃ B) is provable without further assumptions: Assume (A ⊃ B), and assume A. Hence P(that (A ⊃ B), b) and P(that A, b). Hence according to P14, P15, etc.: P(disj(neg(that A), that B), b) and P(that A, b). Hence P(that B, b), because

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of S(that A) and S(that B), according to P13, and because of the theorem ∀x∀y∀z[S(x) ∧ S(y) ∧ P(disj(neg(x), y), z) ∧ P(x, z) ⊃ P(y, z)] (the proof of which I omit). From P(that B, b), finally, it follows on the basis of the definition of the predicate of necessity and the sentence-connective of necessity: (that B), and hence B. However, the attempt to prove B ⊃ B reveals that there is need of further principles: Assume B. Hence P(that B, b). From this – in order to get B – one must derive P(that B, b), which then, purely definitionally, implies B. The step from P(that B, b) to P(that B, b) is, by definition alone, equivalent to the step from P(that B, b) to P(that P(that B, b), b), which, in turn, is a consequence of the following general principle: Principle A

∀x∀y(P(x, y) ⊃ P(that P(x, y), y)).

In other words: if a state of affairs x is a (intensional) part of a state of affairs y, then the state of affairs that x is a part of y is itself a part of y. And we also need a further principle in order to prove ¬B ⊃ ¬B. This principle (which is easily found, following the above procedure for finding Principle A) is the following: Principle B

∀x∀y(S(x) ∧ S(y) ∧ ¬P(x, y) ⊃ P(neg(that P(x, y)), y)).

In other words: if a state of affairs x is not a part of a state of affairs y, then the negation of the state of affairs that x is a part of y is a part of y. Both Principle A and Principle B are easily provable if one assumes the following stronger principle: P22 ∀x∀y(P(x, y) ⊃ t* = that P(x, y)) ∧ ∀x∀y(¬P(x, y) ⊃ k* = that P(x, y)). I adopt this latter principle as an axiom, since a relationship of intensional parthood, or the negation of such a relationship, is clearly (and for conceptual reasons) a matter of logical necessity. Indeed, an equivalent formulation of P22 would be this: ∀x∀y(P(x, y) ⊃ 1P(x, y)) ∧ ∀x∀y(¬P(x, y) ⊃ 1¬P(x, y)). Note that P22 could be proven on the basis of P0 – P6 if the definition D* – that A =Def CONJy(y = y ∧ ¬A) (see Section 3.4) – were adopted. (This fact is important for considerations of consistency; see Section 3.9.)

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Another classical modal principle is not properly speaking a principle if only statements and schemata of statements of the object-language are to be literally called “principles”; rather, it is a special kind of inference-rule, a provability-rule (and therefore a statement of the metalanguage): If B is logically provable,17 then B is also logically provable. On the basis of the definition of (first the definition of the sentence-connective , and then the definition of the predicate ), this provability-rule amounts to: If B is logically provable, then P(that B, b) is also logically provable (b being a state of affairs that is a basis of necessity). I put this, more compactly, in the following manner: ├ B ⇒ ├ P(that B, b) (or in other words, on the basis of the definition of : ├ B ⇒ ├ B), where “├” stands for “logically provable.” The stated provability-rule can easily be derived on the basis of the following other provability-rule: P23 ├ B ⇒ ├ t* = that B, of which an equivalent formulation is this: ├ B ⇒ ├ 1B, or in other words: ├ B ⇒ ├ P(that B, b1). Thus the entirely general provability-rule ├ B ⇒ ├ B (i.e., ├ B ⇒ ├ nB) is implicit in one of its special cases. P23, too, could be proven on the basis of P0 – P6 if D* were adopted (since B ⊃ CONJy(y = y ∧ ¬B) = t* is logically provable on the basis of P0 – P6). Note that the provability-rule EQU* that has already made its appearance in Section 3.4 can be proven on the basis of P23: Assume ├ A ≡ B. Hence ├ (A ⊃ B) and ├ (B ⊃ A). Hence by P23: ├ (t* = that (A ⊃ B)) and ├ (t* = that (B ⊃ A)). Hence ├ (t* = disj(neg(that A), that B)) and ├ (t* = disj(neg(that B), that A)), because ├ (that (A ⊃ B) = disj(neg(that A), that B)) and ├ (that (B ⊃ A) = disj(neg(that B), that A)) (see the beginning of Section 3.4). Hence ├ P(that B, that A) and ├ P(that A, that B), because ├ ∀x∀y[S(x) ∧ S(y) ⊃ (P(x, y) ≡ disj(neg(y), x) = t*)] and ├ S(that A) and ├ S(that B). Hence ├ (that A = that B), because of the conceptually true principle P3.

17

“Logically provable” means: provable purely on the basis of axiomatic principles and inference-rules that are conceptually (broadly logically, analytically) valid (true). Thus, if P8 or P9 are necessary for deducing a certain theorem in the mereology of states of affairs, then that theorem – though provable – is not logically provable, assuming that P8 and P9 are not conceptually true (see Section 3.3).

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When employing provability-rules that involve the concept of logical provability – like P23 and EQU* – in combination with statements or rules that do not involve that concept, there will be occasion, in proofs and deductions, to make use of the step-down rule: ├ B ⇒ B (logical provability implies truth). But use of this rule will, generally, not be explicitly registered. It is worth noting that the modal propositional system S5, that is, classical truth-functional propositional logic, axiomatized (for example) in the following way: ├ A ⊃ (B ⊃ A) ├ (A ⊃ (B ⊃ C)) ⊃ ((A ⊃ B) ⊃ (A ⊃ C)) ├ (¬B ⊃ ¬A) ⊃ (A ⊃ B) ├ A, ├ A ⊃ B ⇒ ├ B, plus (for example) the modal principles ├ B ⊃ B ├ (A ⊃ B) ⊃ (A ⊃ B) ├ B ⊃ B18 ├ ¬B ⊃ ¬B ├ B ⇒ ├ B (where “├” stands for “logically provable”), has now been justified (via logical deduction from a stronger theory) as being the correct system of modal propositional logic for all necessities that (a) can be represented as founded on a basis of necessity and that (b) are, for conceptual reasons alone, faithful to truth. The foundations of this justification of the correctness of S5 are the mereology of states of affairs (in fact, only its conceptually true part) and a conception of necessity that is expressed by the following two definitions: (x) =Def P(x, b), A =Def (that A), where the crucial mereological principle P(b, w*) is either logically provable in the given theory (for the basis b concerned) or is a conceptually true further axiom. These definitions are expressive of a bases-theory of necessity, since “b” refers to a state of affairs that is the basis of the necessity concerned. 18

As is well known, this S5-axiom is not independent of the other ones stated. I nevertheless state it as an S5-axiom, for the sake of explicitness and easier reference.

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The Bases-Theory of Necessity is not a theory of necessity that is applicable to all kinds of necessity. There are some necessities for which the analysis in terms of a basis of necessity is inadequate, for example, epistemic necessity: knowledge. What somebody knows is not what follows from her basis of knowledge (there is no such thing); rather, it is what follows from her basis of belief and satisfies in addition several further conditions. Take the simplest concept of knowledge (which is implied by all other concepts of knowledge): true conviction. In this sense of knowledge, what somebody (at least implicitly) knows is what follows from her basis of belief and is true. The principle ¬B ⊃ ¬B is not valid for knowledge in this sense, for the following situation frequently obtains: B is false, and therefore ¬kB is true; but (in this same situation) that B is an intensional part of the basis of belief of the person concerned, and therefore that person is convinced that B [i.e., cB], hence convinced that she is convinced that B [i.e., ccB, according to P22], hence convinced that she knows that B: ckB [because of ccB ⊃ c(cB ∧ B) and kB =Def cB ∧ B]. Hence k¬kB must be false. For if it were true, then the person concerned would know, and therefore be convinced, that she does not know that B: c¬kB. Hence the person concerned would be both convinced that she knows that B (as we have already seen) and convinced that she does not know that B – what cannot be (at least if rational conviction is intended). Note that conviction – or doxastic necessity – can be treated according to the Bases-Theory of Necessity (and I have done so in the preceding preceding paragraph). Since one will not require that conviction be faithful to truth but only that it be consistent, B ⊃ B needs to be replaced by B ⊃ ¬¬B in the above axiomatic system; this replacement turns it into a system which defines a defensible modal propositional logic for – dispositional, rational – conviction.

The theory of necessity advocated here is, moreover, an ontonomological theory of necessity, since it is grounded in ontological laws for the realm of states of affairs. By being an onto-nomological theory of necessity, it is automatically also an adequate onto-nomological theory of possibility, since possibility can be adequately defined on the basis of necessity (◊A =Def ¬¬A) and is, therefore, completely determined by the latter concept. But the onto-nomological theory of possibility can also be developed independently of the theory of necessity: on the basis of the mereology of states of affairs and the definitions ◊n(x) =Def S(x) ∧ ¬P(neg(x), bn) and ◊nA =Def ◊n(that A). Further on, I will dedicate two chapters (Chapters 5 and 6) to the question how far conditionals (or relational necessities) are treatable within the mereology of states of affairs on the basis of the definitions n→(x, y) =Def P(y, conj(bn, x)) and A n→ B =Def n→(that A, that B).

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3.8

Necessity and Possible Worlds

The theory of modality presented in this book is a theory of modality in which the concept of possible world plays no essential role. Nevertheless, in view of the popularity of the possible-worlds-theory of modality, it will certainly be good to see how that theory is derivable from, and therefore reducible to, the present theory. We have seen above (in Section 3.2) that maximal-consistent states of affairs can be regarded as possible worlds, and we have identified the possible worlds with those states of affairs. (It would hardly make sense to call the maximal-consistent states of affairs possible worlds, but to allow that there are, in addition, some other possible worlds that are not maximal-consistent states of affairs.) We now need to define another important concept of the mereology of states of affairs: D16 O(x, y) =Def P(x, y). O(x, y) is read as “(the state of affairs) x obtains in (the state of affairs) y.” One can then prove the following five theorems: ∀x[S(x) ⊃ (O(x) ≡ O(x, w*))] ∀x[S(x) ⊃ (n(x) ≡ ∀y(MC(y) ∧ O(bn, y) ⊃ O(x, y)))] nA ≡ ∀y(MC(y) ∧ O(bn, y) ⊃ O(that A, y)) ∀x[S(x) ⊃ (◊n(x) ≡ ∃y(MC(y) ∧ O(bn, y) ∧ O(x, y)))] ◊nA ≡ ∃y(MC(y) ∧ O(bn, y) ∧ O(that A, y)). The first theorem states that a state of affairs obtains (simpliciter) if, and only if, it obtains in the world. The second theorem states that a state of affairs is n-necessary if, and only if, it obtains in every possible world in which the basis of the n-necessity obtains. The third theorem is a corollary of the second theorem: for the sentence-connective of necessity (as opposed to the predicate of necessity). The fourth theorem states that a state of affairs is n-possible if, and only if, it obtains in some possible world in which the basis of the n-necessity obtains. The fifth theorem is a

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corollary of the fourth theorem: for the sentence-connective of possibility (as opposed to the predicate of possibility). As an example, here is the (not altogether easy) proof of the second theorem: (1) Suppose: S(x), n(x), MC(y), O(bn, y). Hence P(x, bn) and P(bn, y) (according to the definition of n(x), and D16). Hence by P1: P(x, y), hence by D16: O(x, y). (2) Suppose: S(x), ∀y(MC(y) ∧ O(bn, y) ⊃ O(x, y)). Hence (because of P3, P4, D1): P(x, CONJz∀y(MC(y) ∧ O(bn, y) ⊃ O(z, y))). But CONJz∀y(MC(y) ∧ O(bn, y) ⊃ O(z, y)) = bn, because (applying D16) CONJz∀y(MC(y) ∧ P(bn, y) ⊃ P(z, y)) = bn. Therefore: P(x, bn), and consequently: n(x). But CONJz∀y(MC(y) ∧ P(bn, y) ⊃ P(z, y)) = bn still remains to be proven. It is a consequence of P(bn, CONJz∀y(MC(y) ∧ P(bn, y) ⊃ P(z, y))) and P(CONJz∀y(MC(y) ∧ P(bn, y) ⊃ P(z, y)), bn), according to P3. P(bn, CONJz∀y(MC(y) ∧ P(bn, y) ⊃ P(z, y))) because of ∀y(MC(y) ∧ P(bn, y) ⊃ P(bn, y)) (applying P4, P3, D1, P18). P(CONJz∀y(MC(y) ∧ P(bn, y) ⊃ P(z, y)), bn), finally, is seen to be true as follows: Assume QA(u), P(u, CONJz∀y(MC(y) ∧ P(bn, y) ⊃ P(z, y))); what is in question is established according to P5 if P(u, bn) can be deduced from this assumption. If M(u), then P(u, bn). (S(bn) according to P18.) If ¬M(u), then (according to P6) ∃k´[P(k´, u) ∧ ¬M(k´) ∧ ∃z´(P(k´, z´) ∧ ∀y(MC(y) ∧ P(bn, y) ⊃ P(z´, y)))]. Because of QA(u), P(k´, u), ¬M(k´): k´ = u (according to D9), and also EL(u) (according to D10). Hence: ∃z´(P(u, z´) ∧ ∀y(MC(y) ∧ P(bn, y) ⊃ P(z´, y))). Hence: ∃z´(P(u, z´) ∧ ∀y(S(y) ∧ EL(neg(y)) ∧ ¬P(neg(y), bn) ⊃ ¬P(neg(y), z´))).19 Hence: ∃z´(P(u, z´) ∧ ∀y(S(y) ∧ EL(neg(y)) ∧ P(neg(y), z´) ⊃ P(neg(y), bn))). Hence: ∃z´(P(u, z´) ∧ ∀y(S(y) ∧ EL(neg(neg(y))) ∧ P(neg(neg(y)), z´) ⊃ P(neg(neg(y)), bn))). Hence: ∃z´(P(u, z´) ∧ ∀y(EL(y) ∧ P(y, z´) ⊃ P(y, bn))).20 Hence because of EL(u): P(u, bn).

Note that the conjunct O(bn, y) drops out of the above five theorems for n = 1, since b1 = t* (P18) and ∀y(S(y) ⊃ O(t*, y)). Thus one obtains: ∀x[S(x) ⊃ (1(x) ≡ ∀y(MC(y) ⊃ O(x, y)))]

The theorems employed are ∀y[MC(y) ⊃ ∀x(S(x) ⊃ (P(x, y) ≡ ¬P(neg(y), x)))] and ∀y[MC(y) ≡ EL(neg(y))]. (For getting ∀y[MC(y) ⊃ (P(bn, y) ≡ ¬P(neg(y), bn))] from the first theorem: bn is a state of affairs according to P18.) The additional conjunct “S(y)” is simply a definitional consequence of “MC(y).”) 20 The theorem employed is ∀y(S(y) ⊃ neg(neg(y)) = y). 19

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∀x[S(x) ⊃ (◊1(x) ≡ ∃y(MC(y) ∧ O(x, y)))]. And therefore – in view of the reading these two formulas have – the Leibnizian conception of modality emerges as a pair of theorems of the mereology of states of affairs, which theorems show that the Leibnizian necessity is logical necessity, the Leibnizian possibility logical possibility. I add the following two theorems, which rather strikingly illuminate the relationship between modality and basis of modality21: bn = CONJyn(y). In other words: the basis of n-necessity is the conjunction of all nnecessary states of affairs. bn = DISJy◊n(y). In other words: the basis of n-possibility is the disjunction of all n-possible states of affairs (or in other words: it is what all n-possible states of affairs have intensionally in common).

3.9

Three Final Observations: Another “That”-Principle, the Trickiness of “w*,” and the Consistency of the Mereology of States of Affairs

First observation: In Section 3.6, I left it open whether one can deduce, in the mereology of states of affairs, P(bn, w*) from assuming nB ⊃ B as a general schema, and likewise whether one can deduce bn ≠ k* from assuming nB ⊃ ¬n¬B as a general schema. Let us try to make these deductions. (1) Assume nB ⊃ B as a general schema. Hence nO(bn) ⊃ O(bn), substituting “O(bn)” for “B.” According to the definition of n(x), we have n(bn), because of P(bn, bn), which in turn is true because of P18 and P2. Now, there is another very plausible “that”-principle, a principle complementing P22 and P23: P24 ∀x(S(x) ⊃ x = that O(x)). 21

A basis of necessity bn is also a basis of possibility, in short: it is a basis of modality.

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Applying P24, one obtains n(that O(bn)) from n(bn), and therefore also nO(bn). Hence O(bn), since we already have nO(bn) ⊃ O(bn). Therefore: P(bn, w*) (according to ∀x[S(x) ⊃ (A(x) ≡ P(x, w*))] – the Actuality Principle for States of Affairs proven in Section 3.3 – and D6). (2) Assume nB ⊃ ¬n¬B as a general schema. Hence nO(bn) ⊃ ¬n¬O(bn). Just as in (1), one obtains nO(bn). Therefore: ¬n¬O(bn), and hence (according to definition) ¬P(that ¬O(bn), bn), hence (by applying P14) ¬P(neg (that O(bn)), bn), hence (by applying P24) ¬P(neg(bn), bn), hence bn ≠ k* (because P(neg(k*), k*)). Second observation: The designator “w*” has been defined above as “CONJxO(x)” (D7). The designator “t*”, on the other hand, has been defined as “CONJx¬S(x)” (D2). Though the two definitions have a very similar structure, there is a striking difference between the two defined designators. Of “t*” we have to say that it could not, not even in principle, have denoted a different state of affairs than it denotes in fact. Of “w*,” however, we can say that it could, in principle, have denoted a different state of affairs than it denotes in fact.22 This is a consequence of the occurrence of the predicate “O(x)” in the definition of “w*”: we can say that this predicate could have applied differently than it does in fact apply, because states of affairs could have obtained (i.e., could have been actual) that do not in fact obtain. Therefore, given the definition of “w*”: if those non-obtaining states of affairs had obtained, “w*” would have denoted a different state of affairs than it denotes in fact. But this referential instability of “w*” leads to certain problems: (1´) Let y be an obtaining (therefore, according to D6, actual) state of affairs that is neither w* nor t*. Hence P(y, w*), according to the Actuality Principle for States of Affairs. Hence: t* = that P(y, w*), according to P22, and therefore: 1P(y, w*). (2´) Since y ≠ t*, it follows according to P24: t* ≠ that O(y). But it seems that we also have the following identity: that O(y) = that P(y, w*). Therefore: t* ≠ that P(y, w*), and consequently: ¬1P(y, w*). We are confronted with a contradiction, and it seems that the only way of avoiding it – short of a modification of principles – is to reject the identity 22

Note that these are meta-modal statements. They can, however, be represented in the object-language: ∀y´(t* ≠ y´ ⊃ 1(t* ≠ y´)), ∃y´(w* ≠ y´ ∧ ◊1(w* = y´)).

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statement used in the second deduction, namely, the statement “that O(y) = that P(y, w*).” But, unfortunately, this is not a statement that can easily be rejected. It has intuition on its side, and what is more important: purely on the basis of axiomatic principles and inference-rules that are conceptually valid, one can prove ∀y(O(y) ≡ P(y, w*)) in the mereology of states of affairs, and therefore also the particular case O(y) ≡ P(y, w*). See the proof of ∀y[S(y) ⊃ (A(y) ≡ P(y, w*))] in Section 3.3. One merely needs to consider the axiom P0 and that “O(y)” is defined as “S(y) ∧ A(y)” (according to D6) in order to obtain a proof of ∀y(O(y) ≡ P(y, w*)) from the already proven theorem. No principle or inference-rule that is not conceptually (or broadly logically) valid is used in the proofs.

Using the provability-rule EQU*: “If A ≡ B is logically provable, then that A = that B is also logically provable” (see Section 3.4, and its derivation from P23 in Section 3.7), one obtains: that O(y) = that P(y, w*). There is no escaping this conclusion. Thus, the real source of the contradiction derived above cannot be the statement “that O(y) = that P(y, w*)” (which cannot be rejected, as has just been shown). The real source of the contradiction is an illicit step of inference, which is so unobtrusive as to be easily overlooked. In Section 3.4 an absurd conclusion was obtained by substituting, according to the rule of substitution of identicals, the term “ιx(A ∧ x = b)” into a “that”context. That term is referentially unstable for every sentence A that is neither conceptually true nor conceptually false.23 Above, another absurd conclusion is obtained by substituting another referentially unstable term – “w*” – into another “that”-context. But not by applying the rule of substitution of identicals. Rather, the rule used in the second case is allinstantiation: in (1´), “P(y, w*) ⊃ t* = that P(y, w*)” is obtained from the first conjunct of P22, ∀x∀y(P(x, y) ⊃ t* = that P(x, y)), by all-instantiation (and further, from the already established statement “P(y, w*)” and the newly won material implication “P(y, w*) ⊃ t* = that P(y, w*),” one obtains by applying modus ponens: “t* = that P(y, w*)”). But allinstantiation, just like substitution of identicals, is not a universally valid 23

If A is a sentence that is neither conceptually true nor conceptually false, then “ιx(A ∧ x = b)” could have denoted a different entity then it denotes in fact: if A is true, then the factual referent of “ιx(A ∧ x = b)” is b, but it could have been c*; if A is false, then the factual referent of “ιx(A ∧ x = b)” is c*, but it could have been b. (Concerning c*, see footnote 12.)

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inference-rule: it can lead from a true sentence to a false one when it is applied in such a manner that a substitution of a referentially unstable term into a “that”-context occurs.24 The inference-rule of all-instantiation is entirely safe as long as no referentially unstable instantiating term is substituted into a context ruled by an occurrence of “that.” But if the restrictions on applying that inference-rule are ignored, strange consequences are likely to ensue. For example, P24 can be made to look utterly absurd. From the logically provable (and rather elementary) theorem ∀y(O(y) ≡ P(y, w*)) one obtains the logically provable theorem ∀y(that O(y) = that P(y, w*)) by applying EQU*, as we have seen. From ∀y(that O(y) = that P(y, w*)) one obtains ∀y(S(y) ⊃ y = that P(y, w*)), using P24, and from ∀y(S(y) ⊃ y = that P(y, w*)) one obtains: w* = that P(w*, w*), by all-instantiation and modus ponens, since S(w*) is an elementary theorem. But the statement “w* = that P(w*, w*)” is absurdly false. Third observation: Though P24 can surely not be reduced ad absurdum in the manner presented in the previous paragraph, there is no denying that P24 is a critical principle. An indication of this is the fact that it could not be derived from the principles P0 – P23 even if the definition D* – that A =Def CONJy(y = y ∧ ¬A) – were used. Suppose it could be derived in this manner. Then, because of the elementary theorem ∀x[CONJy(y = y ∧ ¬O(x)) = t* ∨ CONJy(y = y ∧ ¬O(x)) = k*] and because of the definitional consequence of D*, ∀x[that O(x) = CONJy(y = y ∧ ¬O(x))], P24 would imply the following: ∀x(S(x) ⊃ x = t* ∨ x = k*) – contradicting what is provable regarding the number of states of affairs already on the basis of P0 – P9, namely, that there are at least four states of affairs.

24

Note that all modal sentence-connectives Q – ontic, epistemic, or otherwise – have “that”-contexts connected to them, since they all can be defined on the basis of the corresponding predicate for states of affairs, as follows: QB =Def Q(that B). This intrinsic link between modal sentence-connectives and “that”-contexts is the reason why failures of the rule of substitution of identicals and of the rule of all-instantiation were first noticed in connection with the substitution of terms into the scopes of modal operators. But note that “that”-contexts which are connected to the sentenceconnective of negation (as in “it is not the case that”) are safe for the application of both inference-rules, no matter whether the terms substituted in the course of applying the inference-rules are referentially stable or not. In the case of negation, the intensionality of the “that”-context is, so to speak, neutralized.

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A consistency proof of P0 – P23, however, is easily obtained: (a) Identify states of affairs with the subsets of the set of natural numbers: S(x) =Def x is a subset of the set of natural numbers. Identify, moreover, the relation of intensional parthood with the subset relation between sets of natural numbers: P(x, y) =Def x and y are subsets of the set of natural numbers, and x is a subset of y. (b) Identify actuality for states of affairs with being a subset of that set of natural numbers that contains all natural numbers except 19848: A(x) =Def x is a subset of that set of natural numbers that contains all natural numbers except 19848. (c) Let “b1” denote the empty set, and “b2” a proper non-empty subset of that set of natural numbers that contains all natural numbers exept 19848. Moreover, let every term “bn”denote some subset of the set of natural numbers. (d) Use the following definition: that A =Def CONJy(y = y ∧ ¬A). If one acts as is described in (a) – (d), then the principles P0 – P23 can be deduced as theorems of elementary set theory (where the defined terms in P0 – P23 are to be analyzed in the manner given by their definitions, but have the ultimate meaning that can be gathered from (a) – (d)). Therefore, the principles P0 – P23 have a model. They must, therefore, form a consistent set of principles. But, as we have seen, this pleasing picture is shattered if P24 is added to the mereology of states of affairs. There seems to be no easy proof of consistency for P0 – P24. Notice, however, that P24 can be proven true for each state of affairs x that is the intension of some sentence A: Since P10 is a conceptually true principle, we have: ├ A ≡ O(that A). Hence by EQU* (which has been shown to be a consequence of P23 in Section 3.7): ├ that A = that O(that A). There seems to be no reason why only this ought to be true, and not also P24. 3.9.1

Another Apparent Paradox

The Bases-Theory of Necessity can easily appear to lead to paradoxical results. Here is another rather interesting example: As a consequence of P23 (etc.), we have (1) ├ A ⊃ B ⇒ ├ P(that B, that A).

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And apparently it is logically provable that whatever is nomologically necessary is also true. That is, we seem to have (2) ├ 2A ⊃ A. Hence by (1) and (2): (3) ├ P(that A, that 2A), which, according to the Bases-Theory of Necessity, amounts to (4) ├ P(that A, that P(that A, b2)). Suppose now that the state of affairs that A is nomologically necessary, but not logically necessary, that is: (5) P(that A, b2) ∧ t* ≠ that A. And now we are in difficulties. Because of P(that A, b2) [in (5)] and P22, we have (6) t* = that P(that A, b2). Hence, because of (4) (after applying the step-down rule mentioned in Section 3.7), (7) P(that A, t*). But, because of P(t*, that A) and P3, this means (8) t* = that A, contradicting (5). What has gone wrong here? The problem appears to be (2). We can prove 2A ⊃ A (as we have seen; see Section 3.6). But this by itself doesn’t mean that 2A ⊃ A is also logically provable, as is stated by (2), and as is needed to obtain the paradox: if, in the above deduction, we put “2A ⊃ A” in the place of “├ 2A ⊃ A,” then the paradox does not get on its feet

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(since (1) can no longer be applied). The crucial question is: do we in fact have ├ 2A ⊃ A? Or in other words: can 2A ⊃ A be proven in such a manner that it is proven purely on the basis of axiomatic principles and inference-rules that are conceptually (broadly logically, analytically) valid (true)? (See footnote 17.) In proving nA ⊃ A, one cannot do without the assumption P(bn, w*) (see the proof in Section 3.6). Hence, for proving 2A ⊃ A in particular, one needs the assumption P(b2, w*) – which, in fact, is P19, a principle of the mereology of states of affairs. But is P19 a broadly logical principle, a conceptual truth?25 If it is not, then assumption (2) – that is,├ 2A ⊃ A – is seen to be false (though 2A ⊃ A is provably true!), and the paradox is avoided. If, however, P19 is a broadly logical principle – a principle that is true for conceptual reasons alone –, then we do indeed have ├ 2A ⊃ A, since the inference-rules and the other axiomatic principles that are employed in proving nA ⊃ A (and hence 2A ⊃ A) are all of the broadly logical kind. But the paradox can be avoided nonetheless. For “P(b2, w*)” can only be a broadly logical truth if “b2” is – like “w*” – a referentially unstable term; but if “b2” is a referentially unstable term, then the allinstantiation of P22 with “b2” in moving from (5) to (6) is illicit (since in that all-instantiation “b2” – a referentially unstable term – is substituted into a “that”-context). Now, why is it that “P(b2, w*)” can only be a broadly logical truth if “b2” is a referentially unstable term? – We have already seen that “w*,” which designates the conjunction of all obtaining states of affairs, is a referentially unstable term. In turn, the term “b2” designates the conjunction of all (states of affairs that are) laws of nature. The question is: does it designate this conjunction rigidly or not? In the first case, it would be a referentially stable term; in the second case, it would not be such a term. But whether “b2” is a referentially stable or, on the contrary, unstable term is entirely up to us. If “b2” is taken to be a referentially stable term, then “P(b2, w*)” says the following: the conjunction of all states of affairs that are laws of nature in actual fact is intensionally contained in the conjunction of all states of affairs that obtain in the circumstances (whichever states of affairs those states of affairs may happen to be). Understood thus, “P(b2, w*)” is clearly not a conceptual truth.

In other words: is 2 – nomological necessity – for conceptual reasons alone a necessity that is faithful to truth? 25

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For consider: if the obtaining states of affairs were such and such (that is, in certain relevant respects different from the states of affairs that obtain in actual fact), as is certainly logically possible, then the conjunction of all states of affairs that are in actual fact laws of nature would not be intensionally contained in the conjunction of all states of affairs that obtain in the (hypothesized) circumstances. Hence “P(b2, w*)” – given that “b2” is taken to be a referentially stable term – cannot be conceptually true.

Thus, if “P(b2, w*)” is to be a conceptual truth, then “b2” must be taken to be a referentially unstable term. And if one takes it to be a referentially unstable term, then “P(b2, w*)” says the following: the conjunction of all states of affairs that are laws of nature in the circumstances is intensionally contained in the conjunction of all states of affairs that obtain in the circumstances. This, indeed, is conceptually true. For the time being, I leave it open whether “b2” is to be taken as a referentially stable term or not. (A decision is made in Section 7.4, where “b2” is adopted as a referentially stable terms.) In either case, the apparent paradox we have been considering is seen to be not a paradox in fact.

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States of Affairs, Modality, and the Bases-Theory of Necessity

Formulating a theory of states of affairs, and a theory of modality within it, is one thing, answering the many questions the theories raise is another. This chapter addresses some of the most important of these questions.

4.1

States of Affairs, Propositions, and Sets of Possible Worlds

Consider the following linguistic entity: (0) the sentence “There is a talking donkey.” Consider in addition the following non-linguistic entities: (1) the state of affairs that there is a talking donkey; (2) the proposition that there is a talking donkey; (3) the set of possible worlds according to which there is a talking donkey. What is the relationship between the entities (0), (1), (2), and (3)? Surely entity (0) is different from the entities (1), (2), and (3). With great propriety, entity (0), the sentence, can be said to express entity (2), the proposition. With slightly less propriety, entity (0) can also be said to express entity (1), the state of affairs. There is no established propriety for saying that (0) expresses entity (3), the set. We may take these observations regarding usage in philosophical semantics as providing prima facie evidence for assuming that the entities (1), (2), and (3) are not only different from entity (0), but also different from each other. Some philosophers have not been impressed (and, of course, one needn’t be impressed by considerations of linguistic propriety). According to David Lewis, the entities (1), (2), and (3) are identical to each other. This is so, according to Lewis, because the set of possible worlds according to which there is a talking donkey just is the set of worlds that include a talking donkey (as concrete part), and this latter set is, according to him, both the proposition and the state of affairs that there is a talking donkey (see On the Plurality of Worlds, p. 185). In general, Lewis identifies propositions with sets of possible worlds (in his sense), and states of affairs are likewise identified by him with sets of possible worlds (ibid., p. 53 and p. 185).

4 States of Affairs, Modality, and the Bases-Theory of Necessity

According to the mereology of states of affairs, however, there is no good reason to identify states of affairs with any set-theoretical constructions. The concept of a state of affairs is a primitive – and it is well able to play the role of a primitive since our intuitive and axiomatic grasp of states of affairs is hardly less firm than our intuitive and axiomatic grasp of sets. In On the Plurality of Worlds, pp. 184-185, Lewis writes as if we had no solid idea of states of affairs that is independent of their prior identification with sets of possible worlds. Such a position seems strictly false to me. We can present any number of examples for states of affairs without having in mind at all that they are or might be sets of possible worlds, and all of the axioms in Chapter 3 are justifiable without reference to the identification of states of affairs with sets of possible worlds. True, states of affairs may be sets of possible worlds none the less. But this is not the assertion I deny. I deny that we have no solid idea of states of affairs that is independent of their prior identification with sets of possible worlds, and I assert that we have at least as much of a solid idea of states of affairs as primitive entities as we have of sets. We can give examples of sets as primitive entities – we can give examples of states of affairs as primitive entities. We can specify axioms for sets as primitive entities – we can specify axioms for states of affairs as primitive entities. And consider: the concept of state of affairs, such as it is used in philosophy, seems to be rooted to a greater extent in ordinary language than the concept of set, such as it is used in philosophy. On the basis of the mereology of states of affairs, it is, as a matter of fact, provable that each state of affairs is one-to-one correlated with the set of those maximal-consistent states of affairs – my possible worlds, not Lewis’s – of which it is an intensional part. Proof: Let p and q be two different states of affairs. Then the set of maximalconsistent states of affairs of which p is a part is different from the set of maximalconsistent states of affairs of which q is a part, and therefore the set of possible worlds corresponding to p is different from the set of possible worlds corresponding to q. This is seen as follows. Since p and q are different states of affairs, it follows according to P3 and P5: ∃x(QA(x) ∧ P(x, p) ∧ ¬P(x, q)) ∨ ∃x(QA(x) ∧ P(x, q) ∧ ¬P(x, p)). In either case, x must be a non-minimal state of affairs: ¬M(x), since it is not a part of q, respectively p (see D4). Therefore we have according to D10: ∃x(EL(x) ∧ P(x, p) ∧ ¬P(x, q)) ∨ ∃x(EL(x) ∧ P(x, q) ∧ ¬P(x, p)). Hence we have according to the provable principle ∀y[MC(neg(y)) ≡ EL(y)]1: ∃x(MC(neg(x)) ∧ P(x, p) ∧ ¬P(x, q)) ∨ ∀y[MC(neg(y)) ≡ EL(y)] follows from the provable principle ∀y[MC(y) ≡ EL(neg(y))], stated in footnote 19 of Chapter 3 and used in Section 3.8, by employing ∀y[S(y) ⊃ y = neg(neg(y))]. 1

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∃x(MC(neg(x)) ∧ P(x, q) ∧ ¬P(x, p)), and therefore: ∃y(MC(y) ∧ P(neg(y), p) ∧ ¬P(neg(y), q)) ∨ ∃y(MC(y) ∧ P(neg(y), q) ∧ ¬P(neg(y), p)) (employing ∀x[S(x) ⊃ x = neg(neg(x))]). Hence we have according to the provable principle ∀y[MC(y) ⊃ ∀x(S(x) ⊃ (P(x, y) ≡ ¬P(neg(y), x)))]2: ∃y(MC(y) ∧ ¬P(p, y) ∧ P(q, y)) ∨ ∃y(MC(y) ∧ ¬P(q, y) ∧ P(p, y)) – QED.

“A one-to-one correspondence is an opportunity for reduction,” says Lewis (“Events,” p. 245). But the one-to-one correspondence just proven is certainly not an opportunity for reducing states of affairs to sets of possible worlds, these worlds being, according to the present theory, states of affairs themselves. Or one might as well reduce individuals to their singleton sets, that is: identify each individual with the singleton set one-to-one correlated with it – which is not a reasonable, and certainly not a reductive, step to take. It thus turns out that, according to the present theory, states of affairs are not sets of possible worlds. And in particular, the state of affairs that there is a talking donkey – entity (1) – is not the set of possible worlds according to which there is a talking donkey – entity (3) –, although it is indeed one-to-one correlated with that set in the manner described. Nor is, according to the present theory, the proposition that there is a talking donkey – entity (2) – identical with the set of possible worlds according to which there is a talking donkey. This would already be clear on the basis of what has been said above if I, like Lewis, identified propositions and states of affairs. But it seems advantageous to reserve the designator “proposition” for entities that are neither sets of worlds nor states of affairs, because they are more closely connected with sentences than either states of affairs or sets of worlds, and because they are more finely differentiated than either states of affairs or sets of worlds. Instead of identifying states of affairs and propositions like Lewis, Alvin Plantinga merely asserts that “states of affairs are isomorphic to propositions” (“Self-Profile,” p. 90). This, I take it, is supposed to imply that any two sentences that express two different propositions are also about two different states of affairs, namely, the two states of affairs that correspond to the two propositions expressed, according to the supposed isomorphism between states of affairs and propositions. But it seems to me more sensible to suppose that the following is true: while “Dave is on the left of Al at t0” and “Al is on the right of Dave at t0” express two different propositions, they are nonetheless about the very same state of affairs.

2

This principle is stated in footnote 19 of Chapter 3 and used in Section 3.8.

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Plantinga’s isomorphism between states of affairs and propositions, and Lewis’s identification of states of affairs and propositions, can be grounded in two different ways: (a) states of affairs are assimilated to finegrained propositions, or (b) propositions are assimilated to coarse-grained states of affairs. Lewis, clearly, is a follower of (b). Plantinga, in contrast, is a follower of (a). This has a striking – and implausible – consequence. According to Plantinga, there is probably more than one actual world: “Among the possible worlds, there is one such that every state of affairs it includes is actual. In fact there are probably several such worlds.” (“SelfProfile,” p. 89). Plantinga holds that possible worlds are certain states of affairs (see Section 4.2), states of affairs being assimilated by him to finegrained propositions, and therefore (ibid.): “A pair of states of affairs [like a pair of propositions] can be distinct but equivalent in the broadly logical sense; and probably the same goes for possible worlds.” Clearly, if there is an actual world that is distinct from, but in the broadly logical sense equivalent to another possible world (as seems probable to Plantinga), then there are two actual worlds – because actuality is isomorphic to truth under the supposed isomorphism between states of affairs and propositions: just as propositions that are equivalent in the broadly logical sense are either both true or both not true, so states of affairs that are equivalent in the broadly logical sense are either both actual or both not actual. It is somewhat of a misnomer to call states of affairs “propositions”; it is a glaring misnomer to call sets of worlds “propositions.” Lewis has nevertheless opted for this infelicitous façon de parler. But at least Lewis recognizes the entities that I have just set apart from states of affairs and sets of possible worlds, and does not deny that they, instead of sets of worlds, could be called “propositions” (see On the Plurality of Worlds, pp. 57-58, p. 185). How are those entities I have now decided to call “propositions” connected to states of affairs? In the following way: Each proposition determines (in a certain sense) a state of affairs (and hence also a set of possible worlds); in particular, the proposition that A determines the state of affairs that A (where “A” can be replaced by any true or false sentence). But different propositions do not in every case determine different states of affairs (nor different sets of worlds). It sometimes happens that different propositions determine the same state of affairs (and the same set of worlds): although the proposition that a is larger than b is different from the proposition that b is smaller than a (at least if a and b are different), both propositions nevertheless determine the same state of affairs: the state

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of affairs that a is larger than b, which is no other state of affairs than the state of affairs that b is smaller than a.3 The set of propositions is partitioned into equivalence classes by the equivalence relation of stateequality. There is a one-to-one correlation between state-equality classes and states of affairs: (i) the propositions in each state-equality class determine the same state of affairs: they determine it in the sense that they are all about that state of affairs; (ii) propositions in different state-equality classes, being about different states of affairs, determine different states of affairs. “A one-to-one correspondence is an opportunity for reduction,” says Lewis (to repeat the above quotation). Would it be more reasonable, by the lights of the present theory, to reduce states of affairs to state-equality classes than to reduce them to classes of possible worlds? Probably not. Against both envisaged reductions, the case to be made is essentially the same. If possible worlds are special states of affairs, as they are according to the present theory, then the reduction of states of affairs to sets of possible worlds is out of the question, since states of affairs are already presupposed. And if propositions are constructions out of states of affairs, then the reduction of states of affairs to sets of propositions is also out of the question – since states of affairs are already presupposed. Propositions are not a concern of this book, since propositions (in contrast to states of affairs) are not an ontological requirement of a theory of ontic (alethic) modalities. But, indeed, I hold the view that propositions are constructions out of states of affairs.4 What would a construction of propositions out of states of affairs look like? Like this: a proposition is a state of affairs plus structure. Hence, for example, the proposition that a is larger than b is the state of affairs that a is larger than b (= the state of affairs that b is smaller than a) plus the ordered pair , in other words: that proposition is this: ; and the proposition that b is smaller than a is the state of affairs that b is smaller than a (= the state of affairs that a is larger than b) plus the ordered pair , in other words: that 3

Husserl seems to have been the first thinker to become aware of these relationships. See Logische Untersuchungen, vol. II, part 1, pp. 48-49. 4 This is related to, and at the same time quite different from, Lewis’s view that the entities which, he concedes, might be called “propositions” (“structured propositions”) instead of the sets of possible worlds are more complicated set-theoretical constructions out of possibilia than are sets of possible worlds; see On the Plurality of Worlds, p. 57.

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proposition is this: . It is immediately evident why those propositions are different (as intuition requires) if a is different from b, and identical (as intuition also requires) if a is identical with b – and why they determine the same state of affairs in both cases. But how would a more complicated proposition, like the one expressed by the sentence “George is born in 1964 or in 1963” be constructed out of states of affairs? – That proposition is the state of affairs that George is born in 1964 or in 1963 plus the relevant structure. That structure can be coded as follows: . Hence the whole proposition is this: .

However, the sketched theory of propositions does not fit all intuitions regarding propositions. On the one hand, it seems that the proposition that Hesperus is Phosphorus is different from the proposition that Hesperus is Hesperus, no matter whether Hesperus is identical with Phosphorus or not. But this intuition is contradicted by the suggested theory of propositions, since the state of affairs that Hesperus is Phosphorus plus (this combined entity) is certainly identical with the state of affairs that Hesperus is Hesperus plus if Hesperus is identical with Phosphorus – a condition which is in fact fulfilled. On the other hand, it seems that the proposition that George and Mary are human is identical with the proposition that Mary and George are human, no matter whether George is identical with Mary or not. But this intuition, too, is contradicted by the suggested theory of propositions, since the state of affairs that George and Mary are human plus (this combined entity) is certainly different from the state of affairs that Mary and George are human plus if George is different from Mary – again a condition which is in fact fulfilled. The rationale behind these intuitions, which, in effect, leaves the here presented theory of propositions entirely unscathed, is this: (1) the sentence “George and Mary are human” and the sentence “Mary and George are human” have the same cognitive function for us; this makes the propositions they express seem identical to us, although these propositions are as a matter of fact different from each other; (2) the sentence “Hesperus is Phosphorus” and the sentence “Hesperus is Hesperus” have different cog-

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nitive functions for us; this makes the propositions they express seem different to us, although these propositions are as a matter of fact identical to each other. It remains to be added: contrary to what may seem right at first sight, the identity/non-identity of expressed propositions should not invariably follow the sameness/difference of cognitive function of the expressing sentences, since sameness and difference of cognitive function in sentences are contingent and subjective, whereas identity and non-identity of propositions are necessary and objective. Indeed, for most people, “Hesperus is Phosphorus” and “Hesperus is Hesperus” have different cognitive functions; but doubtlessly, for some people, “Hesperus is Phosphorus” and “Hesperus is Hesperus” do have the very same cognitive function, and there might have been more people of that ilk. True, for most people “George and Mary are human” and “Mary and George are human” have the very same cognitive function; but doubtlessly, for some people, “George and Mary are human” and “Mary and George are human” do have different cognitive functions, and again there might have been more people of that ilk. Of course, if identity of expressed propositions does not invariably follow from the sameness of cognitive function in the expressing sentences (this is the Mary-George-case), then the proposition a sentence expresses will not always be determined by its cognitive function; and likewise: if non-identity of expressed propositions does not invariably follow from the difference of cognitive function in the expressing sentences (this is the Hesperus-Phosphorus-case), then the cognitive function of a sentence will not always be determined by the proposition it expresses.5 And so it is. But this does certainly not mean that the cognitive functions of sentences and the propositions expressed by them are to a considerable extent independent of each other. No, normally they co-vary, and a difference in cognitive function is coupled with a difference in expressed proposition, and vice versa.

Consider that ∀S∀S´(cog(S) = cog(S´) ⊃ prop(S) = prop(S´)) is a logical consequence of ∀S[prop(S) = ϕ(cog(S))] (“prop(S) is always determined by cog(S)”), and ∀S∀S´(cog(S) ≠ cog(S´) ⊃ prop(S) ≠ prop(S´)) a logical consequence of ∀S[cog(S) = ϕ´(prop(S))] (“cog(S) is always determined by prop(S)”), where cog(S) is the cognitive function of sentence S, and prop(S) the proposition expressed by it. 5

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4.2

Possible Worlds as States of Affairs

According to the mereology of states of affairs, possible worlds are maximal-consistent states of affairs. According to the modal realism of David Lewis, in contrast, possible worlds are spatiotemporally maximal individuals (see On the Plurality of Worlds, p. 71). Which theory is right? There is no right and wrong in the straightforward sense here, since what is at issue is a conceptual decision. But one may well point out that taking possible worlds to be states of affairs has certain advantages – provided of course one can make sense of states of affairs without already presupposing possible worlds that are not states of affairs. Lewis professes that he cannot make sense of states of affairs without already presupposing possible worlds that are not states of affairs (ibid., pp. 184-185); but, in view of what has been said in the previous section, this need not detain us here. Consider the state of affairs that U.M. is in the city of Regensburg and in no other city on July 31, 1977. Consider also the state of affairs that U.M. is not in the city of Regensburg, but in some other city on July 31, 1977. Both states of affairs are directly about U.M., and there seems to be nothing that contravenes the supposition that U.M. – he himself – is a constituent of both. It is not the usual thing to do, but it is certainly very useful if the concepts of part and of constituent are carefully distinguished. If x is a part of y, then this requires that x and y are of the same ontological category, whereas if x is a constituent of y, then x need not belong to the same ontological category that y belongs to. Thus U.M., being an individual (and therefore not a state of affairs), cannot be a part of the state of affairs that U.M. is in the city of Regensburg on July 31, 1977, or of any other state of affairs; but U.M. can certainly be a constituent of the aforementioned state of affairs, and of others. In contrast, the state of affairs that U.M. is in some city on July 31, 1977, is a part of the state of affairs that U.M. is in the city of Regensburg on July 31, 1977. Whether the first state of affairs is also a constituent of the second state of affairs is a matter of policy. On the one hand, it seems reasonable to postulate that every part of something is also a constituent of it (though not vice versa); hence the in-some-city state of affairs would be a constituent of the in-the-city-of-Regensburg state of affairs. On the other hand, to some people it will undoubtedly seem an improper way of speaking to say that the in-some-city state of affairs is a constituent – or, for that matter, a part – of the in-the-city-of-Regensburg state of affairs. But the reason for their discomfort seems to be a fact that is

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quite irrelevant from the ontological point of view: the sentence that standardly expresses the former state of affairs is neither a part nor a constituent of the sentence that standardly expresses the latter. Since the following state of affairs is logically possible: that U.M. is in the city of Regensburg and in no other city on July 31, 1977, this state of affairs is a part of some possible world (maximal-consistent state of affairs), say, of w. And since also the following state of affairs is logically possible: that U.M. is not in the city of Regensburg, but in some other city on July 31, 1977, this state of affairs, too, is a part of some possible world, say, of w´. Since U.M. is a constituent of the first-mentioned state of affairs, it follows that he is also a constituent of w, and since U.M. is a constituent of the last-mentioned state of affairs, it follows that he is also a constituent of w´. But w and w´ are different worlds. Therefore, U.M. is a constituent of two different worlds, and hence there is, in a clear sense, transworld identity of individuals. Transworld identity of individuals, in the sense described, is no more to be balked at than the familiar situation that two sentences are such that they are both about the same individual, that they can each be true, but that they cannot be true together. Plantinga has argued rather similarly: see “Self-Profile,” p. 90. But for him, presumably, I am not a constituent of two different worlds, although I, like Plantinga, “exist in at least two distinct worlds” (ibid.). For, in Plantinga’s eyes, to “exist in a world w” merely means as much as “necessarily, would have existed [would have had the property of existence] if w had been actual” (see ibid., p. 89).

As is well known, according to Lewis’s view of possible worlds, transworld identity of individuals, in the strict sense, is out of the question. In the place of transworld identity, Lewis has a substitute: the counterpart relation, and he is forced to give an account of de re possibility that more than anything else in Lewis’s philosophy of modality is bound to provoke the “incredulous stare” (concerning which, see On the Plurality of Worlds, pp. 133-135). According to Lewis, “Humphrey might have won the election” is true if there is some counterpart of Humphrey in some possible world that wins the election (see ibid., p. 10, pp. 194-195). Any such a possible world must be different from the actual world (for according to the actual world, Humphrey lost the election), and any such counterpart cannot be Humphrey; for Humphrey, being a spatiotemporal part of the actual world, cannot also be a spatiotemporal part of some other world. Of course not. Hence the relevant counterpart of Humphrey in some other world – the entity which renders it true that Humphrey might have won the election

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(given that Humphrey did not win the election) – is a person that is different from Humphrey. But how can anybody, who is not F, be constituted as being possibly F by somebody else being F in some other possible world? No amount of dialectic will ever soften the utter bizarreness of Lewis’s construal of the harmless, true sentence “Humphrey might have won the election.” The philosophy of modality developed in this book stays clear of counterpartism. There is the state of affairs that Humphrey wins the (presidential) election (in 1969); and there is the state of affairs that Humphrey loses that election. Both states of affairs are about Humphrey, and he is a constituent of both. By being a constituent of both states of affairs, Humphrey is a constituent of different worlds, namely, of the different worlds those states of affairs are part of. According to some worlds (the actual world among them), he loses the election, and since he is a constituent of these worlds, it is true to say that he loses the election in them. According to other worlds, he wins the election, and since he is a constituent of those worlds, too, it is also true to say that he wins the election in them. One might indeed say that Humphrey is leading more than one life (cf. the title of chapter 4 of Lewis’s On the Plurality of Worlds: “Counterparts or Double Lives?”). But as long as only one of those lives is actual, where is the deep inadequacy? I also fail to see what is supposed to be the deep inadequacy in supposing that Humphrey has five fingers on the left hand according to the actual world, of which he is a constituent, and six fingers on the left hand according to some other world, of which he is also a constituent (this is Lewis’s famous “problem of accidental intrinsics”; see On the Plurality of Worlds, pp. 199-201). If Humphrey were a figure cut out of cardboard, in such a way that there are five fingers on his left hand, and this cardboard figure somehow managed to be located in two rooms at once, then, indeed, it would be difficult to conceive how Humphrey could have five fingers on his left hand in one room, and six fingers in the other. But neither is Humphrey a cardboard figure, nor is his relationship to possible worlds, as a constituent of several of them, in any way comparable to being in several rooms at once. Being wholly present at several places in logical space is something entirely different from being wholly present at several places in space or space-time. This may well be the case even if logical space does not consist of maximalconsistent states of affairs, but of space-times isolated from each other (as it does for David Lewis; see ibid., pp. 69-76). Being wholly present in several space-times is cer-

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tainly not obviously comparable to being wholly present at several places in the same space-time. One should, therefore, not conclude that the former kind of multilocation is impossible if, on reflection, one finds that the only reason for concluding this is the unquestioned assumption of an analogy to the (alleged) impossibility of the latter kind of multilocation. Lewis himself gives no reason why one and the same individual might not be as a whole in several space-times at once; it must have seemed just obvious to him that such a thing cannot be.

But should one conclude in each and every case where we have a state of affairs that is about a certain individual, that that individual is a constituent of the state of affairs, and thereby a constituent of every world of which the state of affairs is a part? – One needn’t conclude this, but I also see no reason why one shouldn’t. What one should not do, however, is to run heedlessly through the following sequence of conclusions: “Fa” is a true sentence, hence the state of affairs that Fa is about a, and hence a is a constituent of the state of affairs that Fa, and thereby a is a constituent of every world (most likely several worlds) of which that state of affairs is a part. The reason for being careful about this is this: the initial inference “Fa” is a true sentence ⇒ the state of affairs that Fa is about a is not in every case an uncontroversially valid inference. Many people would agree that “Pegasus [the King of France in the year 2002, the golden mountain, the round square, …] does not exist” is a true sentence, but would deny that the state of affairs that Pegasus [the King of France in the year 2002, the golden mountain, the round square, …] does not exist is about Pegasus [the King of France in the year 2002, the golden mountain, the round square, …]. The following inference, however, is indeed a valid inference in every case (uncontroversially, I should say): “Fa” is a true sentence; a exists ⇒ the state of affairs that Fa is about a. Here it should be remembered what has been pointed out in Section 1.4: “a exists” – Ea – can be not true even if “exists” is understood in such a manner as to make “Everything exists” – ∀xEx – a logical truth: suppose the name “a” does not refer to anything. Hence the extra premise “a exists” is not otiose – even if “exists” is understood in such a manner (not by me, though) as to make ∀xEx a logical truth.

Turning to a different aspect of the question of the proper conception of possible worlds, I note that David Lewis opposes his realism about pos-

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sible worlds to certain other conceptions of them: to those which treat possible worlds (minus, of course, the actual world) in one way or another as abstract entities (see, in particular, chapter 3 of On the Plurality of Worlds). But if realism is what is wanted, then it seems to me one can also be a realist about abstract entities. Frege, for example, was a realist about abstract entities. However, setting aside the marginal question whether it is indeed appropriate to speak of realism with respect to possible worlds when they are taken to be abstract entities, let us rather ask whether possible worlds do in fact turn out to be abstract entities if they are taken to be maximal-consistent states of affairs. Are maximal-consistent states of affairs abstract entities? Lewis apparently assumes that they are, since he counts Plantinga – who, among others, has proposed that possible worlds are maximal-consistent states of affairs (see The Nature of Necessity, pp. 44-45) – among those that put abstract substitutes, ersatz worlds, in the place of the genuine things,6 the concrete spatiotemporal individuals Lewis has in mind when speaking of possible worlds. But states of affairs do not seem to be notedly abstract affairs for Plantinga. Is Socrates’ being snubnosed – Plantinga’s example of a state of affairs (see The Nature of Necessity, p. 45) – something abstract for him? Perhaps not always: perhaps not when he looked at a bust of Socrates (if he ever did); but, according to published records, Plantinga does indeed think that states of affairs – like propositions (to which, according to him, states of affairs are isomorphic; see “Self-Profile,” p. 90) – are abstract entities. This position is implicit in the following quotation: [In The Nature of Necessity] I took a possible world to be a way things could have been or a state of affairs … A possible world, therefore, like a property, proposition or set, is an abstract object: an object that (like God) is immaterial, but (unlike God) is essentially incapable of life, activity or causal relationships. (“Self-Profile,” p. 88.)

I wonder what could be abstract about Socrates’ being shorter than Plato (another Plantingian example of a state of affairs, ibid.). Moreover, for reasons given below in this section, it is not recommendable for someone who holds that possible worlds are states of affairs to maintain at the same time that states of affairs are – all of them – abstract entities.

6

Plantinga supposedly does so in a nondescript way; on “nondescript ersatzism,” see On the Plurality of Worlds, p. 141 and p. 183.

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Plantinga’s approach to possible worlds as states of affairs differs markedly in several respects from the present theory. For one thing, there is no elaborated theory of states of affairs into which his interpretation of possible worlds as states of affairs is embedded. Secondly, as was noted in the previous section, Plantinga, though not positively identifying states of affairs and propositions, assimilates states of affairs to finegrained propositions. Thirdly, and most importantly, Plantinga does not analyze modal statements, including counterfactual conditionals, in terms of a basically modality-free ontology.7 Moreover, apparently with no sense of inherent unclarity, he copiously uses unanalyzed modal meta-formulations, like the following: “if [possible world] W had been actual, x would have existed” (“Self-Profile,” p. 89), “it is necessary that if W had been actual, then God would have strongly actualized [state of affairs] S*” (ibid., p. 50), “there are possible worlds including God’s existence that he could not have weakly actualized” (ibid., p. 52). Because of their inherent unclarity, such modal statements should be avoided as far as possible (though they may well be not completely avoidable – this is one of the possible morals to be drawn from the next section).

No matter what Plantinga thinks, are those state of affairs – Socrates’ being snub-nosed, Socrates’ being shorter than Plato – abstract entities? It seems rather doubtful. But if it seems doubtful that all states of affairs are abstract entities, then it must seem even more doubtful that all maximal-consistent states of affairs are abstract entities. Wittgenstein, in the Tractatus, identified the world with everything that is the case, or in other words: with the totality of facts. Since the world is certainly not an abstract entity, the Wittgensteinian identification would be grotesquely wrong if CONJxO(x) – the totality of facts: a maximal-consistent state of affairs (cf. Section 3.2, D7, etc.) – were an abstract entity. But Wittgenstein’s identification of the world with everything that is the case is certainly not grotesquely wrong. I propose that states of affairs – the irreducible entities that I have in mind when speaking of states of affairs – are sometimes abstract, but mostly concrete. Like other concrete entities, concrete states of affairs are made minimally concrete by (essentially) involving time. If they involve space in addition to time, their concreteness is certainly enhanced. But spatiality – in contrast to temporality – is not a conditio sine qua non of concreteness. An example of a state of affairs that is concrete in a normal way (because involving both space and time) is this: the state of affairs that par7

“[A] possible world is a certain kind of possible states of affairs. Such modal notions as possibility and necessity, then, are not to be defined or explained in terms of possible worlds; the definition or explanation must go the other way around.” (“SelfProfile,” p. 89.)

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ticle a is at time t0 at location l0. And every maximal-consistent state of affairs of which the specified state of affairs is an intensional part is certainly also a concrete state of affairs. Why? – Since the state of affairs that particle a is at time t0 at location l0 essentially involves space and time, every state of affairs of which that state of affairs is an intensional part essentially involves space and time, intensional parthood being an essential relation.

Clearly, maximal-consistent states of affairs – possible worlds, according to the present theory – will typically be concrete entities. This provides sufficient reason for holding that the present theory of modality is not an alternative to modal realism (and therefore no ersatz for it), but an alternative form of modal realism: it is an alternative to modal realism of the Lewisian kind. Whether there are also abstract maximal-consistent states of affairs – abstract possible worlds – is an open question: they would have to be timeless, and yet maximally comprehensive consistent states of affairs. In any case, possible worlds such that all of them are abstract entities do not appear fit to serve as a general foundation of possibility. For possibility that is based on abstractly conceived possible worlds would just be abstract possibility. But, for the most part, possibility does not appear to be abstract. The possibility for me to go left and the possibility for me to go right at a certain juncture do not seem to have anything to do with my, somehow, going left according to one abstract entity and right according to another. Thus, if abstract worlds only are used (presupposed in one’s theory), concrete possibilities do not appear to be adequately establishable. In general, the general possibility of something is established by its being the case according to some possible world – this much seems clear. (The exceptions to the stated rule – not normally thought of – are discussed in the next section.) But not just any manner of being the case according to a world will do for every possibility. A question asked by David Lewis (leaving out the word “ersatz” in it) is highly pertinent (On the Plurality of Worlds, p. 141): “[H]ow is it that the ersatz worlds represent? In other words, how is it that such-and-such is the case according to a certain ersatz world?” Clearly, whatever is the manner in which an abstract world (whether ersatz or not) “represents” (something as being the case according to it), that manner does not appear fit for establishing concrete possibilities. Fortunately there are enough concrete worlds among the possible worlds regarded as states of affairs to establish all of the usual concrete

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possibilities. (Concerning unusual possibilities, see the next section.) And how is it that possible worlds regarded as states of affairs “represent”? In a very simple and direct manner: such-and-such is the case according to a certain world – a maximal-consistent state of affairs – if, and only if, suchand-such is an intensional part of it.

4.3

Supercontingent States of Affairs

The central theme of this section requires a certain amount of preparation. This is the task to which I now turn. One of the axiom-schemata of the mereology of states of affairs is P13: S(that A). Given the other principles of the mereology of states of affairs, the language whose sentences are represented by the schematic letter “A” must contain only sentences that are appropriate for assuming P13 together with these other principles; or alternatively, “A” must be taken to stand only for those sentences of that language that are appropriate for assuming P13 together with those other principles. Which sentences, then, are not intended by the “A” in P13 (and by the schematic letters “B,” “C,” etc., as used in other axiomatic principles)? First of all, no sentence is intended that is not syntactically appropriate for making an assertion. Secondly, no sentence is intended that – without being referred to a given context – makes only an incomplete assertion. Thirdly, as a safeguard against antinomies, no metalinguistic sentence is intended. I take it that P13 and the other principles of the mereology of states of affairs that contain schematic letters apply to every nonmetalinguistic sentence (and, by implication, predicate) of the intended language, which is such that it is syntactically appropriate for making an assertion and which in and by itself makes a complete assertion, even without being referred to a context. The qualification “non-metalinguistic” can be omitted if the intended language is not the metalanguage of any language (and therefore not of itself). Let’s assume it isn’t. The condition of syntactical appropriateness for making an assertion can be omitted if the sentences of the intended language are all syntactically appropriate for making an assertion. Let’s assume they all are (in other words: let’s assume the intended language is purely descriptive). Let the intended language be a fragment of natural language, formally regimented as needed, and at least as rich as our examples require. It is true that the contextdependence of complete assertion is not generally ruled out for the sentences of the intended language; but now let P13 (and the other principles

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which use schematic letters) apply only to those sentences of that language which are not context-dependent in their making a complete assertion. Consider, then, the following three sentences: “U.M. is 175 cm tall at t0” – “U.M. is 175 cm tall at t0 in w*” – “U.M. is 175 cm tall at t0 in w0.” Here “w0” is taken to designate the same entity as “w*”; but the semantic difference between the two names is this: “w0” is stipulated to be unable to designate any other entity than it does in fact (it is determined to be referentially stable), whereas we have seen (in Section 3.9) that “w*” – which is short for “CONJxO(x)” according to D7 – might have designated a different entity than it does in fact (it is not referentially stable). All three sentences make a complete assertion (add tacitly: by themselves, without reference to context), the first and second sentence no less so than the third. This view of the matter is not entirely indubitable. For is not the relationship between “U.M. is 175 cm tall at t0” and “U.M. is 175 cm tall at t0 in w0” analogous to the relationship between “U.M. is 175 cm tall” and “U.M. is 175 cm tall at t0”? Does not in each of these two pairs of sentences the second sentence have a higher degree of assertorial completeness than the first? It seems so, and therefore not only “U.M. is 175 cm tall” but also “U.M. is 175 cm tall at t0” – and together with this latter sentence, also its logical equivalent: “U.M. is 175 cm tall at t0 in w*” – do seem to make an incomplete assertion. But assertorial completeness need not be the same as maximal specificity. Although “U.M. is 175 cm tall at t0 in w0” is maximally specific, and more specific than “U.M. is 175 cm tall at t0,” we need not conclude, on that account, that the latter sentence does not make a complete assertion. We need not draw this conclusion, and I decide that I shall not draw it.

Therefore, I take it that P13 applies to all three sentences (considering them to be included in the intended language), and therefore: it is a state of affairs that U.M. is 175 cm tall at t0; it is a state of affairs that U.M. is 175 cm tall at t0 in w*; and it is a state of affairs that U.M. is 175 cm tall at t0 in w0. Moreover we have: [a] The first and second state of affairs are the same state of affairs. For the state of affairs that U.M. is 175 cm tall at t0 in w* is the state of affairs that P(that U.M. is 175 cm tall at t0, w*), and the latter state of affairs is the state of affairs that O(that U.M. is 175 cm tall at t0) (see Section 3.9), which in turn is the state of affairs that U.M. is 175 cm tall at t0 (according to P24; all-instantiation regarding the singular

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term “that U.M. is 175 cm tall at t0” can be applied, because that singular term is referentially stable). [b] The first and third state of affairs, however, are different states of affairs. For the state of affairs that U.M. is 175 cm tall at t0 in w0 is the state of affairs that P(that U.M. is 175 cm tall at t0, w0), and that state of affairs is either t* or k*, according to P22 (to which allinstantiation regarding the singular terms “that U.M. is 175 cm tall at t0” and “w0” can be applied, because both “w0” and “that U.M. is 175 cm tall at t0” are referentially stable terms). But the state of affairs that U.M. is 175 cm tall at t0 is certainly neither t* nor k*. Instead of the three sentences considered so far, consider now the following three, which are in a relevant respect analogous to the former ones, but are more “philosophical,” so to speak: “w0 is actual,” “w0 is actual in w*,” “w0 is actual in w0.” Since “actual” is to be understood tenselessly, each of these three sentences makes a complete assertion, and therefore P13 can be taken to apply to them. Arguing analogously to [a] and [b] above, the state of affairs that w0 is actual is seen to be identical with the state of affairs that w0 is actual in w*, but different from the state of affairs that w0 is actual in w0. The difference is due to the fact that the latter state of affairs is either t* or k* (cf. the argument in [b]), whereas the state of affairs that w0 is actual is certainly neither t* nor k*. This is intuitively clear, but it can also be proven in the mereology of states of affairs if the axiom “S(w0)” – an axiom of the broadly logical kind – is added to it, and also the axiom “w0 = w*” (which is not an axiom of the broadly logical kind): [c] The state of affairs that w0 is actual – in short: that A(w0) – is identical with the state of affairs that w0 obtains – in short: that O(w0); this is so on the basis of EQU*,8 because A(w0) ≡ O(w0) is logically provable. (Since A(w0) ≡ O(w0) is provable on the basis of D6 and the broadly logical axiom S(w0), it is provable purely on the basis of the broadly logical – i.e., conceptually true – axiomatic part of the mereology of states of affairs, that is: it is logically provable; cf. Section 3.7, footnote 17.) The state of affairs that O(w0), in turn, is identical with w0, according to P24 and the (new) axiom S(w0). 8

Regarding EQU*, see Section 3.4; regarding the derivation of EQU* from P23, see Section 3.7.

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(Since “w0” is a referentially stable term, the instantiation of P24 by “w0” is allowed.) Hence we have: the state of affairs that w0 is actual is identical with w0. Since w0 = w* (which identity-statement we have also assumed as an axiom) and w* is neither identical with t* nor with k* (according to P9 and P7), it finally follows that the state of affairs that w0 is actual is neither identical with t* nor with k*. QED. In [c], it has also been shown: the state of affairs that w0 is actual is identical with w0. In fact, as is easily seen, it is true of every state of affairs x: that x is actual = x. Thus, there is a certain redundancy to actuality if applied to states of affairs. But note that this peculiar redundancy does not make the predicate of actuality, when applied to states of affairs, redundant; it is, of course, not redundant, when applied to states of affairs, since certainly not every state of affairs is actual (rather, some states of affairs are actual, and some not). What has been said so far suffices to show that David Lewis’s doctrine of the indexicality of actuality (see his “Anselm and Actuality,” pp. 18-20, and On the Plurality of Worlds, pp. 92-94) cannot be true for possible worlds. Although Lewis’s possible worlds are individuals, not states of affairs, there is even for Lewis only a very small difference between possible worlds qua individuals and possible worlds qua states of affairs. This is so because states of affairs, according to Lewis (see On the Plurality of Worlds, p. 185), are sets of possible worlds qua individuals. Then the maximal-consistent states of affairs – which are the possible worlds qua states of affairs – must be for Lewis the unit sets of possible worlds qua individuals. Hence, for Lewis, possible worlds qua states of affairs and possible worlds qua individuals are one-to-one correlated, and a possible world qua state of affairs differs from its corresponding possible world qua individual only to the extent that a unit set differs from its sole element.

If the statement “w0 is actual” (or “w1 is actual,” or “w2 is actual,” …) were an indexical statement, as Lewis supposes, then P13 would not apply to it, since it would not by itself make a complete assertion. But “w0 is actual” could not be indexical with respect to persons, times, or spatial locations; it could only be indexical with respect to possible worlds. Hence: if the statement “w0 is actual” were indexical, then the complete assertorial content that is expressed by it in context (and in context only) would depend on the world in which that statement – “w0 is actual” – is made.

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And what, exactly, would this dependence be like? There is only one way for it to be: “w0 is actual” must be taken to mean the same as “w0 is a part of this world.” Then, with respect to w0 (that is, if stated in w0), “w0 is actual” would say as much as “w0 is a part of w0” (or in other words: “w0 is actual in w0”); with respect to another possible world w1, “w0 is actual” would say as much as “w0 is a part of w1” (or in other words: “w0 is actual in w1”); and so on. Consequently: with respect to w0, “w0 is actual” would express the state of affairs that w0 is a part of w0; with respect to w1, “w0 is actual” would express the state of affairs that w0 is a part of w1; and so on. This means: with respect to w0, “w0 is actual” would express t*, and with respect to every other possible world, “w0 is actual” would express k*, t* and k* being different states of affairs. The result reached (stated in the previous sentence) can be proven in the mereology of states of affairs, but does not depend on that background. It is something that also Lewis must accept, t* and k* being interpreted in the way he prefers: for him, t* is the set of all possible worlds qua individuals, k* the empty set.

But we have seen above that “w0 is actual” expresses – simpliciter, and hence with respect to every possible world – only one state of affairs: w0 (because “w0 is actual” expresses – simpliciter, and hence with respect to every possible world – the state of affairs that w0 is actual, and because that state of affairs is identical with w0). The necessary conclusion is this: there is no indexicality in “w0 is actual.” How could Lewis (have) escape(d) this conclusion? For him, w0 is a certain unit set whose sole element is a possible world qua individual. By Lewis’s lights, the indexicality of actuality that Lewis assumes for possible worlds qua individuals (and for other individuals) is doubtlessly inherited by Lewisian possible worlds qua states of affairs (i.e., by the unit sets of possible worlds qua individuals). But this position is incompatible with the following utterly plausible and natural assumption: “w0 is actual” expresses with respect to every possible world the state of affairs that w0 is actual, that is: w0. How could one deny this assumption with any good reason? I see none.

Besides the indexical interpretation of the actuality of possible worlds and the non-indexical interpretation I have opposed to it, there is a third interpretation of the actuality of possible worlds – an interpretation which is also non-indexical, but which is not merely non-indexical but also non-relative. The indexical interpretation and the non-indexical interpretation we have already considered both have the logical consequence that the

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statement “w0 is actual” is true according to one world – namely, according to w0 – and false according to all the other worlds. The same (mutatis mutandis) is true on both considered interpretations for any possible world W that is different from w0: the statement “W is actual” is true according to one possible world – namely, according to W – but false according to all the other possible worlds. This result is established as follows. Consider an arbitrary world W and the statement “W is actual.” This statement is true according to a world w´ – respectively: false – if, and only if, the state of affairs which “W is actual” expresses with respect to w´ is – respectively: is not – an intensional part of w´. According to the indexical interpretation, we have: With respect to W, “W is actual” expresses t*; with respect to any other possible world, “W is actual” expresses k*. Hence the state of affairs which “W is actual” expresses with respect to W is an intensional part of W (for that expressed state of affairs is t*, which is a part of every state of affairs), and the state of affairs which “W is actual” expresses with respect to any other possible world w´ is not an intensional part of w´ (for that expressed state of affairs is k*, which is not a part of any possible world). – The rest is clear. According to the non-indexical interpretation already considered, we have: With respect to any possible world, “W is actual” expresses W. Hence the state of affairs which “W is actual” expresses with respect to W is an intensional part of W (for that expressed state of affairs is W, and every state of affairs is an intensional part of itself), and the state of affairs which “W is actual” expresses with respect to any other possible world w´ is not an intensional part of w´ (for that expressed state of affairs is the world W, and no possible world is an intensional part of any other possible world, as is easily provable in the mereology of states of affairs). – The rest is clear.

However, according to the mentioned third interpretation of the actuality of possible worlds – which I now turn to – the picture is rather different. If that interpretation is adhered to, then “w0 is actual” turns out to be true according to all possible worlds, whereas for any world W that is different from w0, “W is actual” turns out to be false according to all possible worlds (hence: “W is actual” turns out to be false even according to W). The third interpretation of the actuality of possible worlds represents an absolute (or non-relative) understanding of the word “actual” as applied to possible worlds. The following is common ground for all three interpretations: among all the possible worlds, one world is actual, w0, all the other worlds are not actual (but merely possible). But this situation is not only a relative matter, a matter relative to w0 (which relativity might be interpreted by supposing that the statement “w0 is actual” is indexical, varying its full content with respect to different worlds; which relativity, however, is more plausibly interpreted without supposing that “w0 is actual” is in-

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dexical: by allowing it the same full content with respect to every world, in the manner described above). Rather, the actuality of w0 and the nonactuality of every other possible world is – in addition to being a matter relative to w0 – also an absolute matter, a matter that is not relative to any possible world. The distinction in actuality between w0 and all the other possible worlds can also be regarded as an absolute distinction – and it must also be thus regarded, or the telling of the metaphysical truth will be seriously deficient. Make no mistake: absolute actuality is a concept that is also used by David Lewis, although he does declare that absolute actuality is incomprehensible to him (see On the Plurality of Worlds, p. 93). But Lewis, too, would certainly have been ready to acknowledge that the actualitydifference between w0 and the other possible worlds (between these possible worlds qua states of affairs, but also between the corresponding possible worlds qua individuals) is not quite the same as the actuality-difference between, say, w1 and the other possible worlds. He, too, would have acknowledged that the first actuality-difference is in some way special. The correct interpretation of Lewis’s position is, therefore, not that absolute actuality is meaningless for Lewis, but that, for him, absolute actuality is no deep matter. Absolute actuality is, for him, metaphysically shallow: “The actual world is not special in itself, but only in the special relation it bears to the ontological arguer [or to anyone else among us]. … The world an ontological arguer [or anyone of us] calls actual is special only in that the ontological arguer [or anyone of us] resides there” (“Anselm and Actuality,” p. 20). In addition, the distribution of absolute actuality among the possible worlds is for him a broadly logical truth: “Ours [i.e., the world qua individual that corresponds to w0] is the actual world; the rest are not actual. Why so? – I take it to be a trivial matter of meaning. I use the word ‘actual’ to mean the same as ‘this-worldly’.” (On the Plurality of Worlds, p. 92.) The cited assertions may seem to address merely Lewis’s indexical conception of actuality, which can be captured by the following definition: x is actualI =Def x is a part of this world (cf. what has been said above). Lewis’s absolute conception of actuality, in contrast, is captured by the following definition: x is actual* =Def x is a part of the world David Lewis resides in [which is no other world than the world qua individual that corresponds to w0]. How are the two Lewisian concepts of actuality related? Aren’t they utterly different? How can Lewis have both of them? First of all, in ordinary language as used by Lewis, the two concepts are not ex-

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pressed by syntactically different predicates: it’s simply “actual” for both of them. Moreover, in the given context (of utterance made by David Lewis), they are in fact logically equivalent for Lewis; because it is found to be the case – from a Lewisian point of view (to which belongs, in particular, the doctrine of singular world-inhabitation) – that the identitystatement “this world = the world David Lewis resides in” is a broadly logical truth in the given context (but not in all contexts!); i.e., its truth, given the context, is determined solely by the content of the sentence.9 It therefore follows that in the given context – and from a Lewisian point of view – the statement “Ours is the actual* world; the rest are not actual*” (that is: “Our world is a part of the world David Lewis resides in, and no other world is a part of that world”) is as much a broadly logical truth as the statement “Ours is the actualI world; the rest are not actualI” (that is: “Our world is part of this world, and no other world is a part of it”). I shall use the predicate A*(x) – “x is absolutely actual” – for actuality in an absolute sense. How, then, is this new predicate – in the present theory, not Lewis’s – related to the ordinary predicate of actuality, A(x) – “x is actual,” for which I have specified several axioms in the mereology of states of affairs?10 The two predicates are co-extensional, and hence the following is true: ∀x[S(x) ⊃ (A*(x) ≡ A(x))]. But the co-extensionality of the two predicates, although it may be assumed as an axiom (and will then be provable in the trivial sense in which all axioms are regarded as provable), cannot be regarded as logically provable. For although “A*(w0)” is just as true as “A(w0),” although, therefore, “A(w0) ≡ A*(w0)” is true, the identity-statement “that A(w0) = that A*(w0)” is certainly false. And this shows that the co-extensionality of A(x) and A*(x) cannot be assumed to be logically provable. For if that co-extensionality were logically provable, then “A(w0) ≡ A*(w0)” would of course be logically provable, too, and then, by applying EQU*, we would also have: “that A(w0) = that A*(w0)” is logically provable. Hence something would be logically provable which is false – a result that must of course be avoided.

9

The statement “this world = the world David Lewis resides in” thus has, for Lewis, a status that is comparable to the status for him of the statement “I am David Lewis,” which is a broadly logical truth in the context of being uttered by David Lewis (but not in all contexts). 10 The explicit axioms for “A(x)” are P11 and P12, the implicit ones (relating to “A(x)” via “w*” and/or “O(x),” in view of the definitions of these latter expressions) are P7 – P10, P19, P21, and P24.

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Why is it that the statement “that A(w0) = that A*(w0)” false? If it were true, then “w0 = that A*(w0)” would be true, too, since “ that A(w0) = w0” is provably true (given the extra axiom S(w0), and given the semantical information that “w0” is a referentially stable term; see [c] above). But “w0 = that A*(w0)” is certainly false. – Or so it seems, for this forthright assertion leads to certain difficulties, which I am now going to present. If the entity designated by “that A*(w0)” is indeed not w0, which entity, then, is it? A prima facie (but only prima facie) likely candidate for designation by “that A*(w0)” which is different from w0 will soon become apparent. So far, nothing forbids the substitution of “A*(w0)” for the placekeeper “A” in P13, which yields: S(that A*(w0)). If this is correct, then we know: the entity designated by “that A*(w0)” is a state of affairs. If the entity designated by “that A*(w0)” is a state of affairs, then it is clear that the non-indexical statement “A*(w0)” expresses with respect to every possible world the same state of affairs: the state of affairs that A*(w0); then, too, the following truth-rule is valid for “A*(w0)”: For all possible worlds w: “A*(w0)” is true according to w if, and only if, the state of affairs expressed by “A*(w0)” with respect to w is a part of w. Hence: For all possible worlds w: “A*(w0)” is true according to w if, and only if, the state of affairs that A*(w0) is a part of w. Now, we already know that the sentence “A*(w0)” – “w0 is actual” in the absolute interpretation – is true according to all possible worlds (see above). And therefore we have: the state of affairs that A*(w0) is a part of every possible world. But, as can be shown in the mereology of states of affairs, there is only one state of affairs that is a part of every possible world: t*. And therefore we finally obtain: the state of affairs that A*(w0) is identical with t*, in other words: that A*(w0) is identical with the state of affairs that 1 = 1. This result, however, contradicts an intuition which most of us have: that “A*(w0)” is only contingently true, that it is true, but might have been not true. Has this intuition now been shown to be wrong? Is the absolute actuality of w0 (and therefore also the absolute actuality of all the evil deeds w0 is replete with) logically necessary and “A*(w0)” true with logi-

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cal necessity? Are, in other words, “1(that A*(w0))” and “1A*(w0)” true? We might accept this conclusion, although it is truly hard to believe. We might also seek a less doxastically costly way out of the difficulty into which the above argumentation has brought us. The most satisfactory way out, I submit, is the following, which consists in adopting the positions (1) – (8) below: (1) It will not do to deny that there is an absolute understanding of actuality, and in particular of the actuality of possible worlds. That there is an absolute understanding of actuality must be accepted. (2) “w0 is absolutely actual” is true, but might not have been true. (3) Nevertheless, the sentence “w0 is absolutely actual” – “A*(w0)” – is true according to every possible world (every maximal-consistent state of affairs). (Patience!) (4) The entity designated by “that w0 is absolutely actual” – “that A*(w0)” – is not a state of affairs; (3) is not equivalent to the following assertion: “the state of affairs that w0 is absolutely actual is a part of every possible world.” (5) Since “S(that A*(w0))” is false, P13 does not apply to that sentence. Therefore: the restrictions for the sentences that are substitutable for “A” in P13 or for the other schematic letters in the other principles of the mereology of states of affairs go further than hitherto required. (So far, these restrictions merely consisted in this: the substituted sentences must make a complete assertion, the intended language being such that it contains no metalinguistic sentences and only sentences that are appropriate for making an assertion.) (6) Another consequence of the falsity of “S(that A*(w0))” is this: both “w0 = that A*(w0)” and “t* = that A*(w0)” are false (because “S(w0)” and “S(t*)” are both true). (7) Although “that A*(w0)” does not designate any state of affairs in the normal sense (see (5)) – that is, one of the entities which I have simply called “states of affairs” – it designates a higher-order state of affairs,11 and hence something outside the extension of “S(x).” The unattractive alternative would be to assert that “A*(w0)” expresses nothing that is a state of affairs in any sense; for if “A*(w0)” ex11

Note that the notion of higher-order states of affairs here invoked is entirely different from what is meant by “higher-order states of affairs” in David Armstrong’s sense of the phrase (see his A World of States of Affairs, p. 1).

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presses what is a state of affairs in some sense, then it must be a higher-order state of affairs and is rightfully designated by “that A*(w0).” (8) Since “A*(w0)” is contingently true (see (2)), the higher-order state of affairs expressed by it must be contingent. But that higher-order state of affairs is not contingent in the way normal states of affairs – the states of affairs in the extension of “S(x)” – are contingent: by being a part of some possible world (that is: of some maximalconsistent normal state of affairs), and not a part of some other possible world;12 its contingency is a higher-order contingency. Thus, the higher-order state of affairs that is expressed by “A*(w0)” is a supercontingent state of affairs. (At the same time, the truth of “A*(w0)” does not depend on any possible world – which is precisely the fact that yields (3).) Here are two further examples of sentences that express supercontingent states of affairs (in other words: contingent higher-order states of affairs): ¬A*(w1) (“w1” designating in a referentially stable way a world that is different from w0), ∃w(MC(w) ∧ A*(w)). But there are certainly limits to supercontingency regarding the predicate A*(x): “A*(t*)” and “¬A*(k*)” are not contingent truths, and these two sentences do not express supercontingent states of affairs; nor does “A*(w*)” – in contrast to “A*(w0)” (in spite of the extra axiom “w0 = w*,” which, however, does not state a conceptual truth). How so? – In Section 3.3 we have seen that “A(t*)” and “¬A(k*)” are logically provable, that is: provable (in the mereology of states of affairs) purely on the basis of axiomatic principles and inference-rules that are conceptually valid (true) (cf. footnote 17 in Chapter 3). It seems appropriate to adopt the following inference-rules: ├ A(n) ⇒ ├ A*(n), ├ ¬A(n) ⇒ ├ ¬A*(n), where “├” stands for “logically provable.” Then it follows on the basis of ├ A(t*) and ├ ¬A(k*): ├ A*(t*) and ├ ¬A*(k*). And therefore also: ├ A(t*) ≡ A*(t*), ├ ¬A(k*) ≡ ¬A*(k*). Hence by applying EQU* (allowing it – and P23 – to be applicable no matter which sentences that make a complete assertion are substituted for the schematic letters in it): ├ that A(t*) = that A*(t*), According to P0, ∃yP(z, y) is not true for any higher-order state of affairs z (since z, as a higher-order state of affairs, does not belong to the extension of “S(x)”). But z may yet be an intensional part of something: of some higher-order state of affairs or other; for P0 does certainly not imply that the relation of intensional parthood holds only between states of affairs: P0 is also made true if “P(x, y)” is understood to mean as much as “x is an intensional part of y, and S(x) and S(y).” 12

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├ that ¬A(k*) = that ¬A*(k*). But from ├ A(t*) and ├ ¬A(k*) it follows by P23: ├ t* = that A(t*), and ├ t* = that ¬A(k*). And therefore we finally have: ├ t* = that A*(t*), ├ t* = that ¬A*(k*). In other words, the sentences “A*(t*)” and “¬A*(k*)” both express the state of affairs t*. Concerning “A*(w*),” it must first be noticed that “A(w*)” – definitionally equivalent to “A(CONJx(S(x) ∧ A(x)))” on the basis of D6 and D7 – is the conceptually true axiom P12. And therefore we have (trivially): ├ A(w*). From this we obtain by the first of the two inference-rules introduced above: ├ A*(w*). Hence we also have: ├ A(w*) ≡ A*(w*), and hence we obtain by applying EQU*: ├ that A(w*) = that A*(w*). From ├ A(w*) it follows by P23: ├ t* = that A(w*) [and consequently: w* ≠ that A(w*) = that O(w*), since w* ≠ t,* according to P9; no contradiction arises: P24 is not instantiable by “w*,” since “w*” is not a referentially stable designator]. And therefore we finally have: ├ t* = that A*(w*).

We may take it that every sentence (of the intended language) which, (a), makes a complete assertion (by itself, without reference to context) and, (b), is true, but not already true for conceptual reasons (or in other words: not already true because of the content expressed), and whose truth, (c), does nevertheless not depend on possible worlds (i.e., which is nevertheless true according to all possible worlds) expresses a supercontingent state of affairs. However, it is not irrational to suppose that there are no sentences that fulfill all three conditions: (a), (b), and (c). In other words, it is not irrational to suppose that every true sentence that makes a complete assertion and whose truth does not depend on possible worlds – for example, the sentence “A*(w0)” – is conceptually true, that is: expresses the state of affairs t*. This supposition just seems wrong to me on the basis of my metaphysical intuitions. Consider, then, the sentence “QC(w*)”: the axiom P8. This is a sentence which is formulated in the original language of the mereology of states of affairs – without the predicate “A*(x).” Does “QC(w*)” express a supercontingent state of affairs? At the end of Section 3.3, I maintained that this principle, though true, is not a conceptual truth, and it certainly seems to make a complete assertion. For obtaining the supercontingency of “QC(w*),” the only remaining question to be answered positively is, then, the following: Does the truth of “QC(w*)” depend on possible worlds? Is that sentence true according to some possible worlds (maximal-consistent states of affairs), not true according to others? – And in fact there seems to be no plausible way of stating truth-conditions for “QC(w*)” so as to make it true according to some worlds and not true according to others; “QC(w*)” is simply not a sentence that truth-behaves (if I may say so) like “The only son of A.M. is 175 cm tall at t0.” But on the other hand, there are

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two prima facie acceptable truth-rules for the predicate “O(x)” which make the sentence in question true according to every possible world: Given the following truth-rule: (I) “O(x)” is true of y according to a possible world w iff O(y, w) [=Def P(y, w), according to D16], the outcome is that “CONJxO(x)” (“w*”) designates, according to each possible world w, no other state of affairs than w itself. Since every possible world is (by definition) a quasi-complete state of affairs, the further consequence is that “QC(w*)” is true according to every possible world. However, the basis given for this result does not fit the intuition that “QC(w*)” is not a conceptual truth, since the invoked truth-rule seems to be correct for purely conceptual reasons if it is correct at all, and since “O(y, z)” applies (respectively, does not apply) for intrinsic reasons whenever it applies (respectively, does not apply). Consider, therefore, the following alternative truth-rule: (II) “O(x)” is true of y according to a possible world w iff O(y, w*) [=Def P(y, w*), according to D16]. Here the outcome is that “CONJxO(x)” (“w*”) designates, according to each possible world w, no other state of affairs than w*. Since w* is in fact a quasi-complete state of affairs, the further consequence, once more, is that “QC(w*)” is true according to every possible world. And in this case the basis for this result does fit the intuition that “QC(w*)” is not a conceptual truth. True, the invoked truth-rule is again correct for purely conceptual reasons if it is correct at all; but, in contrast to “O(y, z),” it is not the case that “O(y, w*)” applies/does not apply for intrinsic reasons whenever it applies/does not apply. Indeed, for all y and z, we have 1O(y, z) ∨ 1¬O(y, z); however, not for every y we have 1O(y, w*) ∨ 1¬O(y, w*).13 Suppose the contrary: ∀y(1O(y, w*) ∨ 1¬O(y, w*)). From the logically provable theorem ∀y(O(y) ≡ P(y, w*)) we obtain by all-instantiation the logically provable theorem O(that U.M. is 175 cm tall at t0) ≡ P(that U.M. is 175 cm tall at t0, w*), and therefore by applying EQU*: (i) that O(that U.M. is 175 cm tall at t0) = that P(that U.M. is 175 cm tall at t0, w*). By P13 (applying it to “U.M. is 175 cm tall at t0”) and by P24 (we may take it that all designators involved are referentially stable) we have: (ii) that U.M. is 175 cm tall at t0 = that O(that U.M. is 175 cm tall at t0). Therefore (combining (i) and (ii)): (iii) that U.M. is 175 cm tall at t0 = that P(that U.M. is 175 cm tall at t0, w*). Written out in more basic notation, the supposition ∀y(1O(y, w*) ∨ ∀y(1O(y, w*) ∨ 1¬O(y, w*)) cannot be (safely) inferred from ∀z∀y(1O(y, z) ∨ 1¬O(y, z)) via all-instantiation, since “w*” is not a referentially stable term. (Note that substituting a term into the context of 1O(y, z) involves substituting it into a “that”-context, since 1O(y, z) definitionally amounts to P(that P(y, z), t*).) 13

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1¬O(y, w*)) amounts to this: ∀y(P(that P(y, w*), t*) ∨ P(that ¬P(y, w*), t*)), which (since the relevant referential stability is given) logically implies: P(that P(that U.M. is 175 cm tall at t0, w*), t*) ∨ P(that ¬P(that U.M. is 175 cm tall at t0, w*), t*). Hence, using P14 and the identity-statement (iii) obtained above: P(that U. M. is 175 cm tall at t0, t*) ∨ P(neg(that U. M. is 175 cm tall at t0), t*), or in other words (using P14 again): P(that U.M. is 175 cm tall at t0, t*) ∨ P(that U.M. is not 175 cm tall at t0, t*). But this is absurdly false.

We may, therefore, conclude that the sentence “QC(w*)” – axiom P8 – is a case of supercontingency: it expresses a supercontingent state of affairs, though the case is certainly less clear than that of “A*(w0).” How could one escape the conclusion of supercontingency? The best idea, perhaps, is to deny that “QC(w*)” makes a complete assertion – on the ground that “w*” is a referentially unstable term. It is less promising to deny that “QC(w*)” is not a conceptual truth. While I believe that “QC(w*)” is indeed not a conceptual truth, and that it is indeed a case of supercontingency, I do allow that believing the contrary of what I believe is not irrational. As was pointed out above, even assuming that no sentence fulfills the conditions (a), (b) and (c) is not irrational. But it does have a high price: among other things, it means accepting that “A*(w0)” is conceptually true, that the absolute actuality of w0 is a matter of content, that the intrinsic nature of w0 and of absolute actuality determine by themselves that w0 is absolutely actual. Does this seem believable? The alternative: the supercontingency of “A*(w0),” its expressing a supercontingent state of affairs (a contingent higher-order state of affairs), is by no means an idea that is easy to develop. Most questions regarding supercontingent states of affairs – once it is assumed that there are such things – are entirely open. Prima facie, the idea of supercontingency seems to imply an infinite hierarchy of levels of higher-order states of affairs (including higher-order possible worlds). Is such a hierarchy a bad thing, ontologically speaking? Is there a plausible way to avoid it? But asking these questions must suffice. This book is not the place for developing a theory of supercontingent states of affairs – that is, a theory of higher-order states of affairs and higher-order contingency. The purpose of this section was to point out a deep issue that makes it seem unlikely that a reductive theory of modality can be entirely successful.

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4.4

Armstrong’s States of Affairs

It is not amiss to compare some of the views on states of affairs advocated in this book with the very different views of David Armstrong, as presented in his book A World of States of Affairs. The following Armstrongian positions regarding states of affairs strike me as central for his approach: 1. States of affairs and facts are identified. 2. The central motivation for assuming states of affairs is the need for truthmakers of atomic sentences. 3. There are no negative or disjunctive states of affairs, and all states of affairs are contingent. 4. Possibilities are combinations of simple particulars, properties, and relations. Concerning Armstrong’s position 1: This position is contradicted by the present theory of states of affairs. In agreement with a large part of philosophical usage, facts are defined as obtaining states of affairs, and, in agreement with common sense, it is assumed that not every state of affairs obtains. For example, the state of affairs that David Armstrong never met David Lewis does not obtain. Therefore: some states of affairs are not facts. As a matter of fact, Armstrong is less than clear on this issue. The word “fact” seems to mean for Armstrong as much as “existent state of affairs.” In Armstrong’s eyes, the modifier “existent” can be dropped, since for him every state of affairs exists, and hence it comes to pass that Armstrong uses the words “fact” and “state of affairs” interchangeably. But what, exactly, does the word “exist” mean for Armstrong when he applies it to states of affairs? On the one hand, we find him saying that “[s]ince states of affairs are always existents, there is no point in speaking of a disjunction of states of affairs” (A World of States of Affairs, p. 134); this suggests that by “existent state of affairs” Armstrong means what can also be expressed by the phrase “obtaining state of affairs”: if all states of affairs obtained, then, indeed, there would be no point in speaking of a disjunction of states of affairs (for by claiming that a disjunction of states of affairs obtains, one would be saying less than one can say, namely: that both its disjuncts ob-

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tain). That Armstrong interprets “existent state of affairs” to mean the same as “obtaining state of affairs” is also suggested by the following quotation: “A state of affairs exists if and only if a particular ... has a property or, instead, a relation holds between two or more particulars.” (Ibid., p. 1.) The condition Armstrong adduces for the existence of a state of affairs (presumably he means to speak of an atomic state of affairs; cf. ibid., p. 20) is clearly a condition that is necessary and sufficient for the obtaining of a state of affairs (an atomic one). On the other hand, we also find Armstrong “proceed[ing] on the assumption that the merely possible is non-existent” (ibid., p. 149) and saying in the very next sentence: “But, of course, it is often true that some state of affairs is possible, although the state of affairs does not obtain.” This last sentence is utterly puzzling, in view of what has preceded it. In the sentence “The merely possible is non-existent,” the word “non-existent” can be taken either in the quantifier-sense or in the modifier-sense; if “non-existent” is taken in the quantifier-sense, then that sentence says as much as “Nothing is merely possible”; if “non-existent” is taken in the modifier-sense, then that sentence says as much as “Everything that is merely possible is non-existent.” Obviously, the two senses of “non-existent” are not logically equivalent, and they lead to two readings of “The merely possible is non-existent” that are not logically equivalent. Which of the two senses is intended by Armstrong? If it were the modifier-sense, then Armstrong would not embroil himself in contradiction: the assertions “Everything that is merely possible is non-existent” and “Some state of affairs is possible but does not obtain” do not contradict each other. Unfortunately, it does not seem to be the modifier-sense of “non-existent” that is intended by Armstrong when he says that the merely possible is non-existent. On p. 148 of A World of States of Affairs he asserts: “A doctrine of merely possible states of affairs … is incompatible with Naturalism.” In consideration of the fact that naturalism is Armstrong’s strongest metaphysical commitment, his assertion can only be taken to imply his believing that no state of affairs is merely possible. And, in fact, he subsequently (on p. 149) offers general arguments against assuming any merely possible entities. In the arguments, it is quite clear that Armstrong is intending the quantifiersense of “exists” and “existence”: the sense of the usual “there is.” (If an existencepredicate is taken to be necessary for the analysis of this sense, then that predicate can only have the bland meaning of general being; cf. Section 1.4). One of his arguments is this: Since the merely possible is causally inert regarding the actual, and stands in no

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other external relation to the actual, “it becomes very hard to see how we could know or even have any reason to believe in the existence of merely possible entities” (ibid.). The other one of his arguments against merely possible entities is the following: “Whether [the merely possible] exists or whether it does not exist, the course of thought in the actual world is exactly the same” (ibid.). But, concerning the first argument, it suffices to point out that I have very good reasons to believe that the state of affairs that David Lewis never met David Armstrong is merely possible (and David Armstrong has even better reasons than I to believe this), or that the-electron’s-havingthis-mass (which is somewhat greater than the actual mass) is merely possible (to use an example that Armstrong would certainly prefer; cf. ibid., p. 20). Therefore: I have very good reasons to believe that there are merely possible entities. In fact, I can reasonably be said to know that there are such entities. In turn, the claim made in Armstrong’s second argument is simply false. If the merely possible did not exist, if, in other words, there were no merely possible entities, then the state of affairs that David Lewis never met David Armstrong would be actual, and so would be all merely possible states of affairs regarding the mass of the electron – and this situation would certainly make a difference to the “course of thought in the actual world.”

It is, therefore, as good as certain that Armstrong intends the quantifier-sense of “non-existent” in saying “the merely possible is non-existent.” But this gets him into trouble; for the assertions that he then must be taken to make, “Nothing is merely possible” and “Some state of affairs is possible but does not obtain,” do contradict each other. For a state of affairs that is possible but does not obtain is, ipso facto, merely possible, is it not? How might Armstrong escape from contradicting himself here? There seems to be no way out. It seems as clear as anything that x’s being a possible state of affairs and a non-obtaining state of affairs analytically implies that x is a merely possible state of affairs. The only thing Armstrong might say in response (but which is at variance with common philosophical usage) is this: the non-obtaining of a possible state of affairs is not sufficient for its being a merely possible state of affairs; it is required that the state of affairs be non-existent. But this move – in implying that there is no valid broadly logical inference from being a non-obtaining state of affairs to being a non-existing one, in other words: from being an existing state of affairs to being an obtaining one – is at variance with Armstrong’s antecedent identification of the existence (in the modifier-sense) of states of affairs with their obtaining (see above). The conclusion must be that Armstrong is not quite sure of what he means to assert when he says that every state of affairs exists. The majority of his utterances point towards his meaning to assert that every state of affairs is a fact, an actual state of affairs, an obtaining state of affairs. On p. 19 of A World of States of Affairs, he even declares: “The phrase ‘state

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of affairs’ will be used in the same way that Wittgenstein in the Tractatus used the term ‘fact’.” As is well known, Wittgenstein by “fact” (“Tatsache”) meant as much as is meant by “obtaining state of affairs” (“bestehender Sachverhalt”); see Tractatus, proposition 2. But in another place Armstrong has flatly contradicted the interpretation of “Every state of affairs exists” as “Every state of affairs obtains”; we find him saying what was already quoted above: “But, of course, it is often true that some state of affairs is possible, although the state of affairs does not obtain.” Does Armstrong then, in claiming that every state of affairs exists, merely mean to say that every state of affairs is something? If this is Armstrong’s claim, then he is certainly asserting something that is true – but also something that is as uninteresting as it is uncontroversial. (Note that ∀x(S(x) ⊃ ∃y(x = y)) is a trivial theorem of classical first-order predicate logic with identity, and also of free logic.) The most likely diagnosis is that Armstrong, like so many others, is from time to time confusing the two concepts of existence that were distinguished in Section 1.4: existence as actuality and existence as being something. Concerning Armstrong’s position 2: Armstrong declares (A World of States of Affairs, p. 119): “[T]here seems to be no acceptable candidate for a truthmaker for statements that contingently link particulars to universals other than states of affairs.” There is good reason for disagreeing with Armstrong in this respect. There is, of course, a link between states of affairs and truth (of statements); that link is adequately captured by axiom(-schema) P10 of the mereology of states of affairs. But to interpret that link as a link of (literal) truthmaking is more than is philosophically warranted – even in a case that seems ideal for such an interpretation. Consider the following statement, assumed to be true: “The electron e 0 has the mass m0.” We may reasonably say (in accordance with P10) that this statement is true because the state of affairs that e0 has the mass m0 obtains. But does this mean that this state of affairs, or any other state of affairs, makes that statement be true, in the proper (and not the just described harmless) sense of the phrase, in which proper sense it is more than just a façon de parler? I do not think so. For it is entirely reasonable to ask: what, ultimately, makes a state of affairs obtain that, allegedly, makes the statement “The electron e0 has the mass m0” be true? Now, an ultimate factmaker with regard to an alleged factual truthmaker for “The electron e0 has the mass m0” cannot be a state of affairs; for any alleged factual truthmaker for that statement would have to be a con-

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tingent state of affairs (a state of affairs for which it is possible that it does not obtain), and no contingent state of affairs is ultimately made a fact by a state of affairs. This last assertion is seen to be true in the following way: (I) A state of affairs p is ultimately made a fact by a state of affairs x if, and only if, x makes p a fact and is itself a fact ipso facto, that is: makes itself a fact. (II) If a state of affairs makes a contingent state of affairs a fact, then it is itself a contingent state of affairs. (III) No contingent state of affairs makes itself a fact. From the principles (I) through (III) it follows logically: No contingent state of affairs is ultimately made a fact by a state of affairs.

Thus, there either is no ultimate factmaker with regard to an alleged factual truthmaker for the statement “The electron e0 has the mass m0”, or there is one, but it is not a state of affairs. In the first case, one is forced to conclude that the alleged factual truthmaker of the statement is not really a truthmaker for it: in reality, the truth of the statement is not made by anything, it just is; that is: the state of affairs expressed by the statement obtains and is a part of the totality of all obtaining states of affairs – and that’s the end of it. In the second case, one must again conclude that the alleged factual truthmaker of the statement is not really a truthmaker for it: in reality, an ultimate factmaker of the alleged factual truthmaker is a nonfactual truthmaker for the statement. Therefore, searching for a truthmaker (in the proper sense of the word) for a contingently true, atomic statement either leads to no such thing, or to a truthmaker that is not a state of affairs. Hence states of affairs cannot be the truthmakers par excellence for contingently true, atomic statements, like “The electron e0 has the mass m0” – contrary to what is maintained by Armstrong. He says: We are asking what in the world will ensure, make true, underlie, serve as the ontological ground for, the truth that a is F. The obvious candidate seems to be the state of affairs of a’s being F. In this state of affairs (fact, circumstance) a and F are brought together. (A World of States of Affairs, p. 116.)

Armstrong might have profitably asked himself what in the world will ensure, underlie, serve as the ontological ground for, the state of affairs (fact, circumstance) of a’s being F. If there is such a thing X in the world, then it is also the rightful candidate for being the (or a) truthmaker of the true sentence “a is F,” and it is far from obvious that X will turn out to be identical with a’s being F. But if there is no such thing, then there simply is no

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truthmaker for “a is F,” its truth (its expressing an obtaining state of affairs) notwithstanding. (One had better not require that where there is truth there must be a truthmaker in the proper sense of the word.) Being intent on states of affairs as truthmakers (“the truthmaker argument for states of affairs”; ibid., p. 119), Armstrong entirely ignores that there is a second ontological approach to states of affairs – an approach that is both more promising and much broader than the truthmaker approach, which, for the reasons given above, can properly be said to be a non-starter. (And even regarding an attenuated sense of “truthmaking,” one can fairly ask what makes any state of affairs more of a truthmaker for a true statement of the form “a is F” than the entity a itself?) This second, but historically prior, approach to states of affairs was, for example, pointed out by Edmund Husserl (in the Logische Untersuchungen, especially in the Fifth Logical Investigation). According to Husserl, states of affairs (in German: “Sachverhalte”) make their appearance to us as intentional objects: objects of our mental intentionality. Indeed, states of affairs are the objects of our perception, imagination, consideration, judgment, belief, hope, will, doubt, and so on. (One can add that they are also the objects of our non-mental – or rather: not purely mental – intentionality: the intentionality of making and doing.) It should be noted that the intentionality approach to states of affairs does not imply that states of affairs are, somehow, subjective entities; nor does it imply that states of affairs are more finely differentiated – “finegrained” – than they ought to be if they are to be language-independent entities. In the cases where mental intentionality requires as its objects entities that are like states of affairs but more finely differentiated, such entities are at hand: propositions (cf. Section 4.1). Concerning Armstrong’s position 3: As long as one is myopically concentrating on the truthmaker approach to states of affairs (the approach is described above), it can seem as if negative and disjunctive states of affairs have nothing going for them. Thus Armstrong: Since states of affairs are always existents, there is no point in speaking of a disjunction of states of affairs. Suppose, however, that S, though a possible state of affairs, does not in fact obtain. Should we postulate a negative state of affairs, its not being the case that S to be the truthmaker for the true statement that S does not obtain? … [W]e shall not countenance negative states of affairs. (A World of States of Affairs, pp. 134-135.)

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This passage is flawed for two reasons. The minor reason is that Armstrong is using the capital letter “S” both as a name for a state of affairs and as a schematic letter, representing a sentence; the major one is that he is suggesting that “existent state of affairs” and “obtaining state of affairs” mean the same (how else could there be no point in speaking of a disjunction of states of affairs, given that states of affairs are always existents?) and also that they do not mean the same (the state of affairs S, supposedly, does not obtain, yet it must be existent, since states of affairs are always existents according to Armstrong). But let this pass.

The situation is, of course, entirely different if the intentionality approach to states of affairs (also described above) is adopted. One may be able to do without negative and disjunctive states of affairs as truthmakers (in fact, are we not able to do without any states of affairs as truthmakers?), but one cannot do without negative and disjunctive states of affairs as intentional objects. What Jack perceives when he sees that nobody is in the room is a negative state of affairs: its not being the case that somebody is in the room (at t0). Though Armstrong does not countenance negative states of affairs, Jack certainly does. Likewise, what Jill judges to be the case, when, after a process of ratiocination, she arrives at the conclusion that either the butler or the gardener is the killer of Jones, is a disjunctive state of affairs: the butler or the gardener being (at t1) the killer of Jones. Though Armstrong thinks it pointless to speak of disjunctive states of affairs, Jill certainly believes otherwise (since she is speaking seriously of such a state of affairs to others or to herself). I do not believe that Armstrong has any philosophical right to correct either Jack or Jill regarding their implicit ontological views on states of affairs; Jack and Jill are within their philosophical rights. If Armstrong thinks it necessary to cripple the ontology of states of affairs by throwing out negative and disjunctive states of affairs, he may of course do so; but there is no rational obligation at all to follow him in this. Consider finally Armstrong’s assertion that “[a]ll states of affairs are contingent.” (A World of States of Affairs, p. 150.) This is almost correct, since among the infinitely many states of affairs there are, there are indeed only two that are (absolutely) non-contingent: k* and t*;14 k* is a state of affairs whose non-obtaining is not contingent (it is not the case that it might conceivably have obtained), and t* is a state of affairs whose obtaining is not contingent (it is not the case that it might conceivably have not 14

If this seems too small a number, remember that we are talking about states of affairs (taken to be coarsely differentiated), and not about propositions (taken to be finely differentiated).

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obtained). Independently of the mereology of states of affairs here employed, one cannot deny that there are at least two (absolutely) noncontingent states of affairs as soon as one acknowledges (i) S(that Armstrong is self-different), and (ii) S(that Armstrong is self-identical). Concerning Armstrong’s position 4: According to Armstrong, “[t]he idea for possibility … is that all the combinations of simple particulars, properties and relations that respect the form of atomic states of affairs constitute the [that is: all the] possibilities for first-order states of affairs” (A World of States of Affairs, p. 160), firstorder states of affairs being states of affairs that do not have states of affairs as constituents (see ibid., p. 1). And the following quotation provides further perspectives: Given the truth of Logical Atomism, which we are assuming for the present, and abstracting from anything but first-order states of affairs, the world is a vast conjunction of atomic states of affairs. The particulars which are the constituents of these atomic states of affairs may not in every case be atomic … But the universals involved, that is, the state-of-affairs types, will be atomic. (A World of States of Affairs, p. 159.)

Armstrong’s “idea for possibility” cannot be true in the way he intends it: we shall see immediately that some possibilities for first-order states of affairs are not constituted in the way Armstrong envisages. But further below, we shall also see that in a way not intended by Armstrong – in a rather modified way – his “idea for possibility” might perhaps be true. Let S1 and S2 be two contingent atomic states of affairs that are independent of each other. I leave, for the moment, unanswered and undiscussed the crucial question of what is an atomic state of affairs, but I take it that every atomic state of affairs is ipso facto a first-order state of affairs. What, then, are the possibilities for S1 and S2? For each of the two states of affairs, there is the possibility that it obtains, and the possibility that it does not obtain; and among the possibilities for both states of affairs together there certainly are the following four: that both S1 and S2 obtain; that S1 obtains, but S2 does not; that S1 does not obtain, but S2 does; that both S1 and S2 do not obtain. But besides these eight elementary possibilities, there are others, non-elementary ones, for example, the possibility that S1 or S2 does not obtain. The obvious question to ask is this: how are all these possibilities for S1 and S2 constituted – as according to Armstrong they must be – by “combinations of simple particulars, properties and rela-

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tions that respect the form of atomic states of affairs”? The relevant simple particulars, properties and relations can only be the particulars, properties and relations that are the constituents of S1, respectively S2. Say, S1 has merely the simple particular a and the simple property F as constituents, and S2 merely the simple particular b and the simple property G. The elementary possibility that S1 does not obtain is, then, constituted in what way by the combination of the constituents of S1? And in what way is the nonelementary possibility that S1 or S2 does not obtain constituted by the combination of the constituents of S1 and of S2? Let the combination of F and a “that respect[s] the form of atomic states of affairs” be [F, a], and let the combination of G and b “that respect[s] the form of atomic states of affairs” be [G, b]. For more on the dyadic combinatory operator [x, y], see Chapter 7; [x, y] cannot be defined by the ordered-pair-operator , since sometimes [x, y] will be identical with [x, z] despite the fact that y and z are different.

One might say that [F, a] constitutes the possibility that S1 obtains, and [G, b] the possibility that S2 obtains, and that [F, a] and [G, b] together constitute the possibility that both S1 and S2 obtain (and of course also a possibility which is identical with this latter possibility: the possibility that both S2 and S1 obtain). But even these relatively unproblematic cases of Armstrongian possibility-constitution are far from being truly understood. Concentrating on [F, a] and the possibility that S1 obtains, a certain dilemma is manifest, as follows: Either [F, a] is a state of affairs, or it is not. In the first case, [F, a] can only be the atomic state of affairs S1 itself. But how, then, can [F, a] constitute the possibility that S1 obtains? Does S1 itself constitute the possibility of its obtaining? The idea seems prima facie unclear. In the second case, we must ask how a certain combination of the constituents of S1 – a combination which is not a state of affairs – can constitute the possibility of S1’s obtaining. What is it that [F, a] has to do with the possibility of S1’s obtaining? This is not as clear as it may seem at first sight.

What is totally inconceivable, however, is how the possibility that S1 does not obtain might be constituted by [F, a], and how the possibility that S1 or S2 does not obtain might be constituted by [F, a] and [G, b] together. What is absolutely needed here are additional combinatorial elements – combinatorial elements which are neither simple particulars nor simple properties and relations. Hence it is not true that the – that is: all the – possibilities for first-order states of affairs are constituted by “combinations of

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simple particulars, properties and relations that respect the form of atomic states of affairs.” In what combinatorial manner might the possibility that S1 does not obtain and the possibility that S1 or S2 does not obtain be constituted if besides a, b, F and G we allow additional combinatorial elements? – Let these additional elements be N (negation) and D (disjunction). Then the possibility that S1 does not obtain might perhaps be constituted by the following combination: {N, [F, a]}, and the possibility that S1 or S2 does not obtain might perhaps be constituted by the following other combination: {D, {N, [F, a]}, {N, [G, b]}}. (But what, indeed, is it that {N, [F, a]} has to do with the possibility of S1’s not obtaining, and what is it that {D, {N, [F, a]}, {N, [G, b]}} has to do with the possibility that S1 or S2 does not obtain?) I append a general note concerning combination. For finite combinations – combinations of elements x1, …, xn – where, regarding the purposes for which these combinations are employed, the order of the combined elements is not important, one can simply use as combinatorial operator the finite-set-operator {x1, …, xn}. For finite combinations where the order of the combined elements is important, one can often use the finite-sequence-operator . But not always: not in those cases where the order of the combined elements is partly important, and partly not. This is so, for example, regarding the possibility of the obtaining of the state of affairs that c lies between d and e (where c, d, and e are three distinct simple particulars, and where liesbetween – let’s allow this – is a simple triadic relation). This possibility cannot be said to be constituted by {lies-between, c, d, e}, for the order of the combined elements is, partly, important for constituting it. But neither can this possibility be said to be constituted by , for the order of the combined elements is also partly not important for constituting it: the possibility of the obtaining of the state of affairs that c lies between e and d is certainly the same possibility as the possibility of the obtaining of the state of affairs that c lies between d and e.

I come back to the question the answering of which was postponed above: what is an atomic state of affairs? First, what is an atomic state of affairs for Armstrong? A simple definition comes to mind: DA An atomic state of affairs is a state of affairs every constituent of which is a simple [or in another word: atomic] particular, simple property, or simple relation. This definition is, perhaps, not quite what Armstrong has in mind, for in the second of the two quotations which introduce this section he explicitly says that “the particulars which are the constituents of … atomic states of affairs may not in every case be atomic [or in another word: simple].” But

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the matter does not appear to be of great importance. Hence there is no substantial obstacle to accepting DA as Armstrongian. Note that, according to DA, it easily follows that every atomic state of affairs is a first-order state of affairs. If an atomic state of affairs had a state of affairs as constituent, it would have to be a simple particular, simple property, or simple relation; but no state of affairs is any such thing. Note also that according to DA the usual atomic, or simple, state of affairs will not really be simple, or atomic: because it will have at least two constituents that differ from it. Atomic states of affairs are, according to DA, atomic – or simple – only in an analogical sense. This, in itself, is no great matter, but it makes one wonder in what sense the constituents of atomic states of affairs are supposed to be simple. Are they, or are they not, simple – or atomic – in a literal sense? Are they simple in the strictly literal sense of not having any proper constituents (that is: of having no constituents which differ from them)? Are they simple in the less strict but still fairly literal sense of each having at most one proper constituent (the sense of being non-composita)? In precisely what ontological sense, literal or not, are the constituents of atomic states of affairs simple? Or are they, perhaps, simple merely in a non-ontological, epistemological sense: simple in the sense of being not analyzable by us human beings? – These questions, unanswered by Armstrong, circumscribe an inherent unclarity in Armstrong’s combinatorial theory of possibility. Another such unclarity is circumscribed by questions regarding the function atomic states of affairs have in constituting the possibilities for first-order states of affairs. When Armstrong says that “all the combinations of simple particulars, properties and relations that respect the form of atomic states of affairs constitute the possibilities for first-order states of affairs,” is he to be taken to mean that the atomic states of affairs themselves, by being themselves these combinations, constitute the possibilities for first-order states of affairs? In this case, for some first-order states of affairs – namely, all the atomic ones – the possibilities of their obtaining are constituted by the states of affairs themselves. Or are the combinations of simple particulars, properties and relations that respect the form of atomic states of affairs supposed by Armstrong to be something different from the atomic states of affairs themselves, and hence, certainly, from all states of affairs? In that case, it becomes unclear in what sense these combinations can constitute even the possibilities of obtaining for atomic states of affairs. The first alternative seems rather more acceptable than the second (essentially, the dilemma was already exhibited in a note above). Ac-

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cordingly, I will proceed on the assumption that, according to Armstrong, the atomic states of affairs constitute the (i.e., all the) possibilities for firstorder states of affairs. But the next question is already at hand: in what sense do they constitute these possibilities? We have already seen above that the prospects for this constituting are not good, since the possibilities for first-order states of affairs have a complexity to them that cannot be addressed if, in constituting these possibilities, all that one is allowed to avail oneself of are individual atomic states of affairs (as defined by DA) and assemblies of such. But perhaps Armstrong is just adhering to an unhelpful definition of atomic state of affairs? Let’s try a different definition – a definition, it must be admitted, that is totally alien to Armstrong’s ways of thinking: DB An atomic state of affairs is a state of affairs that has exactly one proper intensional part. According to this definition, atomic states of affairs are still simple – atomic – in a fairly literal sense, though not simple in the strictly literal sense of not having any proper intensional part (the only state of affairs which is atomic in that sense is t*). In view of the fact that we are now talking about parts, and not about constituents, when discussing simplicity, the following should be taken note of. Simplicity (atomicity) – even when taken literally – can either refer to constituents or to parts. The relationship between these two mereological notions, in their most general conception, can be taken to be this: all parts are constituents (of what they are parts of), but not all constituents are parts (of what they are constituents of); this is so because parts of X, in contrast to constituents of X, must always have the same ontological category as X (cf. Section 4.2). In the present context, however, we are not talking about parts in their most general conception, but about intensional parts; and we are not here talking about constituents in their most general conception (although, following Armstrong, we have been using the word “constituent” without qualifier), but about combinatorial constituents. An intensional part of something is not usually a combinatorial constituent of it, and a combinatorial constituent of something is not usually an intensional part of it. (But here is an exception: the state of affairs p is a combinatorial constituent of the (Armstrongian) higher-order state of affairs that p is logically necessary, and it is also an intensional part of this latter state of affairs.) Thus, part and constituent have in the present context – in the way they are presently understood – rather less to do with each other than in other contexts.

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According to DB, the atomic states of affairs are precisely the elemental states of affairs, as defined by D10 (on the basis of D9 and D4) in Section 3.2. As such, the atomic states of affairs are precisely the negations of the maximal-consistent states of affairs (or in other words: of the possible worlds, considered as states of affairs). Thus the negations of positive states of affairs – for maximal-consistent states of affairs are positive states of affairs if there are any positive states of affairs – are nonetheless atomic state of affairs. “Negative states of affairs which are atomic” – for Armstrong, these words can only spell absurdity. But there is no compelling reason that they must do so for everybody else.

Atomic states of affairs, as defined by DB, have a remarkable property: one can prove, in the mereology of states of affairs, that every state of affairs is the conjunction of the elemental states of affairs that are its intensional parts, in formal terms: ∀x[S(x) ⊃ x = CONJz(EL(z) ∧ P(z, x))]. And therefore we have, in a very clear sense of “to constitute,” (1) Every state of affairs is constituted by atomic states of affairs (as defined by DB). If we add to this (the totally non-Armstrongian, not to say “antiArmstrongian”) postulate (2) Every possibility for a first-order state of affairs can be identified one-to-one [that is to say: in such a way that the non-identity of possibilities for first-order states of affairs is respected] with a state of affairs, then we obviously obtain: (3) All the possibilities for first-order states of affairs are constituted by atomic states of affairs. By way of making (2) plausible (for those without Armstrongian bias), consider again the two possibilities for the DA-atomic states of affairs S1 (with the constituents F and a) and S2 (with the constituents G and b) which were considered above: the possibility that S1 does not obtain, and the possibility that S1 or S2 does not obtain. Which are the states of affairs with which these possibilities are to be one-to-one identified? The possibility that S1 does not obtain is to be identified with the state of affairs neg([F, a]), and the possibility that S1 or S2 does not obtain with the state of affairs

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disj(neg([F, a]), neg([G, b])), where the functors “neg” and “disj” are to be employed with the meaning given to them by the definitions D11 (based on D1) and D13 (based on D1 and D12) in Section 3.2, and “[x, y]” is a state-of-affairs-forming operator (for appropriate inputs x and y, of course; see Chapter 7). Note that disj(neg([F, a]), neg([G, b])) is identical with neg(conj([F, a], [G, b])), “conj” being used with the meaning given to it by D8 (based on D1). Accordingly, the possibility that S1 or S2 does not obtain is indeed identical with the possibility that not both S1 and S2 obtain.

Now, the concluded statement (3) is something that Armstrong asserts. However, it is such a thing only verbally, since the concept that is employed under the name “atomic state of affairs” – the DB-concept – is totally different from Armstrong’s concept of atomic state of affairs, the DA-concept. Well, not totally. The world is a vast conjunction of atomic states of affairs, as Armstrong says it is (see the above quotation), also if atomic states of affairs are DBatomic states of affairs. In the mereology of states of affairs, it is provable that the world – that is, w* – is the conjunction of all elemental states of affairs except one (namely, neg(w*)).

But the reinterpreted Armstrongian assertion has the vast advantage over Armstrong’s original one that it can be true (and is demonstrably true within the present ontological framework). Moreover, in the semantic respect, the similarity of the reinterpreted assertion to Armstrong’s original assertion might well be increasable. For perhaps it is not a priori unreasonable to suppose that the DB-atomic states of affairs – the elemental states of affairs – are certain DA-atomic states of affairs, namely, combinations of certain simple particulars, properties and relations that respect the form of DA-atomic states of affairs. If it were so, then, because of (3), it would be true that combinations of certain simple particulars, properties and relations that respect the form of atomic states of affairs constitute the possibilities for first-order states of affairs – which is almost what Armstrong says (see above). For convenient comparison, I quote again what Armstrong says: “[T]he idea for possibility … is that all the combinations of simple particulars, properties and relations that respect the form of atomic states of affairs constitute the possibilities for firstorder states of affairs.” (A World of States of Affairs, p. 160.) Note also: if already combinations of certain simple particulars, properties and relations that respect the form of atomic states of affairs constitute the possibilities for first-order states of affairs, then, in a sense, (a fortiori) all the combinations of such things constitute the possibilities for first-order states of affairs.

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But of course the simple particulars, properties and relations that would go into constituting DB-atomic states of affairs as DA-atomic states of affairs would have to be vastly different from any simple particulars, properties and relations Armstrong, or anyone else, has in mind or is capable to imagine – if, indeed, there could be any such simple particulars, properties, or relations.

4.5

Van Fraassen’s Objection against the Basic Idea of a Bases-Theory of Necessity

In his paper “The Only Necessity Is Verbal Necessity,” p. 73, and in a simpler manner in his paper “Essence and Existence,” p. 15, Bas van Fraassen formulates a logical objection against the basic idea of a basestheory of necessity (although he is in deep sympathy with it). In the terminology of this book, van Fraassen’s objection can be put in the following way: If the Bases-Theory of Necessity were correct, then nomological (or “physical”) necessity, 2, could in principle be defined in the following way: 2A =Def 1(P ⊃ A), where 1 is logical (or “pure”) necessity and “P” stands for the appropriate “law sentence.” But this would mean that 2, like 1, is an S5-necessity.15 According to van Fraassen, this is not true for 2, and it is also not true for a host of other necessities i that one might think to be definable according to the definition-schema enjoined by the BasesTheory of Necessity: iA =Def 1(Pi ⊃ A), “Pi” standing for the appropriate basis-sentence. Preliminary to responding to this objection, it needs to be pointed out that the Bases-Theory of Necessity is not meant to apply to every necessity. It is meant to apply only to those ontic necessities that can be represented as founded on a basis of necessity. To every such necessity, if it is what has been called faithful to truth (see Section 3.6), the Bases-Theory of Necessity does indeed assign the principles of the modal system S5.

That is: 2A ⊃ A, 2(A ⊃ B) ⊃ (2A ⊃ 2B), 2A ⊃ 22A, and ¬2A ⊃ 2¬2A are principles for 2. The first principle follows from the definition – 2A =Def 1(P ⊃ A) – only under the assumption that P is true. But, of course, P – the “law sentence” – is assumed to be true, no matter what may be its precise content: not its truth but the content fitting that truth is open to question. 15

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Compare Section 3.7. If the S5-principles are all to be logical principles for the necessity concerned, conceptually true principles for it, then the faithfulness to truth of the necessity concerned must be true of it already for conceptual reasons – a precondition of the logical-provability-result regarding S5 in Section 3.7. Otherwise, A ⊃ A, though still demonstrably true for , will not be logically true (and hence not logically provable) for it.

But it seems that every ontic necessity can be represented as founded on a basis of necessity. Hence the Bases-Theory of Necessity tells us that every ontic necessity that is faithful to truth obeys S5. And it certainly seems that nomological (or “physical”) necessity is an ontic necessity faithful to truth. Hence the Bases-Theory of Necessity tells us that physical necessity obeys S5. What reason does van Fraassen have for believing that nomological necessity does not obey S5? Such a reason can arise when 2 is regarded in such a way that not only the simple truth and falsity of 2S is being considered, or in other words: the truth and falsity of 2S at w*, but also its – possibly variant – truth and falsity at other possible worlds than w*. The consideration of the truth and falsity of 2S at other possible worlds than w* is unavoidable if (a) iterated occurrences of 2, as in 22S or 2¬2S, are allowed and if (b) the following truth-rule is adopted: (A) For all possible worlds w: 2S is true at w if, and only if, for every possible world w´ that is nomologically accessible from w: S is true at w´. This is simply the standard semantic explication of nomological necessity that is given in standard possible-worlds-semantics. The question whether S5 is correct for 2 depends, then, on the nature of the invoked accessibility relation. So far, nothing forbids the following situation: Suppose: (1) For every world w´: w´ is nomologically accessible from w* if, and only if, P is a law (and hence true) at w´. Suppose also: (2) For every world w´: w´ is nomologically accessible from w1 if, and only if, (P ∧ A) is a law (and hence true) at w´. Here P states all the laws of w*: P and all its logical consequences; and (P ∧ A) states all the laws of world w1: (P ∧ A) and all its logical consequences. Hence we have: P and A are laws of w1, and P is true at w*, and (P ∧ A) is true at w1. Suppose finally: (3) A is not true at w*. Hence according to the above truth-rule (and the

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standard truth-rule for negation): ¬2A is true at w*, because w* is obviously nomologically accessible from w* and A is not true at w*; but 2¬2A is not true at w*, since w1 is accessible from w* (P being a law at w1), but ¬2A is not true at w1 (A being true at every world that is nomologically accessible from w1). And hence the necessary conclusion is this: ¬2A ⊃ 2¬2A is not a principle for 2, and therefore S5 is not correct for 2. For obtaining this unfavorable result the interpretation of nomological accessibility is crucial. Obtaining it is not excluded if one chooses the interpretation that is expressed by the following principle: (I) For any worlds w and w´: w´ is nomologically accessible from w if, and only if, all the laws of w are also laws of w´. The above-described situation does not contradict (I). If, however, one chooses the interpretation of nomological accessibility that is expressed by the following principle: (II) For any worlds w and w´: w´ is nomologically accessible from w if, and only if, the laws of w´ are (the very same as) the laws of w, then the outcome is this: the above-described situation contradicts (II), and S5 is after all correct for 2. It is, therefore, not correct to say that 2 (nomological or “physical” necessity) is not an S5-necessity. It is, at most, correct to say that it is not unequivocally an S5-necessity, because it might seem that (II) is no more correct than (I), while (I) is certainly no more correct than (II). But, in fact, one can well argue that a world w´ should only be nomologically accessible from a world w if it is, as far as laws are concerned, indistinguishable from w, which makes (II) the correct interpretation of nomological accessibility. The above-described standard treatment of nomological necessity in possible-worlds-semantics – epitomized by truth-rule (A) – can be said to turn nomological necessity into a world-indexical notion, because it evaluates 2S also with respect to other worlds than w* and because what is asserted by 2S relative to a world varies at least between some worlds (since what the laws are will also vary at least between some worlds). The treatment of nomological necessity according to the Bases-Theory of Necessity, in contrast, is entirely non-indexical if “b2” is taken to rigidly designate, in whatever context, the conjunction of all states of affairs that are the actual laws of nature. (Note that the predicate of intensional parthood between states of affairs, which also goes into the definition of 2, is not a predicate that is itself in any way context-dependent.) Van Fraassen, in effect, urges the Bases-Theory of Necessity to turn indexical regarding nomological (or

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“physical”) necessity: “All assertions of physical necessity are tacitly conditional16 and tacitly indexical.” (“The Only Necessity Is Verbal Necessity,” p. 74.) But there does not seem to be enough motivation for such a momentous move. In the first place, other natural laws for other worlds than w*, and therefore otherworldly nomological necessity that is different in content from nomological necessity at w*, is a highly problematic notion, since we have no non-arbitrary objective way to determine what other laws for other worlds might be. Prima facie, any state of affairs that is an intensional part of a world w that is different from w* may be a law of w, or it may not be a law of w. To be sure, by formulating formal criteria of lawhood we can restrict the list of candidates for laws of w. But let M be the set of states of affairs that are intensional parts of w and satisfy those formal criteria. Then it would still be an objectively arbitrary decision to identify the laws of w with the states of affairs in M, just as it would be to identify them with the states of affairs in any proper subset of M. In the second place, as we have seen (in Section 3.7), one can make sense of iterated nomological necessity without drawing into consideration other natural laws for other worlds than w* (i.e., without considering otherworldly nomological necessity that is different in content from the fairly familiar nomological necessity at w*) – and indeed without troubling one’s mind in any way about the truth-value of 2S at other possible worlds than w*. In a trivial manner, one can, of course, assign a truth-value to 2S at other worlds than w*, although it is only its truth-value at w* that matters. In the idiom of possible-worlds-semantics, the truth-rule for 2 that reflects the non-indexical interpretation of 2 that can be given within the BasesTheory of Necessity is simply this: (B) For all possible worlds w: 2S is true at w if, and only if, for every possible world w´ of which b2 is an intensional part (or in other words: at which P is true): S is true at w´. 16

The following is a clear enough description by van Fraassen of the Bases-Theory of Necessity applied to nomological necessity: “What is physically [nomologically] necessary is the same, on this view, as what is logically implied by some tacit antecedent – say, the laws of physics.” (“The Only Necessity Is Verbal Necessity,” p. 71; cf. Section 6.6.)

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This truth-rule will make 2S either true at all the worlds (including w*), or false at all the worlds (including w*). It remains to be pointed out that the following problem is encountered within standard possible-worlds-semantics: whereas (A) yields the full S5-picture (under interpretation (II) of the accessibility relation), (B) does not. If logical truth is defined as it standardly is: as truth at all the worlds, whatever worlds they are (“at all the worlds in all the models”), then truth-rule (A) yields the logical truth of 2A ⊃ A, 2(A ⊃ B) ⊃ (2A ⊃ 2B), 2A ⊃ 22A, and ¬2A ⊃ 2¬2A – given that (A) is assumed together with (II), no matter which worlds the worlds are (“for all the worlds, according to whatever model”). However, under that same conception of logical truth, truth-rule (B) – even though assumed no matter which worlds the worlds are – does not yield the logical truth of 2A ⊃ A (although it does yield the logical truth of 2(A ⊃ B) ⊃ (2A ⊃ 2B), 2A ⊃ 22A, and ¬2A ⊃ 2¬2A). This is so because nothing forbids the following situation: S is not true at a world w, but S is true in every world of which b2 is an intensional part. Hence, according to (B), 2S is true at w – in spite of the fact that S is not true at w. In order to make (B), like (A), yield the full S5-picture of logical truth, it is not advisable to forbid the situation just described by special decree. One cannot postulate that if S is true at every world of which b2 is an intensional part, that then S is true at every world. For in this manner, obviously, nomological necessity would be conflated with logical necessity. But the problem can be solved by using a different conception of logical truth, though still squarely within possible-worlds-semantics. If logical truth is not defined as truth at all the worlds, whatever worlds they are, but as truth at the actual world, whichever world it is among the worlds, whatever worlds they are, then the difficulty does no longer arise. One merely needs to postulate (mirroring P19), for each relevant model, that w* (the actual world of the model) has b2 (“the law” of the model) as an intensional part.17 The described problem also presents itself, but in a different manner, if the general theoretical background is not model-theoretic possibleworlds-semantics, but the mereology of states of affairs (or an extension 17

If, as usual, a set-theoretical framework is used, then this postulate amounts to postulating, in correspondence to P19, for each relevant model: {w*} ⊆ b2, where b2 is taken to be a certain set of worlds (of the model), a set fulfilling also the formal conditions that correspond to P20 and P21: b2 ≠ W (W being the set of all the worlds of the model), b2 ≠ {w*}.

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thereof). From the point of view of the mereology of states of affairs, 2A ⊃ A is a logical truth if, and only if, P19 – P(b2, w*) – is a conceptually true principle. If, however, “b2” is taken to be a rigid designator, as was suggested above and will be made definite in Section 7.4, P19 is not conceptually true (see Section 3.9.1). We must then rest content with 2A ⊃ A being true for 2, though not logically. 4.6

Metaphysical Necessity? Nomological Necessity?

In recent literature, a certain kind of necessity, supposedly different from both logical and nomological necessity, has figured prominently: metaphysical necessity. It has been the refuge of physicalistically oriented thinkers who, on the one hand, thought the necessity of logical correlation between the mental and the physical at least in some cases too strong (for it does seem to be logically possible for some mental states to occur without corresponding physical states) and who, on the other hand, thought the necessity of nomological correlation between the mental and the physical in all cases too weak (for many dualists hold it to be nomologically impossible for mental states to occur without corresponding physical states). From the point of view of the present theory of modality, metaphysical necessity is, formally considered, no great problem. Like all ontic necessities, metaphysical necessity must have a basis, a certain state of affairs; call it “b3.” It is metaphysically necessary that A if, and only if, the state of affairs that A is an intensional part of b3. The sentence-connective of metaphysical possibility can be defined on the basis of the sentenceconnective of metaphysical necessity in the usual way: it is metaphysically possible that A if, and only if, it is not metaphysically necessary that nonA. The problem with metaphysical necessity (and possibility) is not a formal problem; it is a problem of content. It is not entirely clear which state of affairs is b2, the basis of nomological necessity; it is, at least prima facie, very much unclear which state of affairs is b3 if b3 is supposed to be different from both b1 (= t*) and b2. But the whole point of introducing metaphysical necessity is to have a necessity that is different from both logical and nomological necessity. Hence b3 ought to be different from both b1 and b2. Hence the perplexing question: which state of affairs is b3? David Chalmers, on the contrary, has held in the context of the philosophy of mind – a great field of application for modal notions – that

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metaphysical necessity is just logical necessity (see The Conscious Mind, p. 38). First (in the order of thought), he contends: [C]onsidering the possibility of worlds …, we would … have three objective classes of possible worlds: logically possible worlds, metaphysically possible worlds, and naturally [i.e., nomologically] possible worlds. We have good reason to believe in the first and the last of these classes, but we have very little reason to believe in a third, distinct class as a metaphysical given. (The Conscious Mind, pp. 137-138.)

Since states of affairs can be represented as classes (or sets) of possible worlds, Chalmers, in talking about three classes of possible worlds, is talking about three states of affairs, namely (in the order of mention): b1, b3, and b2. And he is, in effect, asserting that we have no good reason to distinguish b3 (the basis of metaphysical necessity) both from b1 (the basis of logical necessity, which is t*) and from b2 (the basis of nomological necessity, which is the conjunction of all states of affairs that are laws of nature). What are his reasons for this claim? In fact, Chalmers gives reasons for identifying metaphysical possibility/necessity with logical possibility/necessity, that is, for identifying b3 with b1, or in other words: for identifying the class of metaphysically possible worlds with the class of logically possible worlds.18 Since all metaphysically possible worlds are doubtlessly logically possible worlds, he merely needs to show, or at least make plausible, that all logically possible worlds are metaphysically possible. His (relatively) best argument for this conclusion is an epistemological argument: [I]f some worlds are logically possible but metaphysically impossible, it seems that we could never know it. By assumption the information is not available a priori, and a posteriori information only tells us about our world. … Any claims about the added constraints of metaphysical possibility would seem to be a matter of arbitrary stipulation; one might as well stipulate that it is metaphysically impossible that a stone could move upward when one lets go of it. (Ibid., p. 138.) 18

Why does he not also consider the alternative option: the option of identifying b3 with b2, that is: the option of identifying the class of metaphysically possible worlds with the class of nomologically possible worlds? – Perhaps because it does not fit the traditional (i.e., empiricist) separation of epistemic competence. If the metaphysically possible worlds were identified with the nomologically (or naturally) possible worlds, natural science would largely overlap with metaphysics. But is the alternative overlap of metaphysics with logic any better?

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Right, one is tempted to say. But, on closer inspection, serious doubts should arise about this argument. If correct, Chalmers’ argument can (mutatis mutandis) also be turned against separating nomological possibility (and necessity) from logical possibility (and necessity). Consider the following fittingly modified version of Chalmers’ argument: If some worlds are logically possible but nomologically impossible, it seems that we could never know it. By assumption the information is not available a priori, and a posteriori information only tells us about our world. … Any claims about the added constraints of nomological possibility would seem to be a matter of arbitrary stipulation; one might as well stipulate that it is nomologically impossible that there could be a sphere of gold with a diameter of one mile. If Chalmers’ argument for identifying metaphysical possibility with logical possibility is a good argument, then the above modification of that argument must, in turn, be a good argument for identifying nomological possibility with logical possibility. Conversely: if the modified argument is not a good argument for identifying nomological possibility with logical possibility, then Chalmers’ original argument for identifying metaphysical possibility with logical possibility is not a good argument, either. Hence: since most philosophers and common-sensers do in fact reject the modified argument, it is only consistent for them to reject Chalmers’ original argument, too. Therefore: since in this case I adopt the opinion of the majority as reasonable,19 I conclude: Chalmers’ argument presents no good reason for identifying b1 and b3, it presents no good reason for identifying metaphysical possibility (and necessity) with logical possibility (and necessity). If b3 is to be identified with any given basis of necessity, then b2 does seem to be the likely candidate (an option not considered by Chalmers; cf. footnote 18), resulting in the identification of metaphysical possibility/necessity with nomological possibility/necessity. Alternatively, one might base metaphysical modalities on a proper intensional part of b2: by identifying b3 with the conjunction of those states of affairs that are the most basic laws of nature (which determination, it must be admitted, is still rather imprecise). Indeed, this, in the end, seems 19

Actually, I am not sure whether one can rationally adopt the opinion of the majority regarding nomological modalities. In my earlier book Ereignis und Substanz, I, in effect, denied that one can. But this is, admittedly, a radical – a skeptical – position.

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to be the most adequate foundation for metaphysical modalities. According to it, metaphysical necessity and possibility are not as clear as logical possibility and necessity, but also not significantly less clear than nomological possibility and necessity. It must be emphasized that all three kinds of modalities – logical, nomological, and metaphysical – are ontic modalities: they concern the constitution of things, they are not “merely verbal” (cf. Section 6.6). Thus, in a certain (harmless) sense, all that is logically possible is indeed metaphysically possible, that is: possible in an ontic sense. In fact, the expression “metaphysically possible” equivocates between the just indicated harmless meaning of it, which is not the meaning here relevant, and the not so harmless meaning of it that concerns the question at issue (see above). The equivocation is exploited by Chalmers – unintentionally, I presume – in the following fallacious argument for identifying logical and metaphysical possibility: Presumably it is in God’s powers, when creating the world, to do anything that is logically possible. Yet the advocate of metaphysical necessity must say either the possibility [of a zombie world] is coherent, but God could not have created it, or God could have created it, but it is nevertheless metaphysically impossible. (Ibid., p. 138.)

The physicalistically motivated advocate of metaphysical necessity is ready to admit that a zombie world is logically possible, but will also hold that a zombie world is metaphysically (and not merely nomologically) impossible. And this, says Chalmers (as I interpret him), confronts the advocate of metaphysical necessity with a dilemma. Its first horn is to assert that a zombie world is logically possible, metaphysically impossible, and not creatable by an almighty being – but how can what is logically possible be not creatable by an almighty being? Its second horn is to assert that a zombie world is logically possible, metaphysically impossible, and creatable by an almighty being – but how can what is creatable by an almighty being be metaphysically impossible? In criticism of this argument, it must be pointed out that the first horn of the dilemma is perfect (and the reference to metaphysical possibility in it is inessential); but the second horn of the dilemma is not perfect. It is only true in the non-relevant, harmless sense of “metaphysically possible” – the sense of “possible in an ontic sense” – that what is creatable by an almighty being must be metaphysically possible; it is not true in the relevant, not so harmless sense of “metaphysically possible” (which is located

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somewhere between logical and nomological possibility, including the former and included by the latter).

4.7

An Assertion of Plantinga’s: Every State of Affairs Is Actual in Itself

In “Self-Profile,” p. 90, Plantinga maintains that every state of affairs is actual in itself (“actuality in itself, a property had by every states of affairs,” says Plantinga). One might simply render “every state of affairs is actual in itself” as “every state of affairs obtains in itself,” that is, as ∀x(S(x) ⊃ O(x, x)) – which is easily provable on the basis of P2 and D16. But it is more interesting to define actuality in a state of affairs more generally than merely for states of affairs x (for which it amounts to O(x, y)), as follows: A(x, y) =Def P(that A(x), y). Plantinga’s claim, then, is rendered as follows: ∀x(S(x) ⊃ A(x, x)), or in other words: ∀x(S(x) ⊃ P(that A(x), x)). And this is an immediate consequence of P2 and ∀x(S(x) ⊃ x = that A(x)). The latter principle is, intuitively, just as true as P24: ∀x(S(x) ⊃ x = that O(x)), since, for states of affairs, to be actual is to obtain. But is ∀x(S(x) ⊃ x = that A(x)) already provable in the theory presented in Chapter 3? – It is indeed, but the proof is not entirely obvious: Assume S(x). Hence on the basis of P24: x = that O(x), that is (on the basis of D6): x = that (S(x) ∧ A(x)). Hence (because of P15): x = conj(that S(x), that A(x)) [a]. Now: t* = that S(x) [b], which is shown as follows: We have: ├ S(x) ≡ P(x, x) (because of the conceptually true principles P0 and P2). Hence by EQU*, which is provable on the basis of P23 (see Section 3.7), ├ that S(x) = that P(x, x) [c]. Because of the assumption S(x) and P2, we have P(x, x), and therefore: t* = that P(x, x) [d], according to P22. On the basis of [c] and [d]: t* = that S(x) [b]. Applying [b] to [a], we obtain: x = conj(t*, that A(x)) [e]. Because of the theorem ∀y(S(y) ⊃ conj(t*, y) = y) (provable in the mereology of states of affairs) and S(that A(x)) (an instance of P13), [e] implies: x = that A(x) – which is the identity that was to be derived.

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4.8

On Chihara’s Finding of a Contradiction in Plantinga’s Ontological Theory

In his recent book, The Worlds of Possibility, Charles Chihara asserts the following (under the heading A Contradiction in Plantinga’s Ontological Theory): As shown above, we can conclude from Plantinga’s principles that there is a set α whose elements are all the possible states of affairs that obtain. Then, according to Cantor’s Theorem, the power set of α, 2α, has cardinality greater than the cardinality of α. Hence, there cannot be a one-one correspondence between 2α and any subset of α. But, as will be shown below [on the basis of Plantinga’s principles], there is. (The Worlds of Possibility, p. 126.)

This is highly interesting, of course, also for the present theory of modality, because, like Plantinga’s, it relies heavily on a theory of states of affairs. Does Chihara’s reductio ad absurdum of Plantinga’s theory of states of affairs (if it is one) have any force against the present theory of states of affairs? Let’s see. Chihara sets out in the following way: I shall first argue that, if σ and τ are members of 2α, and if σ ≠ τ, then s(σ) ≠ s(τ). (Ibid.)

And we need not go any further. For this principle that Chihara says he will first be arguing for is absolutely crucial for his whole argument. But, though the principle is perhaps true for Plantinga who has a fine-grained conception of states of affairs (see Section 4.1 above) it is glaringly false according to the present theory of states of affairs (the mereology of states of affairs, as I call it), and also according to common sense. Consider: Is it true for all sets σ and τ of obtaining states of affairs that if σ and τ are different, that then also (the state of affairs which is) the sum (or conjunction) of the states of affairs in σ, s(σ), is different from the sum of the states of affairs in τ, s(τ)? Certainly not. Here is an obvious counterexample: the sets {that all human beings are mortal, that Chihara is not a human being or mortal} and {that all human beings are mortal, that Plantinga is not a human being or mortal}. These sets are obviously different sets of obtaining states of affairs. But s({that all human beings are mortal, that Chihara is not a human being or mortal}) – which, in the present theory, can only be conj(that all human beings are mortal, that Plantinga is not a human being or mortal) as defined by D8 and, further, by D1 – and

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s({that all human beings are mortal, that Chihara is not a human being or mortal}) – i.e., conj(that all human beings are mortal, that Chihara is not a human being or mortal) – are one and the same state of affairs, namely: that all human beings are mortal. Why? Let the states of affairs concerned be p [that all human beings are mortal], q [that Chihara is not a human being or mortal], r [that Plantinga is not a human being or mortal]. Then we have: P(q, p) and P(r, p), since the sentence expressing q and the sentence expressing r are logical consequences of the sentence expressing p. And therefore (according to the mereology of states of affairs): conj(p, q) = p and conj(p, r) = p and hence, of course, conj(p, q) = conj(p, r) – in spite of the fact that {p, q} is different from {p, r}.

Consider also the following, more general argumentation. According to the present theory of states of affairs, every usual state of affairs p, obtaining or not, can be divided in countless different ways into states of affairs that are its intensional parts. To each of these ways of dividing it, there corresponds a different set of states of affairs, namely, the complete set of intensional parts of p resulting from dividing p in that particular way. But the conjunction of the states of affairs in any one of these sets is – according to the present theory of states of affairs – always identical to one and the same state of affairs: p. Thus, Chihara’s first principle (quoted above) is not true, according to the present theory of states of affairs.

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5

The Theory of Conditionals in the Onto-Nomological Theory of Modality

In Chapter 3, conditionals – statements of the form “if A, then B” (statements of relational necessity) – were addressed by the following two definition-schemata: →(x, y) =Def P(y, conj(bn, x)). A n→ B =Def n→(that A, that B).

n

But no discussion was provided in that chapter. The present chapter is dedicated to the question to what extent the analysis of conditionals that is implicit in these two schemata can be squared with the phenomenology of ontically understood conditionals. Every conditional involves a relational necessity, and therefore the discussion of conditionals provided in this chapter is at the same time also a discussion of relational necessities. The thesis I will argue for is this: The Bases-Theory of Conditionals In every utterance of an ontically intended statement of the form “if C, then D,” provided that utterance has a determinate truth-value, reference is made to some basis or other of necessity bn such that the full meaning of the utterance can be precisely captured by expressing it in the form “C n→ D” (as defined above). For the sake of simplifying exposition, I will use “conditional(s)” simpliciter for designating ontically intended conditionals only (without wishing to deny that there are conditionals that are not ontically intended). Moreover, I adopt the convention that the form “if C, then D” always indicates a conditional in the indicative mood, or in other words: a simple conditional; if a non-simple conditional is to be indicated, then indices will be added to the “if” and to the “then.” I also adopt the convention that a simple conditional – though always in the indicative mood – is to be an indica-

5 The Theory of Conditionals in the Onto-Nomological Theory of Modality

tive conditional only if it is asserted on its own, and does not, for example, merely occur as a definitional part of another conditional. A very important kind of non-simple conditional is the counterfactual conditional, or for short: the counterfactual. Counterfactuals can be regarded as simple conditionals with the presupposition of the falsity of (at least) the antecedent added (and hence no longer simple). In logical analysis that is done within the framework of classical logic (i.e., that does not allow truth-value gaps), it is usual to represent presuppositions as logically implied necessary conditions. In this book, all of logical analysis is done within the framework of classical logic, and therefore counterfactuals “ifCF A, thenCF B” can here be very well defined simply as follows: (I) ifCF A, thenCF B =Def ¬A ∧ (if A, then B). According to this analysis, “ifCF A, thenCF B” is not without a truth-value in case its antecedent is true; rather, it is trivially false in that case. Most modern theorists of counterfactuals (influenced by David Lewis) would, however, prefer that a counterfactual is not trivially false in case its antecedent is true, but that it has, in that case, the same truth-value as A ∧ B. This idea can be captured by the following definition: (II) ifCF A, thenCF B =Def ¬A ∧ (if A, then B) ∨ A ∧ B.1 Whether one chooses analysis (I) of counterfactuals or analysis (II), it is clear that counterfactuals can be defined in terms of simple conditionals (and truth-functional logical connectives) if either analysis (I) or analysis (II) is correct. I will argue that analysis (II) is correct (having a slight advantage over (I)). Therefore, if the Bases-Theory of Conditionals is correct (as stated above), counterfactuals, too, will fall within the purview of the Onto-Nomological Theory of Modality (or, in other words, the BasesTheory of Modality). But the Bases-Theory of Conditionals needs to be complemented by the following additional thesis: Hidden Indexicality of Conditionals It frequently happens that different utterances of ontically meant conditionals do not refer to the same basis of necessity (in short: In order to save brackets, the binding-strength of ∧ is considered to be greater than the binding-strength of ∨. In general, binding-strength is taken to decrease in the sequence ¬, ∧, ∨, ⊃, ≡ from left to right. 1

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have different bases). This can even be the case if the conditional uttered in these different utterances is the very same statement qua syntactical-semantic entity: a syntactic structure with a certain contextindependent meaning. As will be seen, the hidden indexicality (or context-dependence) of conditionals provides a satisfactory explanation of the many peculiarities of the logic of conditionals, which contrast it rather strikingly with the logic of strict implication and the logic of material implication.

5.1

Relational and Non-Relational Necessity

First of all, a central fact needs to be noted concerning the relationship between relational and non-relational necessity. It is expressed by the following theorem: A n→ B ≡ n(A ⊃ B). Proof: A n→ B amounts to P(that B, conj(bn, that A)), and n(A ⊃ B) amounts to P(disj(neg(that A), that B), bn). The following theorem can be proven in the mereology of states of affairs: ∀x∀y∀z[S(x) ∧ S(y) ∧ S(z) ⊃ (P(y, conj(z, x)) ≡ P(disj(neg(x), y), z))]. What is to be proven follows from this lemma, considering that we have S(that A) and S(that B), according to P13, and S(bn), according to P18. Proof of the lemma: Assume S(x), S(y), S(z). (i) Assume P(y, conj(z, x)), but ¬P(disj(neg(x), y), z). Hence [according to P5]: ∃u(QA(u) ∧ P(u, disj(neg(x), y)) ∧ ¬P(u, z)). Hence [because of P(u, disj(neg(x), y)) ⊃ P(u, neg(x)) ∧ P(u, y), according to a theorem of the mereology of states of affairs]: ∃u(QA(u) ∧ P(u, neg(x)) ∧ P(u, y) ∧ ¬P(u, z)). Hence [because of the assumption P(y, conj(z, x)) and P1]: ∃u(QA(u) ∧ P(u, neg(x)) ∧ P(u, y) ∧ ¬P(u, z) ∧ P(u, conj(z, x))). Hence [because of QA(u) ∧ P(u, conj(z, x)) ⊃ P(u, z) ∨ P(u, x), according to a theorem of the mereology of states of affairs]: ∃u(QA(u) ∧ P(u, neg(x)) ∧ P(u, y) ∧ ¬P(u, z) ∧ (P(u, z) ∨ P(u, x))). Hence ∃u(QA(u) ∧ ¬P(u, z) ∧ P(u, neg(x)) ∧ P(u, x)) – which can be shown to be impossible [using P6 and D11, taking into account: ¬M(u), because of ¬P(u, z)]. (ii) Assume P(disj(neg(x), y), z), but ¬P(y, conj(z, x)). Hence [according to P5]: ∃u(QA(u) ∧ P(u, y) ∧ ¬P(u, conj(z, x))). Hence [because of P(u, z) ∨ P(u, x) ⊃ P(u, conj(z, x)), according to a theorem of the mereology of states of affairs]: ∃u(QA(u) ∧ P(u, y) ∧ ¬P(u, z) ∧ ¬P(u, x)). Hence [according to D11, D1, P4, P3]: ∃u(QA(u) ∧ P(u, y) ∧ ¬P(u, z) ∧ P(u, neg(x))). Hence [because of P(u, neg(x)) ∧ P(u, y) ⊃ P(u, disj(neg(x), y)), according to a theorem of the mereology of states of affairs]:

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∃u(QA(u) ∧ ¬P(u, z) ∧ P(u, disj(neg(x), y))). Hence because of P1 and the assumption P(disj(neg(x), y), z): P(u, z) – contradicting the result ¬P(u, z) already reached.

The above theorem shows that sentences of the forms A n→ B (A 1→ B, A 2→ B, etc.) express relational necessities that are reducible to nonrelational necessities on the pattern of the definition of strict implication. Among the strict implications expressed by sentences of the forms A n→ B, material implications can be included as a special case if we postulate: b10 = w*.

P18b

P18b is intended as a conceptual truth (as, by the way, is P18 that includes “b1 = t*”). Hence “b10” is a term that is referentially unstable, just as is “w*” (defined by “CONJxO(x)”); if it weren’t, “b10 = w*” could not be a conceptual truth. To postulate that, of all basis-terms, “b10” is to denote w* (and in the same manner as “w*”) is, of course, entirely arbitrary; any other basis-term – except the two basis-terms already “occupied”: “b1” and “b2”– would have done just as well. – According to the theorem just proven, we have in particular: A 10→ B ≡ 10(A ⊃ B). And according to P18b and the other principles of the mereology of states of affairs, we have: 10(A ⊃ B) ≡ A ⊃ B.2 Proof: Assume A ⊃ B. Hence O(that (A ⊃ B)), because of P10. Hence P(that (A ⊃ B), w*), because of D6 and the Actuality Principle for States of Affairs (i.e., ∀x[S(x) ⊃ (A(x) ≡ P(x, w*))]; see Section 3.3). Hence 10(that (A ⊃ B)), and therefore 10(A ⊃ B), according to P18b and the definition of 10. It has now been shown: (A ⊃ B) ⊃ 10(A ⊃ B). The reverse, 10(A ⊃ B) ⊃ (A ⊃ B), has already been shown in 3.6, 10 being (for conceptual reasons) a necessity faithful to truth.

Hence:

Brackets around A ⊃ B can be omitted because the binding-strength of ⊃ is supposed to be greater than the binding-strength of ≡. See footnote 1. 2

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A 10→ B ≡ A ⊃ B. This is a conceptual truth just like the two equivalences preceding it, of which it is an immediate logical consequence. In order to distinguish the strict implications by courtesy – material implications – from the other strict implications, I will call these other strict implications “strict implications properly speaking.”

5.2

Conditionals as Strict Implications with Variable Bases

Sometimes what is meant by an utterance of “if C, then D” (filling in for “C” and “D” the appropriate sentences) is just the same as what would be meant by an utterance of “C 10→ D,” or in other words: by an utterance of “C ⊃ D.” Consider the following case. Somebody says: “Either the gardener or the butler did it. Therefore: if the gardener didn’t do it, the butler did.” In this utterance of an inference, the conditional “if ¬A, then B” that is the conclusion of the inference must clearly be semantically tantamount to “A ∨ B,” hence to the material implication “¬A ⊃ B,” if the inference is to be logically valid, as is surely intended. Another example of an utterance of “if C, then D” which is semantically tantamount to an utterance of “C ⊃ D” (or “C 10→ D”) is an utterance (in ordinary circumstances) of “If Oswald didn’t shoot Kennedy, someone else did.” Why is that utterance semantically tantamount to an utterance of “Oswald didn’t shoot Kennedy ⊃ someone else shot Kennedy”? Because, if asked to provide the nearest justification for asserting “If Oswald didn’t shoot Kennedy, someone else did,” one will simply assert: “Someone shot Kennedy. That’s for sure.” (Note that ¬F(a) ⊃ ∃x(x ≠ a ∧ F(x)) is a logical consequence of ∃xF(x).) But, of course, not every utterance of “if C, then D” is semantically tantamount to an utterance of “C ⊃ D.” Consider an utterance of (1) If you flip this fair coin, it will land showing heads or tails, and, under the same circumstances, an utterance of (2) If you flip this fair coin, then it will land showing heads, or: if you flip this fair coin, then it will land showing tails.

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The natural interpretation of (1) and (2) is the interpretation according to which the utterance of (1) turns out to be true in the circumstances, while the utterance of (2) does not. If, however, every utterance of “if C, then D” were semantically tantamount to an utterance of “C ⊃ D,” then – as a matter of truth-functional logic – an utterance of (2) would have to be true in all circumstances in which an utterance of (1) is true. Note also that one cannot infer “If you flipped this coin, then it would land showing heads, or: if you flipped this coin, then it would land showing tails” from “If you flipped this coin, then it would land showing heads or tails.” Nor is the following a valid inference (contrary to Stalnaker): “If you had flipped this coin, then it would have landed showing heads or tails. Therefore: if you had flipped this coin, then it would have landed showing heads, or: if you had flipped this coin, then it would have landed showing tails.”

In most utterances of conditionals, the conditionals are made to carry some proper necessity or other, that is, the conditional nexus is established by appeal to a basis of necessity that is different from w* (cf. Section 3.6). Sometimes what is meant by an utterance of “if C, then D” is the same as what would be meant by an utterance of “C 1→ D,” or in other words (as can easily be shown): by an utterance of “P(that D, that C).” Consider the following case: “If either the gardener or the butler did it, and the gardener didn’t do it, then the butler did it.” An utterance of this conditional is semantically tantamount to an utterance of a strict implication properly speaking, namely, a logical implication: “Either the gardener or the butler did it, and the gardener didn’t do it 1→ the butler did it,” or in other words: “1(either the gardener or the butler did it, and the gardener didn’t do it ⊃ the butler did it).” Often what is meant by an utterance of “if C, then D” is the same as what would be meant by an utterance of “C 2→ D.” Consider the following case: “If no force is acting on this object, then its state of motion does not change regarding either direction or quantity.” An utterance of this conditional, too, is semantically tantamount to the utterance of a strict implication properly speaking; but in this case it is a nomological, not a logical implication (the utterance of the conditional would not be true otherwise): “No force is acting on this object 2→ its state of motion does not change regarding either direction or quantity,” or in other words: “2(no force is acting on this object ⊃ its state of motion does not change regarding either direction or quantity).”

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Suppose somebody says: “If no force were acting on this object, then its state of motion would not change regarding either direction or quantity.” Though he is not exactly expressing a strict implication properly speaking, such a strict implication forms the heart of what he is saying: “Some force is acting on this object, and 2(no force is acting on this object ⊃ its state of motion does not change regarding either direction or quantity)” (according to analysis (I)), or alternatively: “Either some force is acting on this object, and 2(no force is acting on this object ⊃ its state of motion does not change regarding either direction or quantity), or no force is acting on this object, and its state of motion does not change regarding either direction or quantity” (according to analysis (II)). As has been widely discussed, what is meant by an utterance of “If Oswald didn’t shoot Kennedy, then someone else did” is quite different from what is meant by an utterance of “If Oswald had not shot Kennedy, then someone else would have.” Analogously, what is meant by an utterance of “If the gardener didn’t do it, then the butler did” is quite different from what is meant by an utterance of “If the gardener hadn’t done it, then the butler would have.” In both cases, the first sentence is a simple conditional, the second sentence a counterfactual. And counterfactuals and simple conditionals do not mean the same in themselves and when uttered – of course. But, above, I have suggested two possible analyses of counterfactuals in terms of simple conditionals: analysis (I) and analysis (II), and the unwelcome news seems to be that the difference in meaning between the above simple conditional and the above counterfactual escapes both these analyses. While an utterance of “If Oswald didn’t shoot Kennedy, then someone else did” is semantically tantamount to an utterance of (3) Oswald didn’t shoot Kennedy ⊃ someone else shot Kennedy, an utterance of “If Oswald had not shot Kennedy, someone else would have” is certainly not semantically tantamount to an utterance of (4) Oswald shot Kennedy and (Oswald didn’t shoot Kennedy ⊃ someone else shot Kennedy) [analysis (I)], or to an utterance of

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(4´) Either Oswald shot Kennedy and (Oswald didn’t shoot Kennedy ⊃ someone else shot Kennedy), or Oswald didn’t shoot Kennedy and someone else shot Kennedy [analysis (II)]. For (4) is logically equivalent to “Oswald shot Kennedy,” and (4´) is logically equivalent to “Oswald shot Kennedy, or someone else did.” Does this refute analysis (I) and analysis (II)? It does not. For in offering (4) or (4´) as stating the meaning of an utterance of “If Oswald had not shot Kennedy, someone else would have,” it is tacitly – and falsely – assumed that this counterfactual, when uttered, has the very same basis as the simple conditional “If Oswald didn’t shoot Kennedy, then someone else did” has when it is uttered. The basis of an utterance of the latter conditional is b10, that is: w*. The interpretation of an utterance of this simple conditional is accordingly this: “If Oswald didn’t shoot Kennedy, then someone else did” := “(Oswald didn’t shoot Kennedy 10→ someone else shot Kennedy)” := “10(Oswald didn’t shoot Kennedy ⊃ someone else shot Kennedy)” := “Oswald didn’t shoot Kennedy ⊃ someone else shot Kennedy.” And the interpretation of an utterance of the corresponding counterfactual – if it is assumed to have the same basis as that simple conditional and if analysis (II) of counterfactuals is chosen – is this: “If Oswald had not shot Kennedy, someone else would have” := “IfCF Oswald did not shoot Kennedy, thenCF someone else shot Kennedy” := “Oswald shot Kennedy ∧ (if Oswald did not shoot Kennedy, then someone else did) ∨ Oswald did not shoot Kennedy ∧ someone else did” := “Oswald shot Kennedy ∧ (Oswald did not shoot Kennedy ⊃ someone else did) ∨ Oswald did not shoot Kennedy ∧ someone else did.”

But an utterance of “If Oswald had not shot Kennedy, someone else would have” does not have the basis b10. In fact, it does not seem to have, in itself, any precise basis at all; rather, such a basis must be provided externally: by the context of utterance. Nevertheless, one can give – even without knowledge of the context of utterance – some indication of what an utterance of that counterfactual will most likely be based on; for its basis – call it “b#” – can to some extent be described independently of the context of utterance: b# is a state of affairs that comprises that section of the past that is prior to the actual shooting of Kennedy at t1, the laws of nature, the general character of human affairs, etc.; though rather comprehensive, b# is certainly different from w*.

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Of some such state of affairs b# as has just been described, the state of affairs that someone shoots Kennedy at t1 or at a later time in the past – or in other words: the state of affairs that Oswald shoots Kennedy at t1 or at a later time in the past, or someone else shoots Kennedy at t1 or at a later time in the past – is asserted to be an intensional part by anyone (believing that Oswald shot Kennedy at t1) who asserts the counterfactual “If Oswald had not shot Kennedy, someone else would have.” Therefore, the correct rendering of the meaning of an utterance of this counterfactual in terms of analysis (II) is not (4´) but the following: (5´) Either Oswald shot Kennedy [at t1 or at some later time in the past] and #(Oswald didn’t shoot Kennedy ⊃ someone else shot Kennedy), or Oswald didn’t shoot Kennedy and someone else shot Kennedy. And the correct rendering of the meaning of an utterance of the presently considered counterfactual in terms of analysis (I) is not (4) but the following: (5) Oswald shot Kennedy [at t1 or at some later time in the past] and #(Oswald didn’t shoot Kennedy ⊃ someone else shot Kennedy). Let (the letter) “A” stand for (the sentence) “Oswald shot Kennedy [at t1 or at a later time in the past]” and “B” for “Someone else shot Kennedy [at t1 or at a later time in the past].” Then (5) is obtained from the schema “A ∧ (if ¬A, then B)” that defines “ifCF ¬A, thenCF B” according to analysis (I), and (5´) from the alternative schema A ∧ (if ¬A, then B) ∨ ¬A ∧ B” that defines “ifCF ¬A, thenCF B” according to analysis (II), in the following way: b# is the basis of utterance of the counterfactual “ifCF ¬A, thenCF B” (meaning what it has just been specified to mean). Therefore, the simple conditional “if ¬A, then B” that according to analyses (I) and (II) is implicitly involved in making the meaning of that utterance is semantically tantamount, in this context, to “(if ¬A, then B)b#”, and we have in the mereology of states of affairs: P(that (A ∨ B), b#) ≡ P(that (¬A ⊃ B), b#) =Def #(that (¬A ⊃ B)) =Def #(¬A ⊃ B) ≡ ¬A #→ B =Def #→(that ¬A, that B) =Def P(that B, conj(b#, that ¬A,)) =Def (if ¬A, then B)b#. We can pick “#(¬A ⊃ B)” from this series and substitute it for “if ¬A, then B” in the

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schema “A ∧ (if ¬A, then B),” which gives us (5); and of course we can also substitute it for “if ¬A, then B” in the schema “A ∧ (if ¬A, then B) ∨ ¬A ∧ B,” which gives us (5´). The simple conditional “if ¬A, then B” that according to analyses (I) and (II) is implicitly involved in making the meaning of an utterance of the counterfactual “If Oswald had not shot Kennedy, someone else would have” must be understood in the sense of “(if ¬A, then B)b#”, since b# is the basis of utterance of that counterfactual. But due to implicitly accepted conventions, an assertion of “if ¬A, then B” on its own – i.e., an independent assertion of “If Oswald did not shoot Kennedy, someone else did” – is not understood in this sense unless a comment is added; for, if uttered without comment, “if ¬A, then B” is automatically understood in the sense of “(if ¬A, then B)b10”, i.e., in the sense of “¬A ⊃ B.” The mentioned conventions are such that they automatically determine the basis b10 for every unexpanded utterance of “if ¬A, then B.” If we want to make “if ¬A, then B” say in an utterance what “(if ¬A, then B)b#” says, then our wording must be expanded, as follows: “If Oswald did not shoot Kennedy, then – already on the basis of the past up to t1 (the time of Kennedy’s actual shooting), the laws of nature, the general character of human affairs, etc. – someone else shot Kennedy.” An utterance of the in this way commented simple conditional “if ¬A, then B” is indeed semantically tantamount to an utterance of “(if ¬A, then B)b#”, and hence also to an utterance of “¬A #→ B” and of “#(¬A ⊃ B).” If we assume that # is a necessity faithful to truth, then “If Oswald had not shot Kennedy, someone else would have” logically implies – according both to analysis (I) and analysis (II) – the material implication “Oswald didn’t shoot Kennedy ⊃ someone else shot Kennedy.” Under that same assumption of # being a necessity faithful to truth, “#(Oswald didn’t shoot Kennedy ⊃ someone else shot Kennedy)” logically implies “If Oswald had not shot Kennedy, someone else would have” if we choose analysis (II), but not if we choose analysis (I). In general, analysis (II) preserves the following logical inference-schema, while analysis (I) does not preserve it (but only its second part): CF(C ⊃ D) ⇒ ifCF C, thenCF D ⇒ C ⊃ D, where CF – the necessity that corresponds to the basis of utterance of the counterfactual “ifCF C, thenCF D” – is a necessity faithful to truth.

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This gives analysis (II) a slight advantage over analysis (I), and I shall from now on stick exclusively to analysis (II). But note that analysis (I) makes an assertion of “ifCF C, thenCF D” stronger than an assertion of “if C, then D,” given a common basis of utterance for both; whereas analysis (II) makes an assertion of “ifCF C, thenCF D” weaker than an assertion of “if C, then D” if the common basis for both assertions is an obtaining state of affairs. For then “C ⊃ D” is implied by “if C, then D,” and hence “C ∧ D” implied by “C” and “if C, then D,” leading to the result that “if C, then D” implies “¬C ∧ (if C, then D) ∨ C ∧ D,” that is: “ifCF C, thenCF D.” In this regard, analysis (I) might be said to have an intuitive advantage over analysis (II). 5.2.1

Intrinsic Vagueness of Basis and Conditions for Bases

As I have said, an utterance of “If Oswald had not shot Kennedy, someone else would have” does not have a basis that is precisely determinable without taking into account the context of utterance (and hence “#” in “b#” and “#” cannot be replaced by any numeral out of “1”, “2”, etc. without considering the context of utterance). This can be highlighted by looking also at the counter-counterfactual of “If Oswald had not shot Kennedy, someone else would have,” i.e., at “If Oswald had not shot Kennedy, no one else would have.”3 If an utterance of either one of these two counterfactuals, which are counter-counterfactuals of each other, had a basis that is precisely determinable without considering the context of utterance, then an utterance of the corresponding counter-counterfactual would certainly have the very same precisely determinable basis; and then it should be possible in principle to find out what the truth-values of those utterances are without considering their context of utterance.4 But, on the contrary, it seems im3

Note that this is a counterfactual without presupposition of the falsity of the consequent. The counterfactual is fine, but its consequent, in contrast to its antecedent, is (presumably) true: no one not identical with Oswald shot Kennedy. 4 Can they be both true? Even if they have the same basis? Yes, they can: (a) “Either Oswald shot Kennedy and #(Oswald didn’t shoot Kennedy ⊃ someone else shot Kennedy), or Oswald didn’t shoot Kennedy and someone else shot Kennedy,” and (b) “Either Oswald shot Kennedy, and #(Oswald didn’t shoot Kennedy ⊃ no one else shot Kennedy), or Oswald didn’t shoot Kennedy and no one else shot Kennedy” are not formally incompatible: if “#(Oswald shot Kennedy)” is true and # a necessity that is faithful to truth, then both (a) and (b) are true. Conversely, if both (a) and (b) are true, then “#(Oswald shot Kennedy)” is true.

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possible in principle to find out what their truth-values are without considering their context of utterance. Intrinsic vagueness of basis is a very common phenomenon regarding utterances of conditionals. (But it can also be found regarding utterances of other modal statements.) Here are two more pairs of counterfactual and corresponding counter-counterfactual such that the in-principleimpossibility of assigning a truth-value to an utterance of either one of them without taking into consideration the context of utterance indicates intrinsic vagueness of basis: “If Wagner had been French, he would not have written an opera in German” – “If Wagner had been French, he would have written an opera in German”; “If Alaska were still in the possession of Russia, Russia would today be bigger than 17 075 400 square kilometers” – “If Alaska were still in the possession of Russia, Russia would not be bigger today than 17 075 400 square kilometers.” Bases of utterances of conditionals are often intrinsically vague,5 and this leads us to the question of how utterances of conditionals manage to indicate their bases in most cases precisely – though usually not without taking into account the context of utterance. Of prime importance in the mechanism of indication must be certain restricting conditions. But so far no restricting conditions for bases of utterances of conditionals have been formulated. Which restricting conditions, then, should be imposed on bases of utterances of ontically meant conditionals, of such utterances as are (objectively) acceptable? The main restricting condition is this: (i) The basis of any acceptable utterance of a conditional is an obtaining state of affairs. Restriction (i) guarantees that, under the Bases-Theory of Conditionals, for every acceptable utterance of a simple conditional “if C, then D” the following inference-schema is valid: S(C ⊃ D) ⇒ if C, then D ⇒ C ⊃ D, where S is the necessity that corresponds to the basis of utterance of “if C, then D.”

5

Note that vagueness is not an ontological property, but a semantical one. Properly speaking, only signals, like gestures and linguistic expressions, can be (semantically) vague. But an entity that is not a signal can be called “vague” in a secondary sense if it is vaguely referred to, meant, or indicated by a signal.

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Restriction (i) also guarantees that, under the Bases-Theory of Conditionals, for every acceptable utterance of a counterfactual “ifCF C, thenCF D” the following inference-schema is valid: CF(C ⊃ D) ⇒ ifCF C, thenCF D ⇒ C ⊃ D, where CF is the necessity that corresponds to the basis of utterance of “ifCF C, thenCF D.” Without restriction (i), too many utterances of conditionals could be made to turn out true. In fact, for every simple conditional “if C, then D” a state of affairs can be found that would make every utterance of it true if it were used as a basis of every utterance of it, namely, the states of affairs that D. According to the Bases-Theory of Conditionals, the meaning of any utterance of “if C, then D” amounts to the meaning of “P(that D, conj(bS, that C)),” where bS is the basis of the respective utterance; but “P(that D, conj(bS, that C))” is trivially true if bS = that D. And, of course, the state of affairs k*, if allowed to be a basis, could serve as the universal “truthmaker” for all utterances of simple conditionals, since “P(that D, conj(k*, that C))” is always true, no matter which statements C and D are. That the basis of an utterance of a simple conditional is an obtaining state of affairs is a presupposition of that utterance (more precisely: of its objective acceptability) that may be violated by the utterer: by her taking as the basis of the utterance a state of affairs that, beknownst or unbeknownst to her, is not obtaining. For this case, we adopt the convention that the utterance of the simple conditional is automatically false, no matter whether that conditional is asserted on its own (as an indicative conditional), or as an explicit or implicit (i.e., definition-induced) part of another assertion (say, the assertion of the corresponding counterfactual conditional). Thus, an unacceptable utterance of a simple conditional is false if its unacceptability is due to its having a non-obtaining basis. Consider the following case. Anne believes that the closed box before her has been filled only with black balls for the last five years. But, in fact, for more than a year now the box has been containing both black balls and white balls. She confidently asserts: “If Jack opened this box five minutes ago, then he found it filled with black balls only.” Clearly, as the basis of this utterance of a conditional, Anne is taking a state of affairs that intensionally includes the non-obtaining state of affairs that the box was filled only with black balls five minutes ago. Once Anne discovers that Jack never opened the box, not five minutes ago and not at any other time of the past, this will make her change the mood of her conditional, and she

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will assert just as confidently: “But if Jack had opened this box five minutes ago, then he would have found it filled with black balls only.” The ease with which Anne is shifting from an assertion of an indicative, simple conditional to an assertion of the corresponding subjunctive, counterfactual conditional shows that both utterances are given the same basis by Anne.6 For her, merely conditions regarding propriety of assertion have been changed by the new information, and not any conditions that touch on the truth of either utterance. She still believes that the box has been filled only with black balls for the last five years. And this – no further relevant information having reached her – makes her give the same basis to her utterance of the counterfactual which she gave to her utterance of the simple conditional. Hence both her utterances are false, because the common basis on which she is grounding them is a non-obtaining state of affairs.7 If, however, that state of affairs were an obtaining state of affairs – as Anne believes it is – then, indeed, both her utterances would be true. Are there any restrictions other than restriction (i) for bases of acceptable utterances of (ontic) conditionals? There do not seem to be. One 6

Compare how strikingly different is the behavior of the utterances of two (already mentioned) conditionals that are structurally very much like the two conditionals involving Jack: “If Oswald did not shoot Kennedy, someone else did” – “If Oswald had not shot Kennedy, someone else would have.” Why, in the case of these two conditionals, is there no easy shift from asserting the first conditional to asserting the second (on being informed of the falsity of the antecedent, i.e., of “Oswald did not shoot Kennedy”)? Because convention forbids that utterances of the two Oswald-conditionals have the same basis. According to convention, the basis of an utterance of “If Oswald did not shoot Kennedy, someone else did” and the basis of an utterance of “If Oswald had not shot Kennedy, someone else would have” must be different states of affairs. In contrast, convention does not forbid that utterances of the two Jack-conditionals have the same basis. There is no convention requiring that the basis of an utterance of “If Jack opened this box five minutes ago, then he found it filled with black balls only” and the basis of an utterance of “If Jack had opened this box five minutes ago, then he would have found it filled with black balls only” must be different states of affairs. 7 If the utterance of the counterfactual is evaluated according to analysis (II), then the contextual falsity of “Jack opened this box five minutes ago and found it filled with black balls only” is also needed for the falsity of that utterance. But we have already supposed the contextual falsity of the mentioned conjunction above: as we have supposed, Jack never opened the box. – Note that according to analysis (II) it may happen that the utterance of a counterfactual conditional is unacceptable but true: if the basis of the utterance does not obtain, but antecedent and consequent of the simple conditional that is definitionally incorporated in the counterfactual are both true in the context of utterance.

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might think of postulating a restriction that guarantees the relevance of the antecedent of an uttered conditional for its consequent: (ii?) The state of affairs expressed by the consequent of a conditional is not an intensional part of the basis of any acceptable utterance of that conditional. But it may easily be the case that there are acceptable utterances of “if C, then D” with a true consequent that are semantically tantamount to utterances of the material implication “C ⊃ D.” If there are such utterances, then it follows – contradicting restriction (ii?) – that the state of affairs expressed by D (by the consequent of “if C, then D”) is an intensional part of the basis b10 of all those utterances of “if C, then D”; for if D is true, then: P(that D, w*), and hence because of b10 = w*: P(that D, b10). Moreover, every utterance of “If there is a prime number greater than 7, then there is a prime number greater than 8” is obviously an acceptable utterance of a conditional. But all these utterances have the basis b1 (= t*), and of course we have: P(that there is a prime number greater than 8, b1) – contradicting restriction (ii?). Another plausible-seeming restriction for bases of acceptable utterances of conditionals is stated by the following postulate: (iii?) The negation of the state of affairs expressed by the antecedent of a conditional is not an intensional part of the basis of any acceptable utterance of that conditional. If (iii?) is maintained, then the state of affairs expressed by the antecedent of any conditional is in – a minimal sense – compossible with the basis of any acceptable utterance of the conditional (and if the basis should happen to be expressible by a sentence, then that sentence is cotenable – in a minimal, logical sense – with that antecedent8). However, it does not seem desirable to postulate (iii?). For, under (iii?), no utterance of a counterfactual conditional of the form “IfCF A, thenCF B ∧ ¬B” could be acceptable; but it certainly seems to be the case that some utterances of such conditionals are acceptable. Consider an utterance of the conditional “If the unrestricted Comprehension Principle of set theory were true, then some set would be both an element of itself and not an element of itself,” which is 8

Cotenability plays a prominent role in the so-called metalinguistic theory of conditionals. For some comments on that theory, see the next chapter.

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true. In this example the antecedent is not only false but also (logically) impossible (and therefore the negation of the state of affairs expressed by the antecedent is an intensional part of the basis of the utterance, since that negation is simply t*). But there are even examples of acceptable and true utterances of counterfactuals of the form “IfCF A, thenCF B ∧ ¬B” in which the antecedent is only false and not also (logically) impossible; consider an utterance of the conditional “If the portion of light emitted at t0 from location l1 were at t0 in location l2 10 miles away, then it would be both emitted from location l1 at t0 and not emitted from location l1 at t0.” “How come?” – “Supposing that the light is emitted at t0 from location l1, it is of course emitted at t0 from location l1. And supposing also that the same light is in location l2 at t0, 10 miles away from l1, it is not emitted at t0 from location l1, since light does not cover the distance of 10 miles instantaneously.”

The truth of an utterance of a conditional like our last example is due to the non-trivial incompatibility of the state of affairs expressed by the antecedent with the (obtaining) basis of the utterance (where an incompatibility is non-trivial if, and only if, neither state of affairs participating in the incompatibility is identical with k*, i.e., with the logically impossible state of affairs). We do assert non-trivial incompatibility conditionals (and therefore it seems unacceptable to call such assertions “unacceptable”), since we sometimes assert “IfCF A, thenCF B ∧ ¬B” instead of simply asserting “¬A” (where, nevertheless, it is logically possible that A!) in order to convey the additional information to the hearer or reader that we have very good justification for denying A, namely, that the state of affairs expressed by A – though not logically impossible in itself – is logically incompatible with a certain state of affairs we take to be a fact. It should be noted that neither Stalnaker’s nor Lewis’s theory of counterfactual conditionals makes room for true non-trivial incompatibility conditionals, although these theories do make room for trivial incompatibility conditionals: counterfactual conditionals with logically impossible antecedent (i.e., an antecedent that is not true in any possible world).9 5.2.2

Determining a Basis

The following question is of particular interest: Do the meanings contextindependently expressed by the antecedent and the consequent of an ut9

Concerning Stalnaker’s and Lewis’s theory of conditionals, see the next chapter.

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tered conditional determine the basis of the utterance, be it by themselves or if taken together with various parameters of the context of utterance that are different from the basis of the utterance? Or is that basis an independent parameter of the context of utterance? In some cases, certainly, the latter alternative will be the one that is the case: the basis of the utterance of the uttered conditional is an independent parameter of the context of utterance. This can be made manifest by considering the utterance of a conditional in its context without counting the basis of the utterance as a separate parameter – and finding that the utterance could, then, still have two different truth-values and hence two different utterance-meanings. Consider again one of the pairs of conditionals mentioned in Section 5.2.1: “If Wagner had been French, he would not have written an opera in German” and “If Wagner had been French, he would have written an opera in German.” The following is (presumably) an obtaining state of affairs: that Wagner is a famous opera-composer of the past and that no famous French opera-composer of the past ever wrote an opera in German. Let this state of affairs be denoted by “b11”. But the following is also an obtaining state of affairs: that Lohengrin is an opera in German and that Wagner wrote Lohengrin. Let this state of affairs be denoted by “b12”. As was said above, the basis of an utterance of either Wagner-counterfactual is intrinsically vague: the basis cannot be precisely determined without considering the context of utterance (and therefore the truth-value of an utterance of either Wagner-counterfactual is undetermined if the context of utterance is not taken into account). But, note, even if the context of utterance is taken into account, the basis of either utterance is still undetermined if the basis is supposed to be determined by the context-independent meaning of the sentence that is uttered and those parameters of the context of utterance that are different from the basis. For suppose “If Wagner had been French, he would not have written an opera in German” is asserted by someone at a certain time and place in the real world – call his assertion “assertion 1” –, while someone else asserts, at (roughly) the same time and (roughly) the same place in the real world, “If Wagner had been French, he would have written an opera in German” – call her assertion “assertion 2.” Which one of the two asserters is right? The answer can only be: even if it is known when the assertions are made, and where, and by whom, and in which world, and what their respective context-independent sentence-meaning is, it is nevertheless quite impossible to say which one of the two asserters is

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right. We need additional, independent information: information that tells us what is the basis of assertion 1 and what is the basis of assertion 2. The basis of assertion 1 may easily be b11. In this case assertion 1 is true. For if the basis of assertion 1 is b11, then assertion 1 is semantically tantamount (according to the Bases-Theory of Conditionals) to an assertion of the following: “Wagner was not French ∧ P(that Wagner did not write an opera in German, conj(b11, that Wagner was French)) ∨ Wagner was French ∧ Wagner did not write an opera in German” – which is true. The basis of assertion 2, however, may easily be b12. In this case assertion 2 is true. For if the basis of assertion 2 is b12, then assertion 2 is semantically tantamount (according to the Bases-Theory of Conditionals) to an assertion of the following: “Wagner was not French ∧ P(that Wagner wrote an opera in German, conj(b12, that Wagner was French)) ∨ Wagner was French ∧ Wagner wrote an opera in German” – which is also true.10 Thus, it may easily be the case that assertion 1 and assertion 2 are both true. If they are both true in the way just described: by each assertion having a different basis that makes it true, then – despite appearances to the contrary – they are not logically incompatible; logical incommensurability seems an appropriate description of their relationship. (If both assertions had the same basis, then they could be both true only at the price of the necessity – relative to that basis – of the negation of the antecedent; see footnote 4.) But it can also be the case that both assertions are false. For this effect, assertion 1 and assertion 2 need merely exchange their bases, b12 becoming the basis of assertion 1 and b11 the basis of assertion 2. And it can also be the case that assertion 1 is true and assertion 2 false. For this effect, assertion 1 and assertion 2 need merely have b11 as their common basis. And finally, it can also be the case that assertion 2 is true and assertion 1 false. For this effect, assertion 2 and assertion 1 need merely have b12 as their common basis. Our perplexity as to the truth-values of assertion 1 and assertion 2 can be dispelled by a simple expedient: Ask the utterer of assertion 1 on what matter of fact he bases his judgment, and ask the utterer of assertion 2 on what matter of fact she bases her judgment. Most probably, the matter of fact on which he bases his judgment will amount to b11 or to a fact that properly includes b11, whereas the matter of fact on which she 10

The example under consideration provides an additional reason for not accepting condition (ii?), discussed above, since assertion 2 with b12 as basis obviously violates that condition – in view of the fact that P(that Wagner wrote an opera in German, b12) – and is nevertheless a perfectly good utterance of a conditional.

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bases her judgment will amount to b12 or to a fact that properly includes b12. Given this information, we must conclude that both assertion 1 and assertion 2 are true, and hence that both utterers, in their different ways, are right. Question to the reader (regarding another pair of mutual countercounterfactuals that was mentioned in Section 5.2.1): Are there two obtaining states of affairs such that an assertion of “If Alaska were still in the possession of Russia, Russia would today be bigger than 17 075 400 square kilometers” is true if it has one of the two states of affairs as basis, and an assertion of “If Alaska were still in the possession of Russia, Russia would not be bigger today than 17 075 400 square kilometers” is true if it has the other one of the two states of affairs as basis? Does perhaps only convention militate against accepting both states of affairs as possible bases?

5.3

The Logic of Conditionals

The analyses of the previous sections of this chapter suggest the following truth-conditions for utterances of simple conditionals and utterances of counterfactual conditionals: Let “utt#” designate an utterance of “if C, then D,” “utt*” an utterance of “ifCF C, thenCF D,” “cont(utt#)” the context of utt#, “cont(utt*)” the context of utt*: (I) utt# is true iff b# obtains in cont(utt#), and P([D]#, conj(b#, [C]#)). (II) utt* is true iff either (1) [¬C]* obtains in cont(utt*), and b* obtains in cont(utt*), and P([D]*, conj(b*, [C]*)), or (2) [C]* obtains in cont(utt*) and [D]* obtains in cont(utt*). Here b# is the basis of utt#, which may or may not be an independent parameter of cont(utt#). [S]# is the state of affairs that is expressed by sen-

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tence S in cont(utt#). Substitute “*” for “#” uniformly in the previous two sentences, and the resulting statements are also true. It is widely believed that the logic of conditionals to which the stated truth-conditions give rise is not correct, because it includes inferenceforms as correct that seem to have clear counterexamples. But all alleged counterexamples can be criticized on the following account: they disregard that in a concrete inference which instantiates an inference-form that is included as correct in the envisaged logic of conditionals, all utterances of conditionals involved must be taken to have one and the same basis; for only under this presupposition is the inference-form asserted to be correct (and only under this presupposition can it be shown to be correct). It is truly no surprise if concrete inferences turn out to be invalid which involve utterances of conditionals that do not have the same basis. No drastic consequences – like formulating a special logic of conditionals – should be drawn from acknowledging the unsurprising fact that concrete inferences which involve utterances of conditionals that do not have the same basis often turn out to be invalid although they – in a sense – instantiate an inference-form one would have expected to be correct. No drastic consequences should be drawn, because those inferences do not properly instantiate that inference-form. Consider the following analogy: a = b, a = c ⇒ b = c is generally thought to be a correct inference-form of the logic of identity. Suppose somebody attacks this view on the following ground: There is a true utterance of “I am Uwe Meixner” and there is a true utterance of “I am Hans Rott”; but there is no true utterance of “Uwe Meixner is Hans Rott.” Nobody would be fazed by this, and quite rightly so. Because a concrete inference consisting of a true utterance of “I am Uwe Meixner” and a true utterance of “I am Hans Rott” as premises and a false utterance of “Uwe Meixner is Hans Rott” as conclusion does not properly instantiate the inference-form a = b, a = c ⇒ b = c, notwithstanding the fact that it instantiates it merely syntactically. Why does that concrete inference not properly instantiate the inference-form a = b, a = c ⇒ b = c? Because it is not semantically uniform, where an inference is semantically uniform if, and only if, there are no two occurrences of the same expression in it that differ regarding meaning or reference or in any other semantically relevant way (for example, regarding presupposition). Whatever inference-form is asserted as (logically) correct, it is always asserted to be correct with the tacit proviso that its asserted correctness extends only to all its proper instances,

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not to all its instances simpliciter. If the correctness of an inference-form were asserted for all its instances (proper and improper ones), then just about every inference-from would turn out to be incorrect, the trivial inference-form A ⇒ A not excluded. I will now go through a list of inference-forms that come out as (logically) correct according to the stated truth-conditions for utterances of conditionals, but that are standardly thought to have counterexamples. In every case I will show that the alleged counterexample is not semantically uniform, namely: not semantically uniform with regard to the expression “if, then” (whether occurring in a simple or a counterfactual conditional), that it is, therefore, not in fact a counterexample of the inference-form of which it is thought to be a counterexample (since it is not even a proper instance of that inference-form). Regarding counterfactual conditionals “IfCF A, thenCF B” in the putative counterexamples presented below, it is generally assumed that the sentence A is false in the relevant situation of utterance. It is also generally assumed that all considered bases of utterances of conditionals are obtaining states of affairs. (Otherwise it would be all too easy to explain the truth of the premise-utterances and the falsity of the conclusionutterance by a shift of basis.)

(1)

If A, then B; ¬A ⇒ IfCF A, thenCF B.

Putative counterexample: True: (an assertion, at t0 – now – in the actual world, of the sentence) “If Oswald didn’t shoot Kennedy, someone else did.” True: (an assertion, at t0 in the actual world, of the sentence) “It is not the case that Oswald did not shoot Kennedy.” Likely to be false: (an assertion, at t0 in the actual world, of the sentence) “If Oswald had not shot Kennedy, someone else would have.” Comment: Due to conventions ruling the simple conditional and the counterfactual conditional involved in this putative counterexample, there is a clear change of basis between the assertion of the first premise and the assertion of the conclusion. (The nature of that change has already been discussed; see Section 5.2 above.) Therefore, no counterexample of inferenceform (1) has in fact been stated. (2)

IfCF A, thenCF B ⇒ IfCF A ∧ C, thenCF B.

Putative counterexample: True: (an assertion, at t0 in the actual world, of the sentence) “If I were to pinch you, you wouldn’t die.” False: (an asser-

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tion – by the same speaker, addressed to the same person, at roughly the same time in the actual world – of the sentence) “If I were to pinch you and cut your head off, you wouldn’t die.” Comment: The putative counterexample is taken from Daniel Bonevac’s, Deduction. Introductory Symbolic Logic, from the chapter on counterfactuals. That this textbook, intended for the instruction of undergraduates, is the source of the alleged counterexample shows that inference-form (2) is quite generally accepted to be logically invalid on the basis of putative counterexamples like the one stated. But the offered putative counterexample of inference-form (2) is not in fact a counterexample of it,11 since it clearly involves a change of basis in going from the premise to the conclusion. This change is recognizable purely on the basis of linguistic convention, governing what can be sensibly uttered and what cannot, even without any more specific information regarding the context of utterance: It is, due to linguistic convention (which is not arbitrary, but informed by how things are in this world), an intensional part of the basis of any actual assertion of “If I were to pinch you, you wouldn’t die” that the utterer does not both pinch [at t0] the addressee and cut his head off; in other words: the disjunctive state of affairs that the utterer does not pinch the addressee or does not cut his head off is part of the basis of any actual assertion of the said conditional. In other words: linguistic convention has it that whoever – in the actual world – asserts “If I were to pinch you, you wouldn’t die” must also be ready to assert, on the same basis, “If I were to pinch you, I wouldn’t simultaneously cut your head off.”

But also due to linguistic convention, there is no actual assertion of “If I were to pinch you and cut your head off, you wouldn’t die” of whose basis the state of affairs that the utterer does not both pinch the addressee and cut his head off is an intensional part. For otherwise some actual assertion of “If I were to pinch you and cut your head off, you wouldn’t die” would be trivially true, due to incompatibility of the antecedent state of affairs with the (obtaining) basis; but, due to linguistic convention (as long as it is 11

Nor are the other two putative counterexamples of inference-form (2) that Bonevac offers in fact counterexamples of it. See Deduction, pp. 396-397: True: If Miami had beaten Denver, they’d be in the Super Bowl. False: If Miami had beaten Denver, but had lost every other game this season, they’d be in the Super Bowl. True: If Beethoven hadn’t written the Fifth Symphony, he’d still be remembered as a great composer. False: If Beethoven hadn’t written the Fifth Symphony, and, in fact, hadn’t written any music at all, he’d still be remembered as a great composer.

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observed, and we can assume, in the case at hand, that it is invariably observed), there is no such assertion.12 In other words: linguistic convention has it that whoever – in the actual world – asserts “If I were to pinch you and cut your head off, you wouldn’t die” is not making an assertion that is trivially true.

Thus, an actual assertion of “If I were to pinch you, you wouldn’t die” cannot – due to linguistic convention – have the same basis as a roughly concurrent actual assertion (of the same speaker, addressed to the same person) of “If I were to pinch you and cut your head off, you wouldn’t die.” (3)

IfCF A, thenCF B; IfCF B, thenCF C ⇒ IfCF A, thenCF C.

Putative counterexample: Since (2) is a consequence of (3), a putative counterexample of inference-form (3) has already been presented. (2) is a consequence of (3): IfCF A ∧ C, thenCF A; IfCF A, thenCF B ⇒ IfCF A ∧ C, thenCF B is a specialization of (3) [obtainable from it by uniform substitution: A for B, A ∧ C for A, B for C], and the schema IfCF A ∧ C, thenCF A can be eliminated from the premises of the specialization because it is an entirely uncontroversial logically valid schema.

But here is a more direct putative counterexample of (3) (taken from Deduction, p. 397): True: (an assertion at t0 in the actual world of the sentence) “If the U.S. were to withdraw from the United Nations, Israel would too.” True: (an assertion at t0 in the actual world of the sentence) “If Israel were to withdraw from the United Nations, the U.N. budget would be little affected.” False: (an assertion at t0 in the actual world of the sentence) “If the U.S. were to withdraw from the United Nations, the U.N. budget would be little affected.” Comment: That this putative counterexample of (3) is not in fact a counterexample of (3) (since it does not constitute a proper instance of (3)) can be seen when we ask ourselves: Is there a common basis of an assertion of the first counterfactual, of an assertion of the second counterfactual, and of an assertion of the third counterfactual – in the actual world at t0, where neither Israel nor the U.S. show any signs of withdrawing from the U.N. – 12

This criticism can also be applied (mutatis mutandis) to the putative counterexamples of (2) that are cited in the preceding footnote.

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which is such that the first-mentioned assertions are both true but the lastmentioned false? – There is not. For suppose three such assertions had the same basis, in other words: b1 = b2 = b3. Since the assertion of the second counterfactual is to be true, the state of affairs that it is not both the case that Israel withdraws [at t0] from the U.N. and that the U.N. budget is not little affected must be an intensional part of the basis of the utterance of the second counterfactual. For suppose that state of affairs – that is, disj(that Israel does not withdraw from the U.N., that the U.N. budget is little affected) – were not an intensional part of the basis of the assertion of the second counterfactual, i.e., of b2. It is not to be seen, then, how the sentence “P(that the U.N. budget is little affected, conj(b2, that Israel withdraws from the U.N.))” could be true (as is required if the assertion of the second counterfactual is to be true, this assertion being an utterance in the actual world at t0); in fact, that sentence could not, then, be true, since we have in the mereology of states of affairs: P(that B, conj(b, that A)) ≡ P(that (¬A ∨ B), b) ≡ P(disj(that ¬A, that B), b). Since b3 is supposed to be identical with b2, we have: (a) disj(that Israel does not withdraw from the U.N., that the U.N. budget is little affected) is also an intensional part of b3, i.e., of the basis of the assertion of the third counterfactual. Moreover, since the assertion of the first counterfactual is supposed to be true, we must have: P(that Israel withdraws from the U.N., conj(b1, that the U.S. withdraws from the U.N.)), and therefore: (b) P(that Israel withdraws from the U.N., conj(b3, that the U.S. withdraws from the U.N.)), since b3 is supposed to be identical with b1. But the consequence of (a) and (b) in the mereology of states of affairs is this: P(that the U.N. budget is little affected, conj(b3, that the U.S. withdraws from the U. N.)).13 Hence the utterance of the third counterfactual turns out to be true – contrary to fact. The initial assumption: that there is a common basis to three respective assertions – in the actual world, at t0 – of the three considered counterfactuals, must therefore be false. (4)

If A, then B ⇒ If ¬B, then ¬A.

P(disj(that ¬A, that B), b) and P(that A, conj(b, that C)) together imply P(that B, conj(b, that C)). 13

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Putative counterexample: True: (an utterance at t0 in the actual world of the sentence) “If it rains, the street gets wet.” False: (an utterance at t0 in the actual world of the sentence) “If the street doesn’t get wet, it doesn’t rain.” Comment: (4) has not been refuted by this putative counterexample to it, because the utterance of the premise and the utterance of the conclusion do not have the same basis. The basis of the premise-utterance contains the state of affairs that the street is not insulated [at t0] in any way from the effects of rain falling on it. The basis of the conclusion-utterance, clearly, does not contain that state of affairs (otherwise, the conclusion-utterance would be true). (5)

IfCF A, thenCF B ⇒ IfCF ¬B, thenCF ¬A.

Putative counterexample: True: (an utterance at t0 in the actual world of the sentence) “If Hitler had died in the assassination attempt of July 20, 1944, the Second Word War would (nevertheless) have ended in 1945.” False: (an utterance at t0 in the actual world of the sentence) “If the Second World War had not ended in 1945, then Hitler would (nevertheless) not have died in the assassination attempt of July 20, 1944.” Comment: (5) has not been refuted by this putative counterexample to it. The basis of the utterance of the counterfactual conditional in the conclusion cannot be the basis of the utterance of the counterfactual conditional in the premise. Both bases are vague. But the basis of the premiseutterance, which is constituted by certain political and military 1944-facts (mainly on and after July 20, 1944; but that Hitler did not die in the assassination attempt of July 20 is not a part of them) and by natural and historical laws, is a basis that is in itself sufficient for World War II ending in 1945; that is, World War II ending in 1945 is an intensional part of it. The basis of the conclusion-utterance, however, does not have the state of affairs that World War II ended in 1945 as an intensional part. If it had, the conclusion-utterance would be trivially true: because the state of affairs expressed by the antecedent of that conditional would be incompatible with the (obtaining) basis of the utterance; but, on the contrary, the utterance is supposed to be false. (6)

IfCF A, thenCF B ⊃ C ⇒ IfCF A ∧ B, thenCF C.14

14

This inference-schema and the previous one (number (5)) are cited as fallacies by David Lewis in “Counterfactuals and Comparative Possibility,” p. 17.

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Putative counterexample: True: (an utterance at t0 in the actual world of the sentence) “If John F. Kennedy had not been murdered in 1963, then he would have been reelected President in 1964 or he would have been reelected President in 2004.” False: (an utterance at t0 in the actual world of the sentence) “If John F. Kennedy had not been murdered in 1963 and not been reelected President in 1964, then he would have been reelected President in 2004.”15 Comment: (6) is not refuted by this putative counterexample to it. The basis of the utterance of the counterfactual conditional in the premise, b6, is the same basis as the basis of the utterance (at t0 in the actual world) of the counterfactual conditional “If John F. Kennedy had not been murdered in 1963, then he would have been reelected President in 1964.” Both utterances are true on the same basis. (This means that the disjunct in the consequent of the counterfactual conditional in the premise, “or he would have been reelected President in 2004,” has no function in determining that basis – such is the situation.) Therefore we have: P(that Kennedy is reelected President in 1964, conj(b6, that Kennedy is not murdered in 1963)). Suppose now that the putative counterexample to (6) is really a counterexample to it. Then the basis of the conclusion-utterance of the counterexample must be identical with the basis of its premise-utterance, b6. But then the conclusion-utterance must be true (according to analysis (II)), since not only “Kennedy is murdered in 1963 or reelected President in 1964” but also “P(that Kennedy is reelected President in 2004, conj(b6, that Kennedy is not murdered in 1963 and not reelected President in 1964))” is true. The latter statement follows from “P(that Kennedy is reelected President in 1964, conj(b6, that Kennedy is not murdered in 1963)),” according to the mereology of states of affairs. The nearest general principle used for establishing this last assertion is this: ∀y(S(y) ⊃ [P(that B, conj(y, that ¬A)) ⊃ P(that C, conj(y, that (¬A ∧ ¬B)))]). The principle is provable in the mereology of states of affairs.

But, on the contrary, the conclusion-utterance must be false if the putative counterexample to (6) is to be really a counterexample to (6). The assumption that the putative counterexample to (6) is really a counterexample to (6) has, therefore, been reduced ad absurdum.

(6) is putatively refuted by the putative refutation of a special case of it: IfCF ¬A, thenCF ¬B ⊃ C [i.e., B ∨ C] ⇒ IfCF ¬A ∧ ¬B, thenCF C. 15

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In concluding this section, I observe that the inference-schemata (4) and (5) above match each other: (4) is the inference-schema of contraposition for simple conditionals, (5) is the inference-schema of contraposition for counterfactual conditionals; both seem prima facie valid, both have putative counterexamples that, in the end, cannot be maintained. This particular correspondence between inference-schemata is, in fact, an indication of a general pattern. For (2) – the inference-schema of strengthening the antecedent for counterfactual conditionals – is matched by (2´): the inferenceschema of strengthening the antecedent for simple conditionals, and (3) – the inference-schema of transitivity for counterfactual conditionals – is matched by (3´): the inference-schema of transitivity for simple conditionals. Moreover, (6) – the inference-schema of importation for counterfactual conditionals – is matched by (6´): the inference-schema of importation for simple conditionals. Just as there are putative counterexamples to (2), (3), (5), and (6), so there are putative counterexamples to (2´), (3´), (4), and (6´).16 The latter putative counterexamples, too, must all be rejected due to a lack of semantical uniformity. Analysis reveals that there is no constancy of basis regarding the uttered conditionals in the inferences adduced as counterexamples: invariably, when one moves from the uttered premise(s) to the uttered conclusion, a change of basis occurs. 5.3.1

Further Logical Difficulties for the Bases-Theory of Conditionals and Their Solutions

Having, in the previous section, considered and defused arguments against the Bases-Theory of Conditionals that aim to settle that theory with the charge that it does not provide us with the correct logic of conditionals, let us now look at other (broadly) logical difficulties for the Bases-Theory of Conditionals. Difficulty: From the true utterance (at t0, in the actual world) of “Every solid ball of uranium 235 has a diameter of less than 1 mile” one can validly infer the truth of the utterance (at t0, in the actual world) of “If the earth were a solid ball of uranium 235, it would have a diameter of less than 1 mile.” From the true utterance (at t0, in the actual world) of “Every solid ball of gold has a diameter of less than 1 mile,” on the other hand, 16

In view of this fact, one wonders why some philosophers have thought that the logic of the indicative conditional is entirely different from the logic of the counterfactual conditional. Concerning this, see the next chapter.

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one cannot validly infer the truth of the utterance (at t0, in the actual world) of “If the earth were a solid ball of gold, it would have a diameter of less than 1 mile.” It seems that the Bases-Theory of Conditionals cannot explain that striking difference. Solution: Indeed, according to the Bases-Theory of Conditionals, both inferences must turn out to be valid if in both cases the premiseutterance is taken to state the basis of the conclusion-utterance (which is an utterance of a counterfactual conditional). But if in the first inference the premise-utterance states the basis of the conclusion-utterance, whereas in the second inference it does not, then the de facto difference in validity between the two inferences can very well find its explanation also in accordance with the Bases-Theory of Conditionals. Suppose that in a certain context of utterance the overall aim is to draw conclusions from laws of nature. In such a context, the state of affairs expressed by “Every solid ball of uranium 235 has a diameter of less than 1 mile” can serve as the basis, or as part of the basis, of the utterance of a conditional, since “Every solid ball of uranium 235 has a diameter of less than 1 mile” is (or expresses) a law of nature. “Every solid ball of gold has a diameter of less than 1 mile,” on the other hand, cannot serve – in that context of drawing conclusions from laws of nature – as the basis, or as part of the basis, of the utterance of a conditional, since “Every solid ball of gold has a diameter of less than 1 mile” is not a law of nature. Therefore, in the supposed context, the utterance of “Every solid ball of gold has a diameter of less than 1 mile” cannot be taken to state the basis of the utterance of “If the earth were a solid ball of gold, it would have a diameter of less than 1 mile,”17 and therefore the truth of the latter utterance (made in the supposed context) cannot be validly inferred from the truth of the former utterance (also made in the supposed context). Now, I submit that some context of drawing conclusions from laws of nature must in fact be presupposed when the validity of inferring “If the earth were a solid ball of uranium 235, it would have a diameter of less than 1 mile” from “Every solid ball of uranium 235 has a diameter of less than 1 mile” is contrasted with the invalidity of inferring 17

Which state of affairs could be that basis? – In fact, given the conditional “If the earth were a solid ball of gold, it would have a diameter of less than 1 mile” and the supposed context of drawing conclusions from laws of nature, there is no state of affairs that is the basis of an utterance of that conditional in that context. But we can arrange that there is an ersatz basis if context and conditional are such that they exclude (or merely do not determine) a basis of utterance for the conditional in the context. That ersatz basis is simply the state of affairs t*.

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“If the earth were a solid ball of gold, it would have a diameter of less than 1 mile” from “Every solid ball of gold has a diameter of less than 1 mile.” That presupposition explains that anomaly. Difficulty: Consider Goodman’s counterfactuals: “If New York City were in Georgia, New York City would be in the South” and “If Georgia included New York City, Georgia would not be entirely in the South.”18 Both counterfactuals are true. But how can this be if the Bases-Theory of Conditionals is correct? Solution: An utterance of the first counterfactual is true, because its basis consists in the following obtaining state of affairs: that Georgia is [at the time of the utterance] entirely in the South. An utterance of the second counterfactual is true, because its basis consists in the following obtaining state of affairs: that New York City is [at the time of the utterance] not in the South. Let me explain. The following statements are all true: (0) That Georgia is entirely in the South is an obtaining state of affairs, and that New York City is not in the South is also an obtaining state of affairs. (1) New York is not in Georgia, and P(that New York City is in the South, conj(that Georgia is entirely in the South, that New York City is in Georgia)), and (2) Georgia does not include New York, and P(that Georgia is not entirely in the South, conj(that New York City is not in the South, that Georgia includes New York)). Given a context of utterance in which the state of affairs that Georgia is entirely in the South serves as the basis of an utterance of “IfCF New York City is in Georgia, thenCF New York City is in the South,” and another in which the state of affairs that New York City is not in the South serves as the basis of an utterance of “IfCF Georgia includes New York City, thenCF Georgia is not entirely in the South,” we can conclude from (0) – (2), in accordance with the Bases-Theory of Conditionals, that both utterances are true.19 18

Cf. David Lewis, Counterfactuals, p. 43. Lewis (see Counterfactuals, p. 43) gives an entirely different and utterly complicated explanation of the simultaneous truth of “If New York City were in Georgia, New York City would be in the South” and “If Georgia included New York City, Georgia would not be entirely in the South” – an explanation in terms of a sophisticated version of his counterpart theory. As far as Goodman’s counterfactuals are concerned, I believe we can very well do without such complications.

19

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Difficulty: How, then, could it have seemed that the Bases-Theory of Conditionals has a problem with the fact of their truth? Solution: Not only the Bases-Theory has an apparent problem with this fact, and the apparent problem is not simply the simultaneous truth of actual utterances of Goodman’s counterfactuals. For the conjunction of “IfCF A, thenCF B” and “IfCF A, thenCF ¬C” seems to be logically equivalent with “IfCF A, thenCF B ∧ ¬C,” and this seeming logical equivalence is not only endorsed by the Bases-Theory of Conditionals but also by other theories of counterfactuals (e.g., Stalnaker’s). Hence if both an utterance of “IfCF A, thenCF B” and an utterance of “IfCF A, thenCF ¬C” are true, then an utterance of “IfCF A, thenCF B ∧ ¬C” (assuming it to be uttered) ought also to be true. But an utterance (at t0, in the actual world) of “If Georgia included New York City [in other words: if New York City were in Georgia], New York City would be in the South and Georgia would not be entirely in the South” just does not seem to be true, although both an utterance (at t0, in the actual world) of “If New York City were in Georgia, New York City would be in the South” and an utterance (at t0, in the actual world) of “If Georgia included New York City [i.e., if New York City were in Georgia], Georgia would not be entirely in the South” are certainly true. Apparently, the inference-schema (7)

IfCF A, thenCF B; IfCF A, thenCF ¬C ⇒ IfCF A, thenCF B ∧ ¬C

stands refuted. If it were really refuted, this would spell trouble not only for the Bases-Theory of Conditionals but also for all other analyses of counterfactuals that make (7) a valid inference-schema. But at least from the point of view of the Bases-Theory of Conditionals (and of what has been said in the previous section) it is easily seen that our putative counterexample to (7) is not really a counterexample to it (from the point of view of other theories of counterfactual conditionals, matters are not as clear20). There is no uniformity of bases throughout that putative counterexample: the basis of the utterance (at t0, in the actual world) of the sentence “IfCF A, thenCF B” (here used as an abbreviation of the relevant ordinary language sentence) is that Georgia is [at t0] entirely in the South; the basis of the ut20

With them, it is certainly not as easy as with the Bases-Theory of Conditionals to show that even if assertions of “IfCF A, thenCF B” and “IfCF A, thenCF ¬C” are true and an assertion of “IfCF A, thenCF B ∧ ¬C” false, no refutation of the logical equivalence – as stated by inference-schema (7) and its converse – of “IfCF A, thenCF B ∧ ¬C” with the conjunction of “IfCF A, thenCF B” and “IfCF A, thenCF ¬C” has been effected.

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terance (at t0, in the actual world) of “IfCF A, thenCF ¬C” is that New York City is [at t0] not in the South. These bases are different states of affairs, and no matter which of the two bases is the basis of the utterance (at t0, in the actual world) of “IfCF A, thenCF B ∧ ¬C,” that utterance turns out to be false. But, of course, we have no real counterexample to (7). For obtaining a counterexample to (7) that is not merely putative, we must have a true actual utterance of “IfCF A, thenCF B” with the basis X, and a true actual utterance of “IfCF A, thenCF ¬C” with that same basis X, and a false actual utterance of “IfCF A, thenCF B ∧ ¬C” with that same basis X again. No such basis X can exist, as is easily provable. “A” stands for the sentence “New York City is in Georgia,” “B” stands for the sentence “New York City is in the South,” “C” stands for the sentence “Georgia is entirely in the South.” Suppose we have a true actual utterance of “IfCF A, thenCF ¬C” with the basis X, and a true actual utterance of “IfCF A, thenCF B” with that same basis X, and a false actual utterance of “IfCF A, thenCF B ∧ ¬C,” again with the basis X. Since A is a false statement, the considered utterances of “IfCF A, thenCF ¬C” and “IfCF A, thenCF B” can only be true if we have: P(that B, conj(X, that A)) and P(that ¬C, conj(X, that A)), where X is an obtaining state of affairs (that A, that B, that ¬C being the states of affairs which “A,” “B,” and “¬C” express in the context of utterance). But from this it can be deduced in the mereology of states of affairs: P(that (B ∧ ¬C), conj(X, that A)), X being an obtaining state of affairs. Hence the considered utterance of “IfCF A, thenCF B ∧ ¬C” is true – contradicting the supposition. Could we at least have a true actual utterance of “IfCF A, thenCF ¬C” with the basis X, and a true actual utterance of “IfCF A, thenCF B” with that same basis X? If so, we must also have: P(that B, conj(X, that A)) and P(that ¬C, conj(X, that A)), where X is an obtaining state of affairs. If X is conj(that C, that ¬B), then X is an obtaining state of affairs and we do have: P(that B, conj(X, that A)) and P(that ¬C, conj(X, that A)) – but only trivially so, since conj(that C, that ¬B) (= X) is incompatible with the state of affairs that A (i.e., conj(X, that A) = k*). This is so because the set {A, C, ¬B} is an inconsistent triad: it is impossible that all three statements in the set are true. But can we have P(that B, conj(X, that A)) and P(that ¬C, conj(X, that A)), where X is an obtaining state of affairs and compatible with the state of affairs that A? Consider that we have P(that (B ∨ ¬C), that A), ¬P(that A, w*), ¬P(that B, w*), ¬P(that ¬C, w*), k* ≠ that (A ∧ B ∧ ¬C); and from these premises and the additional premises P(that B, conj(X, that A)) and P(that ¬C, conj(X, that A)), we cannot deduce in the mereology of states of affairs: P(neg(X), that A) ∨ ¬P(X, w*). Hence the set of premises plus ¬P(neg(X), that A) and P(X, w*) must have a model, and there seems to be no reason why it should not also have a realistic model (with “A,” “B,” and “C” standing for the sentences they are above taken to stand for, and with w* being all that is the case). But this, of course, doesn’t tell us which state of affairs X might be.

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It remains to be added that there is exactly as much reason to be impressed by our putative counterexample to inference-schema (7) as there is reason to be impressed by the putative counterexamples to (2) (strengthening of the antecedent for counterfactual conditionals; see the previous section) and to (3) (transitivity for counterfactual conditionals; see the previous section). Why, therefore, should the putative counterexamples to (2) and (3) (and (5) and (6)) be allowed to have an impact on the logic of conditionals, while the putative counterexample to (7) is not allowed to have such an impact? The point, of course, I would like to drive home is that none of these putative counterexamples should be allowed to have an impact on the logic of conditionals. With some imagination one can presumably think up “counterexamples” of the kind we have seen – all of them defeatable from the point of view of the Bases-Theory of Conditionals – for just about every inference-schema for conditionals that is not entirely trivial. If this “counterexampling” is uncritically accepted as cogent, the outcome must be that there remains no interesting logic of conditionals to speak of.

5.4

Might-Conditionals, Explanatory Conditionals, and Causation

We have so far considered simple conditionals and counterfactual conditionals, indicative conditionals being simple conditionals that are asserted on their own. I herewith present a few more types of conditionals, all of them – in the final analysis – reducible to simple conditionals. An unqualified might-conditional is a conditional of the following form: “If A, then it might be the case that B,” or in short: “If A, thenM B.” An example of an unqualified might-conditional is the statement “If he goes to Paris, then he might see the Conciergerie.” If that statement is asserted on its own, it is an example of an indicative might-conditional. If the basis of an utterance of “If A, thenM B” is a non-obtaining state of affairs, then that utterance is automatically false. If that basis is an obtaining state of affairs, then the utterance of “If A, thenM B” is semantically tantamount to uttering “¬(If A, then ¬B)” with that same basis. Presupposing that “If A, thenM B” is uttered with an obtaining basis, one can define: If A, thenM B =Def ¬(If A, then ¬B).

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A counterfactual might-conditional is a counterfactual conditional of the following form: “If it were [had been] the case that A, then it might be [might have been] the case that B,” or in short: “IfCF A, thenM B.” If “IfCF A, thenM B” is uttered with a non-obtaining basis, then the utterance of “IfCF A, thenM B” is semantically tantamount to uttering “A ∧ B.” If “IfCF A, thenM B” is uttered with an obtaining basis, then the utterance of “IfCF A, thenM B” is semantically tantamount to uttering “¬A ∧ ¬(If A, then ¬B) ∨ A ∧ B” with that same basis. (The basis, of course, is specifically a basis for uttering the part “If A, then ¬B” of “¬A ∧ ¬(If A, then ¬B) ∨ A ∧ B.”) Presupposing that “IfCF A, thenM B” is uttered with an obtaining basis, one can define: IfCF A, thenM B =Def ¬A ∧ ¬(If A, then ¬B) ∨ A ∧ B. An explanatory conditional is a conditional of the following form: “B, because A.” If “B, because A” is uttered with a non-obtaining basis, then the utterance of “B, because A” is semantically tantamount to uttering “¬A ∧ ¬B.” If “B, because A” is uttered with an obtaining basis, then the utterance of “B, because A” is semantically tantamount to uttering “A ∧ (If A, then B) ∨ ¬A ∧ ¬B” with that same basis. Presupposing that “B, because A” is uttered with an obtaining basis, one can define: B, because A =Def A ∧ (If A, then B) ∨ ¬A ∧ ¬B. Given explanatory conditionals, it is merely a short way to a satisfactory analysis of causation. Since all modalities considered in this book are ontic modalities, whether conditional (relational) or non-conditional (nonrelational) ones, explanatory conditionals are here already taken to be explanatory in an ontic, objective sense – as is necessary if they are to be employed in the analysis of causation. Consider, then, the following definition: E causes E´ =Def E and E´ are events, E occurs before E´ occurs, and E´ occurs, because E occurs. In this definition, causation is treated as a sufficiency-relation (not as a sine-qua-non or probabilistic relation) between events, with the temporal priority of causes definitionally integrated into it. These decisions in the theory of causation are not self-evident, of course, and need arguing for.

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This is done in great detail in my book Theorie der Kausalität, and more compendiously (and in English) in my paper “Causation in a New Old Key.” Moreover, for giving full content to the above definition of causation, three questions need to be answered. (i) What are events? (ii) What does it mean that an event E occurs before an event E´? (iii) Do assertions of causation always have the same basis (the basis of an assertion of causation being identical with the basis for asserting the specific explanatory conditional which is definitionally involved in the assertion of causation)? For questions (i) and (ii), whose subject matter does not fall within the purview of this book, the reader is referred to my above-mentioned book and paper. Regarding question (iii), one plausible answer to it is to maintain that indeed all assertions of causation have (or after philosophical analysis ought to have) the same basis, namely, b2: the conjunction of all laws of nature (taken to be certain states of affairs). Whatever basis or bases for assertions of causation are drawn into consideration, the Bases-Theory of Conditionals enables a satisfactory solution to the epistemological problem of causal necessity, forcefully pointed out by Hume in the Treatise and Enquiry. Suppose we have a normal assertion of “E1 causes E2,” i.e., an assertion of “E1 causes E2” with an obtaining basis. Suppose, in addition, that this assertion is true. Hence we also have a true and normal implicit assertion of “E2 occurs, because E1 occurs,” and a true and normal implicit assertion of “If E1 occurs, then E2 occurs.” In fact, all three assertions, the explicit one and the two implicit ones, have the same obtaining basis, call it “bcausal”. Necessity is involved in the true assertion of “E1 causes E2” by being involved in the (logically implied) true assertion of “E2 occurs, because E1 occurs.” In turn, necessity is involved in the true assertion of “E2 occurs, because E1 occurs” by being involved in the true assertion of “If E1 occurs, then E2 occurs” (which follows from the former assertion, in view of this assertion having an obtaining basis and in view of the truth of “E1 occurs”). According to the BasesTheory of Conditionals, necessity is involved in the true assertion of “If E1 occurs, then E2 occurs” in virtue of that assertion being semantically tantamount to an assertion of “causal(E1 occurs ⊃ E2 occurs),” or in other words: to an assertion of “P(that E2 occurs, conj(bcausal, that E1 occurs)).” This is what the necessity that is involved in “E1 causes E2” – i.e., causal necessity – boils down to. And there is, in the end, nothing deeply problematic about causal necessity; for the only necessity involved in an assertion of causation is, in the last analysis, logical necessity: “P(that E2 occurs, conj(bcausal, that E1

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occurs))” is, in the mereology of states of affairs, provably equivalent to “1(O(bcausal) ∧ E1 occurs ⊃ E2 occurs).” As will be remembered, not even Hume considered the notion of logical necessity to be problematic: he had no objections against the kind of necessity that is grounded in “relations of ideas.” It should, however, be appreciated at this point that the intended logical necessity is not logical necessity in a narrow sense. It is not formal logical necessity, nor is it logical necessity which, though not formal, is nevertheless firmly connected to language. Suppose we have two states of affairs, p and q, which are each too big – have too much content – to be expressed by any sentence (of a normal language) in which they are not also named (referred to by singular terms). Suppose q is an intensional part of p, that is: P(q, p). Then it can be shown in the mereology of states of affairs: 1(O(p) ⊃ O(q)), i.e., that it is logically necessary that the obtaining of p implies the obtaining of q. [P(q, p) iff P(that O(q), that O(p)) iff P(that O(q), conj(t*, that O(p))) iff P(disj(neg(that O(p)), that O(q)), t*) iff P(disj(that ¬O(p), that O(q)), t*) iff P(that (¬O(p) ∨ O(q)), t*) iff 1(that (¬O(p) ∨ O(q))) iff 1(¬O(p) ∨ O(q)) iff 1(O(p) ⊃ O(q)).] This is the same relationship that also holds between the state of affairs that John is a bachelor at t0 and the state of affairs that John is not married at t0: it is logically necessary that the obtaining of the former state of affairs implies the obtaining of the latter. But in contrast to p and q, the two states of affairs that John is a bachelor at t0 and that John is not married at t0 are readily expressible by sentences (of a normal language) which do not name them: “John is a bachelor at t0” and “John is not married at t0”.

According to the Bases-Theory of Modality, the notion of causal necessity – like any other non-logical notion of necessity – can be separated into two conceptual parts: logical necessity + relevant basis, which basis, in the case at hand, is bcausal. The reason for this is the following provable theorem: for all states of affairs q: causal(q) ≡ 1(O(bcausal) ⊃ O(q)). The problem of causal necessity, therefore, reduces to the question which state of affairs, exactly, is bcausal. The rest is a matter of verifying or falsifying certain logically necessary implications, for example, whether 1(O(bcausal) ⊃ (E1 occurs ⊃ E2 occurs)) (in other words: whether E1 causally necessitates E2). If bcausal is a complex state of affairs, perhaps a state of affairs which is so complex that it cannot be grasped completely by any human being, then the verification and falsification of causal necessity is to some extent an epistemological problem, namely, to exactly the extent that we fail to grasp bcausal. If we fail completely to grasp bcausal, if we have no idea which state of affair is bcausal, then, and only then, we are totally in the dark regarding causal necessity.

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Hume has definitely shown the (hardly surprising) fact that bcausal is not t* (= b1); otherwise, the occurrence of E1 would, by itself, with logical necessity imply the occurrence of E2 – which it surely doesn’t. And bcausal cannot be w* (= b10); otherwise, the causation of E2 by E1 would amount to no more than event E1 occurring before event E2 – which is absurd. It is plausible to assume that bcausal is b2 (in the case of the causation of E2 by E1, and in all other cases of causation). But, of course, this does not answer all the questions regarding bcausal. For which state of affairs is b2? We can be said to have a certain grasp of b2, but certainly not a complete grasp.21

5.5

The Solution to the Epistemological Problem of Ontic Modalities

What has been said with respect to causation in the previous section can easily be generalized to provide a solution to the Epistemological Problem of Ontic Modalities stated in Section 1.1: How can we know, in principle, that a true / false sentence of the form “It is possible that A,” or the form “It is necessary that A,” or the form “If A, then B” is true / is false? The answer to this question is this: Finding out whether an assertion of “It is possible that A” is true amounts to checking whether or not we have ¬P(that ¬A, b), where b is the basis of the assertion; finding out whether an assertion of “It is necessary that A” is true amounts to checking whether or not we have P(that A, b´), b´ being the basis of the assertion; finding out whether an assertion of “If A, then B” is true amounts to checking whether or not we have P(that B, conj(b´´, that A)), b´´ being the basis of the assertion. Determining the basis of a given modal assertion – answering the question: which state of affairs is its basis? – is the first hurdle on the way to modal knowledge. Finding out whether or not that basis is an obtaining state of affairs, is – in most, though not in all cases – the second hurdle on the way to modal knowledge. The third hurdle on the way to modal knowledge consists in checking whether a certain relationship of intensional parthood obtains / does not obtain between the basis of the modal assertion 21

In my paper “Laws of Nature – A Skeptical View,” I argue that we do not know which items are the laws of nature if the status of law of nature is entirely objective and not at least partly dependent on us (the actual human beings).

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on the one hand, and the state(s) of affairs the modal assertion is about on the other. Each of the three hurdles on the way to modal knowledge may in certain cases turn out to be difficult to negotiate. But there can hardly be any good reason for advocating wholesale or retail modal skepticism. According to the Bases-Theory of Modality, the heart of (ontic) modal matters is the objective relation of intensional parthood between states of affairs: the relation which is mirrored by the relation of broadly logical implication between sentences. We have: “A” broadly logically (in one word: analytically) implies “B” iff 1(A ⊃ B) iff P(that B, that A). For example: “Anne is smaller than Tom” broadly logically implies “Tom is not smaller than Anne” iff 1(Anne is smaller than Tom ⊃ Tom is not smaller than Anne) iff P(that Tom is not smaller than Anne, that Anne is smaller than Tom).

The Bases-Theory of Modality – taken at its most ambitious – proposes that all ontic modal notions22 are ultimately reducible to an appropriate basis (of necessity, a certain state of affairs) and to the relation of intensional parthood between states of affairs, which relation, it should be noted, is just the relational equivalent of (broadly) logical necessity.23 The proposed reducibility-thesis I hope to have made to some extent plausible in the previous two chapters, but especially, in this chapter, by the development of the Bases-Theory of Conditionals (it being intended as a highly comprehensive theory of ontic conditionals). Further corroboration of the reducibility-thesis will follow in the next chapter, and also (though less visibly) in Chapter 7. If the thesis is true, then the epistemology of ontic modality centers on the following double question: Of which states of affairs p and q can we know that q is / is not an intensional part of p, and in which ways can we obtain this knowledge? Or in other words: Of which states of affairs p and q can we know that the obtaining of p implies / does not imply with logical necessity the obtaining of q, and in which ways can we obtain this knowledge? 22

Here “modal” is taken to imply “non-probabilistic.” Modal probabilistic notions – like “it is in the degree 0.75 necessary that A” – require a different approach. 23 Logical necessity (as expressed by the predicate of logical necessity) and intensional parthood are interdefinable: 1(x) =Def P(x, t*); P(x, y) =Def S(x) ∧ S(y) ∧ 1(that (O(y) ⊃ O(x))). (If the latter definition is to be non-circular, then the background-theory for it can of course not be the mereology of states of affairs, but must be a theory that is formulated in terms of the basic predicates O(x) and 1(x), and the name-forming operator “that A.”)

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Two further ways to formulate the first part of the above double question: Of which states of affairs p and q can we know that disj(neg(p), q) is / is not identical with t*, or in other words: that t* is / is not identical with the state of affairs that (O(p) ⊃ O(q))?

The many linguistically mediated instances where we are certain of knowing that a certain state of affairs is an intensional part of another state of affairs suggests that the posed question is to be answered in the following way: Of very many states of affairs p and q we can know whether q is an intensional part of p, namely, of those states of affairs p and q which are linguistically represented; ascertaining intensional parthood is simply a matter of our understanding of language. But this answer does not give as much extent to our knowledge as we would like to have, and perhaps need. Most states of affairs do not have a linguistic representation (since there are superdenumerably many states of affairs, but only denumerably many names and sentences). This fact notwithstanding, we may take it that there are relationships of intensional parthood also between states of affairs that have no linguistic representation. It is not language that makes it so that one state of affairs is an intensional part of another: even if language had never evolved, it would still be the case that the state of affairs that Mount Everest is at t0 not smaller than Mount Edith Cavell is an intensional part of the state of affairs that Mount Edith Cavell is at t0 smaller than Mount Everest. Moreover, even in some cases where we perfectly understand two sentences, the answer to the question whether the state of affairs expressed by one of the two is an intensional part of the state of affairs expressed by the other is far from perfectly clear. Consider the following two sentences: “Napoleon is alive in 2003” and “Every even number is either a prime or the sum of two primes.” The relationship between language and intensional parthood is accidental, because the relationship between language and states of affairs is accidental. Sentences do express states of affairs, but if there were no language to express them, states of affairs and their relationships of intensional parthood would still be around. Therefore, our knowledge of intensional parthood between states of affairs is not simply a matter of our understanding of language. On the basis of experiencing states of affairs, we judge them to be obtaining – and thereby we also judge them to be logically possible. On the basis of imagining states of affairs, we do not (generally) judge them to be obtaining, but we still judge them to be logically possible. By imagining the state of affairs p without the state of affairs q – i.e., by imagining the

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state of affairs that O(p) ∧ ¬O(q), or in other words: by imagining the state of affairs conj(p, neg(q)) – we judge it to be logically possible that p obtains without q, and hence that q is not an intensional part of q. Experiencing and imagining are our a posteriori methods for gaining knowledge of logical possibilities. These methods are fallible, since neither imagining a state of affairs nor even experiencing it guarantee that the state of affairs is logically possible. Only if imagination or experience grasp a state of affairs p in sufficient intrinsic detail and depth (the presence of which state of the imagination, respectively experience, may not be subjectively decidable), will imagining or experiencing p guarantee its being logically possible.24 (Compare Sections 2.3 and 2.4.) It is fallible to conclude the truth of “p is a logically possible state of affairs” from the truth of “p is experienced/imagined by person X.” It is even more fallible to conclude the truth of “p is not a logically possible state of affairs” from the truth of “person X has not succeeded in experiencing/imagining p even after having made several serious efforts.” The described a posteriori methods are not only fallible, they are also restricted in application, since certainly not all logically possible states of affairs are imaginable, let alone experienceable. Viewed from a certain perspective, gaining knowledge of logical possibility/impossibility can seem very easy. There is only one logically impossible state of affairs: k*. Therefore, finding out whether or not a state of affairs is logically possible amounts to finding out whether or not it is identical with k*. For many states of affairs it is immediately evident that they are not identical – or, on the contrary, identical – with k* (though it is difficult to say which of our epistemic faculties makes this identity or nonidentity immediately evident; it is certainly not the imagination). However, for all too many states of affairs, their identity or non-identity with k* is not immediately evident (i.e., not evident for us human beings). In those cases, there is yet another epistemic tool to work with in order to ascertain logical possibility, respectively impossibility: 24

Note that even a completely detailed experience of p does not guarantee that p obtains. One may, for example, have a completely detailed visual experience of p without p obtaining (in other words: one may have a completely detailed visual hallucination of p). Seeing p, indeed, does guarantee that p obtains and, a fortiori, that it is logically possible; but having a visual experience of p is not seeing p. (Seeing a state of affairs, on the other hand, cannot be an epistemic lever for its obtaining or its logical possibility, since one must already know that a state of affairs obtains and that it is logically possible before one can know that one is seeing it.)

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A very common method for finding out whether a state of affairs p is identical with k* is the following. One starts with assuming a sentence A that expresses p. Then, one proceeds to draw out the broadly logical (analytical) consequences of A. If, in the process of doing so, both B and ¬B have been concluded for some sentence B, then it has been shown a priori: k* = that A = p, and therefore: p is not logically possible. If, however, in the process of drawing out the broadly logical consequences of A, B and ¬B have not been concluded for any sentence B and there is yet room for drawing out further consequences, then the method is, so far, inconclusive: it may be that B and ¬B will yet be concluded, sooner or later; it may also be that B and ¬B will never be concluded, no matter how long the deductive process is continued. If B and ¬B will never be concluded, then there are two possible reasons for this: p (= that A) is not identical to k*, or p is, in fact, identical with k*, but the deductive machinery used for drawing out the broadly logical consequences of A is too weak for deducing a contradiction from it. If one has diligently tried to deduce a contradiction from a sentence A that expresses p, and the effort, though serious and prolonged, has failed, and if one does not see how strengthening the means of deduction will lead to a different outcome, then one can take this as fallible a posteriori evidence for p being not identical with k*, and hence for p being logically possible. It is easy to adapt the deductive method just described to the case of determining whether state of affairs q is an intensional part of state of affairs p, in view of the following fact (provable in the mereology of states of affairs): P(q, p) ≡ conj(p, neg(q)) = k*. If A expresses p and C expresses q, then A ∧ ¬C expresses conj(p, neg(q)), and A ∧ ¬C will be the assumption from which, in applying the deductive method, one will attempt to deduce a contradiction. If one is successful in this attempt, one has shown a priori: k* = that (A ∧ ¬C) = conj(p, neg(q)), and therefore: P(q, p).

The range of the reductio-ad-absurdum method can be extended by considering that not only single sentences express states of affairs, but also sets of sentences. In fact, even infinite sets of sentences express states of affairs: Let M be an infinite set of sentences; then CONJx∃s(s ∈ M ∧ s expresses x) is the state of affairs that is expressed by M. Thus, besides starting the reductio-ad-absurdum method for determining whether a state of affairs is identical with k* by assuming a single sentence A expressing a state of affairs p, one might also start that method by assuming a whole set

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of sentences M that expresses a state of affairs q.25 This additional option constitutes an extension of the range of the reductio-ad-absurdum method, since more states of affairs can be expressed by sets of sentences in which they are not named than by single sentences in which they are not named. The considerations in this section show that even though all ontic modalities are reducible, according to the Bases-Theory of Modality, to the broadly logical modalities (logical possibility, impossibility, necessity), this does not solve all epistemic difficulties regarding ontic modalities. For our epistemic hold on logical modalities is somewhat tenuous. What we can be absolutely certain of, however, is this: Finding out about logical possibility (impossibility, necessity) is not a matter of inspecting in our mind’s eye (as David Lewis would have it) as best we can spatiotemporal totalities which differ from the spatiotemporal totality that we belong to (and which differ regarding actuality from the spatiotemporal totality that we belong to only by our not belonging to them). Finding out about logical possibility (impossibility, necessity) is a matter of ascertaining intrinsic relationships of intensional containment between states of affairs – as best we can. 5.5.1

But There Is a Puzzle Concerning Logical Modalities

As best we can. But the best we can do in this respect may not be good enough. Our understanding of logical modalities is not very firm. This is strikingly illustrated by the following bewildering situation: Let M0 be the set of axioms that are assumed in a standard system S of axiomatic (pure) set theory. Let M1 be the set of axioms that are assumed in S + the continuum hypothesis. Let M2 be the set of axioms that are assumed in S + the negation of the continuum hypothesis. Consider CONJx∃s(s ∈ M0 ∧ s expresses x), CONJx∃s(s ∈ M1 ∧ s expresses x), and CONJx∃s(s ∈ M2 ∧ s expresses x), or in short: m0, m1, and m2. We have strong a posteriori evidence for: (1) m0 ≠ k*. Moreover, it has been proven (by Gödel and by Cohen): (2) m0 ≠ k* ⊃ m1 ≠ k*, and (3) m0 ≠ k* ⊃ m2 ≠ k*. Since S is plausibly regarded as a broadly logical theory, we also have: (4) m0 ≠ k* ≡ m0 = t*. Finally, we doubtlessly have: (5) conj(m1, m2) = k*. From these

25

But of course one will only then have a deductive grasp on M that is comparable with the deductive grasp one has on a single sentence if M can be specified in some finite way, for example, by a finite set of schemata.

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five assumptions one can deduce: m0 = t* ∧ (m1 ≠ k* ∧ m1 ≠ t*) ∧ (m2 ≠ k* ∧ m2 ≠ t*). Proof: From (1) and (4) we obtain: m0 = t*. From (1) and (2) we obtain: m1 ≠ k*. Assume m1 = t*; hence by (5): conj(t*, m2) = k*, and therefore: m2 = k* – contradicting (1) and (3). From (1) and (3) we obtain: m2 ≠ k*. Assume m2 = t*; hence by (5): conj(m1, t*) = k*, and therefore: m1 = k* – contradicting (1) and (2).

And this result means the following: While S is a logically non-contingent theory, both S + CH and S + ¬CH are logically contingent theories: there are possible worlds (worlds being maximal-consistent states of affairs, as was determined in Chapter 3) in which they are true, and possible worlds in which they are not true. S + CH is true in any world w iff M1 is true in w iff m1 obtains in w iff P(m1, w). Since m1 is not identical with k*, there is a world w´ such that P(m1, w´); and since m1 is not identical with t*, there is a world w´´ such that ¬P(m1, w´´). Hence S + CH is true in w´ and not true in w´´. Entirely analogous considerations hold for S + ¬CH.

But this contradicts our intuition that S + CH, and also S + ¬CH, is necessarily true if it is true (i.e., true simpliciter: true in w*), or in other words, our intuition that the following is true: (6) P(m1, w*) ⊃ m1 = t*, and P(m2, w*) ⊃ m2 = t*. One might hope to escape the contradiction between (1) – (5) and (6) by simply accepting the consequence that neither S + CH nor S + ¬CH are true, i.e., that ¬P(m1, w*) ∧ ¬P(m2, w*). But, in fact, one cannot accept this consequence, whether or not one wishes to accept it: Since w* is a maximal-consistent state of affairs (see Chapter 3) and m1 = conj(m0, ch) and m2 = conj(m0, neg(ch)) (ch being the state of affairs that is expressed by the continuum hypothesis), either m1 or m2 must be an intensional part of w* (because m0 – being identical with t* – is an intensional part of w* and because either ch or neg(ch) is an intensional part of w*). Therefore, at least one statement out of the statements (1) – (6) must be given up. But which one? Each of the six statements is very well argued for. This is also true of (6). The continuum hypothesis CH and its negation ¬CH are as impeccable sentences of pure set theory as any other sentences of pure set theory: they can be formulated by employing exclusively the minimal set-theoretical vocabulary. And therefore ch and neg(ch) should not be treated differently from other states of affairs expressed by sentences of pure set theory. That is, one should assume: (7) ch = t* ∨ neg(ch) = t*. But then it follows inexorably that (6) is true.

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Proof: Suppose P(m1, w*), that is: P(conj(m0, ch), w*). If ch ≠ t*, then according to (7): neg(ch) = t*, and therefore: ch = k*. But if ch were k*, P(conj(m0, ch), w*) could not be true (contrary to the initial supposition), since w* ≠ k*. Therefore: ch = t*, and consequently – since m0 = t* – conj(m0, ch) = t*, that is: m1 = t*. Suppose P(m2, w*), that is: P(conj(m0, neg(ch)), w*). If neg(ch) ≠ t*, then according to (7): ch = t*, and therefore: neg(ch) = k*. But if neg(ch) were k*, P(conj(m0, neg(ch)), w*) could not be true (contrary to the initial supposition), since w* ≠ k*. Therefore: neg(ch) = t*, and consequently – since m0 = t* – conj(m0, neg(ch)) = t*, that is: m2 = t*.

Thus, instead of the contradiction between (1) – (5) and (6), one can also consider the contradiction between (1) – (5) and (7). In fact, it makes no difference which of the contradictions one considers, since (6) and (7) can be proven to be equivalent to each other in the mereology of states of affairs, given m1 = conj(m0, ch), m2 = conj(m0, neg(ch)), (1), and (4). What, then, should one do about this situation? Should one merely deny (7)? This is certainly sufficient for avoiding contradiction. But it seems entirely arbitrary to deny (7) – as long as one does not change anything else and, in particular, keeps holding on to (4). If (7) is denied, then (4) must be denied, too (and vice versa). To see this, consider that (4) is equivalent in the mereology of states of affairs to “m0 = t* ∨ m0 = k*” (since “m0 ≠ t* ∨ m0 ≠ k*” is a theorem of that mereology), and that (7) is equivalent in the mereology of states of affairs to “ch = t* ∨ ch = k*” (since “neg(ch) = t* ≡ ch = k*” is a theorem of that mereology). In view of the fact that CH is of one kind with the sentences in M0 (that is, CH and the sentences in M0 are sentences of pure set theory), denying “ch = t* ∨ ch = k*” cannot be more plausible than denying “m0 = t* ∨ m0 = k*.” But if (4) is denied (together with (7)), then this makes standard (pure) set theory – as axiomatized in S – a logically contingent theory: a theory that is true in some worlds and not true in others, which is a result that is totally alien to prevailing habits of thought. Suppose (4) is indeed denied. Then S turns out to be what, in Kantian terminology, one would call a synthetic theory. Since we presumably know a priori that S is true, S, therefore, is revealed to be a synthetic a priori theory, a perfectly respectable piece of general metaphysics. Or do we, perhaps, not know a priori that S is true? After all, nobody has ever even proven that S is consistent; we merely have a posteriori evidence for the consistency of S (though fairly strong evidence). Since we do not know a priori that S is consistent, how can we know a priori that S is true? But if we do not know a priori that S is true, how, then, do we know

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at all that S is true? Observation and experiment, certainly, have no say in this matter. Could it be that we do not know that S is true, that we merely believe it to be true for the reason that it helps us to do mathematics in a fruitful, theoretically unified way? If so, how exactly does S differ in epistemological status from a theological system S´ that we also do not know to be true, but believe to be true for the reason that it helps us to live an essentially happy life?26 The difficulty of finding a satisfactory answer to this last question (without any unwarranted shortcuts, like taking for granted that set theory is rational, while theology is not) may make one strongly inclined to reassume (4), and thereby to reassume that S is a broadly logical theory. But if (4) is reinstated, so must be (7), and therewith returns the perplexity – caused by antinomy – which one wanted to escape from. What else can one do? There are two not entirely implausible possibilities: (i) One might deny (5) on the following grounds: If S is combined with ¬CH the resulting theory treats of different entities than are treated in the theory which results if S is combined with CH (say, S + CH treats of sets*, while S + ¬CH treats of sets#). In consequence, there is no logical conflict between S + CH and S + ¬CH, and therefore: conj(m1, m2) ≠ k* – the negation of (5). Note that according to (i) the state of affairs expressed by S + CH + ¬CH is not the conjunction of the state of affairs m1 (expressed by S + CH) and the state of affairs m2 (expressed by S + ¬CH). For S + CH + ¬CH certainly expresses k*. Note also that according to (i) the following is not true: m1 = conj(m0, ch) ∧ m2 = conj(m0, neg(ch)). For “conj(m1, m2) = k*” is a logical consequence of this.

In response it must be pointed out that certainly not in all cases where we can consistently add either B or ¬B to a theory T, we irenically assume that T + B is logically consistent with T + ¬B. On the contrary, in the vast majority of such cases, we assume that T + B is logically inconsistent with T + ¬B, exactly as syntactical appearance suggests. In other words, we assume that no matter whether B or ¬B is added to T, this does nothing to change the meaning of B; thus, if ¬B is added to T, the state of affairs that is expressed by ¬B is exactly the negation of the state of affairs that is expressed by B if B is added to T. Why should we see things differently in the case of S + CH and S + ¬CH? Because doing so is the only way to get out of the above-described antinomy? It certainly seems to be a somewhat 26

In fact, for the founder of set theory, Georg Cantor, set theory and theology were not so very different.

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desperate measure, notwithstanding the fact that it has a famous precedent everybody seems to be satisfied with: the reconciliation, by semantic separation, of non-Euclidean geometry with Euclidean geometry. (ii) One might reject (2) (respectively (3)) on the following grounds: perhaps Gödel’s result (respectively, Cohen’s) does not in fact prove “m0 ≠ k* ⊃ m1 ≠ k*” (respectively, “m0 ≠ k* ⊃ m2 ≠ k*”). How could this be? Gödel proved that if S is consistent, then S + CH is also consistent. This may not be translatable into “m0 ≠ k* ⊃ m2 ≠ k*,” or in other words: “m0 ≠ k* ⊃ conj(m0, ch) ≠ k*.” Consider an analogy: Let M0´ – the set of all predicate-logical formulas of the form F(a), where a is a name-constant – be the set of the axioms of T. Consider the theories T + ∀xF(x) and T + ¬∀xF(x). The corresponding sets of axioms are M1´ (= M0´∪{∀xF(x)}) and M2´ (= M0´∪{¬∀xF(x)}). One can prove: if no contradiction can be predicate-logically deduced from T, then no contradiction can be predicatelogically deduced from T + ∀xF(x), and none from T + ¬∀xF(x); or equivalently, in other words: if T is made true by some predicate-logical interpretation, then T + ∀xF(x) is made true by some predicate-logical interpretation, and so is T + ¬∀xF(x). But suppose now that the predicatelogical language L has an intended interpretation: L is to be used under all circumstances for speaking about the real numbers and nothing else, and the predicate “F(x)” of L is supposed to express the property of being named by a name-constant of L. Given this interpretation, m0´ (=Def CONJx∃s(s ∈ M0´ ∧ s expresses x)) is clearly not identical with k*, since all sentences in M0´ are true if they are understood as intended. According to the intended interpretation, “F(a)” means – whatever nameconstant a is being considered – as much as “a is a real number that is named by a name-constant of L.” Clearly, according to the intended interpretation, every sentence of L that has the form F(a) is true. In consequence, the state of affairs expressed (according to the intended interpretation) by any sentence in M0´ is an intensional part of w*. Hence also the conjunction of all states of affairs that are expressed by some sentence in M0´ is an intensional part of w*. Hence m0´ is an intensional part of w*, and therefore: m0´ ≠ k*.

The sentence “∀xF(x),” however, is not true if understood as intended. For according to the intended interpretation, “∀xF(x)” means as much as “Every real number is named by a name-constant of L,” and this is false – false, in fact, in every possible world. Therefore, the state of affairs which (according to the intended interpretation) is expressed by “∀xF(x)” is iden-

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tical with k*. Consider now the state of affairs m1´ expressed (in the intended interpretation) by M1´. Clearly, m1´ = conj(m0´, k*) = k*. Hence, while m0´ is different from k*, m1´ is not – in spite of the fact that M1´ is (provably) just as consistent as M0´. Thus, if – one the basis of this fact – we had assumed (2´) m0´ ≠ k* ⊃ m1´ ≠ k* (in analogy to (2): m0 ≠ k* ⊃ m1 ≠ k*), we would have been wrong. The point to be made is this: the consistency proven in a proof of consistency (be it absolute or relative consistency) is always the consistency according to the logic or model theory used in the proof. That consistency might not be the consistency that is relevant from the point of view of the intended interpretation of the theory whose consistency is proven. If T is the theory, and s(T) the state of affairs expressed by T according to the intended interpretation of it, then the consistency of T that is relevant from the point of the intended interpretation of T is this: that s(T) be different from k*, or in other words: that s(T) be a logically possible state of affairs. It may not be the case that s(T) is a logically possible state of affairs, although according to the logic used no contradiction is deducible from T, and although according to the model theory used there is an interpretation of T in which it is true. The state of affairs m2´, on the other hand, which is expressed (in the intended interpretation) by M2´ (=M0´∪{¬∀xF(x)}) is t*, since the state of affairs expressed by “¬∀xF(x)” (in the intended interpretation) is t* and since the state of affairs expressed by M0´ is also t*. The state of affairs expressed by M0´ is t*, since (in the intended interpretation) every sentence in M0´ expresses t*. Suppose, for some name-constant a of L, “F(a)” does not express t*. Hence “F(a)” is not true in every possible world. Hence “a is a real number named by a name-constant of L” is not true according to some possible world w. But if so, then L (since “a” is a name-constant of L and names something according to w) is not used under all circumstances for speaking about the real numbers and nothing else – contrary to the intended interpretation of L.

Thus, for the states of affairs m0´, m1´, m2´ the following statements are true (on the basis of the intended interpretation of L): (1´) m0´ ≠ k*; (3´) m0´ ≠ k* ⊃ m2´ ≠ k*; (4´) m0´≠ k* ≡ m0´ = t*; (5´) conj(m1´, m2´) = k*; (6´) P(m1´, w*) ⊃ m1´ = t*, and P(m2´, w*) ⊃ m2´ = t*; (7´) s(“∀xF(x)”) = t* ∨ neg(s(“∀xF(x)”)) = t*, s(“∀xF(x)”) being the state of affairs expressed by “∀xF(x)” (in the intended interpretation). The one statement out of (1´) – (7´) which is not true is (2´): m0´ ≠ k* ⊃ m1´ ≠ k*. For letting (3´) (instead of (2´)) be the one statement out of (1´) – (7´) which is not true, con-

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sider the following alternative intended interpretation of L: L is to be used under all circumstances for speaking about the name-constants of L and nothing else, each name-constant rigidly naming itself, and “F” is supposed to express the property of being named by a name-constant of L. Might it not be the same way with m0, m1, m2 and the statements (1) – (7) as it is with m0´, m1´, m2´ and the statements (1´) – (7´): that either (2) or (3) is the one statement in the series (1) – (7) which is not true, just like (2´) or (3´) is the one statement in the series (1´) – (7´) which is not true, though initially it seemed as if both (2) and (3), (2´) and (3´) have to be accepted (for similar reasons)? – It might be the same way. However, while it is easy to find an intended interpretation of L that makes T + ¬∀xF(x) true in all possible worlds, or an alternative intended interpretation of L that makes T + ∀xF(x) true in all possible worlds, it is not so easy to find an intended interpretation of the set-theoretical language that makes S + ¬CH true in all possible worlds, or an alternative intended interpretation that makes T + CH true in all possible worlds. The reason for this is that S – standard axiomatic set theory, taken by itself – already has an intended interpretation that must be respected. That interpretation, however, is silent about CH (if S is consistent), and no natural mere extension of that interpretation suggests itself. Even logical possibility and necessity for states of affairs are not the epistemologically easy matters they seem to be. This much, at least, should be clear at the end of this section and this chapter.

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Other Theories of Conditionals versus the Bases-Theory of Conditionals

The title of this chapter well explains what it is about. But completeness regarding its subject matter was not to be reached or even approached, and there is an exception to its thematic tenor (similar to the one in the previous chapter; cf. Section 5.5 and 5.5.1): since the concept of logical necessity is central to my treatment of states of affairs and their ontic modalities, the last section of this chapter (Section 6.6) is dedicated to yet another scrutiny of that concept. (While presupposing what has gone before, the thematic focus of the remaining two chapters of the book will no longer be on states of affairs, but on properties.)

6.1

Are Indicative Conditionals (Always) Material Implications?

In his paper “Indicative Conditionals,” Robert Stalnaker formulates the following puzzle (for which he then offers a solution): ‘Either the butler or the gardener did it. Therefore, if the butler didn’t do it, the gardener did.’ This piece of reasoning – call it the direct argument – may seem tedious, but it is surely compelling. Yet if it is a valid inference, then the indicative conditional conclusion must be logically equivalent to the truth-functional material conditional [the converse reasoning being logically valid uncontroversially?], and this conclusion has consequences that are notoriously paradoxical. (Ibid., p. 136; Stalnaker’s italics.)

Stalnaker’s strategy for solving the puzzle is this: I will argue that, although the premiss of the direct argument does not semantically entail its conclusion, the inference is nevertheless a reasonable inference. My main task will be to define and explain a concept of reasonable inference which diverges from semantic entailment, and which justifies this claim. (Ibid., p. 137; Stalnaker’s italics.)

6 Other Theories of Conditionals versus the Bases-Theory of Conditionals

From the point of view of the Bases-Theory of Conditionals, no such momentous steps as Stalnaker suggests are necessary. An indicative conditional is sometimes logically equivalent to the corresponding truthfunctional material implication, namely, if an utterance of it has the basis b10 (= w*), as in the example Stalnaker adduces; and sometimes it is not logically equivalent to the corresponding material implication, namely, if an utterance of it has an obtaining basis that is different from b10, as in the following example of someone saying: “The butler did it. Therefore, if the butler didn’t do it, the gardener did.” Since this inference is not valid in the implied context of utterance, the conditional “If the butler didn’t do it, the gardener did” cannot have the basis b10 in that context (for otherwise the inference would be valid). An interesting alternative explanation – also within the Bases-Theory of Conditionals – of the validity of inferring “If the butler didn’t do it, the gardener did” from “Either the butler or the gardener did it” is this: the utterance of the premise states the basis of the utterance of the conditional which is the conclusion. If so, then the utterance of the conclusion semantically amounts to an utterance of “Either the butler or the gardener did it, and P(that the gardener did it, conj(that either the butler or the gardener did it, that the butler didn’t do it)),”1 which, of course, is a logical consequence of “Either the butler or the gardener did it,” since “P(that B, conj(that (A ∨ B), that ¬A))” reflects the logical truth “(A ∨ B) ∧ ¬A ⊃ B,” no matter which sentences A and B are being considered.2 The dependence on variable bases that indicative conditionals exhibit when uttered prohibits a general semantical identification of such conditionals with material implications. Many uttered indicative conditionals have a nexus of (genuine) necessity to them, which the corresponding material implications (material conditionals) lack. This nexus of necessity becomes apparent in the following way: A material implication “A ⊃ B” is denied by asserting “A ∧ ¬B,” and an indicative conditional “If A, then B” can be denied in the same way. But at least some indicative conditionals “If A, then B” can also be 1

It is also correct to say that the utterance of the conclusion semantically amounts to an utterance of “P(that the gardener did it, conj(that either the butler or the gardener did it, that the butler didn’t do it))” – but only if the obtaining of the basis of the utterance of the conclusion (i.e., of the state of affairs that either the butler or the gardener did it) is presupposed instead of being considered a part of what is being asserted. 2 In order to capture “either, or” strictly speaking, replace “∨” by “≡ ¬.” Then “P(that B, conj(that (A ≡ ¬B), that ¬A))” reflects the logical truth “(A ≡ ¬B) ∧ ¬A ⊃ B.”

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denied, according to the semantic content they have in certain utterances of them, by merely asserting “It might be that (A ∧ ¬B),” without also asserting “A ∧ ¬B” – which is not a way to deny “A ⊃ B.” It should also be noted that there is no logically consistent way to deny both “A ⊃ B” and “A ⊃ ¬B”; for if both are denied, then both “B” and “¬B” are asserted. But in many cases of utterance there is a consistent way to deny both “If A, then B” and “If A, then ¬B”: by asserting both “It might be that (A ∧ ¬B)” and “It might be that (A ∧ B).” Proponents of the view that, as far as truth-conditions are concerned, indicative conditionals are material implications3 need to interpret the assertion of “It might be that (A ∧ ¬B),” when made in opposition to an assertion of “If A, then B” (without also asserting “A ∧ ¬B”), not as a denial of its truth (for the truth of asserting “If A, then B” could not be denied in that way if indicative conditionals were – as far as truth conditions are concerned – material implications), but merely as a denial of its assertability. According to them, “It might be that A ∧ ¬B,” if maintained in opposition to “If A, then B,” must be understood subjectively: as meaning either (1) that the subjective probability of the state of affairs that A ∧ ¬B is significantly higher than zero, or (2) that the conditional subjective probability of the state of affairs that ¬B, given the state of affairs that A, is significantly higher than zero.4 “It might be that A ∧ ¬B,” when accorded the subjective interpretation (1), certainly betokens, if true, the unassertability of “If A, then B.” In addition, many philosophers hold that “It might be that A ∧ ¬B,” when accorded the subjective interpretation (2), betokens, if true, the unassertability of “If A, then B” even if the subjective probability p(A ∧ ¬B) is, though still higher than 0, not significantly higher than zero. The reason they adduce for assuming such a position is this: the assertability of “If A, then B” invariably requires a conditional subjective probability p(B/A) not signifi3

David Lewis favors the view in Counterfactuals, p. 72, footnote. He reaffirms it in “Probabilities of Conditionals and Conditional Probabilities,” and also in the later postscript to (a reprint of) that paper. Another well-known proponent of the view that indicative conditionals are material implications (as far as truth-conditions are concerned) is Frank Jackson; see his “On Assertion and Indicative Conditionals.” Assigning to indicative conditionals and material implications the same linguistic meaning (in addition to letting them have the same truth-conditions) is, however, out of the question. 4 Note that (1) implies (2), for 0 < p(A), since p(¬B/A) ≥ p(A ∧ ¬B), for 0 < p(A).

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cantly lower than 1, as Ernest Adams has argued.5 This seems plausible.6 But it is far from plausible that “It might be that A ∧ ¬B” does not state an alethic (but states merely a doxastic) possibility if it is held against “If A, then B.” If somebody asserts, “If this radium atom exists at t1, then it decays within a year from t1,” and I reply, “No. It might be that this radium atom exists at t1 and does not decay within a year from t1,” am I then merely claiming that my subjective probability for this radium atom to decay within a year from t1, given that it exists at t1, is significantly less than 1? I do not think so.

6.2

The Assertability of Indicative Conditionals

In this section, in order to let p(B/A) be defined in the usual way, it is (where not mentioned as a supposition, implicitly) assumed that p(A) > 0. It is also assumed that metalinguistic “if, then” has merely the force of material implication. Let us suppose, then, that p(If A, then B) is not significantly lower than 1, but that p(B/A) is significantly lower than 1. Let us also suppose: (i) “If A, then B” is assertable if p(If A, then B) is not significantly lower than 1, and (ii) “If A, then B” is not assertable if p(B/A) is significantly lower than 1.7 5

See Ernest Adams, “The Logic of Conditionals” and “Probability and the Logic of Conditionals.” David Lewis accepts Adams’s views in “Probabilities of Conditionals and Conditional Probabilities,” p. 134. 6 But it is not entirely correct, because for some assertable utterances of indicative conditionals there is no conditional subjective probability at all (let alone one that is not significantly lower than 1). One sometimes hears utterances of conditionals like the following: “If George is born in 1945, he is 58 in 2003. True. But I am sure he was not born in 1945.” Here the relevant conditional subjective probability p(B/A) is not defined, since the probability of the antecedent, A, is 0 for the speaker. Nevertheless, there seems to be nothing wrong with his asserting the conditional. 7 (ii) might be called “Adams’s Thesis.” Note that the converse of (ii), stating that “If A, then B” is assertable if p(B/A) is not significantly lower than 1, is not in general plausible: In many cases p(¬A ∨ B/A) is not significantly lower than 1 (in fact, in

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But these assumptions lead to a straight contradiction: “If A, then B” turns out to be both assertable and not assertable. The natural reaction to this “antinomy” is to deny that the supposed situation can occur: it simply cannot be the case that p(If A, then B) is not significantly lower than 1, whereas p(B/A) is significantly lower than 1. This sounds plausible, and certainly we would not wish to deny that p(B/A) = 1 if p(If A, then B) = 1. (The Bases-Theory of Conditionals complies: according to it, it is provable that p(B/A) = 1 if p(If A, then B) = 1; see below.) But David Lewis has shown that there are no truth conditions for “If A, then B” such that – with sufficient generality – p(If A, then B) = p(B/A), where p is subjective probability.8 This, of course, greatly reduces the plausibility of what I have called the “natural reaction,” and we had better abstain from it. Should we give up (i)? Perhaps. But can we give up (i) if we adhere to the Bases-Theory of Conditionals? Suppose that “If A, then B” is uttered by a person P, and that it is interpreted, as uttered, according to the BasesTheory of Conditionals. Suppose also that the obtaining of the basis of utterance of the conditional “If A, then B” is expressed by “O(b).” Then, according to the Bases-Theory of Conditionals, the utterance of “If A, then B” is true in the context of utterance under consideration if, and only if, “O(b) ∧ 1(O(b) ∧ A ⊃ B)”9 is true in that context (where 1 stands for logical necessity). Therefore: p(If A, then B) = p(O(b) ∧ 1(O(b) ∧ A ⊃ B)), where p is (now) the subjective probability of person P in the context under consideration. It is a consequence of this equation (according to standard logic of probability) that p(B/A) = 1 if p(If A, then B) =1. Another consequence is this: p(If A, then B) is either p(O(b)) or 0, depending on whether 1(O(b) ∧ A ⊃ B) is true or not. If 1(O(b) ∧ A ⊃ B) is true, then p(1(O(b) ∧ A ⊃ B)) every case in which p(B/A) is not significantly lower than 1); but “If A, then ¬A ∨ B” is certainly is not made an assertable conditional merely by this fact. Though the converse of (ii) is not plausible, David Lewis by using the vague phrase “assertability [for conditionals] goes by conditional probabilities” for his own position (which he takes to be identical with Adams’s; see “Probabilities of Conditionals and Conditional Probabilities,” p. 134 and passim) certainly suggests that he believes otherwise. 8 See “Probabilities of Conditionals and Conditional Probabilities,” p. 135. 9 In order to read this correctly, remember that “∧” binds more strongly than “⊃.”

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= 1 (according to standard rationality principles for subjective probability), and therefore: p(O(b) ∧ 1(O(b) ∧ A ⊃ B)) = p(O(b)). But if 1(O(b) ∧ A ⊃ B) is not true, then p(1(O(b) ∧ A ⊃ B)) = 0 (according to standard rationality principles for subjective probability), and therefore: p(O(b) ∧ 1(O(b) ∧ A ⊃ B)) = 0. This suggests that there are at least two factors that are needed for the assertability of “If A, then B” (by the speaker P, in the context of utterance): (a) p(O(b)) is not significantly lower than 1, (b) p(If A, then B) is not 0. And it is easily seen that p(If A, then B) is not significantly lower than 1 if, and only if, both (a) and (b) are fulfilled. Suppose p(If A, then B) is not significantly lower than 1; hence p(If A, then B) is not 0, and hence p(If A, then B) = p(O(b)) [since p(If A, then B) is either p(O(b)) or 0]. Therefore: (a) p(O(b)) is not significantly lower than 1, and (b) p(If A, then B) is not 0. Suppose (a) and (b); hence p(If A, then B) is not significantly lower than 1 [since p(If A, then B) is either p(O(b)) or 0].

But then, if (a) and (b) are the only factors needed for the assertability of “If A, then B,” (i) is vindicated within the Bases-Theory of Conditionals – and we have a serious problem. For the only still remaining premise that could be given up in order to escape from the above derivation of a contradiction is (ii) – and who would want to give up (ii) in the face of the evidence Adams has adduced in its support? Therefore, there must be another factor, (c), needed for the assertability of “If A, then B,” a factor in addition to (a) and (b), or in other words: in addition to p(If A, then B) being not significantly lower than 1. This additional factor is neglected in (i), which neglect renders it false, and the falsity of (i), in turn, allows us to escape from the difficulty we are in. But what is this additional factor? It is this: (c) b is a basis that is regarded (by the speaker P) as relevantly connecting the state of affairs that A with the state of affairs that B. What it means for a basis b to be regarded as relevantly connecting that A with that B is best seen by considering a basis that connects that A with that B, but is not regarded as doing this relevantly. Let “A” and “B” represent two sentences that say entirely heterogeneous things (for example, “U.M. is born in 1969” and “London has less than 4 million inhabitants in 1956”), and let “b´” designate the state of affairs that (¬A ∨ B). As a matter of fact, “¬A ∨ B” is true, and therefore: b´ is an obtaining state of affairs and the sentence “O(b´)” is true. Hence the sentence “O(b´) ∧

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1(O(b´) ∧ A ⊃ B)”10 is also true. Therefore, if “If A, then B” is uttered (in the actual world) with the basis b´ at a time t by a speaker P´, then this utterance of “If A, then B” is true. But “If A, then B” is not likely to be uttered with the basis b´ by any speaker at any time as an assertion, even if p´(O(b´) ∧ 1(O(b´) ∧ A ⊃ B)) = 1 (for the speaker P´) at time t. The basis b´ connects that A with that B, but b´ is certainly not likely to be considered as connecting the two states of affairs relevantly. And if it isn’t considered by the speaker P´ as connecting those states of affairs relevantly, then this blocks the assertability of “If A, then B” (for the speaker, in the given context), even though all other conditions of assertability are fulfilled. In Section 6.1, we have seen that a basis of the form that (C ∨ D) is sometimes regarded as relevantly connecting that ¬C and that D (see the inference involving the butler and the gardener that is analyzed in Section 6.1). Nor is it invariably the case that a basis of the form that (¬C ∨ D) is not regarded as relevantly connecting that C with that D. Consider: “Either he is not smart, or he does not work hard enough. Therefore, if he is smart, then he does not work hard enough” (said the father to the mother when puzzling over the unsatisfactory scholastic performance of their son). Moreover, even given that “A” stands for “U.M. is born in 1969,” and “B” for “London has less than 4 million inhabitants in 1956,” one can easily imagine not entirely unlikely circumstances in which that (¬A ∨ B) would, in fact, relevantly connect that A with that B. Suppose there is a book of two thousand pages, compiled by an eccentric millionaire, called “The Book of Truths,” which is a collection of miscellaneous irrelevancies, all of which, however, are certainly true and known to be so. Entry no. 28574 in that book happens to be this: “Either U.M. is not born in 1969 or London has less than 4 million inhabitants in 1956.” (Make up your own story that explains how this sentence got into “The Book of Truths.”) Suppose now that there is a reader of “The Book of Truths” who has no idea who is U.M., and no idea how many millions of inhabitants London had in 1956, but who knows that all sentences in “The Book of Truths” are true and happens to have just read entry no. 28574. Such a one is likely to assert in a conversation on the next day that happens to touch the subject of the number of inhabitants London had in 1956: “If U.M. is born in 1969, then Here are two other formulations of what is stated by that sentence: “P(that (¬A ∨ B), w*) ∧ P(that B, conj(that (¬A ∨ B), that A))” and “(¬A ∨ B) ∧ 1((¬A ∨ B) ∧ A ⊃ B).” 10

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London has less than 4 million inhabitants in 1956” – and doing so he would be entirely within his rights (as he could easily make evident to his startled conversation partners). Summing up what can be determined within the Bases-Theory of Conditionals concerning the assertability of indicative conditionals, we have: Assertability of indicative conditionals Let “If A, then B” be uttered with the basis b by a rational speaker P, whose subjective probability function in the context of utterance is designated by “p().” Then: “If A, then B” is assertable (its utterance is a permissible assertion) if, and only if, (a) p(O(b)) is not significantly lower than 1, (b) p(If A, then B) is not 0, and (c) b is regarded (by P) as relevantly connecting the state of affairs that A with the state of affairs that B. From the point of view of the principle of assertability just stated, it is clear why assumption (i) above is not correct in all cases. According to that principle, it is always necessary, but not always sufficient for the assertability of “If A, then B” that p(If A, then B) is not significantly lower than 1. This is so, because the statement “p(If A, then B) is not significantly lower than 1” is equivalent – according to the Bases-Theory of Conditionals – with the conjunction of the conditions (a) and (b) (see above), but not with the conjunction of the conditions (a), (b), and (c). From the point of view of the stated principle of assertability, it is also clear why assumption (ii) above (“Adams’s Thesis”) is correct. If “If A, then B” is to be assertable (by the speaker, in a given context), then conditions (a), (b), and (c) must be fulfilled for some basis b, and then p(B/A) is bound to be not significantly lower than 1. To see this, consider again our example: If the speaker is fairly convinced that either U.M. is not born in 1969 or London has less than 4 million inhabitants in 1956 (fulfillment of (a)), and regards this fact as relevantly connecting the state of affairs that U.M. is born in 1969 with the state of affairs that London has less than 4 million inhabitants in 1956 (fulfillment of (c)), then p(London has less than 4 million inhabitants in 1956/U.M. is born in 1969) will certainly be not significantly lower than l. For this result, it must, of course, be presupposed that the relevant conditional probability is defined, which is guaranteed by supposing that the speaker does not exclude that U.M. is born in 1969. But it should be

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noted that the above principle of assertability for indicative conditionals sustains – as is desirable – the assertability of the conditional that was exhibited in footnote 6, even if that conditional is combined with the sequel also exhibited in that footnote: “If George is born in 1945, he is 58 in 2003. True. But I am sure he was not born in 1945.” There is nothing wrong with taking an utterance of the conditional “If George is born in 1945, he is 58 in 2003” to be a permissible assertion – even if the very next assertion made by the speaker P´´ is “But I am sure he was not born in 1945” – if, and only if, conditions (a), (b), and (c) are fulfilled for that conditional (relative to the given context of utterance). Suppose, then, that the basis b´´ of the utterance is expressed by the following statement: “George has not died in or before 2003.” Suppose the speaker is rational and completely convinced – rightly – that the state of affairs expressed by this sentence obtains. Clearly, the statement “George has not died in or before 2003 and is born in 1945 ⊃ he is 58 (years old) in 2003” is a (broadly) logical truth, and hence is accorded the subjective probability 1 by the speaker – due to his being rational. Then the outcome according to the Bases-Theory of Conditionals and the theory of rational subjective probability is this: First, the b´´-based utterance of the conditional “If George is born in 1945, he is 58 in 2003” is true. Second, the assertability conditions (a) and (b) are fulfilled for that conditional (in the relevant context of utterance). Moreover, it cannot be rationally otherwise but that the speaker regards b´´ as relevantly connecting the state of affairs that George is born in 1945 with the state of affairs that George is 58 in 2003 – although p´´(George is 58 in 2003/George is born in 1945) is undefined! Hence the assertability condition (c) is also fulfilled for “If George is born in 1945, he is 58 in 2003,” and therefore this conditional is assertable (by the speaker, in the given context) – although p´´(George is born in 1945) = 0.

6.3

The Metalinguistic Theory of Conditionals and the Bases-Theory of Conditionals

The Bases-Theory of Conditionals is closely related to the so-called metalinguistic theory of conditionals,11 which David Lewis discusses in 11

The Bases-Theory of Conditionals is also related – though not as closely as to the metalinguistic theory – to what Lewis calls “premise semantics,” its main proponent being Angelika Kratzer. See Lewis’s “Ordering semantics and premise semantics for counterfactuals.”

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Counterfactuals, pp. 65-77, as a predecessor to his own theory of counterfactual conditionals. The first thing to be said about the metalinguistic theory of conditionals is that there need not be anything specifically metalinguistic about it. David Lewis gives a helpful schematic statement in terms of possible-worlds-semantics of the necessary and sufficient truthcondition of “If A, then B” according to a metalinguistic theory: [/If φ, then ψ/]12 is true ... on a metalinguistic theory if and only if ψ holds at all φ-worlds of a certain sort: φ-worlds at which some further premises, suitable for use with the antecedent φ, hold. (Counterfactuals, p. 66.) In this, the only indication that a metalinguistic theory of conditionals truly deserves the designation “metalinguistic” is the quantificational phrase “some further premises” (since premises are linguistic entities). But this instance of metalinguistic quantification is inessential. The above necessary and sufficient truth-condition for conditionals can just as well be stated in the following manner: /If φ, then ψ/ is true if and only if ψ holds at all φ-worlds of a certain sort: φ-worlds which belong to some state of affairs (set of worlds) suitable for use with the state of affairs expressed by the antecedent φ.13 There are several ways to interpret the “some” in this statement and in the above, original metalinguistic statement, quoted from Lewis: (i) the scope of the “some” may be the entire biconditional; (ii) its scope may be the entire right-hand side of the biconditional; (iii) it may not be meant in a quantificational sense at all, but may merely serve, in the phrase to which it be12

Lewis uses this own notation, which, moreover, is meant by him to represent counterfactual conditionals only. It is more in the spirit of the metalinguistic theory, however, to understand the schematically stated necessary and sufficient truth-condition as being the truth-condition of what I have called a “simple conditional” (and represented by “If A, then B”), which can be used for defining the corresponding counterfactual conditional, or, if asserted on its own, becomes an indicative conditional. (For Lewis, of course, all proper conditionals are counterfactual conditionals, since according to him indicative conditionals, as far as truth-conditions are concerned, amount to material implications. See Section 6.1.) 13 The state of affairs expressed by φ is simply the set of possible worlds at which φ holds.

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longs, to emphasize the generality of what is intended to be the most general description of a particular state of affairs, respectively, a particular set of premises. Especially under interpretation (iii) of the “some” the closeness to the Bases-Theory of Conditionals is manifest. For under interpretation (iii) of the “some” we can reformulate: /If φ, then ψ/ is true if and only if ψ holds at all φ-worlds belonging to the state of affairs b+, which is suitable for use with the state of affairs expressed by the antecedent φ. Or in metalinguistic terms: /If φ, then ψ/ is true if and only if ψ holds at all φ-worlds at which the statements in β+ hold – a set of statements that is suitable for use with the antecedent φ. The Bases-Theory of Conditionals is almost (but not quite yet) attained via the so-called metalinguistic theory of conditionals if “true” is replaced by “true in a context #” – or, alternatively, if “/If φ, then ψ/ is true ...” is replaced by “An utterance of /If φ, then ψ/ is true ...,” where # is the context of that utterance – and “b+” and “β+” are understood as abbreviations of functional terms: “b+(φ, ψ, #)” and “β+(φ, ψ, #).” “b+(φ, ψ, #)” designates the state of affairs that the ontological basis-function b+ assigns in the context # to the conditional-connected states of affairs expressed (in #) by φ and ψ. “β+(φ, ψ, #)” designates the set of statements that the linguistic basis-function β+ assigns in the context # to the conditional-connected statements φ and ψ. The two functions are connected in the following way: the state of affairs expressed (in #) by β+(φ, ψ, #)– that is: the conjunction of all states of affairs expressed (in #) by some statement or other in β+(φ, ψ, #) – is identical to b+(φ, ψ, #). – Then we have: /If φ, then ψ/ is true in context # if and only if ψ [in #] holds at all φ[in-#-]worlds belonging to the state of affairs b+(φ, ψ, #), which is suitable for use with the state of affairs expressed [in #] by the antecedent φ.

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/If φ, then ψ/ is true in context # if and only if ψ holds at all φ-worlds at which the statements in β+(φ, ψ, #) hold – a set of statements that is suitable for use with the antecedent φ. [Alternatively: An utterance of /If φ, then ψ/ is true if and only if ψ holds at all φworlds belonging to the state of affairs b+(φ, ψ, #), which is suitable for use with the state of affairs expressed by the antecedent φ. (# is the context of the utterance.) An utterance of /If φ, then ψ/ is true if and only if ψ holds at all φworlds at which the statements in β+(φ, ψ, #) hold – a set of statements that is suitable for use with the antecedent φ. (# is the context of the utterance.)] The final question is this: How must “suitable for use with the antecedent φ” be made precise? In other words: What is cotenability (with the antecedent)? David Lewis writes: I shall use Goodman’s term ‘cotenability’ for the present, simply to name whatever relation it is that must obtain between a truth and an antecedent to make that truth eligible to enter into a backing argument with that antecedent. ... What is cotenability? That is the problem of counterfactuals for a metalinguistic theorist, and counterfactuals remain mysterious to him just to the extent that cotenability remains mysterious. (Counterfactuals, p. 68, p. 69.)

Well, what is cotenability? – We must distinguish between two senses of the term: one, which is objective and relevant for truth, and another, which is subjective and relevant for assertability. Only in the first sense can cotenability enter into the truth-conditions of ontic (or objective) conditionals. Then we have: A statement σ is objectively cotenable with the antecedent φ of a conditional /If φ, then ψ/ (that is, objectively “suitable for use with the antecedent φ”) if, and only if, σ is true.14 And a state of affairs s is objectively According to Lewis, all statements cotenable with a counterfactual supposition φ are true (i.e., true at w*), but not all true statements are cotenable with a counterfactual supposition. (Cf. Counterfactuals, p. 57.) Contemplate the following statements: (1) there is no φ-world that is closer to w* than any ¬χ-world, i.e., for every φ-world there 14

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cotenable with the state of affairs expressed by the antecedent φ of a conditional /If φ, then ψ/ if, and only if, s obtains.15 These definitions of objective cotenability are the simplest possible – and simplicity is a virtue in itself. Nonetheless, arguments for the conclusion that nothing more than the truth of the “cotained” statement (respectively, the obtaining of the “cotained” state of affairs) is necessary for objective cotenability (albeit without mention of the term “cotenability”) can be found in Section 5.2.1. By employing the above definitions of being – as statement / as state of affairs – objectively cotenable with the antecedent φ of a conditional (i.e., of being objectively “suitable for use with the antecedent φ / with the state of affairs expressed by the antecedent φ”), the Bases-Theory of Conditionals itself is finally attained, after an act of analysis that takes its departure from Lewis’s formulation of the metalinguistic theory of conditionals; it is seen that the Bases-Theory of Conditionals can indeed be regarded as a certain interpretation of that theory: /If φ, then ψ/ is true in context # if and only if ψ holds at all φ-worlds belonging to the state of affairs b+(φ, ψ, #), which is an obtaining is a ¬χ-world that is at least as close to w* as the φ-world; (2) χ is not true at some world accessible from w*. According to Lewis (ibid.), (1) and (2) together are necessary and sufficient for making χ not cotenable (at w*) with φ. Conditions (1) and (2) are clearly fulfilled if χ is not true at w* (since they are implied by χ being not true at w*; note that w* is accessible from w*); but what is their import if χ is true at w*? Suppose (1) and (2) are fulfilled. But, given the truth of χ, even the logical incompatibility of χ with φ [implying (1) and – together with (3), presented below – (2)] does not necessarily make χ not cotenable with φ: it may yet be the basis on which an utterance of a counterfactual conditional of the form “IfCF φ, thenCF ¬φ” or “IfCF φ, thenCF ψ ∧ ¬ψ” is true and even assertable. And there certainly seem to be utterances of counterfactual conditionals of these forms that are true and assertable – even if (3) φ is true at some world accessible from w*. So (1), (2) and (3) can be fulfilled, and yet χ be cotenable (at w*) with φ. In consequence, both the definition of cotenability Lewis offers in Counterfactuals, p. 57, and the somewhat different definition he offers in “Counterfactuals and Comparative Possibility,” p. 11, are open to serious doubts. 15 There are two linguistic discomforts in the sentences to which this footnote refers: (a) the word “cotenable” suggests a subject that holds various opinions; it therefore seems strange so suppose that something could be cotenable with something else in a purely objective sense; (b) the word “cotenable” is meant to apply to statements; it therefore seems strange to apply it also to state of affairs. But (a) and (b) describe linguistic discomforts, nothing more serious than that: nothing one could not learn to accept simply by getting used to it.

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state of affairs [i.e., a state of affairs objectively suitable for use with the state of affairs expressed by the antecedent φ]. /If φ, then ψ/ is true in context # if and only if ψ holds at all φ-worlds at which the statements in β+(φ, ψ, #) hold, which is a set of true statements [i.e., a set of statements that is objectively suitable for use with the antecedent φ]. There are two obvious options still left open: The phrases “which [i.e., b+(φ, ψ, #)] is an obtaining state of affairs” and “which [i.e., β+(φ, ψ, #)] is a set of true statements” in the above rules of truth can be regarded (i) as stating a conjunct of the necessary and sufficient truth-condition stated, or (ii) as stating a condition of the entire biconditional that provides the truthcondition. In my presentation of the Bases-Theory of Conditionals in the previous chapter, I have already opted for (i), although (ii) does more justice to ordinary language, because under (i) truth-value gaps for conditionals are definitively avoided and the logic of conditionals is kept classical and simple. But now, what about subjective cotenability? – A statement σ is subjectively cotenable (by the speaker, in the given context) with the antecedent φ of a conditional /If φ, then ψ/ (that is, subjectively “suitable for use with the antecedent φ”) if, and only if, σ is believed (with complete conviction or just a little less than that) to be true by the subject (the speaker) and is regarded by the subject as relevantly connecting φ with ψ. And a state of affairs s is subjectively cotenable with the state of affairs expressed by the antecedent φ of a conditional /If φ, then ψ/ if, and only if, s is believed to obtain by the subject and is regarded by the subject as relevantly connecting the state of affairs expressed by φ with the state of affairs expressed by ψ. Once the distinction between objective and subjective cotenability has been drawn and filled with content, by giving necessary and sufficient conditions for each kind of cotenability, it becomes clear that many a conditional is on some occasion #, though asserted, not assertable (by the speaker) because the state of affairs b +(φ, ψ, #) – the basis of the utterance of /If φ, then ψ/ on the occasion # – is not subjectively cotenable (by the

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speaker) with the state of affairs expressed by the antecedent φ.16 But this is no reason to believe that every such conditional is not true on such occasions. Whether or not b+(φ, ψ, #) is subjectively cotenable (in #) with the state of affairs expressed by the antecedent φ is irrelevant for the truth of /If φ, then ψ/ (in #). What matters for the truth of /If φ, then ψ/ is whether or not b+(φ, ψ, #) is objectively cotenable with the state of affairs expressed by the antecedent φ. The rest is logic. Yet these matters are far from being generally clear to philosophers. The conflation of subjective cotenability, which is relevant for assertability, with objective cotenability, which is relevant for truth, is, I surmise, the principal source of the perplexities that beset the philosophical analysis of conditionals. Subjective cotenability follows subjective criteria (sometimes highly idiosyncratic and situationally dependent ones), which simply cannot be turned into objective criteria (i.e., into parts of truth-conditions). Treating subjective criteria of cotenability as if they were objective ones is bound to make one utterly skeptical about conditionals having any objective content at all, and, deplorably, it has made the metalinguistic theory of conditionals (so-called17) – which is essentially the correct idea – seem less plausible than it really is.

6.4

The Argument of David Lewis against the Idea of a Bases-Theory of Conditionals

In his book Counterfactuals, chapter 1.2, David Lewis uses the terminology of possible worlds and of accessibility relations between possible worlds (in a set-theoretical framework) to discuss and reject the idea that counterfactual conditionals are strict conditionals. The different terminology should not prevent us from seeing that Lewis, implicitly, also discusses and rejects the idea of a bases-theory of conditionals. Let us review his argument. First, some preparations. Let S(w) be a function that assigns to every possible world w the set of all possible worlds that are – in some sense – accessible from w. The strict conditional that corresponds to the accessibil16

Given the above definition of subjective cotenability, failure of subjective cotenability of the utterance-basis with the state of affairs expressed by the antecedent implies failure of assertability for the conditional; this much can be gathered from Section 6.2. 17 As I have indicated, the metalinguistic aspect of the metalinguistic theory of conditionals is not essential to it.

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ity-function S is truth-defined (i.e., specified by a necessary and sufficient truth-condition) as follows: A S→ B is true at world w iff for every world w´ which is an element of S(w) and at which A is true: B is true at w´.18 This rule of truth can also be put in the following manner if we let “[A]” designate the set of all worlds at which A is true, and “[B]” the set of all worlds at which B is true: A S→ B is true at world w iff S(w) ∩ [A] ⊆ [B]. The idea that counterfactual conditionals are strict conditionals can, then, be formulated in the following precise way: from “IfCF A, thenCF B” and “¬A” one can logically infer “A S→ B,” given an appropriate accessibilityfunction S. (Here arbitrary sentences can be substituted for the schematic letters. The extra premise “¬A” besides “IfCF A, thenCF B” could be omitted if one stipulated that, for every appropriate accessibility-function S, S(w) = {w} if A should happen to be true at w.) The idea that counterfactual conditionals are, at bottom, strict conditionals is also endorsed by the Bases-Theory of Conditionals, and therefore Lewis, in arguing against the idea that counterfactual conditionals are strict conditionals, argues against that theory (though not known to him under the name I gave it). The relevance of his arguments becomes especially clear if we consider that the sets of possible worlds designated by “S(w),” “[A],” “[B]” are standardly taken to be states of affairs (or at least settheoretical representations of states of affairs), and that “S(w) ∩ [A] ⊆ [B]” expresses in the set-theoretical idiom that the state of affairs that B is an intensional part of the conjunction of a basis – the state of affairs S(w) – with the state of affairs that A. Consider the following sequence (see Counterfactuals, p. 10), consisting of several concatenated counterfactual conditionals: “If Otto had come, it would have been a lively party; but if both Otto and Anna had come, it would have been a dreary party; but if Waldo had come as well, it would have been lively; but …”

18

Cf. Counterfactuals, p. 7.

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All assertions in this sequence might easily be true together (and known to be true together by anyone who knows Otto, Anna, Waldo, …, and the particular party that is being talked about). But how could the assertions in the quoted sequence of sentences be all true if counterfactual conditionals were strict conditionals? This is David Lewis’s challenge. Let “IfCF A, thenCF B” abbreviate the first counterfactual conditional in the above sequence, and “IfCF A ∧ C, thenCF ¬B” the second (taking “dreary” to be defined as “not lively”). And let us assume what easily might be: every assertion in the sequence is true. We may take it that Otto did not come to the party; therefore, both A and A ∧ C are false, and hence, assuming that counterfactual conditionals are strict conditionals, we can infer from the above text: A S→ B is true for an appropriate accessibilityfunction S, and A ∧ C S´→ ¬B is true for an appropriate accessibilityfunction S´. Now, the value of the accessibility-function S for w* (the actual world, which is our world of reference: truth is truth at w*) cannot be identical with the value of the accessibility-function S´ for w*; we must have S(w*) ≠ S´(w*). For if S(w*) = S´(w*), then A S→ B and A ∧ C S´→ ¬B could only be true together if S´(w*) ∩ [A] ∩ [C] were identical to ∅, the empty set. But S´(w*) ∩ [A] ∩ [C] must certainly be different from the empty set if the accessibility-function S´ is to be appropriate for the counterfactual conditional “IfCF A ∧ C, thenCF ¬B.” Thus, a proponent of the idea that counterfactual conditionals are strict conditionals can make the following reply to David Lewis’s challenge: All assertions in the text quoted above can be true together if the accessibility-function changes when one is going from the first counterfactual to the second (and again when one is moving from the second counterfactual to the third). The accessibility-function appropriate for the second counterfactual conditional, S´, is simply different from the accessibilityfunction appropriate for the first, S; in fact, they are different at w*: S(w*) ≠ S´(w*). In the terminology of the Bases-Theory of Conditionals: In going from the first to the second counterfactual conditional, a shift of basis takes place: the second counterfactual conditional is based on a state of affairs (i.e., S´(w*)) which is different from the state of affairs (i.e., S(w*)) on which the first counterfactual conditional is based. This is how they can be true together. Why is it that this shift takes place? This is a matter (1) of the meaning of the two sentences “IfCF A, thenCF B” and “IfCF A ∧ C, thenCF ¬B,” and (2) of any context in which the conjunction of the two sentences might be appropriately uttered. But Lewis objects:

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Neither will it help to plead dependence on context. … [O]ur problem is not a conflict between counterfactuals in different contexts, but rather between counterfactuals in a single context. It is for this reason that I put my examples in the form of a single run-on sentence, with the counterfactuals of different stages conjoined by semicolons and ‘but’. (Counterfactuals, p. 13.)

But, when several conditionals are uttered in a single context of utterance (same world, same speaker, same interval of time), why should not, then, that single context and the various pure sentence-meanings of the uttered conditionals determine a sequence of accessibility-functions, one function for each of the uttered conditionals, such that each one of these accessibility-functions has a different value at w* (in other words: such that each one of them determines a different state of affairs as basis for the uttered conditional it belongs to)? Lewis himself concedes that counterfactual conditionals might be regarded as being dependent on “a very local context: the antecedent itself” (Counterfactuals, p. 13; the antecedent alone, of course, will not do: see the example discussed in 5.2.2). “That is not altogether wrong,” he says, “but it is defeatist. It consigns to the wastebasket of contextually resolved vagueness something much more amenable to systematic analysis than most of the rest of the mess in that wastebasket.” (Ibid.) I submit that the Bases-Theory of Conditionals offers as much of systematic analysis of counterfactual conditionals as Lewis’s theory does. At the same time, it is simpler and fits better the evidence that can be gathered from our linguistic practice. (Evidence for the latter assertion can be found further down in this section.) As a matter of fact, Lewis can be said to have hit on the basic concept of the Bases-Theory of Conditionals – without making any use of it. In his discussion of various forms of strict conditionals, he mentions (after logical necessity and its corresponding strict conditional, physical necessity and its strict conditional, inevitability at time t and its strict conditional) “necessity in respect of facts of so-and-so kind, and its strict conditional” (ibid., p. 7; Lewis’s italics). Necessity in respect of facts of so-andso kind is nothing else than the basic concept of the Bases-Theory of Conditionals – as becomes particularly clear once it is appreciated that there are just as many necessities in respect of facts of so-and-so kind, and strict conditionals corresponding to them,19 as there are facts. When he or she is A strict conditional, →, corresponds to a certain necessity, , if it is definable as follows: A → B =Def (A ⊃ B). 19

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asserting a counterfactual conditional “IfCF A, thenCF B,” a speaker intends to assert one instance of one of those many necessities: an instance that consists in that necessity being prefixed to “A ⊃ B,” an instance that is an implicit (definitional) part of the assertion of “IfCF A, thenCF B.” Every true (non-vague) utterance of a counterfactual conditional with false antecedent has at its heart a true instance of the strict conditional that corresponds to a (specific) necessity in respect of (specific) facts of so-and-so kind. Those facts constitute the basis of the utterance of the conditional – facts which establish the necessitating nexus between the antecedent and the consequent of the uttered counterfactual. Consider again our example: “If Otto had come, it would have been a lively party; but if both Otto and Anna had come, it would have been a dreary party; but if Waldo had come as well, it would have been lively; but …”

The most plausible solution to the difficulty posed by this commonplace sequence of counterfactual conditionals is not a Lewisian semantics of counterfactuals; rather, it is the Bases-Theory of Conditionals. The BasesTheory of Conditionals can not only solve the difficulty (as was already argued above), it can solve it better – more realistically, so to speak – than its competitor, as becomes very palpable if we expand our example into a dialog: “If Otto had come, it would have been a lively party.” “Assuming, of course, that Anna didn’t come to the party.” “Correct. In fact, she didn’t come. But if both Otto and Anna had come, it would have been a dreary party.” “Assuming that Waldo didn’t come to the party.” “Right. In fact, he didn’t come. But if Otto, Anna, and Waldo had come, it would have been a lively party.”

This dialog strongly suggests that the Bases-Theory of Conditionals is correct. In the dialog, the state of affairs that Anna didn’t come to the party is revealed as part of the basis of the first uttered counterfactual conditional, and the state of affairs that Waldo didn’t come to the party is revealed as part of the basis of the second uttered counterfactual conditional. (Or how else could the data be interpreted?) The state of affairs that Waldo didn’t come to the party may or may not be a part of the basis of the first uttered counterfactual conditional; but the state of affairs that Anna didn’t come to the party is certainly not a part of the basis of the second uttered counterfactual conditional (since one shouldn’t assume without good reason that

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the speaker is making an assertion that is vacuous if true). Hence the two bases are seen to be different, and hence the two uttered conditionals based on them can be both true, non-vacuously true, from the point of view of the Bases-Theory of Conditionals, which treats conditionals (indicative conditionals and counterfactual conditionals with false antecedent) in effect as strict implications. No comparative similarities between possible worlds are drawn into consideration by the speakers, no such similarities need to be drawn into consideration by the semanticist.

6.5

Comparative Similarity (Closeness) of Worlds, Lewis’s and Stalnaker’s Interpretations of Counterfactual Conditionals, and the Bases-Theory of Conditionals

How does the Bases-Theory of Counterfactual Conditionals compare with Lewis’s approach? In his paper “Causation,” p. 164, Lewis states the following rule of truth for counterfactual conditionals: [1] “IfCFL1 A, thenCFL1 B” is true (at a world w) iff either (1) there are no possible A-worlds, or (2) some A-world where B holds is closer (to w) than is any A-world where B does not hold.20 We can also gather from “Causation,” p. 164,21 the rule of truth for somewhat differently interpreted counterfactual conditionals: [2] “IfCFL2 A, thenCFL2 B” is true (at a world w) iff either (1) there are no possible A-worlds, or (2) B holds at all the closest A-worlds. We take as our world of reference w*, the actual world, and we identify states of affairs with sets of possible worlds, letting (as in the previous section) “[A],” “[B]” designate the sets of possible worlds at which A, respectively B, are true, in other words: we let “[A]” and “[B]” designate the states of affairs expressed by A, respectively B. For simplicity’s sake, let “w,” “w´”, “w´´,” … be variables whose range is restricted to possible worlds. Note, finally, that the isolated capital letters – A, B, C, ... – func20

This is only almost a literal quotation. Lewis symbolizes counterfactual conditionals, as interpreted by him, in a different way than I do, and I left out an – inessential – parenthesis of Lewis’s after (1). Moreover, I have replaced Lewis’s “C” by “B.” 21 See also David Lewis, Counterfactuals, p. 1.

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tion (as they have done previously) both as schematic letters, with sentences being substitutable for them, and as variables for sentences, with names of sentences being substitutable for them; no confusions should arise from this convenience. Then [1], specifically for w*,22 can be put in the following way: [1] “IfCFL1 A, thenCFL1 B” is true (at w*) iff either (1) ¬∃w(w ∈ [A]), or (2) ∃w(w ∈ [A ∧ B] ∧ ∀w´(w´ ∈ [A ∧ ¬B] ⊃ w is closer to w* than w´)). And [2], specifically for w*, can be put in the following way: [2] “IfCFL2 A, thenCFL2 B” is true (at w*) iff either (1) ¬∃w(w ∈ [A]), or (2) ∀w(w ∈ [A] ∧ ¬∃w´(w´∈ [A] ∧ w´ is closer to w* than w) ⊃ w ∈ [B]). Note that condition (1) can be omitted in [2], since (1) logically implies (2) in [2]. As Lewis notes (ibid., p. 164), under the supposition that ∃w(w∈[A] ∧ ¬∃w´(w´ ∈ [A] ∧ w´ is closer to w* than w)), “IfCFL1 A, thenCFL1 B” and “IfCFL2 A, thenCFL2 B” have equivalent truth-conditions. Proof: (i)

Assume: ∀w(w ∈ [A] ∧ ¬∃w´(w´∈ [A] ∧ w´ is closer to w* than w) ⊃ w ∈ [B]). Hence with the supposition ∃w(w∈[A] ∧ ¬∃w´(w´ ∈ [A] ∧ w´ is closer to w* than w)): w ∈ [B] (for the w in that supposition). Hence w ∈ [A ∧ B]. Assume now for reductio: w´ ∈ [A ∧ ¬B] ∧ w is not closer to w* than w´. Hence w´ ∈ [A] ∧ w´ ∉ [B]. Hence with the initial assumption: ∃w´´(w´´∈ [A] ∧ w´´ is closer to w* than w´). Since w is not closer to w* than w´, w´ is at least as close to w* as w. Hence, since w´´ is closer to w* than w´, w´´ is closer to w* than w – contradicting what has been supposed of w. Therefore: ∀w´(w´ ∈ [A ∧ ¬B] ⊃ w is closer to w* than w´). It has now been derived from the initial assumption, on the basis of the supposition ∃w(w∈[A] ∧ ¬∃w´(w´ ∈ [A] ∧ w´ is closer to w* than w)): ∃w(w ∈ [A ∧ B] ∧ ∀w´(w´ ∈ [A ∧ ¬B] ⊃ w is closer to w* than w´)).

22

The rules of truth [1] and [2] are, subsequently, stated for w* (the actual world), but one can read them as being stated for every possible world by letting “w*” function as a variable for possible worlds. The more general reading becomes necessary if one is considering counterfactual conditionals that are embedded in other counterfactual conditionals.

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(ii)

Assume: ∃w(w ∈ [A ∧ B] ∧ ∀w´(w´ ∈ [A ∧ ¬B] ⊃ w is closer to w* than w´)). Assume for reductio: w´´ ∈ [A] ∧ ¬∃w´(w´∈ [A] ∧ w´ is closer to w* than w´´) ∧ w´´ ∉ [B]. Hence: w´´ ∈ [A ∧ ¬B]. Hence with the initial assumption: w is closer to w* than w´´. Since w ∈ [A], we therefore obtain: ∃w´(w´∈ [A] ∧ w´ is closer to w* than w´´) – contradicting the assumption for reductio. Therefore: ∀w(w ∈ [A] ∧ ¬∃w´(w´∈ [A] ∧ w´ is closer to w* than w) ⊃ w ∈ [B]).

Given [1], there is another interpretation of counterfactual conditionals close by, which, stated for w*, is this: [3] “IfCFL3 A, thenCFL3 B” is true (at w*) iff either (1) ¬∃w(w ∈ [A]), or (2) ∃w(w ∈ [A ∧ B]) ∧ ∀w(w ∈ [A ∧ B] ⊃ ∀w´(w´ ∈ [A ∧ ¬B] ⊃ w is closer to w* than w´)). It is immediately evident that the following inference is valid: IfCFL3 A, thenCFL3 B ⇒ IfCFL1 A, thenCFL1 B. But consider now the following, additional interpretation of counterfactual conditionals, which is in the spirit of the Bases-Theory of Conditionals: [4] “IfCFL4 A, thenCFL4 B” is true (at w*) iff either (1) w* ∉ [A] ∧ w* ∈ bL1 ∧ bL1∩[A] ⊆ [B], or (2) w* ∈ [A ∧ B]. Here the basis bL1 is the following set of possible worlds (i.e., the following state of affairs): {w: ∀w´(w´ ∈ [A ∧ ¬B] ⊃ w is closer to w* than w´)}. Hence the statement “w* ∈ bL1” is logically equivalent to the statement “∀w´(w´ ∈ [A ∧ ¬B] ⊃ w* is closer to w* than w´),” which, in turn, logically implies “w* ∉ [A ∧ ¬B],” i.e., w* ∈ [A ⊃ B]; for suppose that ∀w´(w´ ∈ [A ∧ ¬B] ⊃ w* is closer to w* than w´), and suppose that w* ∈ [A ∧ ¬B]. Hence w* is closer to w* than w* – which is (logically) impossible. The following theorem, however, is central: bL1∩[A] ⊆ [B]. Proof: bL1∩[A] ⊆ [B], that is: {w: ∀w´(w´ ∈ [A ∧ ¬B] ⊃ w is closer to w* than w´)}∩[A] ⊆ [B]. And this is shown to be true as follows. Assume: ∀w´(w´ ∈ [A ∧ ¬B] ⊃ w is closer to w* than w´) ∧ w ∈ [A], and assume for reductio: w ∉ [B]. Hence: w ∈ [A ∧ ¬B]. Hence: w is closer to w* than w – which is absurd.

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As a consequence of this theorem, “[A ∧ B]” in [1] can be replaced by “[A].”23 As another consequence of this theorem, (2) in [3] can be replaced by: ∃w(w∈ [A ∧ B]) ∧ ∀w(w ∈ [A] ⊃ (w ∈ [B] ≡ ∀w´(w´ ∈ [A ∧ ¬B] ⊃ w is closer to w* than w´))). Yet another consequence of the theorem is that the conjunct “bL1∩[A] ⊆ [B]” can simply be omitted from [4]. The latter consequence is particularly interesting: “either (1) w* ∉ [A] ∧ w* ∈ bL1 ∧ bL1∩[A] ⊆ [B], or (2) w* ∈ [A ∧ B]” is, according to [4], the necessary and sufficient condition for the truth of “IfCFL4 A, thenCFL4 B.” But given the theorem just stated and proven, [4] can also be formulated (logically equivalently) in the following manner: [4] “IfCFL4 A, thenCFL4 B” is true (at w*) iff: w* ∈ [A ⊃ B] and (w* ∈ bL1 or w* ∈ [A ∧ B]). This is obtained as follows. Because of the above theorem, “bL1∩[A] ⊆ [B]” can be omitted from [4], leaving us with “either (1) w* ∉ [A] ∧ w* ∈ bL1, or (2) w* ∈ [A ∧ B]” as the necessary and sufficient condition for the truth of “IfCFL4 A, thenCFL4 B.” All of the following formulas result from “either w* ∉ [A] ∧ w* ∈ bL1, or w* ∈ [A ∧ B]” by transformations that preserve logical equivalence [for brevity’s sake, I use the same symbols for object-language and meta-language truth-functional connectives]: ¬(w* ∉ [A] ∧ w* ∈ bL1) ⊃ w* ∈ [A ∧ B], (w* ∈ [A] ∨ w* ∉ bL1) ⊃ w* ∈ [A ∧ B], (w* ∈ [A] ⊃ w* ∈ [A ∧ B]) ∧ (w* ∉ bL1 ⊃ w* ∈ [A ∧ B]), w* ∈ [A ⊃ B] ∧ (w* ∈ bL1 ∨ w* ∈ [A ∧ B]). The last formula is the formula we were looking for.

Since “w* ∈ [A ⊃ B]” is a logical consequence of “w* ∈ bL1” – as we have seen – and also of “w* ∈ [A ∧ B],” we obtain another formulation of [4] which is logically equivalent to the previous ones: [4] “IfCFL4 A, thenCFL4 B” is true (at w*) iff: w* ∈ bL1 or w* ∈ [A ∧ B].

23

The replacement provides an alternative (but equivalent) formulation of condition (2) in Lewis’s interpretation [1] of counterfactual conditionals: (21) ∃w(w ∈ [A] ∧ ∀w´(w´ ∈ [A ∧ ¬B] ⊃ w is closer to w* than w´)). Here are two more such formulations, obviously equivalent with (21): (22) ∃w(w ∈ [A] ∧ ¬∃w´(w´ ∈ [A ∧ ¬B] ∧ w´ is at least as close to w* as w)), (23) ∃w(w ∈ [A] ∧ ∀w´(w´ is at least as close to w* as w ⊃ w´ ∈ [A ⊃B])). (Concerning formulation (23), compare Counterfactuals, p. 49.)

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Is “IfCFL4 A, thenCFL4 B,” then, equivalent to “A ⊃ B”? For this result, one merely needed to show that “w* ∉ bL1” logically implies “w* ∉ [A ⊃ B],” or in other words: that it logically implies “w* ∈ [A ∧ ¬B].” For then “w* ∈ bL1” would be logically equivalent to “w* ∈ [A ⊃ B]” (since we have already seen that “w* ∈ bL1” logically implies “w* ∈ [A ⊃ B]”), and therefore “w* ∈ bL1 or w* ∈ [A ∧ B]” would be logically equivalent to “w* ∈ [A ⊃ B] or w* ∈ [A ∧ B],” which is logically equivalent to “w* ∈ [A ⊃ B].”

Suppose, therefore, w* ∉ bL1; hence: ∃w´(w´ ∈ [A ∧ ¬B] ∧ w* is not closer to w* than w´), hence ∃w´(w´ ∈ [A ∧ ¬B] ∧ w´ is at least as close to w* as w*). But from this one can conclude “w* ∈ [A ∧ ¬B]” only if no world different from w* is at least as close to w* as w* – and that’s not an obviously correct assumption.24 Consider that some differences between w* and other possible worlds may be indifferent as far as the comparative similarity relation between worlds is concerned. At least, nothing prevents interpreting the comparative similarity relation in that way. Such differences would make a world different from w*, while they do not prevent that it is as similar to w* as w* is (that is: maximally similar; but not even maximal similarity automatically implies identity: not if the range of respects of similarity is restricted – as it normally is). We have now before us four different interpretations of counterfactual conditionals: [1], [2], [3] and [4], yielding counterfactual conditionals with four different meanings: “IfCFL1 A, thenCFL1 B,” “IfCFL2 A, thenCFL2 B,” “IfCFL3 A, thenCFL3 B,” and “IfCFL4 A, thenCFL4 B.” What is common to all of these interpretations is that they make use of the concept of comparative closeness (or comparative similarity) between possible worlds. Three of them (the exception being [2]) also make use of the set bL1. Here once more, for the sake of easy comparison, the interpretations that make use of bL1 (now formulated by making use of the designator “bL1”): 24

The comparative closeness (similarity) relation between worlds has the property of centeredness iff for any (possible world) w: (a) w is at least as close to w as any w´ [i.e., no w´ is closer to w than w], and (b) any w´ that is at least as close to w as w is identical to w [i.e., w is closer to w than any w´ different from w]. (Cf. Counterfactuals, pp. 14-15.) The fulfillment of condition (a) by a world w makes w an element of the set of worlds that are at least as close to w as w. The additional fulfillment of condition (b) by a world w make the set of worlds that are at least as close to w as w a singleton set: {w}. While (a) is uncontroversial for any world w, (b) is not. Hence: the property of centeredness cannot be unproblematically assumed for the comparative closeness relation between worlds.

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[1] “IfCFL1 A, thenCFL1 B” is true iff: ¬∃w(w ∈ [A]) or ∃w(w ∈ [A ∧ B] ∧ w ∈ bL1). [3] “IfCFL3 A, thenCFL3 B” is true iff: ¬∃w(w ∈ [A]) or (∃w(w ∈ [A ∧ B]) ∧ ∀w(w ∈ [A ∧ B] ⊃ w ∈ bL1)). [4] “IfCFL4 A, thenCFL4 B” is true iff: w* ∈ bL1 or w* ∈ [A ∧ B]. Of these three interpretations, [4] is the one that is in the spirit of the Bases-Theory of Conditionals (although this is no longer recognizable in the present guise of [4]). Note that, despite appearances, “bL1” does not denote a universal basis that connects states of affairs expressed by any two sentences A and B, whatever world is the world of reference; for “bL1” does not denote the same state of affairs for all sentences A and B, independently of the world of reference. It is more accurate to write “bL1(A, B, w*)” instead of “bL1”; in this way, it becomes apparent that we have before us a function that assigns, at w*, a state of affairs to sentences A and B – a state of affairs that serves as a basis for (an utterance of) a counterfactual conditional connecting those sentences: “IfCFL4 A, thenCFL4 B.” Here are some considerations subsequent to envisaging salient (because extreme) values of the function bL1(A, B, w*) [= {w: ∀w´(w´ ∈ [A ∧ ¬B] ⊃ w is closer to w* than w´)}]: If B is true at every possible world at which A is true (in other words: if [A] ⊆ [B], or: if A logically implies B), then bL1(A, B, w*) = W (the set of all possible worlds), and therefore: w* ∈ bL1(A, B, w*) (since w* ∈ W), and therefore according to [4]: “IfCFL4 A, thenCFL4 B” is true (at w*). A corollary: If A is true at no possible world or B true at every possible world, then bL1(A, B, w*) = W, and therefore according to [4]: “IfCFL4 A, thenCFL4 B” is true. Another corollary: If B is true at exactly the same possible worlds at which A is true (which is trivially the case if B is the same sentence as A), then bL1(A, B, w*) = W, and therefore according to [4]: “IfCFL4 A, thenCFL4 B” is true. The other extreme value of bL1(A, B, w*) is ∅, the empty set, which is obtained, for example, when [A ∧ ¬B] = W (for which both [A] and [¬B] must be W). In that case, “IfCFL4 A, thenCFL4 B” is false. And in any case, bL1(A, B, w*) is ∅ if, and only if, for every world w there is a world w´ in [A ∧ ¬B] which is at least as close to w* as w.

The following principles are provable for “IfCFL4 A, thenCFL4 B”: (I) IfCFL4 A, thenCFL4 A [is true at w* – whichever world is designated by “w*”].

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Proof: According to [4]: “IfCFL4 A, thenCFL4 A” is true at w* iff w* ∈ bL1(A, A, w*) or w* ∈ [A ∧ A]. But w* ∈ bL1(A, A, w*), since bL1(A, A, w*) = W (the set of all possible worlds).

(II) (IfCFL4 ¬A, thenCFL4 A) ⊃ IfCFL4 B, thenCFL4 A. Proof: Assume “IfCFL4 ¬A, thenCFL4 A” is true at w*. Hence (according to [4]): w* ∈ bL1(¬A, A, w*) or w* ∈ [¬A ∧ A]. Hence w* ∈ bL1(¬A, A, w*). Suppose now (for reductio): “IfCFL4 B, thenCFL4 A” is not true at w*. Hence (according to [4]): w* ∉ bL1(B, A, w*), and hence: ∃w´(w´ ∈ [B ∧ ¬A] ∧ w* is not closer to w* than w´). At the same time (because of w* ∈ bL1(¬A, A, w*)): ∀w´(w´ ∈ [¬A ∧ ¬A] ⊃ w* is closer to w* than w´). But [B ∧ ¬A] is a subset [¬A ∧ ¬A] (= [¬A]), and therefore we obtain a contradiction (completing the reductio).

(III) (IfCFL4 A, thenCFL4 B) ⊃ (A ⊃ B). Proof: Assume “IfCFL4 A, thenCFL4 B” is true at w*. Hence (according to [4]): w* ∈ bL1(A, B, w*) or w* ∈ [A ∧ B]. In both cases: w* ∈ [A⊃B].

(IV) (A ∧ B) ⊃ (IfCFL4 A, thenCFL4 B). Proof: Obvious.

The converse of (III) is not provable on the basis of [4], as we have already seen above; note that it is necessary for this result that it is not assumed that no world different from w* is at least as close to w* as w* is. Nor is the Principle of Conditional Excluded Middle, for which Stalnaker’s logic of conditionals is notorious, provable on the basis of [4]. The attempt to prove it is instructive: Assume that neither “IfCFL4 A, thenCFL4 B” nor “IfCFL4 A, thenCFL4 ¬B” is true at w*. That is (according to [4]): w* ∉ bL1(A, B, w*) and w* ∉ [A ∧ B], and w* ∉ bL1(A, ¬B, w*) and w* ∉ [A ∧¬B]. That is: ∃w´(w´ ∈ [A ∧ ¬B] and w* is not closer to w* than w´) and ∃w´´(w´´ ∈ [A ∧ B] and w* is not closer to w* than w´´) and w* ∉ [A]. This situation cannot be excluded if we allow the indifference (the making of no difference) of the states of affairs [A ∧ ¬B] and [A ∧ B] with regard to the relation of comparative closeness to w* of at least some worlds w´ and w´´. Let [A] be the state of affairs that radium atom r decays at time t, and let it be a state of affairs that does not obtain in w*; let [B] be the state of affairs that radium atom r´ decays at time t´, and suppose: r´ ≠ r, t´≠ t. Then the relation of comparative closeness can apparently be understood in

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such a way that, for some worlds w´ and w´´, both w´ and w´´ are at least as close to w* as w* itself is, although w´ is a world in which [A] obtains and [B] does not, and w´´ a world in which [A] obtains and also [B]. This is borne out by intuition, according to which neither “If r had decayed at t, then r´ would have decayed at t´” nor “If r had decayed at t, then r´ would not have decayed at t´” is true. Obviously, the non-provability of the Principle of Conditional Excluded Middle on the basis of [4] requires – like the non-provability of the converse of (III) – that it is not assumed that no world different from w* is at least close to w* as w* is – and intuition supports the non-assumption of this (as long as one does not allow oneself to take the geometric representation of comparative similarity more literally than it ought to be taken). It seems that counterfactual conditionals of the brand “IfCFL4 A, thenCFL4 B” – or in other words: counterfactual conditionals in the interpretation they have according to [4] – are well-behaved, and can take their place beside the [1]-, [2]-, and [3]-conditionals. Now, by using the term “bL2” – or more accurately: “bL2(A, w*)” –, which is taken to designate the set {w: w ∈ [A] ∧ ¬∃w´(w´ ∈ [A] ∧ w´ is closer to w* than w)}, [2] can be put very compactly in the following way: [2] “IfCFL2 A, thenCFL2 B” is true iff bL2 ⊆ [B]. But note that the basis bL2 (in contrast to bL1) cannot be used in the spirit of the Bases-Theory of Conditionals to specify yet another reasonable interpretation of counterfactual conditionals. An interpretation of counterfactual conditionals in the spirit of the Bases-Theory of Conditionals that uses basis bL2 looks like this: [5] “IfCFL5 A, thenCFL5 B” is true (at w*) iff either (1) w* ∉ [A] ∧ w* ∈ bL2 ∧ bL2∩[A] ⊆ [B], or (2) w* ∈ [A ∧ B]. But [5] is totally inadequate, for according to it “IfCFL5A, thenCFL5 B” is true if, and only if, A and B is true (as is easily seen). Consider finally a third basis: bL6 = {w: ¬∃w´(w´ is closer to w* than w)}. bL6 is the set of all worlds that are closest to w*. This basis can indeed be used in the spirit of the Bases-Theory of Conditionals to specify another reasonable interpretation of counterfactual conditionals, as follows:

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[6] “IfCFL6 A, thenCFL6 B” is true (at w*) iff either (1) w* ∉ [A] ∧ w* ∈ bL6 ∧ bL6∩[A] ⊆ [B], or (2) w* ∈ [A ∧ B]. This, since “w* ∈ bL6” (i.e., “no world is closer to w* than w*”) is logically true, can be simplified, yielding: [6] “IfCFL6 A, thenCFL6 B” is true (at w*) iff either (1) w* ∉ [A] ∧ bL6∩[A] ⊆ [B], or (2) w* ∈ [A ∧ B]. Just like “IfCFL4 A, thenCFL4 B,” “IfCFL6 A, thenCFL6 B” is a reasonable counterfactual conditional as long as the centeredness25 of the comparative similarity relation between worlds is not assumed. If, however, centeredness is assumed, then bL6 is {w*}, and the condition that according to [6] is necessary and sufficient for the truth of “IfCFL6 A, thenCFL6 B” would amount to this: either w* ∉ [A] ∧ {w*}∩[A] ⊆ [B], or w* ∈ [A ∧ B]; which is logically equivalent to this: w* ∈ [A ⊃ B] (because {w*}∩[A] = ∅ if w* ∉ [A], and “∅ ⊆ [B]” is a logical truth). Interpretation [6] of counterfactual conditionals can easily be made to encompass a large part of the Bases-Theory of Counterfactual Conditionals: that part in which merely conditionals are considered that have an obtaining basis (in the context in which they are uttered). How can this be done? In the following way: For every obtaining state of affairs p (in the presently used set-theoretical idiom: for every set of possible worlds of which w* is an element) a relation of comparative closeness (similarity) between worlds can be specified such that p is the set of worlds closest (most similar) to w*. There is a set of properties P such that p is the set of worlds that have all the properties in P, and we can define: w is closer to w´ than w´´ (is) =Def w has more properties out of P in common with w´ than w´´ (has). (i) Suppose: w ∈ p; hence: w has all the properties in P. Suppose in addition that there is a world w´ that is closer to w* than w, in other words: that there is a world w´ that has more properties out of P in common with w* than w. But since w has all the properties out of P in common with w* (since both w and w* have all the properties in P, both being elements of p), this cannot be. Therefore: there is no world w´ that is closer to w* than w. (ii) Suppose conversely that there is no world w´ that is closer to w* than w. Hence no world w´ has more properties out of P in common with w* than w has. But 25

See the preceding footnote.

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suppose w has not all the properties in P. Then w* itself, having all the properties in P (since it is an element of p), would be a world that has more properties out of P in common with w* than w has – contradicting the result already reached. Therefore: w has all the properties in P, and hence: w ∈ p.

Is there some criterion that allows one to rationally prefer some of the five viable interpretations of counterfactual conditionals considered to the rest of the five? (Five, because interpretation [5] has already been excluded as inadequate.) There is indeed such a criterion. Sometimes counterfactuals conditionals of the forms “IfCF A, thenCF ¬A” and “IfCF A, thenCF B ∧ ¬B” can be true, even if A is true at some possible world: Suppose somebody, using a contingently true theory, deduces ¬A from A, or B ∧ ¬B from A. Wouldn’t he, then, be saying something true and assertable if he uttered “IfCF A, thenCF ¬A” and “IfCF A, thenCF B ∧ ¬B”? The BasesTheory of Conditionals says “yes,” and this answer seems reasonable and the correct thing to say (for more on this, see Section 5.2.1, concerning (iii?)). But what do the above interpretations of counterfactual conditionals determine regarding the question just asked? “IfCFL1 A, thenCFL1 ¬A” and “IfCFL1 A, thenCFL1 B ∧ ¬B” are always false if A is true at some possible world. For then the first disjunct of (the right-hand side of biconditional) [1] is not fulfilled, and it is evident that the second disjunct of [1] cannot be fulfilled in the cases of “IfCFL1 A, thenCFL1 ¬A” and “IfCFL1 A, thenCFL1 B ∧ ¬B.” “IfCFL2 A, thenCFL2 ¬A” and “IfCFL2 A, thenCFL2 B ∧ ¬B”, however, can be true, even if A is true at some possible world. Suppose they are true, with A being true at some possible world. According to [2], the assumption of truth for the two counterfactual conditionals requires no more and no less than that bL2 – {w: w ∈ [A] ∧ ¬∃w´(w´ ∈ [A] ∧ w´ is closer to w* than w)} – is the empty set. That is: ∀w(w ∈ [A] ⊃ ∃w´(w´ ∈ [A] ∧ w´ is closer to w* than w)). Because A is true at some world, this condition is not trivially fulfilled, but only by an infinite set [A], containing, without end, worlds ever closer and closer to w*. “IfCFL3 A, thenCFL3 ¬A” and “IfCFL3 A, thenCFL3 B ∧ ¬B”, in turn, are always false if A is true at some possible world; this follows according to [3], as is easily seen. “IfCFL4 A, thenCFL4 ¬A” and “IfCFL4 A, thenCFL4 B ∧ ¬B” can be true, even if A is true at some possible world. Suppose they are true. According to [4], this means: w* ∈ {w: ∀w´(w´ ∈ [A ∧ ¬¬A] ⊃ w is closer to w* than w´)} or w* ∈ [A ∧ ¬A]; w* ∈ {w: ∀w´(w´ ∈ [A ∧ (¬B ∨ ¬¬B)] ⊃

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w is closer to w* than w´)} or w* ∈ [A ∧ B ∧ ¬B]; that is: w* ∈ {w: ∀w´(w´ ∈ [A] ⊃ w is closer to w* than w´)}, or in other words: ∀w´(w´ ∈ [A] ⊃ w* is closer to w* than w´). And this can be the case, and if A is true at some possible world, it can even be the case non-trivially.26 “IfCFL6 A, thenCFL6 ¬A” and “IfCFL6 A, thenCFL6 B ∧ ¬B” can be true, even if A is true at some possible world. According to [6], these two counterfactual conditionals are true if, and only if, w* ∉ [A] and bL6∩[A] is the empty set, which is equivalent to: bL6∩[A] is the empty set (since w* ∈ bL6). In other words: the two counterfactual are true if, and only if, there is no world closest to w* at which A is true, which can be the case even if A is true at some world. We have now seen that the criterion suggested above is not satisfied by David Lewis’s favorite interpretation of counterfactual conditionals, which is [1], and a fortiori not by any stronger version of that interpretation, for example [3]. Interpretations [4] and [6], in contrast, do satisfy that criterion – interpretations that are, though formulated within the framework of comparative closeness (similarity) of worlds, nevertheless in the spirit of the Bases-Theory of Conditionals. Interpretation [2] satisfies the criterion, too – an interpretation, however, that Lewis disfavors in “Counterfactuals and Comparative Possibility,” p. 7 and p. 9. Interpretation [2] is a modification of the original interpretation of counterfactual conditionals by Robert Stalnaker, which interpretation can be put in the following way: [7] “IfCFL7 A, thenCFL7 B” is true iff either (1) ¬∃w(w ∈ [A]) or (2) ιw(w ∈ bL2(A, w*)) ∈ [B], where bL2(A, w*) is {w: w ∈ [A] ∧ ¬∃w´(w´ ∈ [A] ∧ w´ is closer to w* than w)}.27 There are three serious problems with [7]. The first problem is its rather dubitable implicit presupposition: that in every non-empty set of worlds there is exactly one world which, among all the worlds in that set, is closest to w*. The second problem is a consequence of this presupposition: the Principle of Conditional Excluded Middle turns out to be valid according to [7]; but there are plausible counterexamples to that principle.28 The third Note that “∀w´(w´ ∈ [A] ⊃ w* is closer to w* than w´)” implies “w* ∈ [¬A],” but is not equivalent to “w* ∈ [¬A],” since we are not assuming that w* is closer to w* than every world w´ that is different from w*. 27 Cf. Lewis, “Counterfactuals and Comparative Possibility,” p. 5. 26

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problem is that – implausibly – neither “IfCF A, thenCF ¬A” nor “IfCF A, thenCF B ∧ ¬B” can be true according to [7] if A is true at some possible world, since, in that case, “IfCFL7 A, thenCFL7 ¬A” and “IfCFL7 A, thenCFL7 B ∧ ¬B” can only be false. The first problem and the second disappear if we use the following interpretation of counterfactual conditionals, which is still Stalnakerian in spirit: [8] “IfCFL8 A, thenCFL8 B” is true iff either (1) ¬∃w(w ∈ [A]) or (2) ∃!w(w ∈ bL2(A, w*)) ∧ ∃w(w ∈ bL2(A, w*) ∧ w ∈ [B]).29 But the third problem of interpretation [7] remains a problem also of interpretation [8]. This leaves us with interpretations [2], [4] and [6] being the most recommendable ones among the eight interpretations of counterfactual conditionals considered. Are there further grounds for choosing among the remaining three interpretations? We have seen that [4] and [6] are viable only if centeredness of the comparative similarity (closeness) relation between worlds is not assumed. Is this something to be held against these interpretations? If (and only if) the similarity of worlds to w* is defined on the basis of all the properties w* has, then it is indeed very plausible that there is no world different from w* which is at least as similar to w* as w* is. Presumably only w* has all the properties w* has, and therefore no world different from w* is at least as similar to w* as w* is – if (and only if) all the properties of w* count in the comparison of similarity. Suppose w´ is different from w*. Then there is a property that w* has, but that w´ does not have (for if w´ had all the properties that w* has – for each property w* does not have, w* has its negation –, then it would be identical with w*). And therefore: w´ is less similar to w* than w* – if the similarity of worlds to w* is defined on the basis of all the properties w* has. If, on the other hand, the similarity of worlds to w* is not defined on the basis of all the properties of w*, but on the basis of the properties in some proper subset, P*, of the set of w*’s properties, then w´ can easily be different from w* (because w* has some property, not in P*, that w´ does not have) and yet as similar to w* as w* is: because w´, like w*, has all the properties in P*.

28

See also Lewis, Counterfactuals, pp. 79-80. “∃!w” means as much as “there is exactly one w.” A logically equivalent alternative formulation of (2) in [8] is this: ∃w(w ∈ [A] ∧ w is closer to w* than any other world in [A] ∧ w ∈ [B]). 29

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But, in fact, the similarity of worlds to w* cannot be usefully defined on the basis of all the properties w* has, since it is humanly impossible to carry out a comparison of worlds to such a comprehensive extent. What we can do is this: We compare worlds regarding their similarity to w* on the basis of some finite (preferably smallish) set of w*-properties, P*; w* has all the properties in P*, and other worlds are more or less similar to w* according to how many properties in P* they have. If they have all the properties in P*, then their degree of (P*-)similarity to w* is 1. If they do not have all the properties in P*, then their degree of (P*-)similarity to w* is less than 1; if they do not have any property in P*, then their degree of (P*-)similarity to w* is 0. In fact we can define (under the presupposition that P* is finite): s[P*, w*](w) =Def the ratio of the number of properties in P* that are had by w to the number of the properties in P*. s[P*, w*](w) is the degree of w’s similarity to w*, as defined on the basis of the properties in the set P* of w*-properties. In general, we can define for any worlds w´ and w: s[Pw´, w´](w) =Def the ratio of the number of properties in Pw´ that are had by w to the number of properties in Pw´ (where Pw´ is a finite subset of the set of all the properties of w´), that is: the degree of w’s similarity to w´, as defined on the basis of the properties in the finite set Pw´ of chosen similarity-relevant w´properties. And we can define a comparative similarity (closeness) relation between worlds as follows: w is at least as similar [as close] to w´ as w´´ =Def s[Pw´, w´](w) ≥ s[Pw´, w´](w´´). But the defined relation of comparative similarity between worlds – a realistic relation: a relation people can really make use of – is not likely to have the property of centeredness. In particular, some world w, different from w*, is likely to have all the properties in Pw* (= P*), and therefore: s[Pw*, w*](w) = s[Pw*, w*](w*). The “Centering Assumption” – “that each world i is … closer to itself than any other world is to it” (“Counterfactuals and Comparative Possibility,” p. 10) is one of the assumptions regarding comparative similarity of worlds that Lewis wants to hold on to. The Centering Assumption may be true for Lewis’s “comparative overall similarity of worlds” (Counterfactuals, p. 91; my emphasis), but that comparative similarity relation between worlds is certainly not the one we have in mind (or can, as human beings, have in mind) when we assert counterfactual conditionals – if indeed we have in mind any comparative similarity relation between worlds when we assert counterfactual conditionals. (The Bases-Theory of Conditionals, as stated within the mereology of states of affairs, implies that,

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fundamentally, we do not have in mind any comparative similarity relation between possible worlds when we are asserting counterfactual conditionals; but, for the moment, I set this aside.) About comparative overall similarity Lewis says the following: Overall similarity consists of innumerable similarities and differences in innumerable respects of comparison, balanced against each other according to the relative importances we attach to those respects of comparison. (Counterfactuals, p. 91.)

But this is just to say that we cannot have the slightest inkling of how overall similar a world is to some other world, that, therefore, we decide questions of overall similarity between worlds in an objectively arbitrary way: on the myopic grounds of what we happen to think important. The disproportion between what is judged (objective overall similarity between worlds) and the criteria used in the judging (subjective importance of a few respects of comparison) couldn’t be any greater. I do not believe that the scenario which is suggested by Lewis’s interpretation of counterfactual conditionals comes in any way near to what really goes on when we (sincerely) assert a counterfactual conditional “IfCF A, thenCF B,” with an antecedent we take to be false but possible. When making such an assertion, we do not more or less haphazardly assess the comparative overall similarity of certain possible worlds to the actual world, and then determine that some [A ∧ B]-world is more similar to the actual world than any [A ∧¬B]-world. What we really do is what the Bases-Theory of Conditionals suggests we do: we pick a basis, a certain state of affairs that we believe to obtain, which basis we consider as relevantly connecting the antecedent of the conditional with the consequent (where this relevant connection includes, of course, that the basis logically implies together with the antecedent the consequent of the conditional, but also comprises – albeit only on the subjective side – more than just that).30 But staying, for the time being, within the framework of comparative similarity, we cannot but come to the conclusion (as we have seen) that the Centering Assumption, which Lewis believes in, is not to be upheld for a 30

The Bases-Theory of Conditionals is far from denying that there is such a thing as the vagueness of conditionals. But the vagueness of a conditional is due to the vagueness (for us) of its basis (usually manifested when the conditional is not considered within some context of utterance, but sometimes manifested also within such a context); it is not due to the vagueness (for us) of some relation of comparative similarity between worlds.

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realistic comparative similarity relation between worlds as defined above. Therefore, the incompatibility of the interpretations [4] and [6] with the Centering Assumption (or more accurately speaking: their being made inadequate by that assumption since, under it, they can no longer distinguish counterfactual conditionals from material implications) cannot be urged against them. The “Limit Assumption,” however, which Lewis does not believe in, can be maintained for a realistic comparative similarity relation between worlds. Indeed, it must be maintained for such a relation if it is defined in the way presented above. The Limit Assumption assumes that for any world w´ and any non-empty set of worlds X there is at least one world w in X such that there is no world in X which is closer (more similar) to w´ than w.31 Consider, then, an arbitrary non-empty set of worlds X and an arbitrary possible world w´. Since Pw´ is finite, s[Pw´, w´](w) can only assume finitely many values (because s[Pw´, w´](w) =Def the ratio of the number of properties in Pw´ that are had by w to the number of properties in Pw´). Hence the value of s[Pw´, w´] for some world w in X must be the largest among the values of s[Pw´, w´] for worlds in X, or in other words: ∃w(w ∈ X ∧ ¬∃w´´[w´´ ∈ X ∧ s[Pw´, w´](w´´) > s[Pw´, w´](w)]). Hence: There is a world w in X such that no world in X is more similar (closer) to w´ than w. QED.32 Lewis maintains that interpretation [2] (or “Analysis 2,” as he calls it) “is not yet satisfactory” (“Counterfactuals and Comparative Possibility,” p. 9), his reason being that “it depends on an implausible assumption.” That assumption, he says, is the Limit Assumption – which, however, we have just seen to be not at all implausible, at least not for the realistic relation of comparative similarity between worlds that I have been envisaging (though it seems to be indeed implausible for comparative overall similarity between worlds). But what is perhaps more important: the Limit Assumption is in fact not a part of interpretation [2] (or of “Analysis 2,” formulated by Lewis himself; see “Counterfactuals and Comparative Possibil-

31

Cf. “Counterfactuals and Comparative Possibility,” p. 9. The following assertion – which we might call “Stalnaker’s Assumption” – is logically stronger than the Limit Assumption and (as is desirable) cannot be proven on the basis of the definitions given: For any world w´ and any non-empty set of worlds X there is at least one world w in X which is closer to w´ than any other world in X. Or in other words: For any world w´ and any non-empty set of worlds X there is exactly one world w in X which, of all the worlds in X, is closest to w´. 32

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ity,” p. 7). What, then, can Lewis mean by asserting that this interpretation “depends on an implausible assumption”? The answer seems to be this: If there are A-worlds closer and closer to i without end, then [according to Analysis 2, i.e., interpretation [2]] any consequent you like holds at every closest A-world to i, because there aren’t any. (“Counterfactuals and Comparative Possibility,” p. 9.)

This suggests that interpretation [2] is thought by Lewis to depend on the Limit Assumption in the sense that only this assumption can save that interpretation from generating absurdities. But is it absurd that “IfCF A, thenCF B” is true for any consequent B if there are no worlds in [A] that among all the worlds in [A] are closest to w*? It is not absurd if [A] is empty (i.e., if A is not true at any world). But if [A] is non-empty and there are no worlds in [A] that of all the worlds in [A] are closest to w*, then, indeed, it seems absurd that this by itself should make “IfCF A, thenCF B” true, as it must according to interpretation [2], no matter which sentence the consequent B is. Interpretations [4] and [6], too, can render “IfCF A, thenCF B” true for all consequents B, even if [A] is non-empty. But, in their cases, this behavior is much more understandable. If “IfCF A, thenCF B” is true for all consequents B (and hence also for the consequents ¬A and B ∧ ¬B), even if the antecedent A is true at some world, then this is due according to interpretation [4], and also according to interpretation [6], to an incompatibility of the antecedent with the obtaining basis. If “IfCFL6 A, thenCFL6 B” is true for all consequents B (A being held constant), then “IfCFL6 A, thenCFL6 ¬A” is true, and the latter is true if, and only if, bL6∩[A] is the empty set (as we have already seen). And in any case: w* ∈ bL6. Hence: if – and only if – “IfCFL6 A, thenCFL6 B” is true for all consequents B, we have: bL6∩[A] = ∅ ∧ w* ∈ bL6 – which, from the point of view of interpretation [6], means precisely that the antecedent of the counterfactual is incompatible with its obtaining basis. If “IfCFL4 A, thenCFL4 B” is true for all consequents B (A being held constant), then “IfCFL4 A, thenCFL4 ¬A” is true, and the latter is true if, and only if, w* ∈ bL1(A, ¬A, w*) [= {w: ∀w´(w´ ∈ [A] ⊃ w is closer to w* than w´)}]. And in any case: bL1(A, ¬A, w*)∩[A] = ∅, since bL1(A, ¬A, w*)∩[A] ⊆ [¬A]; the latter assertion is simply an instance of the above central theorem: bL1∩[A] ⊆ [B], or more explicitly: bL1(A, B, w*)∩[A] ⊆ [B], where bL1(A, B, w*) = {w: ∀w´(w´ ∈ [A ∧ ¬B] ⊃ w is closer to w* than w´)}. Hence: if – and only if – “IfCFL4 A, thenCFL4 B” is true for all consequents B, we have: bL1(A, ¬A, w*)∩[A] = ∅ ∧ w* ∈ bL1(A, ¬A, w*) – which, from the point of

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view of interpretation [4], means precisely that the antecedent of the counterfactual is incompatible with its obtaining basis.

Summing up: Interpretation [2] without the Limit Assumption can render counterfactual conditionals true for every consequent combined with a given antecedent, even if the antecedent is true at some world; but this behavior is not well understood in the case of [2] (whereas it is well understood in the case of [4] and of [6]). Interpretation [2] with the Limit Assumption, on the other hand, cannot render counterfactual conditionals true for every consequent combined with a given antecedent if the antecedent is true at some world; nor can it, if the antecedent is true at some world, render “IfCF A, thenCF ¬A” or “IfCF A, thenCF B ∧ ¬B” true, thus showing the very same failing that interpretations [1], [3], [7], and [8] are afflicted with (but not interpretations [4] and [6]). Hence, with or without the Limit Assumption, interpretation [2] is at a disadvantage when compared with interpretations [4] and [6]. Therefore, among the eight interpretations that have been considered in this section within the (set-theoretical) framework based on comparative similarity between worlds, interpretations [4] and [6], which are in the spirit of the Bases-Theory of Conditionals, emerge as the relatively best interpretations of counterfactual conditionals. An even better – because more general and conceptually simpler – interpretation of counterfactual conditionals is provided by the Bases-Theory of Conditionals itself. For understanding counterfactual conditionals and for assigning truth-values to them, we need not compare possible worlds (least of all the huge physical objects David Lewis thought possible worlds are). The truth or falsity of a counterfactual conditional (as uttered in a certain context) is a matter of intensional and (in a broad sense) mereological relationships between four states of affairs: the state of affairs expressed by its antecedent, the state of affairs expressed (in the context) by its consequent, the state of affairs w*, and the state of affairs which is the basis of the conditional (in the given context of utterance).

6.6

Is Logical Necessity Verbal Necessity?

I subscribe to the following credo: There is no irreducible ontic necessity other than logical necessity (and Hume is its prophet). My reason is this: every ontic necessity other than logical necessity reduces to logical-

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necessity-cum-a-certain-basis; the exact manner of reduction is spelled out in the Bases-Theory of Modality, of which the Bases-Theory of Conditionals is a proper part. Logical necessity, in turn, might be taken to be reducible to intensional parthood between states of affairs (given the mereology of states of affairs; see Chapter 3). But, in fact, I do not believe that either of the following two definitions reflects the order of conceptual priority more faithfully than the other: 1(x) =Def P(x, t*); P(x, y) =Def S(x) ∧ S(y) ∧ 1(that (O(y) ⊃ O(x))). In the second definition, “1” – the predicate of logical necessity – is a primitive term, and the theoretical background must be modified accordingly; that background cannot remain the mereology of states of affairs as presented in Chapter 3.

The first definition does without primitive modality. But that does not make it reflect the order of conceptual priority more faithfully than the second. Therefore, though logical necessity is formally reducible to intensional parthood, and has been reduced to the latter notion in Chapter 3, there is no deeper philosophical motivation for this reduction than the mere fact that it is formally feasible. My position in the philosophy of modality is in some respects similar to the stance that is adopted by the nominalists of Bas van Fraassen (who are not so very unlike van Fraassen himself): Are there necessities in nature? The nominalists, and subsequently the empiricists, answered that all necessities are reducible to logical necessity (taken broadly, to include necessity ex vi terminorum). What is physically necessary is the same, on this view, as what is logically implied by some tacit antecedent – say, the laws of physics. (“The Only Necessity Is Verbal Necessity,” p. 71.) The nominalists’ first and basic move … is to say that all natural necessities are elliptic for conditional verbal necessities. This sheet on which I write must burn if heated, because it is paper – yes. But the only necessity that is really there is that all paper must burn when heated. This is so, but means only that we would not call something ‘paper’ if it behaved differently. … There are technical difficulties for logicians in making sense of this move; but when sufficiently refined, the position that all non-verbal necessities are ellipses for conditional necessities ex vi terminorum can be held. (“Essence and Existence,” p. 1.) At [the] bottom level the only necessity we can countenance [according to the paradigm nominalist] is purely logical or verbal necessity which, like God, is no respecter of persons. In this modality, whatever Peter can do, Paul can do also. … According to this nominalist then, pure logical necessity is no respecter of

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persons; physical necessities and necessities de re are impurities produced by our theorizing activity. (“Essence and Existence,” pp. 14-15.)

Indeed, all non-logical ontic necessities are ellipses for conditional logical necessities,33 since the following is provable in the mereology of states of affairs: ∀x[S(x) ⊃ (n(x) ≡ 1(that (O(bn) ⊃ O(x))))], with the following corollary (in view of P13, P10, and the definitions of the sentenceconnectives nC, 1C corresponding to the predicates of necessity n(y), 1(y)): nA ≡ 1(O(bn) ⊃ A), or in other words: nA ≡ O(bn) 1→ A. But I do not agree with van Fraassen’s nominalists that logical necessity is verbal necessity. In fact, “logical necessity” is something of a misnomer for what is meant by that term; for the necessity called “logical necessity” is only accidentally connected with the logoi, the words. Better terms for designating that necessity are “inner necessity,” “intrinsic necessity,” or “contentnecessity”; the chances that these designations might catch on and gain a general currency are, however, infinitesimal. Howsoever it is called, logical necessity is, in my view, not a product of convention, as the nominalists would have it; it is not of our making but objectively given in rebus (i.e., in individuals, properties, relations, states of affairs). In other words, I am a realist about logical necessity. And I am a reductionist (in the above-described manner) about all other ontic necessities – which does not preclude my being a realist about them, too. If b2 – the conjunction of all the laws of nature – can be objectively selected from the universe of states of affairs, then nomological necessity is certainly a necessity in rerum natura, and at least as real as electrons. But the big question is whether b2 is indeed precisely specifiable in an objective way, or whether, on the contrary, it can only be specified under the influence of a large dosage of convention, produced by human interests.

My reasons for distinguishing logical from verbal necessity are the following:

33

This is also true of nomological necessity, in spite of the difficulty raised in Section 4.5. But certainly not all necessities are ellipses for conditional logical necessities: knowledge – epistemic necessity – is not. For this reason, it is not treatable within the Bases-Theory of Necessity; see Section 3.7. In fact, no necessity that does not satisfy A ⊃ A or that does not satisfy ¬A ⊃ ¬A is an ellipsis for a conditional logical necessity; accordingly, no such necessity is treatable within the Bases-Theory of Necessity. Note that a non-ontic necessity – doxastic necessity – satisfies the two principles and is treatable within the Bases-Theory of Necessity (see Section 3.7).

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(1) Unlike verbal necessity (and, in fact, like God), logical necessity is a respecter of persons, and unlike verbal necessity, logical necessity is de re. It is logically necessary that Peter has the property of being identical with Peter, whereas it is not logically necessary that Paul has the property of being identical with Peter (Peter and Paul being different persons). In fact, it is logically necessary that Paul does not have the property of being identical with Peter. Thus, even in the modality of logical necessity, not everything that Peter can do (so to speak), Paul can do also. But perhaps verbal necessity is in fact also de re and also a respecter of persons? If so, van Fraassen’s nominalists have promulgated falsehoods about verbal necessity, but the chances that logical necessity is verbal necessity have become better. Further reasons for distinguishing logical necessity from verbal necessity seem to be needed. Here are two other such reasons (the first presented under (2), the second under (3)): (2) It is logically necessary that Peter has the property of being identical with Peter. Does this merely mean that we would not call somebody “Peter” if he did not have the property of being identical with Peter? It is also logically necessary that what is entirely red is not entirely blue. Does this merely mean that we would not call something “entirely red” if it were entirely blue? That we would not call somebody “Peter” if he did not have – meaning: if we knew him to lack – the property of being identical with Peter, and that we would not call something “entirely red” if it were – meaning: if we knew it to be – entirely blue are convention-contingent linguistic consequences of the stated logical necessities; but they do not constitute those logical necessities. For linguistic convention can change, logical necessity cannot. Suppose Paul were now called “Peter” by all of us. Would the fact that all of us now call somebody “Peter” who lacks the property of being identical with Peter abolish the logical necessity of Peter (i.e, this person) having that property? Surely not. And suppose that some objects that are entirely blue are now called “entirely red” by all of us. Would the fact that all of us now call some things “entirely red” which are entirely blue abolish the logical necessity that whatever is entirely red (i.e., has this color) is not entirely blue? Again, surely not. If there were no such thing as language and attendant conventions of calling things by this or that term (be it a singular or a general term), it would still be logically necessary (i.e., content-necessary) that Peter has the property of being identical with Peter, and that what is entirely red is not entirely blue.

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(3) Sometimes logical necessity, though normally conveyed in full by words, goes further than the words go: sometimes it cannot be fully conveyed by words – cannot be entirely unfolded in them – though enough of it can be conveyed by words as to make a sentence recognizable as true by logical necessity. Let “b*” (unlike “w*”34) stably (or rigidly) designate a certain possible word (a maximal-consistent state of affairs), and let A be an arbitrary sentence. Then we have (that is, it is provable in the mereology of states of affairs): 1(O(b*) ⊃ A) ∨ 1(O(b*) ⊃ ¬A). Suppose, by choosing b* in the right way, ¬1(O(b*) ⊃ Al Gore is not elected President of the United States in the year 2000). Hence: 1(O(b*) ⊃ Al Gore is elected President of the United States in the year 2000). So here we have a logical necessity: the sentence “O(b*) ⊃ Al Gore is elected President of the United States in the year 2000” is a logically necessary truth (in the broad sense here intended). But is it a verbal necessity? It is certainly not a necessity ex vi terminorum in a literal sense. The words in the sentence “O(b*) ⊃ Al Gore is elected President of the United States in the year 2000” and the particular way in which they are concatenated do not make it somehow necessary that that sentence is true, nor is there any special convention to the effect that that sentence is necessarily true. It is correct to say that the sentence is made necessarily true by what its words mean, by how they are concatenated in it, and by what the names in it refer to (in virtue of their meaning). However, this does not imply that the sentence is a verbal necessity: that it is necessarily true ex vi terminorum in a literal sense. It might be said to be necessarily true ex vi terminorum in a non-literal sense (in a sense that takes the words themselves – metaphorically – for what the words mean and designate). But it would be altogether better to avoid this catchy Latin phrase – or to replace it by another, more accurate one: the sentence we have been considering is necessarily true ex repletione terminorum. It needs to be added: the repletio – or content – of part of that sentence cannot be made fully explicit in words, and therefore that sentence is not only not a verbal necessity, but its logical necessity cannot even be fully conveyed by words. The reason is this: a possible world – for example, b* – can be named, but not fully expressed (in spite of the fact that, being a state of affairs, it has the right category for being expressed); it is just too big for that. In other words, we cannot replace “O(b*)” in “O(b*) ⊃ Al Gore is elected President of the United States in the year 2000” by a logi34

See Section 3.9.

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cally equivalent sentence that does not contain at least implicitly the name “b*” or any other name that refers to what “b*” refers to. The content of “O(b*)” – i.e., the state of affairs that O(b*), which state of affairs is identical with b* itself according to P24 – cannot be made fully explicit in words (in contrast to the content of the entire aforementioned material implication, of which “O(b*)” is the antecedent; for this latter content is simply t*). And therefore: the logical necessity of “O(b*) ⊃ Al Gore is elected President of the United States in the year 2000” cannot be fully conveyed by words (in contrast to the logical necessity of “Mary is taller than Anne at t0 ⊃ Anne is not taller than Mary at t0”).

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The Onto-Nomological Theory of Modality Extended: Adding Properties and Individuals

The theory of the modal aspects of properties of individuals is a natural outgrowth of the Bases-Theory of Necessity (of which the Bases-Theory of Conditionals is a special part), once the mereology of states of affairs has been extended by certain axioms and definitions regarding properties and individuals. I begin by stating that extension. But first of all, before language and axiom system will be significantly enlarged, something that is implicit in the system already presented (see Chapter 3) must be made explicit. (So far, it did not need to be made explicit.) Those axiomatic principles of that system which involve schematic letters are supplemented by the corresponding fully quantified schematic principles. In this way, the schemata are made to fully exhibit their implicitly intended generality. Thus we have: P4

∃z[S(z) ∧ ∀x(S(x) ∧ A[x] ⊃ P(x, z)) ∧ ∀y(S(y) ∧ ∀x(S(x) ∧ A[x] ⊃ P(x, y)) ⊃ P(z, y))]; ∀u1...∀un∃z[S(z) ∧ ∀x(S(x) ∧ A[u1, ..., un, x] ⊃ P(x, z)) ∧ ∀y(S(y) ∧ ∀x(S(x) ∧ A[u1, ..., un, x] ⊃ P(x, y)) ⊃ P(z, y))].

P6

∀x[P(x, CONJyA[y]) ∧ ¬M(x) ⊃ ∃k´(P(k´, x) ∧ ¬M(k´) ∧ ∃z(P(k´, z) ∧A[z]))]; ∀u1...∀un∀x[P(x, CONJyA[u1, ..., un, y]) ∧ ¬M(x) ⊃ ∃k´(P(k´, x) ∧ ¬M(k´) ∧ ∃z(P(k´, z) ∧ A[u1, ..., un, z]))].

P10 A ≡ O(that A); ∀u1...∀un(A[u1, ..., un] ≡ O(that A[u1, ..., un])). P13 S(that A); ∀u1...∀unS(that A[u1, ..., un]). P14 that ¬A = neg(that A); ∀u1...∀un(that ¬A[u1, ..., un] = neg(that A[u1, ..., un])).

7 The Onto-Nomological Theory of Modality Extended

P15 that (A ∧ B) = conj(that A, that B); ∀u1...∀un∀u´1...∀u´n´(that (A[u1, ..., un] ∧ B[u´1, ..., u´n´]) = conj(that A[u1, ..., un], that B[u´1, ..., u´n´])), where either the u-sequence or the u´-sequence may also be empty. P16 follows suit. All instances of these schemata are taken to be sentences proper, that is, sentences without free variables. Alternatively, one can let letters “A,” “B,” etc. represent not only sentences proper, but also so-called open sentences, which contain at least one free variable. (In consequence, “A[x]” and “A[y]” in P4 and P6 need not be taken to contain only the free variable “x,” respectively “y.”) Then the formulation of P4, P6, P10, P13 – P16 (and also of P23) can stay as it was in Chapter 3; but in order to exhibit the full content of these schemata, one needs to add a special rule: if A[y] is an axiom, then ∀yA[y] is also one. Note that this rule is not already a consequence of the following rule: if A[y] is logically provable, so is ∀yA[y], that is: ├ A[y] ⇒ ├ ∀yA[y], nor (a fortiori) of the following more general rule (which we have been employing all along): ├ T ⊃ A[y] ⇒ ├ T ⊃ ∀yA[y], if y does not occur free in T.

If one reserves the use of letters “A,” “B,” etc. (without an indicated free variable in square brackets, as in “A[x]”) for the representation of sentences proper – which is the practice observed in this book –, then, in addition to supplementing P4, P6, P10 and P13 – P16 (as shown above), one needs to supplement the rule-schema P23 (introduced in Chapter 3) in the following way, thus making explicit its full content: P23 ├ B ⇒ ├ t* = that B; ├ B[y1, …, yn] ⇒ ├ t* = that B[y1, …, yn]. In consequence of this, the most important corollary of P23, EQU*, becomes derivable in its supplemented form, which version of it I also present here (in view of the frequency EQU* is being made use of): EQU*

├ A ≡ B ⇒ ├ that A = that B; ├ A[z1, …, zk] ≡ B[z´1, …, z´k´] ⇒ ├ that A[z1, …, zk] = that B[z´1, …, z´k´], where either the zsequence or the z´-sequence may also be empty.

Now we are ready for further construction of ontological theory, focusing on properties of individuals.

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7.1

Individuals, Properties, and Exemplification

The language in which the mereology of states of affairs is formulated (see Chapter 3) – called “the intended language”; cf. the beginning of Section 4.3 – is extended by adding a new formal predicate to it: I(x) – “x is an individual.” Moreover, the intended language is extended by adding to its basic expressions the dyadic functor [x, y], called “the saturator.” The saturator forms singular terms from singular terms; “[x, y]” is read, in full, as “the predicative saturation of x by y,” or, abbreviated, as “the saturation of x by y.” The function [x, y] will yield a value if x is predicatively saturable by y; it will also yield a value, albeit an artificial one, in those cases where x is not predicatively saturable by y. That artificial value is c*, which is not an individual (nor a state of affairs, nor a property, as we shall soon see; in fact, let c* be an entity that is as alien to the present purposes as it can be). The constant “c*” is another new basic expression (herewith officially adopted, after making its first appearance in footnote 12 of Chapter 3). Then, in continuation of the numbered basic principles (axioms) in Chapter 3, we have: P25 ∀x∀y(¬F(x, y) ⊃ [x, y] = c*). P26 ¬I(c*). It is not necessary to add the conjunct ¬S(c*) to ¬I(c*): given the remainder of the theory, ¬S(c*) is provable. See the remarks below D17.

The unfamiliar predicate “F(x, y)” occurring in P25 and P26 is to be read as “x is predicatively saturable by y,” or in other words: “x fits y predicatively,” or again in other words: “x is predicable of y.” Usually we shall read “F(x, y),” in abbreviation, simply as “x fits y.” Note that “F(x, y)” is not a basic predicate; it will be defined below. Meanwhile, we have as further axioms: P27 ∀x(I(x) ⊃ ¬S(x)). P28 ∀x(S(x) ∨ I(x) ∨ x = c* ⊃ ¬∃yF(x, y)).1 1

It should be remembered that, in order to save brackets, the binding-strength of sentence-connectives is taken to diminish in the sequence ¬, ∧, ∨, ⊃, ≡ from left to right.

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According to P27, no individual is a state of affairs. According to P28, neither c* nor states of affairs nor individuals fit any entity predicatively. The following definition – in continuation of the numbered definitions in Chapter 3 – gives definite content to a claim of predicative fit: according to that definition, x fits y predicatively if, and only if, the predicative saturation of x by y is a state of affairs. D17 F(x, y) =Def S([x, y]). ¬S(c*) is a logical consequence of the given principles. This is seen as follows. According to P28: ¬F(c*, c*), that is (according to D17): ¬S([c*, c*]). According to P25, we have on the basis of ¬F(c*, c*): [c*, c*] = c*. Hence: ¬S(c*). Given this and D17, it is easily seen that we have: ∀x∀y(F(x, y) ⊃ [x, y] ≠ c*). Therefore, using P25, we obtain the theorem ∀x∀y(F(x, y) ≡ [x, y] ≠ c*). We are now in a position to define the concept of a property, and the concept of a property of individuals: D18 PR(x) =Def ∃yF(x, y). D19 PRI(x) =Def ∃yF(x, y) ∧ ∀y(F(x, y) ⊃ I(y)). According to D18, a property is an entity that fits some entity (predicatively); and according to D19, a property of individuals is a property that fits only individuals. It is an immediate consequence of P28 and D18 that states of affairs, individuals and c* are not properties, and a fortiori not properties of individuals. That a property x fits an entity y (predicatively), that x is predicable of y – this does not yet mean that x applies (as a property) to y, that y has x (as a property), that y instantiates or exemplifies (property) x. But the definition of exemplification (of a property) is close at hand: D20 EXM(y, x) =Def O([x, y]). Moreover, in ∧-sequences and ∨-sequences, brackets will also be omitted, as will be all outer brackets. Usually, the brackets around the identity-predicate will also be omitted, writing “x = y” instead of “(x = y).”

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In view of D6, the definiens of D20 is definitionally equivalent to “S([x, y]) ∧ A([x, y]),” and therefore, in view of D17, also definitionally equivalent to “F(x, y) ∧ A([x, y]).” Hence, in view of D18, we have as an elementary theorem: ∀x∀y(EXM(y, x) ⊃ PR(x) ∧ F(x, y)) – “If y exemplifies x, then x is a property that fits y.” Should one assume additional axiomatic principles for predicative fit? Once particular properties are introduced (see Section 7.4 below), it will be necessary to characterize their predicative fit. But these locally assumed characterizations, though in a way first principles, will not be elevated to the status of numbered principles (axioms). Moreover, there will have to be principles that describe how the predicative fit of complex properties depends on the predicative fit of the properties that enter into their composition. But I will not here develop this important theme to any considerable extent. Finally, the following general principle of predicability (among others) suggests itself: ∀x∀y(F(x, y) ∨ ∃z1...∃zn(F(x, z1) ∧ ... ∧ F(zn, y)) ⊃ ¬F(y, x)), for all n ≥ 1. Particularly salient consequences of the envisaged principle of predication are the following: ∀x∀y(F(x, y) ⊃ ¬F(y, x)), ∀x¬F(x, x), and ∀x∀y∀z(F(x, z) ∧ F(z, y) ⊃ ¬F(y, x)). In view of ∀x∀y(EXM(y, x) ⊃ PR(x) ∧ F(x, y)), these three consequences imply three further ones: ∀x∀y(EXM(y, x) ⊃ ¬EXM(x, y)), ∀x¬EXM(x, x), and ∀x∀y∀z(EXM(y, z) ∧ EXM(z, x) ⊃ ¬EXM(x, y)). If these statements were true, then the predicate F(x, y) would be, with regard to the logical structure of the statements, like the predicate “x is father of y,” and the predicate EXM(y, x) like the predicate “y is fathered at t0 by x,” and consequently the relations of exemplification and predicability would induce a certain hierarchy in their respective relational fields (the relational field of exemplification being included in the relational field of predicability). But although the Principle of Predicability has much that can be said in its favor, I shall refrain from assuming it here, noting that the property of self-identity seems to be a clear case of a self-exemplifying and hence self-predicable entity. ∀x∀y∀z(F(x, z) ∧ F(z, y) ⊃ ¬F(y, x)) and ∀x∀y∀z(EXM(y, z) ∧ EXM(z, x) ⊃ ¬EXM(x, y)) are each contradicted by EXM(si, si) (where “si” is short for “selfidentity”). From EXM(si, si) – hence F(si, si) – and those general statements, one ob-

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tains: ∀z(F(si, z) ⊃ ¬F(z, si)) and ∀z(EXM(si, z) ⊃ ¬EXM(z, si)), and hence: ¬F(si, si) and ¬EXM(si, si) – contradiction.

One might, of course, deny that the name “self-identity” refers to what it seems to refer to (namely, to a certain self-exemplifying property). But I prefer to leave both the question of the reference of that name – even regarding the intended universe of discourse, which, in view of the present purposes, can be a rather restricted one – and the question of the truth of the Principle of Predicability open questions. Consider, however, that there do seem to be universally predicable properties (see the next section), that is: properties x for which ∀yF(x, y) is true. If there are indeed such properties, then they are self-predicable (which does not automatically mean that they are also self-exemplifying!) and the Principle of Predicability will be contradicted. Hence we have another reason – and a stronger one – not to assume that principle. Before moving on to property-theoretic particulars, I add a note of limitation. In this book, I will stick to the ontological treatment of properties, not extending it to a treatment of relations. The reason for this is that properties are absolutely central to a theory of modality, while relations are not, and I do not wish to make matters more complex than they need be. It should be noted, however, that the basic property-theoretic concepts can easily be generalized to apply to relations: the 1+1-adic saturator [x, y] is, obviously, just one member of an infinite family of 1+n-adic saturators [x, y1, …, yn]; the monadic abstractor λoA[o] (see Section 7.4 below) is, obviously, just one member of an infinite family of n-adic abstractors λo1…onA[o]. By using the larger conceptual toolbox, the purely propertytheoretic principles and definitions presented in this book can easily be replaced by principles and definitions that in addition to properties apply to relations of any adicity.

7.2

The Identity of Properties, Essential Properties, and Sets

The following principle – which is of central importance – states the (sufficient) identity condition for properties: P29 ∀x∀y(PR(x) ∧ PR(y) ∧ ∀z(F(x, z) ≡ F(y, z)) ∧ ∀z(F(x, z) ∧ F(y, z) ⊃ [x, z] = [y, z]) ⊃ x = y).

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According to P29, properties are identical if they fit the same entities and if their respective predicative saturations by any entity that they both fit are identical. P29 is a principle that will be needed again and again. Concerning essential properties, I will first address the logical essentiality of properties. Consider the following definition: D21 PRE1(x) =Def PR(x) ∧ ∀y(F(x, y) ⊃ [x, y] = t* ∨ [x, y] = k*). D21 defines what it is to be a logically essential property absolutely. D21 can also be read as the definition of what it is to be an extensional property. That this double interpretation of PRE1(x) (as defined in D21) is possible will become apparent below and in Section 7.4. D22, in contrast, defines what it is to be a logically essential property of an entity, or in other words: what it is to exemplify a property in a logically essential manner: D22 EXME1(y, x) =Def [x, y] = t*. The following is an obvious consequence of D20 and D22 and of the mereology of states of affairs (developed in Chapter 3): ∀x∀y(EXME1(y, x) ⊃ EXM(y, x)). The converse of this is not provable, and shouldn’t be. In fact, the negation of the converse, ∃x∃y(EXM(y, x) ∧ ¬EXME1(y, x)) [“There is exemplification that is not logically essential exemplification”], will become provable further on. In this book, the essentiality-predicates used in philosophy: “x is an essential property” and “x is an essential property of y” – essentialitypredicates simpliciter, containing the word “essential” without qualification – are taken (insofar as they have a sufficiently clear meaning at all) to be synonymous with the predicates “x is a logically essential property” and “x is a logically essential property of y.” Fitting the intentions of many philosophers, one might also posit “x is an [simpliciter] essential property” and “x is an [simpliciter] essential property of y” to be synonymous with “x is a metaphysically essential property” and “x is a metaphysically essential property of y.” But the basis of metaphysical necessity may well be said to be not sufficiently clear for making such a move advisable, if, corresponding to the intentions of those who use the notion, metaphysical necessity is taken to be different from both nomological and logical necessity. (For more on metaphysical necessity and its basis, see Section 4.6.)

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It is an unusual perspective on them, but sets can be identified with universally predicable logically essential properties: D23 SET(x) =Def ∀yF(x, y) ∧ PRE1(x). The corresponding definition of elementhood is not far to seek: D24 y ∈ x =Def SET(x) ∧ EXME1(y, x). Using D21 – D24 and P29, we can prove the identity principle for sets, the so-called principle of extensionality: ∀x∀y(SET(x) ∧ SET(y) ∧ ∀z(z ∈ x ≡ z ∈ y) ⊃ x = y). Proof: Assume SET(x), SET(y), ∀z(z ∈ x ≡ z ∈ y), that is (by employing the relevant definitions): ∀zF(x, z) ∧ PR(x) ∧ ∀z(F(x, z) ⊃ [x, z] = t* ∨ [x, z] = k*) ∧ ∀zF(y, z) ∧ PR(y) ∧ ∀z(F(y, z) ⊃ [y, z] = t* ∨ [y, z] = k*) ∧ ∀z([x, z] = t* ≡ [y, z] = t*]). From this, we can gather: (a) PR(x) ∧ PR(y), (b) ∀z(F(x, z) ≡ F(y, z)), (c) ∀z(F(x, z) ∧ F(y, z) ⊃ [x, z] = [y, z]) [employing the theorem t* ≠ k*]. Hence by applying P29: x = y.

Clearly, in view of D23, what sets there are depends on what universally predicable properties there are. Note that to every universally predicable property x1 there corresponds a set x2 in the following sense: ∀z(EXM(z, x1) ⊃ [x2, z] = t*), ∀z(¬EXM(z, x1) ⊃ [x2, z] = k*). This relationship – or rather the existence-assumption implicit in it – will be formally codified in Section 7.4 (it is a consequence of P31). Besides sets simpliciter, as defined in D23, one can also consider sets with restricted predicability, for example, sets of individuals, understood (near enough) in the sense of the theory of types. Sets of individuals in that sense are not (defined as) universally predicable sets whose only elements are individuals; rather, sets of individuals in the type-theoretic sense and the corresponding notion of elementhood are captured by the following three definitions: D25 PRIE1(x) =Def PRI(x) ∧ ∀y(F(x, y) ⊃ [x, y] = t* ∨ [x, y] = k*). D26 SETI(x) =Def ∀y(I(y) ⊃ F(x, y)) ∧ PRIE1(x).

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D27 y ∈I x =Def SETI(x) ∧ EXME1(y, x). D25 defines what it is to be a logically essential property of individuals – where, note, a property of individuals is not (defined as) a universally predicable property that is exemplified only by individuals, and is not a property exemplified by at least some individuals, but must be considered precisely in the sense of PRI(x), as defined by D19. D26 identifies sets of individuals (in the type-theoretic sense) with those logically essential properties of individuals that are predicable of all individuals. D27 defines elementhood that is specific for sets of individuals. The definitions D25, D26 and D27 make it possible to prove the following principle of extensionality that is specific for sets of individuals: ∀x∀y(SETI(x) ∧ SETI(y) ∧ ∀z(z ∈I x ≡ z ∈I y) ⊃ x = y). In analogy to the relationship between sets and universally predicable properties, we have: what sets of individuals there are depends on which properties of individuals are predicable of all individuals. Note that to every property of individuals x1 that is predicable of all individuals – i.e., to every “totally defined” property of individuals – there corresponds a set of individuals x2 in the following sense: ∀z(EXM(z, x1) ⊃ [x2, z] = t*), ∀z(I(z) ∧ ¬EXM(z, x1) ⊃ [x2, z] = k*). (This, too, is a consequence of P31 in Section 7.4.) Thus, the relationship between sets of individuals (in the type-theoretic sense) and properties of individuals that are predicable of all individuals is an obvious analog of the relationship (stated above) between sets (simpliciter) and universally predicable properties. On the basis of these two relationships and the corresponding principles of extensionality (stated above), we can prove: ∀x1[PRI(x1) ∧ ∀y(I(y) ⊃ F(x1, y)) ⊃ ∃!x2(SETI(x2) ∧ ∀z(z ∈I x2 ≡ EXM(z, x1)))]. ∀x1[PR(x1) ∧ ∀yF(x1, y) ⊃ ∃!x2(SET(x2) ∧ ∀z(z ∈ x2 ≡ EXM(z, x1)))].2

∃!xA[x] means: there is exactly one x such that A[x], which means in turn: ∃xA[x] ∧ ∀x∀y(A[x] ∧ A[y] ⊃ x = y).

2

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Logical essentiality is not the only kind of essentiality. In fact, there are as many modal kinds of essentiality as there are bases of necessity that are referred to in a stable way. I presuppose that all basis-terms “bn” which are used in connection with essence and essentiality (either overtly or in a hidden manner: in defined expressions) are referentially stable (thus “b10” cannot be used in that connection; see Section 5.1). Consider, then, the following two definition-schemata: D28 PREn(x) =Def PR(x) ∧ ∀y(F(x, y) ⊃ P([x, y], bn) ∨ P(neg([x, y]), bn)) (for [appropriate] n > 1). D29 EXMEn(y, x) =Def P([x, y], bn) (for n > 1). “1” is excluded from being substitutable into these definition-schemata for the sole reason that the predicates PRE1(x) and EXME1(y, x) have already been defined above: in D21 and D22. (The substitution of “1” in D28 and D29 would yield definitions that are provably equivalent to the definitions D21 and D22.) PREn(x) is read as “x is an n-essential property,” for n > 1 and also for n = 1. The 1-essential properties are here also called “logically essential properties” (see above), and the 2-essential properties are here also called “nomologically essential properties.” EXMEn(y, x), in turn, is read as “x is an n-essential property of y” or, in other words, as “y exemplifies x nessentially” (for n ≥ 1). Instead of “x is a 1-essential property of y” and “y exemplifies x 1-essentially,” one can also say: “x is a logically essential property of y” and “y exemplifies x in a logically essential manner.” Instead of “x is a 2-essential property of y” and “y exemplifies x 2essentially,” one can also say: “x is a nomologically essential property of y” and “y exemplifies x in a nomologically essential manner.” Here follows an important example of an application of the predicates EXME1(y, x), EXME2(y, x), and EXM(y, x). Consider the individual U.M. and the property of being physical, φ. One might hold EXME1(U.M., φ), and therefore also EXME2(U.M., φ) and EXM(y, x) – option 1; or one might hold EXME2(U.M., φ), and therefore also EXM(y,x), and deny EXME1(U.M., φ) – option 2; or one might deny EXME2(U.M., φ), and therefore also EXME1(U.M., φ), and nevertheless hold EXM(U.M., φ) – option 3; or, finally, one might deny EXM(U.M., φ), and therefore also EXME2(U.M., φ) and EXME1(U.M., φ) – option 4. Only the options 1 and 4

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remain adoptable if the following principle is taken to be true: ∀y(EXM(y, φ) ⊃ EXME1(y, φ)). Option 1 does not automatically conflict with (psychophysical) dualism if dualism is taken in a certain sense: one might hold that U.M., though physical in a logically essential manner, has necessarily (according to some necessity faithful to truth) also a non-physical side (just like all other human beings). And option 4 does not automatically conflict with (psychophysical) physicalism if physicalism is taken in a certain sense: one might hold that U.M., though not a physical entity, has necessarily (according to some necessity faithful to truth) also a physical side (just like all other human beings). If, however, “physical” is construed as “entirely physical,” then option 1 conflicts unconditionally with dualism; and, alternatively, if “physical” is construed as “at least partly physical,” then option 4 conflicts unconditionally with physicalism.

7.3

Essences – and Parthood for Properties

Essence is a relative concept: an essence is always the essence of something. I shall consider essences of two categorial kinds: essences which are universally predicable properties, in short: U-essences, and essences which are properties of individuals and predicable of all individuals, in short: Iessences. In each of these two categorial kinds of essence, there are as many modal kinds of essence as there are modal kinds of essentiality (and there are as many modal kinds of essentiality as there are bases of necessity). The general definition-schema for all modal kinds of U-essence is this: D30 ESSUn(x, y) =Def EXMEn(y, x) ∧ ∀zF(x, z) ∧ ∀x´(EXMEn(y, x´) ∧ ∀zF(x´, z) ⊃ PPR(x´, x)) (for n ≥ 1). In other words: x is a n-U-essence of y if, and only if, (1) x is an nessential property of y, (2) x is universally predicable, and (3) all universally predicable n-essential properties of y are intensional parts of x. 1-Uessence – its definition is obtained by substituting in D30 “1” for “n” – is also called logical U-essence, or universal logical essence; 2-U-essence – its definition is obtained by substituting in D30 “2” for “n” – is also called nomological U-essence, or universal nomological essence.

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The general definition-schema for all modal kinds of I-essence (including the logical and the nomological kind: 1-I-essence, 2-I-essence) is this: D31 ESSIn(x, y) =Def EXMEn(y, x) ∧ ∀z(F(x, z) ≡ I(z)) ∧ ∀x´(EXMEn(y, x´) ∧ ∀z(F(x´, z) ≡ I(z)) ⊃ PPR(x´, x)) (for n ≥ 1). The difference between D31 and D30 is the range of predicability (or extent of predicative fit) of the defined essence of x, and of the essential properties of x that are encompassed by that essence; this range of predicability is stated, respectively, by ∀zF(x, z) and ∀z(F(x, z) ≡ I(z)). The latter condition specifies a range of predicability that is clearly less than the universal range specified by the former condition; this is even provable, it being already provable that there are some entities (namely, states of affairs) that are not individuals. Both D30 and D31 contain an as yet undefined predicate: PPR(x´, x) – “property x´ is an intensional part of property x.” Here is the definition of that predicate: D32 PPR(x´, x) =Def PR(x´) ∧ PR(x) ∧ ∀y(F(x´, y) ≡ F(x, y)) ∧ ∀y(F(x´, y) ⊃ P([x´, y], [x, y])). According to D32, intensional parthood between properties is reducible, via the saturation-function (expressed by the saturator), to intensional parthood between states of affairs. For illustration, consider the properties being a human being and being a woman. These two properties are undoubtedly predicable of the same entities (which, to repeat, does not mean that they are exemplified by the same entities – which, of course, they are not). And for every entity y of which being a human being is predicable, we have: the saturation of being a human being by y is an intensional part of the saturation of being a woman by y: P([being a human being, y], [being a woman, y]). Thus the property of being a human being is an intensional part of the property of being a woman.

Another important feature of D32 is that intensional parthood can only hold between properties x´ and x that are equivalent regarding predicability, i.e., which are such that ∀y(F(x´, y) ≡ F(x, y)). The justification for this is the following: parthood requires sameness of ontological category of its

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relata (see Section 4.2), and properties have the same ontological category if, and only if, they are equivalent regarding predicability. Here are some important theorems that are provable on the basis of D32: ∀x´∀x(PPR(x´, x) ∧ PPR(x, x´) ⊃ x´ = x). Proof: Assume PPR(x´, x) and PPR(x, x´). According to D32 we can conclude from the assumption: PR(x´), PR(x), ∀y(F(x´, y) ≡ F(x, y)), ∀y(F(x´, y) ⊃ P([x´, y], [x, y])), and ∀y(F(x, y) ⊃ P([x, y], [x´, y])). Hence: ∀y(F(x´, y) ∧ F(x, y) ⊃ P([x´, y], [x, y]) ∧ P([x, y], [x´, y])), and therefore by applying P3: ∀y(F(x´, y) ∧ F(x, y) ⊃ [x´, y] = [x, y]). We can now apply P29 to obtain: x´ = x.

The theorem just proven is the obvious analog of P3. According to it, properties inherit the coarse-grainedness of states of affairs. This is as it should be: since properties and states of affairs belong together (via the saturation-function), neither of them should be more finely differentiated than the other. ∀x´∀x∀y(ESSUn(x´, y) ∧ ESSUn(x, y) ⊃ x´ = x), for n ≥1. ∀x´∀x∀y(ESSIn(x´, y) ∧ ESSIn(x, y) ⊃ x´ = x), for n ≥ 1. Proof: Assume ESSUn(x´, y) ∧ ESSUn(x, y). According to D30 we can conclude from this assumption: EXMEn(y, x´) ∧ ∀zF(x´, z) ∧ ∀x´´(EXMEn(y, x´´) ∧ ∀zF(x´´, z) ⊃ PPR(x´´, x´)) ∧ EXMEn(y, x) ∧ ∀zF(x, z) ∧ ∀x´´(EXMEn(y, x´´) ∧ ∀zF(x´´, z) ⊃ PPR(x´´, x)). Therefore: PPR(x, x´) ∧ PPR(x´, x). Therefore (according to the preceding theorem): x´ = x. – The proof of the second statement is entirely analogous to the proof of the first.

According to the two theorems just proven, any entity has at most one Uessence of whatever modal kind, and at most one I-essence of whatever modal kind (logical, nomological, or otherwise). ∀x∀x´∀y(EXMEn(y, x) ∧ PPR(x´, x) ⊃ EXMEn(y, x´)), for n ≥ 1. Proof: Assume EXMEn(y, x) ∧ PPR(x´, x). Hence P([x, y], bn) ∧ PR(x´) ∧ PR(x) ∧ ∀y´(F(x´, y´) ≡ F(x, y´)) ∧ ∀y´(F(x´, y´) ⊃ P([x´, y´], [x, y´])). [For n > 1, P([x, y], bn) is straightway obtained from EXMEn(y, x) by D29. For n = 1, we first obtain [x, y] = t* from EXME1(y, x) by D22. But, according to P2 and P18, we also have P(b1, b1) and b1 = t*. And therefore: P([x, y], b1).]

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Since S([x, y]) [by P0 from P([x, y], bn)], F(x, y) by D17, and therefore F(x´, y) [because of ∀y´(F(x´, y´) ≡ F(x, y´))]. Hence: P([x´, y], [x, y]) [because of ∀y´(F(x´, y´) ⊃ P([x´, y´], [x, y´]))]. Hence: P([x´, y], bn) by P1 [because of P([x´, y], [x, y]) and P([x, y], bn)]. Hence finally: EXMEn(y, x´). [For n > 1, EXMEn(y, x´) is straightway obtained from P([x´, y], bn) by D29. For n = 1, we first obtain P([x´, y], t*) from P([x´, y], b1) by P18. And because of P([x´, y], t*) we have [x´, y] = t* by the theorem ∀z(P(z, t*) ⊃ z = t*), which is provable in the mereology of states of affairs. Hence: EXME1(y, x´) by D22.]

According to the theorem just proven, any property which is an intensional part of a property that is n-essentially exemplified by some entity is likewise n-essentially exemplified by that same entity. Thus the intensional parts of a n-U-essence of x are all n-essential properties of x, and so are the intensional parts of a n-I-essence of x.

7.4

Regarding the Question What Properties There Are – and More on Essence and Essentiality

It may have escaped notice, but the sum of the principles that have been introduced so far is compatible with there being no properties and no individuals at all. The ontological framework regarding properties and individuals that has been proposed up to this point has great definitional power, but it is utterly weak in existential assumptions (regarding properties and individuals, not regarding states of affairs). This situation can be changed more or less drastically. Regarding properties, the central (and, in part, familiar) idea is to introduce an operator of abstraction, λ (also called “the abstractor”), together with special variables (called “round variables”): o, o´, o´´, …; one or another of these variables is bound by λ when that operator is made use of. (The sole reason for using round variables with λ – instead of the usual variables – is to make formulas more readable.) λ is syntactically made use of in the following way: if the sentence A[c] contains the name c at the places referred to by the square brackets, then the expression λoA[o] – which is formed by substituting o for c at those places and by prefixing λo to the result – is a name if, and only if, o does not already occur in A[c]. (The same holds, of course, for every other round variable: o´, o´´, … . In what follows, I will use “o” as a representative of any round variable.)

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The axiomatic schematic principle that governs λ and relates the abstractor to two other name-forming operators – to the saturator and to the that-operator – seems to be obviously the following: ∀y([λoA[o], y] = that A[y]) (y not in λoA[o]). According to this principle, every name λoA[o] designates a property (because of P13, D17, and D18) – a universally predicable property f whose predicative saturation by y yields the very same state of affairs from which f can be regarded as having been abstracted by leaving out the constituent y in it. Illustration: Consider the sentence “Eliza is a woman.” Assuming that this sentence is a sentence of the intended language and is covered by the “A” in P13 (that is, in the schema S(that A)), we have: that Eliza is a woman is a state of affairs. According to the principle we are presently discussing, [λo(o is a woman), Eliza] – the saturation of the property of being a woman by the individual Eliza – is that state of affairs.

And the following theorem-schema – the unrestricted principle of property-exemplification – is an easy consequence of ∀y([λoA[o], y] = that A[y]): ∀y(EXM(y, λoA[o]) ≡ A[y]). Proof: EXM(y, λoA[o]) is (according to D20) equivalent to O([λoA[o], y]), which (because of [λoA[o], y] = that A[y]) is equivalent to O(that A[y]), which, in turn, is equivalent to A[y] (according to P10).

But unfortunately the unrestricted principle of property-exemplification has a well-known self-contradictory instance: ∀y(EXM(y, λo¬EXM(o, o)) ≡ ¬EXM(y, y)), implying: EXM(λo¬EXM(o, o), λo¬EXM(o, o)) ≡ ¬EXM(λo¬EXM(o, o), λo¬EXM(o, o)).

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We are forced to conclude that ∀y([λoA[o], y] = that A[y]) cannot be maintained. But which principle should replace it? The best candidate seems to be this: P30 PR(λoA[o]) ∧ ∀y(F(λoA[o], y) ⊃ [λoA[o], y] = that A[y]) (y not in λoA[o]); ∀u1...∀un(PR(λoA[u1, ..., un, o]) ∧ ∀y(F(λoA[u1, ..., un, o], y) ⊃ [λoA[u1, ..., un, o], y] = that A[u1, ..., un, y]) (y not in λoA[u1, ..., un, o]). Following the lead of the axiomatic schematic principles at the beginning of this chapter, the simple form of P30 has been supplemented by the corresponding fully quantified form – in order to exhibit the full intended strength of P30. Correspondingly, every instance of P30 is taken to be a proper sentence.

P30 enables the derivation of the restricted principle of propertyexemplification: ∀y(F(λoA[o], y) ⊃ (EXM(y, λoA[o]) ≡ A[y])). Proof: Assume F(λoA[o], y). Hence by P30: [λoA[o], y] = that A[y]. For the rest of the proof, see the above derivation of EXM(y, λoA[o]) ≡ A[y] on the basis of [λoA[o], y] = that A[y].

From this principle, the contradiction exhibited above can no longer be derived. All one gets is this: ¬F(λo¬EXM(o, o), λo¬EXM(o, o)), and therefore (according to P25): [λo¬EXM(o, o), λo¬EXM(o, o)] = c*. Here are four important corollaries of the restricted principle of property-exemplification: ∀y(EXM(y, λoA[o]) ⊃ A[y]) ∧ ∀y(F(λoA[o], y) ∧ A[y] ⊃ EXM(y, λoA[o])) ∀yF(λoA[o], y) ⊃ ∀y(EXM(y, λoA[o]) ≡ A[y]) ∀y(I(y) ⊃ F(λoA[o], y)) ∧ ∀y(A[y] ⊃ I(y)) ⊃ ∀y(EXM(y, λoA[o]) ≡ A[y]) ∀y(A[y] ⊃ F(λoA[o], y)) ⊃ ∀y(EXM(y, λoA[o]) ≡ A[y]).

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Proof: The derivation of the second theorem is obvious. Regarding the third theorem, assume ∀y(I(y) ⊃ F(λoA[o], y)) ∧ ∀y(A[y] ⊃ I(y)), and assume A[y]. These assumptions imply: F(λoA[o], y), and therefore by the restricted principle of propertyexemplification (because of A[y]): EXM(y, λoA[o]). Assume, conversely, EXM(y, λoA[o]). Hence by the theorem ∀x∀y(EXM(y, x) ⊃ PR(x) ∧ F(x, y)) (stated below D20): F(λoA[o], y), and therefore by the restricted principle of propertyexemplification (because of EXM(y, λoA[o])): A[y]. The proof of the fourth theorem is obvious, given the pattern of the proof of the third theorem. The proof of the first theorem is likewise obvious, in view of the theorem that is invoked in the proof of the third theorem. (By the way: the third theorem is a logical consequence of the fourth theorem, which in turn is a logical consequence of the first.)

Consider also the following beautiful theorem, for the obtaining of which the central principles P24, P29, P30 and the central definition D20 work together. ∀x[PR(x) ∧ ∀y(F(x, y) ≡ F(λoEXM(o, x), y)) ⊃ x = λoEXM(o, x)]. Proof: Assume (1) PR(x), and (2) ∀y(F(x, y) ≡ F(λoEXM(o, x), y)). We have: (3) λoEXM(o, x) is a property (according to P30, but also according to the assumptions (1) and (2)). Assume moreover: F(x, y) ∧ F(λoEXM(o, x), y). Hence (according to P30, D20, and P24): [λoEXM(o, x), y] = that EXM(y, x) = that O([x, y]) = [x, y]. [P24 can be applied to obtain the last identity in this chain, since S([x, y]) is true because of the assumption F(x, y) and D17, and since “[x, y]” is a referentially stable term (and therefore substitutable into P24), both “x” and “y” being referentially stable terms (qua variables).] Thus we have: (4) ∀y(F(x, y) ∧ F(λoEXM(o, x), y) ⊃ [λoEXM(o, x), y] = [x, y]), and therefore – together with (1), (2), and (3) – we have all we need to conclude (according to P29): x = λoEXM(o, x) – “the property x is the property of exemplifying x.”

In view of the theorem just proven, it seems a plausible step to postulate ∀x[PR(x) ⊃ ∀y(F(x, y) ≡ F(λoEXM(o, x), y))] and to move on to ∀x[PR(x) ⊃ x = λoEXM(o, x)] (cf. P24), which step, however, I will not take (due to remaining doubts: it seems that there is some property x such that λoEXM(o, x) is universally predicable, but x is not). The number of named properties is still the same with P30 as with its untenable predecessor, since according to P30 there still is a property λoA[o] corresponding to each of the intended monadic predicates A[x], that is, to the monadic predicates corresponding to appropriate sentences intended by the schematic letter “A” in the original formulation of P13 and the other axiom-schemata in Chapter 3 (concerning this, see Section 4.3).

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To each property λoA[o], in turn, there corresponds its extensional (or logically essential) modification, which in most cases will be different from λoA[o]. The extensional modification of a property x1 is the property x2 which fits the same entities as x1 and which is such that [x2, y] = t* in case EXM(y, x1), and [x2, y] = k* in case F(x1, y) and ¬EXM(y, x1). The following principle postulates the existence of an extensional modification for each property: P31 ∀x1[PR(x1) ⊃ ∃x2(PR(x2) ∧ ∀y(F(x2, y) ≡ F(x1, y)) ∧ ∀y(EXM(y, x1) ⊃ [x2, y] = t*) ∧ ∀y(F(x1, y) ∧ ¬EXM(y, x1) ⊃ [x2, y] = k*))]. On the basis of P29 and P31, a theorem is provable which is exactly like P31, except for the fact that “∃x2” is replaced by “∃!x2” (“there is exactly one x2”). Hence the following definition has all that it needs for being entirely justified: D33 exmo(x) =Def ιx´(PR(x´) ∧ ∀y(F(x´, y) ≡ F(x, y)) ∧ ∀y(EXM(y, x) ⊃ [x´, y] = t*) ∧ ∀y(F(x, y) ∧ ¬EXM(y, x) ⊃ [x´, y] = k*)). D33 formally defines the extensional modification of x. For properties x, exmo(x) is the entity described in the definiens of D33 (since the conditions of existence3 and uniqueness for the defining definite description are provably fulfilled); for non-properties x, the definite description that defines “exmo(x)” is stipulated to designate some suitable entity (c* is suitable), and hence also “exmo(x)” will designate that entity. The next definitional step to take is this: D34 λEoA[o] =Def exmo(λoA[o]). While λoA[o] is read as “the property of being a y such that A[y],” λEoA[o] is read as “the extensional property of being a y such that A[y].” If λoA[o] is a universally predicable property – ∀yF(λoA[o], y) –, then λEoA[o] is the set – SET(λEoA[o]) – corresponding to λoA[o]; if λoA[o] is property that is predicable exactly of the individuals – ∀y(F(λoA[o], y) ≡ I(y)) –, then λEoA[o] is the set of individuals – SETI(λEoA[o]) – corre3

In expressing what I mean to say, I here follow the usual way of speaking. It would be more accurate, but also non-standard, to use the word “someness” instead of the word “existence.” (This is noted in view of Section 1.4.)

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sponding to λoA[o]. Of all extensional (or logically essential) properties (their definition is D21), sets – in the sense of “SET(x)” – are nowadays the ones that get practically all of the attention. This would be unproblematic if all properties were universally predicable; then all extensional properties would be sets. But, as we have seen, not all properties are universally predicable: it is, for example, provable (see above) that λo¬EXM(o, o), though a property, is not a universally predicable property. Some important theorems concerning extensional modifications are the following: ∀x[PR(x) ⊃ ∀y(EXM(y, x) ≡ EXM(y, exmo(x)))] ∀x(PRE1(x) ⊃ x = exmo(x)) ∀x(PR(x) ⊃ exmo(exmo(x)) = exmo(x)). Moving on to other considerations within the global theme of what properties there are, here follows a short list of properties which are of particular importance for the present theoretical purposes, or are otherwise salient: (1) λo(o = a0) – the property of being a0 (“a0” standing for a referentially stable, syntactically simple name); (2) λo(o = a0 ∧ O(w0)) – the property of being a0 while w0 obtains (“w0” being a referentially stable name for w*; cf. Section 4.3); (3) λo(o = a0 ∧ O(b2)) (“b2” designating rigidly the basis of nomological necessity). With the parenthesis connected to (3), I make explicit the stipulation (already in effect in Section 7.2) that the term “b2” (for designating the basis of nomological necessity) is referentially stable, thus resolving an issue that remained unresolved in Section 3.9.1. Hence it is safe to substitute “b2” into “that”-contexts when applying allinstantiation (or to replace it in a “that”-context by a bound variable when applying, so-called, existential instantiation). Note that it would not do to simply stipulate that all designators of the form bn are referentially stable, because it was found in Section 5.1 (see the considerations below P18b) that “b10” is as referentially unstable as “w*,” the term with which “b10” is co-referential (according to P18b, taken to be a conceptual truth). And it may, of course, be necessary to countenance other basis-terms that, regarding referential stability, behave like “b10”. In fact, it is not a priori implausible to consider “b2” to be such a term. Thus, what has here been determined regarding “b2” is

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indeed a stipulation. (For avoiding arbitrariness, it may be advisable to take serious the idea that some necessities – nomological necessity being the first candidate – have two versions: one with a stable basis, and another with an unstable one, it being contextdependent which one of the two versions is meant.)

(4) λo(o is a human being) – the property of being a human being; (5) λo(o is actual) – the property of being actual, λoA(o);4 (6) λo(o is actual ⊃ o is physical) – the property of being physical if actual. Concerning property (1): The property λo(o = a0) can safely be regarded as being universally predicable: ∀zF(λo(o = a0), z). Moreover, it is an extensional (or logically essential) property: ∀y(F(λo(o = a0), y) ⊃ [x, y] = t* ∨ [x, y] = k*) (and therefore: λo(o = a0) = λEo(o = a0)). Hence: SET(λo(o = a0)). Note that the usual (set-theoretical) designation of λo(o = a0) is “{a0}.” The following, however, is the most interesting thing that can be said about λo(o = a0): The Central Principle of Essence ESSU1(λo(o = a0), a0) – “The universal logical essence (or logical Uessence, or 1-U-essence) of a0 is the property of being identical with a0.” For proving the Central Principle of Essence, we need the following obvious counterpart of P22: P32 ∀x∀y(x = y ⊃ t* = that (x = y)) ∧ ∀x∀y(x ≠ y ⊃ k* = that (x =y)). For applying P32 in the proof of ESSU1(λo(o = a0), a0) (which follows below), it is important to remember that “a0” is supposed to represent a referentially stable term. Proof of the Central Principle of Essence: For proving ESSU1(λo(o = a0), a0), we must prove (according to D30): EXME1(a0, λo(o = a0)) ∧ ∀zF(λo(o = a0), z) ∧ ∀x´(EXME1(a0, x´) ∧ ∀zF(x´, z) ⊃ PPR(x´, λo(o = a0))). [This proven, the reading of “ESSU1(λo(o = a0), a0)” as “the universal logical essence of a0 is the property of being 4

“λoA(o)” must not be confused with “λoA[o].” The former is a name of a property, the latter merely a schema for standard names of properties.

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identical with a0” is justified, because there is at most one universal logical essence of anything, as was already proven in Section 7.3.] We already know: ∀zF(λo(o = a0), z). [Technically, this is an additional “axiom,” specifically characteristic of λo(o = a0).] For EXME1(a0, λo(o = a0)), we must prove: [λo(o = a0), a0] = t* (see D22). Since a0 is a referentially stable term and therefore safely substitutable into a “that”-context, we have according to P30: [λo(o = a0), a0] = that (a0 = a0). Since a0 = a0 is logically provable (├ a0 = a0), we have according to P23: ├ t* = that (a0 = a0). Therefore: t* = [λo(o = a0), a0]. Now assume: EXME1(a0, x´) ∧ ∀zF(x´, z). We need to derive: PPR(x´, λo(o = a0)), that is (according to D32): PR(x´) ∧ PR(λo(o = a0)) ∧ ∀y(F(x´, y) ≡ F(λo(o = a0), y)) ∧ ∀y(F(x´, y) ⊃ P([x´, y], [λo(o = a0), y])). PR(x´) is a consequence of EXME1(a0, x´). ∀y(F(x´, y) ≡ F(λo(o = a0), y)) is a consequence of ∀zF(x´, z) and ∀zF(λo(o = a0), z). For deriving ∀y(F(x´, y) ⊃ P([x´, y], [λo(o = a0), y])), assume finally: F(x´, y). There are two cases: (1st) y = a0, and (2nd) y ≠ a0. In the first case: [x´, y] = t* since [x´, a0] = t* because of the assumption EXME1(a0, x´), and [λo(o = a0), y] = t* since [λo(o = a0), a0] = t*, as proven above. Therefore: P([x´, y], [λo(o = a0), y]) (because of the theorem P(t*, t*)). In the second case: [λo(o = a0), y] = k* because, according to P30, [λo(o = a0), y] = that (y = a0) [as a variable, “y” is automatically a referentially stable term] and because, according to P32 (because of y ≠ a0), k* = that (y = a0) [P32 is applicable because both “y” and “a0” are referentially stable terms]. Since [λo(o = a0), y] = k*, it follows, whatever state of affairs [x´, y] is, that we have: P([x´, y], [λo(o = a0), y]) (because of the theorem ∀z(S(z) ⊃ P(z, k*))).

According to the Central Principle of Essence, my universal logical essence is simply the property of being identical with U.M., and the universal logical essence of each reader is simply the property of being identical with him, respectively her. What may make some readers balk at this conclusion is this: the property λo(o = a0) may seem to have too little content for being the universal logical essence of a0. But this impression is an illusion. As is well known, content is inversely proportional to extension, and in any possible world (i.e., maximal-consistent state of affairs) the extension of λo(o = a0) is as small as it can be short of being nil. This is implied by the following theorem: ∀w(MC(w) ⊃ ∀y(P([λo(o = a0), y], w) ≡ y = a0)). Proof: Assume MC(w). There are two cases: (1st) y = a0, and (2nd) y ≠ a0. In the first case: t* = that (y = a0) (according to P32), and therefore: [λo(o = a0), y] = t* (according to P30). Hence: P([λo(o = a0), y], w) [because of S(w), which is a definitional consequence of MC(w), and the theorem ∀x(S(x) ⊃ P(t*, x))].

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In the second case: k* = that (y = a0) (according to P32), and therefore: [λo(o = a0), y] = k* (according to P30). Hence: ¬P([λo(o = a0), y], w) [because of ¬T(w), which is a definitional consequence of MC(w), and the theorem ∀y(T(y) ≡ P(k*, y))]. Thus: y = a0 ⊃ P([λo(o = a0), y], w), and y ≠ a0 ⊃ ¬P([λo(o = a0), y], w), that is: P([λo(o = a0), y], w) ≡ y = a0, which is precisely what needed to be shown.

Given the definition of relative exemplification (“y exemplifies x in w,” “x is a property of y in w”), D35 EXM(y, x, w) =Def O([x, y], w) (=Def P([x, y], w), according to D16), and read in view of the Central Principle of Essence, the above theorem states that the universal logical essence of a0 – that is, λo(o = a0) – is exemplified in every possible world by a0, and by a0 only. And this is exactly as it should be. Concerning property (2): The property λo(o = a0 ∧ O(w0)) is as safely universally predicable as is the property λo(o = a0). Hence we have according to P30: ∀y([λo(o = a0 ∧ O(w0)), y] = that (y = a0 ∧ O(w0))), and therefore according to P15: ∀y([λo(o = a0 ∧ O(w0)), y] = conj(that (y = a0), that O(w0))), hence according to P24 (“w0” being a referentially stable term): ∀y([λo(o = a0 ∧ O(w0)), y] = conj(that (y = a0), w0)). If y = a0, then the term “that (y = a0)” designates t* (according to P32), and hence we have: conj(that (y = a0), w0) = w0. If y ≠ a0, then the term “that (y = a0)” designates k* (according to P32), and hence we have conj(that (y = a0), w0) = k*. Thus the predicative saturation of λo(o = a0 ∧ O(w0)) yields in all cases either w0 or k*. The most interesting thing, however, that can be said about λo(o = a0 ∧ O(w0)) is this: EXM(a0, λo(o = a0 ∧ O(w0))) ∧ ∀x(EXM(a0, x) ∧ ∀yF(x, y) ⊃ PPR(x, λo(o = a0 ∧ O(w0)))). According to this assertion, λo(o = a0 ∧ O(w0)) is the sum (or conjunction) of a0’s universally fitting properties: λo(o = a0 ∧ O(w0)) is exemplified by a0 (and a universally fitting property; see above) and it has all universally fitting properties that are exemplified by a0 as intensional parts; moreover, λo(o = a0 ∧ O(w0)) is the only such property (as a consequence of P29).

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Proof of the assertion: First, EXM(a0, λo(o = a0 ∧ O(w0))). On the basis of D20, EXM(a0, λo(o = a0 ∧ O(w0))) amounts to O([λo(o = a0 ∧ O(w0)), a0]), that is, according to P30, to O(that (a0 = a0 ∧ O(w0))). This latter statement is a consequence of a0 = a0 ∧ O(w0), according to P10. a0 = a0 ∧ O(w0), in turn, is a consequence of P12 (i.e., A(w*)) and of w0 = w* and S(w0) (the principles characterizing w0; cf. Section 4.3): together they yield O(w0) (according to D6). Second, ∀x(EXM(a0, x) ∧ ∀yF(x, y) ⊃ PPR(x, λo(o = a0 ∧ O(w0)))). Assume EXM(a0, x) ∧ ∀yF(x, y). According to D32, we need to show: PR(x) ∧ PR(λo(o = a0 ∧ O(w0))) ∧ ∀y(F(x, y) ≡ F(λo(o = a0 ∧ O(w0)), y)) ∧ ∀y(F(x, y) ⊃ P([x, y], [λo(o = a0 ∧ O(w0)), y])). PR(λo(o = a0 ∧ O(w0))) is already established, PR(x) is a consequence of EXM(a0, x), ∀y(F(x, y) ≡ F(λo(o = a0 ∧ O(w0)), y)) is a consequence of ∀yF(x, y) and ∀yF(λo(o = a0 ∧ O(w0)), y) (the latter is already established; see above). Assume finally F(x, y). We need to derive: P([x, y], [λo(o = a0 ∧O(w0)), y]). There are two cases: (1) y = a0, (2) y ≠ a0. In the first case: [λo(o = a0 ∧ O(w0)), y] = w0 [see the above saturation-description of λo(o = a0 ∧O(w0))]. And because of EXM(a0, x): EXM(y, x); that is (according to D20), O([x, y]), that is (according to D6), S([x, y]) ∧ A([x, y]), and hence, according to the theorem ∀x´[S(x´) ⊃ (A(x´) ≡ P(x´, w*))] (the Actuality Principle for States of Affairs, proven in Section 3.3), P([x, y], w*), i.e., P([x, y], w0) (since w0 = w*). Therefore: P([x, y], [λo(o = a0 ∧ O(w0)), y]). In the second case: [λo(o = a0 ∧ O(w0)), y] = k* [see the above saturation-description of λo(o = a0 ∧ O(w0))], and hence, according to the theorem ∀x´(S(x´) ⊃ P(x´, k*)) [making use of S([x, y]), which is a definitional consequence of the assumption F(x, y)], P([x, y], k*). Therefore once again: P([x, y], [λo(o = a0 ∧ O(w0)), y]).

In fact, one can prove the equivalence of being a universally predicable property that is exemplified by a0 and being a property that is an intensional part of λo(o = a0 ∧ O(w0)): ∀x[EXM(a0, x) ∧ ∀yF(x, y) ≡ PPR(x, λo(o = a0 ∧ O(w0)))]. Proof: Given the proof of the previous theorem, what remains to be proven is this: ∀x[PPR(x, λo(o = a0 ∧ O(w0))) ⊃ EXM(a0, x) ∧ ∀yF(x, y)]. Assume, therefore, PPR(x, λo(o = a0 ∧ O(w0))). ∀yF(x, y) is a consequence of this assumption, on the basis of D32, because of ∀yF(λo(o = a0 ∧ O(w0)), y). Another consequence of the assumption, on the basis of D32, is this: ∀y(F(x, y) ⊃ P([x, y], [λo(o = a0 ∧ O(w0)), y])), and hence because of ∀yF(x, y): ∀yP([x, y], [λo(o = a0 ∧ O(w0)), y]). Hence: P([x, a0], [λo(o = a0 ∧ O(w0)), a0]). But we also have: [λo(o = a0 ∧ O(w0)), a0] = w0 [see the above, initial saturation-description of λo(o = a0 ∧ O(w0))]. Hence: P([x, a0], w0). Therefore (because of w0 = w*): P([x, a0], w*), and this implies on the basis of the Actuality Principle for States of Affairs: A([x, a0]), since we have S([x, a0]). Therefore: O([x, a0]) (according to D6), and hence finally: EXM(a0, x) (according to D20).

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It is easy to prove that the universal logical essence of a0 is an intensional part of the sum of a0’s universally fitting properties. More interesting is the provable fact that the universal logical essence of a0 is not identical with the sum of a0’s universally fitting properties. Thus there is a limit to logical essentialism. The axiomatic principle mainly responsible for this result is P9, that is, w* ≠ t*. Independently of P9, one can prove: λo(o = a0 ∧ O(w0)) = λo(o = a0) ≡ w0 = t*. Proof: Assume λo(o = a0 ∧ O(w0)) = λo(o = a0). Hence: [λo(o = a0 ∧ O(w0)), a0] = [λo(o = a0), a0]. According to the above saturation-descriptions of λo(o = a0 ∧ O(w0)) and λo(o = a0), we have: [λo(o = a0 ∧ O(w0)), a0] = w0, [λo(o = a0), a0] = t*. Therefore: w0 = t*. Assume conversely: w0 = t*. Since we have PR(λo(o = a0 ∧ O(w0))), PR(λo(o = a0)), ∀yF(λo(o = a0 ∧ O(w0)), y), ∀yF(λo(o = a0), y), all that remains to be shown for λo(o = a0 ∧ O(w0)) = λo(o = a0) is this (according to P29): ∀y([λo(o = a0 ∧ O(w0)), y] = [λo(o = a0), y]). There are two cases: (1) y=a0, (2) y≠a0. In the second case, we have [λo(o = a0 ∧ O(w0)), y] = k* = [λo(o = a0), y]. In the first case, we have [λo(o = a0 ∧ O(w0)), y] = w0 and t* = [λo(o = a0), y], and therefore because of the assumption w0 = t*: [λo(o = a0 ∧ O(w0)), y] = [λo(o = a0), y].

Concerning property (3): λo(o = a0 ∧ O(b2)) is a universally fitting property whose saturation by an entity y is b2 in case y = a0, and k* in case y ≠ a0 (the derivation of this saturation-description is analogous to the above derivation of the saturation-description for property (2)). λo(o = a0 ∧ O(b2)) is the universal nomological essence (or nomological U-essence, or 2-Uessence) of a0. We can prove: ∀x[EXME2(a0, x) ∧ ∀yF(x, y) ≡ PPR(x, λo(o = a0 ∧ O(b2)))]. ESSU2(λo(o = a0 ∧ O(b2)), a0). Proof: First, EXME2(λo(o = a0 ∧ O(b2)), a0). For this we must show (according to D29): P([λo(o = a0 ∧ O(b2)), a0], b2). But this is easily shown, since [λo(o = a0 ∧ O(b2)), a0] = b2 (according to the saturation-description of the presently considered property), S(b2) (according to P18), and ∀x(S(x) ⊃ P(x, x)) (P2). Second, ∀x[EXME2(a0, x) ∧ ∀yF(x, y) ⊃ PPR(x, λo(o = a0 ∧ O(b2)))]. For showing this, assume EXME2(a0, x) ∧ ∀yF(x, y); we need to derive (according to D32): (a) PR(x), (b) PR(λo(o = a0 ∧ O(b2))), (c) ∀y(F(x, y) ≡ F(λo(o = a0 ∧ O(b2)), y)), (d) ∀y(F(x, y) ⊃ P([x, y], [λo(o = a0 ∧ O(b2)), y])).

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(a) PR(x) is a consequence of assumption EXME2(a0, x) [for according to D29 and P0, EXME2(a0, x) implies S([x, a0]), and therefore, according to D17, F(x, a0), hence ∃yF(x, y), and therefore, according to D18, PR(x)]. (b) PR(λo(o = a0 ∧ O(b2))) we have according to P30. (c) ∀y(F(x, y) ≡ F(λo(o = a0 ∧ O(b2)), y)) is a consequence of assumption ∀yF(x, y) and the fact that λo(o = a0 ∧ O(b2)) is a universally fitting property. (d) Consider finally [x, y] and [λo(o = a0 ∧ O(b2)), y]. There are two cases: (1) y = a0, (2) y ≠ a0. In the second case, [λo(o = a0 ∧ O(b2)), y] = k*, and therefore P([x, y], [λo(o = a0 ∧ O(b2)), y]) [because of the theorem ∀z(S(z) ⊃ P(z, k*)) and because of S([x, y]), which is a consequence of the assumption ∀yF(x, y) and D17]. In the first case, [λo(o = a0 ∧ O(b2)), y] = b2. But the assumption EXME2(a0, x) means according to D29: P([x, a0], b2); hence we have (because of y = a0): P([x, y], b2). Therefore again: P([x, y], [λo(o = a0 ∧ O(b2)), y]). We now have what is sufficient for concluding: ∀y(F(x, y) ⊃ P([x, y], [λo(o = a0 ∧ O(b2)), y])). Third, ∀x[PPR(x, λo(o = a0 ∧ O(b2))) ⊃ EXME2(a0, x) ∧ ∀yF(x, y)]. Assume PPR(x, λo(o = a0 ∧ O(b2))). Hence ∀yF(x, y) because of ∀yF(λo(o = a0 ∧ O(b2)), y) and ∀y(F(x, y) ≡ F(λo(o = a0 ∧ O(b2)), y)), the latter being a consequence of the assumption according to D32. Moreover, we have as a consequence of the assumption (and ∀yF(x, y)): P([x, a0], [λo(o = a0 ∧ O(b2)), a0]). Therefore we have: P([x, a0], b2), because of [λo(o = a0 ∧ O(b2)), a0] = b2. Therefore finally (according to D29): EXME2(a0, x). The first of the two theorems to be proven is logically equivalent to the conjunction of the second and third result in this proof. The second of the above two theorems follows from the first and second result and the fact ∀yF(λo(o = a0 ∧ O(b2)), y) on the basis of D30.

Thus λo(o = U.M. ∧ O(b2)), for example, is the sum of all the universally predicable properties I have merely in virtue of the laws of nature being what they are. Whether this sum is different from the sum of all the universally predicable properties I have – that is, different from λo(o = U.M. ∧ O(w0)) – depends on whether or not b2 is identical with w0. According to P21, b2 is not identical with w*, and therefore not identical with w0 (because of w0 = w*). Hence λo(o = U.M. ∧ O(b2)) is different from λo(o = U.M. ∧ O(w0)) (though the former property is an intensional part of the latter), and hence there is some universally predicable property I have that I do not have in virtue of the laws of nature (or in other words: there is some universally predicable property I have that I am not determined to have by the laws of nature). Suppose on the contrary: ∀x(∀yF(x, y) ∧ EXM(U.M., x) ⊃ EXME2(U.M., x)). Hence – because of ∀yF(λo(o = U.M. ∧ O(w0)), y) ∧ EXM(U.M., λo(o = U.M. ∧ O(w0))) (previously shown) – we have: EXME2(U.M., λo(o = U.M. ∧ O(w0))), and

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therefore (according to what has been demonstrated above): PPR(λo(o = U.M. ∧ O(w0)), λo(o = U.M. ∧ O(b2))). Since we also have PPR(λo(o = U.M. ∧ O(b2)), λo(o = U.M. ∧ O(w0))) [the principle P19 – P(b2, w*) – is crucial for this], it follows: λo(o = U.M. ∧ O(b2)) = λo(o = U.M. ∧ O(w0)) [applying the theorem ∀x´∀x(PPR(x´, x) ∧ PPR(x, x´) ⊃ x´ = x)]. But λo(o = U.M. ∧ O(b2)) = λo(o = U.M. ∧ O(w0)) has just been seen to be false. Hence the initial assumption must be false, too.

λo(o = U.M. ∧ O(b2)) is also different from λo(o = U.M.), because of P20. Thus λo(o = U.M.), λo(o = U.M. ∧ O(b2)) and λo(o = U.M. ∧ O(w0)) have now be seen to be three different properties. But, note, they all have the same extension: they are exemplified by U.M. only. Concerning property (4): When considering the property of being a human being under the name “λo(o is a human being),” we are presupposing that the predicate “x is a human being,” in its natural interpretation, is a predicate of the intended theoretical language, and by applying the principles of our theory to it, we are presupposing that this predicate falls within their purview. These presuppositions seem unproblematic, but there is one not entirely uncontroversial consequence of them: the application of “x is a human being” must not be context-dependent, in particular, it must not be dependent on time. Otherwise, we would not be making a complete assertion by saying, for example, “U.M. is a human being,” whereas only sentences that make complete assertions are indicated by the schematic letters in the principles of our theory. (For more on this, see Section 4.3.) λo(o is a human being) is for us human beings a more usual property than the three properties previously considered, but its nature, in fact, is far less clear than theirs. First, it is unclear whether λo(o is a human being) is a universally fitting property. Does, for example, λo(o is a human being) fit λo(o is a human being)? Is, in other words, [λo(o is a human being), λo(o is a human being)] a state of affairs? If it is a state of affairs, then – according to P30, and because λo(o is a human being) is a referentially stable term – it will be the state of affairs that λo(o is a human being) is a human being – which entity is a state of affairs according to P13, and is therefore a state of affairs quite independently of the question whether [λo(o is a human being), λo(o is a human being)] is a state of affairs or not. It is, therefore, evident that the correct answer to this latter question does not depend on the correct answer to the question whether the sentence “λo(o is a human being) is a human being” is meaningful, or on the correct answer to the question whether that λo(o is a human being) is a human being is a

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state of affairs. The correct answer to both these questions is “yes” – and nevertheless this does not answer the question whether [λo(o is a human being), λo(o is a human being)] is a state of affairs. What is really at issue here is whether the state of affairs that λo(o is a human being) is a human being can be structured as the saturation of the property of being a human being – λo(o is a human being) – by that same property. If yes, then [λo(o is a human being), λo(o is a human being)] is a state of affairs; if no, then [λo(o is a human being), λo(o is a human being)] is not a state of affairs, since there is no other state of affairs for it to be than the state of affairs that λo(o is a human being) is a human being. Consider, for comparison, a much more famous case: ¬EXM(λo¬EXM(o, o), λo¬EXM(o, o)) is a meaningful sentence, and that ¬EXM(λo¬EXM(o, o), λo¬EXM(o, o)) is a state of affairs. But this has nothing to do with the question whether [λo¬EXM(o, o), λo¬EXM(o, o)] is a state of affairs. The answer to this question can only be “no,” as we have seen. On pain of self-contradiction, the state of affairs that ¬EXM(λo¬EXM(o, o), λo¬EXM(o, o)) cannot be structured as the saturation of the property λo¬EXM(o, o) by the property λo¬EXM(o, o). Hence [λo¬EXM(o, o), λo¬EXM(o, o)] is not a state of affairs, since there is no other state of affairs for it to be than the state of affairs that ¬EXM(λo¬EXM(o, o), λo¬EXM(o, o)). Unfortunately, matters are less clear in the case of λo(o is a human being). One might say, yes, [λo(o is a human being), λo(o is a human being)] is a state of affairs; one might also say, no, it isn’t. In fact, the whole question of the extent of predicative fit for λo(o is a human being) seems to be, to a large extent, a question of arbitrary stipulation. This is not a happy situation, but one that is only all too common among the properties corresponding to commonly employed predicates. However, one can well avoid being arbitrary in the following way. The I-restriction of an entity x1 is the entity x2 such that [x2, y] = [x1, y] if y is an individual, and [x2, y] = c* if y is not an individual. The following principle postulates the existence of an I-restriction for each entity (this given, the uniqueness of I-restriction for each property that fits at least one individual5 is a consequence of P29). If x1 is not a property (i.e., ∀y([x1, y] = c*), according to D18 and the theorem ∀x∀y(F(x, y) ≡ [x, y] ≠ c*)), then I-restriction is obviously not unique for it: every individual and every state of affairs is an I-restriction of x1 (because of P28 and P25). If x1 is a property, but does not fit any individual (that is, we have ∀y(I(y) ⊃ [x1, y] = 5

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P33 ∀x1∃x2[∀y(I(y) ⊃ [x2, y] = [x1, y]) ∧ ∀y(¬I(y) ⊃ [x2, y] = c*)]. The following definition formalizes the definition of the I-restriction given informally above: D36 ires(x1) =Def ιx2[∀y(I(y) ⊃ [x2, y] = [x1, y]) ∧ ∀y(¬I(y) ⊃ [x2, y] = c*)]. If the conditions of existence and uniqueness for the defining definite description are not fulfilled, ιx2[∀y(I(y) ⊃ [x2, y] = [x1, y]) ∧ ∀y(¬I(y) ⊃ [x2, y] = c*)], and hence ires(x1), will be some suitably stipulated entity (for example, c*).

Some interesting theorems concerning I-restriction are the following: ∀x[PR(x) ∧ ∃y(I(y) ∧ F(x, y)) ⊃ ∀y(I(y) ⊃ (EXM(y, x) ≡ EXM(y, ires(x))))] ∀x(PRI(x) ⊃ x = ires(x)) ∀x(PR(x) ∧ ∃y(I(y) ∧ F(x, y)) ⊃ ires(ires(x)) = ires(x)). And the following definition makes the matter complete: D37 λIoA[o] =Def ires(λoA[o]). Note that in some cases λIoA[o] cannot be shown to be a property: if λoA[o] does not fit any individual, then λIoA[o] cannot be shown to be a property. In fact, things should be regulated in such a manner that, in the envisaged case, λIoA[o] can be shown not to be a property. This is easily effected by stipulating that a definite description designates c* if its existence- or uniqueness-condition is not fulfilled (as, in effect, was already stipulated in footnote 12 of Chapter 3).

Consider now λIo(o is a human being) vis-à-vis λo(o is a human being). Though the extent of predicative fit for λo(o is a human being) is, without further stipulation, undetermined, it is certain that, for nonindividuals y, [λo(o is a human being), y] is either always c* or always k*. In contrast, for non-individuals y, [λIo(o is a human being), y] is unequivoc*)), then again I-restriction is not unique: every individual and every state of affairs is an I-restriction of x1.

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cally always c*. Again, though the extent of predicative fit for λo(o is a human being) is undetermined, it is certain that, for individuals y, [λo(o is a human being), y] is always a state of affairs, and this, due to the definition of λIo(o is a human being), must be also true of λIo(o is a human being). Thus the extent of predicative fit for λIo(o is a human being) is completely determined: we have ∀y(F(λIo(o is a human being), y) ≡ I(y)) – no stipulation needed –, whereas, to repeat, the extent of predicative fit for λo(o is a human being) is, without further stipulation, undetermined. At the same time, λIo(o is a human being) is utterly similar to λo(o is a human being): within the domain of individuals, the two properties are identical; and outside the domain of individuals, they differ only insofar as [λIo(o is a human being), y] is always c*, while [λo(o is a human being), y] is undetermined between being always c* and being always k* (other saturationcourses for λo(o is a human being) outside the domain of individuals are out of the question). It should be noted that this way of describing the matter is not quite the proper way of describing it. The proper way is this: outside the domain of individuals, [λo(o is a human being), y] is either always c* – in that case λo(o is a human being) is identical with λIo(o is a human being) – or always k* – in that case λo(o is a human being) is different from λIo(o is a human being) –, and it is undetermined whether [λo(o is a human being), y] is always c* or always k* for non-individuals y. The point I wish to make is that properties cannot properly speaking differ by the fact that the predicative fit of the one property is determined within a certain region, whereas the predicative fit of the other is not. (Vagueness of predicative fit is, in the end, not a matter of ontology, but a matter of language; in the case in point, it is a matter of the indeterminacy of the predicate “x is a human being” regarding its proper applicability.)

This means that we can safely shift our attention from λo(o is a human being) to λIo(o is a human being) and that, for practical theoretical purposes (so to speak), we need no longer worry about the indeterminacy of the extent of predicative fit for λo(o is a human being). λo(o is a human being) is representative of many other properties λoB[o]. If outside the domain of individuals the extent of predicative fit for λoB[o] is indeterminate in the way that is paradigmatically represented by λo(o is a human being), but is entirely determinate inside the domain of individuals, then λIoB[o] is a very good substitute for λoB[o] and by shifting our attention from λoB[o] to λIoB[o] we can entirely dodge the arbitrariness necessary for determining the extent of predicative fit for λoB[o].

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The problem, however, is still there, and it remains true that ontological intuition is silent about it. We still do not know precisely which property is λoB[o]. Is it the property whose saturation by any non-individual is k* and which otherwise is exactly like λIoB[o], or is it the property whose saturation by any non-individual is c* and which, in fact, is identical with λIoB[o]? There seems to be no objective ontological fact one way or the other. Moreover, if we consider λIo(o is a human being) instead of λo(o is a human being), we have in fact not avoided all the problems of objective specification that are relevant for this case. There is some plausibility that the following statement is true of λIo(o is a human being): PRIE1(λIo(o is a human being)), that is (according to D25): PRI(λIo(o is a human being)) ∧ ∀y(F(λIo(o is a human being), y) ⊃ [λIo(o is a human being), y] = t* ∨ [λIo(o is a human being), y] = k*). In other words, there is some plausibility that λIo(o is a human being) is a logically essential property of individuals – some plausibility, but the matter is far from obvious. Might one not as well hold that it is a property of individuals, but not a logically essential one? There seems to be no objective ontological fact one way or the other for deciding the matter; in other words, there seem to be no objective ontological grounds for telling whether λIo(o is a human being) is identical with the property exmo(λIo(o is a human being)) – see D33 – or not. This, of course, is a problem that λIo(o is a human being) inherits from λo(o is a human being); for the latter property, too, it is indeterminate whether its extensional modification – that is, exmo(λo(o is a human being)), or λEo(o is a human being), according to D34 – is identical with it or not. One thing, however, seems very clear, at least to me: that λIo(o is a human being) is a logically essential property of me: EXME1(λIo(o is a human being), U.M.), that is (according to D22): [λIo(o is a human being), U.M.] = t*. And as a consequence of this (U.M. being an individual), we must also have: [λo(o is a human being), U.M.] = t*; λo(o is a human being), too, and not only its I-restriction, is a logically essential property of U.M., of me. Since ESSI1(λIo(o = U.M.), U.M.) is provable – in other words: that the I-restriction of λo(o = U.M.) is a (hence the) logical Iessence of U.M. (for what is implied in this, see D31) – and since the Irestriction of λo(o is a human being) – λIo(o is a human being) – is a logically essential property of U.M. and is predicable precisely of the individuals, we have: PPR(λIo(o is a human being), λIo(o = U.M.)) – the I-restriction of λo(o is a human being) is a part of my logical I-essence. But the two

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properties are different, of course: my logical I-essence goes beyond λIo(o is a human being). Concerning property (5): A universally predicable property that, though a property of me, is safely not a logically essential property of me is λo(o is actual). Hence λo(o is actual) is also not simpliciter a logically essential property: ¬PRE1(λo(o is actual)), because F(λo(o is actual), U.M.) and [λo(o is actual), U.M.] ≠ t* and [λo(o is actual), U.M.] ≠ k* (see D21). Regarding the latter non-identity: If [λo(o is actual), U.M.] were k*, EXM(U.M., λo(o is actual)) would not be true: Assume [λo(o is actual), U.M.] is k*, and assume, for reductio, also EXM(U.M., λo(o is actual)), that is [because of D20], O([λo(o is actual), U.M.]). Therefore: O(k*), that is [because of D6], S(k*) and A(k*). Hence according to the Actuality Principle for States of Affairs: P(k*, w*). But P(k*, w*) [because of P3 and the theorem P(w*, k*)] implies w* = k* – contradicting P7. But of course EXM(U.M., λo(o is actual)) is true: EXM(U.M., λo(o is actual)) iff [because of D20] O([λo(o is actual), U.M.]) iff [because of P30, “U.M.” being a referentially stable term] O(that U.M. is actual) iff [according to P10] U.M. is actual – and there is no doubt about this.

That λo(o is actual) is neither a logically essential property absolutely nor a logically essential property of me does not preclude that λo(o is actual) is a logically essential property of something. In fact, it is a logically essential property of something, for we can prove: EXME1(t*, λo(o is actual)). Proof: According to D22, we need to show: [λo(o is actual), t*] = t*. If A(t*) is logically provable, that is, ├ A(t*), then it follows according to the rule-schema P23: ├ t* = that A(t*), and hence: t* = that A(t*), and therefore (according to P30, “t*” being a referentially stable term): [λo(o is actual), t*] = t*. All that remains to be shown is that A(t*) is logically provable, that is, we must deduce A(t*) purely on the basis of axiomatic principles and inference-rules that are valid purely for conceptual reasons. But this deduction has already been produced; see Chapter 3, footnote 9.

Concerning property (6): Consider finally the property λo(o is actual ⊃ o is physical), or more briefly: λo(A(o) ⊃ ϕ(o)). What is the relationship of this property to the property λoϕ(o)? It must first be noted that λoϕ(o), like λoA(o), appears to be universally predicable.

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Note that this implies that [λoϕ(o), λoϕ(o)] is a state of affairs. But of course it does not yet follow – on the basis of the restricted principle of propertyexemplification – that EXM(λoϕ(o), λoϕ(o)). For this result, one would need the truth of ϕ(λoϕ(o)), which – in contrast to the truth of A(λoA(o)) (see P35, (i)) – is somewhat doubtful.

Given that both λoA(o) and λoϕ(o) are universally predicable, it is plausible to assume that λo(A(o) ⊃ ϕ(o)) is also universally predicable. But the universal predicability of λo(A(o) ⊃ ϕ(o)) cannot be deduced from the universal predicability of λoA(o) and λoϕ(o). If λoA(o) and λoϕ(o) are universally predicable, then we have for an arbitrary entity x: S([λoA(o), x]) and S([λoϕ(o), x]), and therefore: S(disj(neg([λoA(o), x]), [λoϕ(o), x])). Thus: ∀xS(disj(neg([λoA(o), x]), [λoϕ(o), x])). There seems to be no reason against assuming ∀x([λo(A(o) ⊃ ϕ(o)), x] = disj(neg([λoA(o), x]), [λoϕ(o), x])); but this is an extra assumption that cannot be discharged (it will, of course, follow if the universal predicability of λo(A(o) ⊃ ϕ(o)) is already presupposed). Making that assumption, we obtain ∀xS([λo(A(o) ⊃ ϕ(o)), x]), and λo(A(o) ⊃ ϕ(o)), too, turns out to be universally predicable.

Given the universal predicability of λoϕ(o) and λo(A(o) ⊃ ϕ(o)), it is easy to show: PPR(λo(A(o) ⊃ ϕ(o)), λoϕ(o)). But PPR(λoϕ(o), λo(A(o) ⊃ ϕ(o))) is false if there is – as is rather likely – a non-actual entity x that is not physical. Because of ¬A(x), we have A(x) ⊃ ϕ(x), and therefore according to the restricted principle of property-exemplification: (i) EXM(x, λo(A(o) ⊃ ϕ(o)). Because of ¬ϕ(x), we have according to the restricted principle of property-exemplification: (ii) ¬EXM(x, λoϕ(o)). Hence we have because of S([λoϕ(o), x]) (by applying D20 and D6 to (i) and (ii)): S([λo(A(o) ⊃ ϕ(o)), x]) and A([λo(A(o) ⊃ ϕ(o)), x]) and S([λoϕ(o), x]) and ¬A([λoϕ(o), x]). It therefore follows according to the Actuality Principle for States of Affairs: P([λo(A(o) ⊃ ϕ(o)), x], w*) and ¬P([λoϕ(o), x], w*). Therefore: ¬P([λoϕ(o), x], [λo(A(o) ⊃ ϕ(o)), x]); otherwise we would have P([λoϕ(o), x], w*) because of P1. Hence finally (according to D32): ¬PPR(λoϕ(o), λo(A(o) ⊃ ϕ(o))).

Why also accord attention to λo(A(o) ⊃ ϕ(o)), and not only to λoϕ(o)? The reason becomes apparent once we have introduced certain additional, important concepts of essentiality: D38 EXMAEn(y, x) =Def P([x, y], conj(bn, that A(y))) (for n ≥ 1).

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According to D38, x is an existence-n-essential property of y if, and only if, the saturation of x by y is an intensional part of the conjunction of bn (the n-th basis of necessity) with the state of affairs that y is actual. At first sight separate from the concepts defined by D38, there is also the following concept: D39 EXMAE(y, x) =Def P([x, y], that A(y)). According to D39, x is a (simpliciter) existence-essential property of y if, and only if, the saturation of x by y is an intensional part of the state of affairs that y is actual. But, in fact, the concept introduced by D39 is not separate from the concepts introduced by the definition-schema D38, for we can easily prove (on the basis of conceptually true principles only): ∀x∀y(EXMAE(y, x) ≡ EXMAE1(y, x)). Instead of “existence-essential” one can also say “actuality-essential” – and perhaps, for the sake of clarity (in view of the ambiguity of the word “exist” pointed out in Section 1.4), one should say “actuality-essential,” notwithstanding my having stipulated (in Section 1.4) that the predicate of existence is to be considered synonymous with the predicate of actuality. However, the fact remains that existence-talk comes easier to the mind, and hence is more convenient, than actuality-talk. In the philosophy of modality, existence-essentiality (simpliciter) is often identified with essentiality – essentiality simpliciter, which here was stipulated to be identical with logical essentiality (see Section 7.2); David Lewis, for example, can be regarded as identifying essentiality and existence-essentiality with respect to all properties of individuals (as they are conceived of in his framework; note that, for him, the actuality of individuals is not a property of individuals; the matter is treated in Section 8.2). The two concepts, however, are very different. It is (trivially) true that λoA(o) is one of my existence-essential properties: EXMAE(U.M., λoA(o)) – because of P([λoA(o), U.M.], that A(U.M.)). Because of F(λoA(o), U.M.) and P30 we have: [λoA(o), U.M.] = that A(U.M.). Because of P2 and S([λoA(o), U.M.]) we have: P([λoA(o), U.M.], [λoA(o), U.M.]). Hence: P([λoA(o), U.M.], that A(U.M.)).

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But it is certainly not true that λoA(o) is one of my essential (i.e., logically essential) properties: [λoA(o), U.M.] = t* is not true, since it is certainly logically possible for me to be not actual. Existence-essentiality and essentiality are different. But existenceessentiality can be expressed in terms of essentiality to the extent that properties λoB[o] and λo(A(o) ⊃ B[o]) have the same predicative fit. This is shown by the following theorem: ∀y(F(λoB[o], y) ≡ F(λo(A(o) ⊃ B[o]), y)) ⊃ ∀y[EXMAE(y, λoB[o]) ≡ EXME1(y, λo(A(o) ⊃ B[o]))]. Proof: EXMAE(y, λoB[o]), according to D39, is P([λoB[o], y], that A(y)), which – according to the mereology of states of affairs – is equivalent to [1] P([λoB[o], y], conj(that A(y), t*)). And EXME1(y, λo(A(o) ⊃ B[o])), according to D22, is [λo(A(o) ⊃ B[o]), y] = t*, which – according to the mereology of states of affairs – is equivalent to [2] P([λo(A(o) ⊃ B[o]), y], t*). Now, assuming ∀y(F(λoB[o], y) ≡ F(λo(A(o) ⊃ B[o]), y)) and [1], we have S([λoB[o], y]) [because of the assumption and P0], hence F(λoB[o], y) [according to D17], hence F(λo(A(o) ⊃ B[o]), y) [according to assumption]. Hence (according to P30): [3] [λoB[o], y] = that B[y], and [4] [λo(A(o) ⊃ B[o]), y] = that (A(y) ⊃ B[y]). And assuming ∀y(F(λoB[o], y) ≡ F(λo(A(o) ⊃ B[o]), y)) and [2], we again reach (in a completely analogous way): [3] and [4]. Consider, then, that the following is a general theorem of the mereology of states of affairs: ∀z∀y(P(that B[y], conj(that C[y], z)) ≡ P(that (C[y] ⊃ B[y]), z)), and therefore we also have: [5] ∀y(P(that B[y], conj(that A(y), t*)) ≡ P(that (A(y) ⊃ B[y]), t*)). Therefore: By assuming ∀y(F(λoB[o], y) ≡ F(λo(A(o) ⊃ B[o]), y)) and [1], we first obtain P(that B[y], conj(that A(y), t*)) (employing [3]), and therefore (employing [5]): P(that (A(y) ⊃ B[y]), t*). Hence (employing [4]): P([λo(A(o) ⊃ B[o]), y], t*), that is: [2]. By assuming ∀y(F(λoB[o], y) ≡ F(λo(A(o) ⊃ B[o]), y)) and [2], we first obtain P(that (A(y) ⊃ B[y]), t*) (employing [4]), and therefore (employing [5]): P(that B[y], conj(that A(y), t*)). Hence (employing [3]): P([λoB[o], y], conj(that A(y), t*)), that is: [1]. – This completes the proof.

Therefore: since λoϕ(o) and λo(A(o) ⊃ ϕ(o)) have the same predicative fit (being both universally predicable), it follows that λoϕ(o) – being physical – is one of my existence-essential properties if, and only if, λo(A(o) ⊃ ϕ(o)) – being physical if actual – is one of my logically essential (simpliciter essential) properties. If λoϕ(o) is not one of my logically essential properties, it still might be one of my existence-essential properties. But

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this, as we have seen, would mean that λo(A(o) ⊃ ϕ(o)) is one of my logically essential properties. Is this latter property a logically essential property of mine? To me, it seems obvious that it is not. Others will doubtlessly disagree. But the discussion of this issue is not a matter to be included in this book (see, however, my The Two Sides of Being). It should be noted that the above result can be supplemented. We also have the following theorem: ∀y(F(λoB[o], y) ≡ F(λo(A(bn) ⊃ B[o]), y)) ⊃ ∀y[EXMEn(y, λoB[o]) ≡ EXME1(y, λo(A(bn) ⊃ B[o]))]. This theorem and the one preceding it together imply that every kind of essentiality is reducible to logical essentiality – at least for a great many properties. Note, finally, that there is such a thing as the universal existenceessence (or actuality-essence) of something. Being a universal existenceessence of something is defined as follows (analogous to the definition of universal n-essence, D30): D40 ESSUA(x, y) =Def EXMAE(y, x) ∧ ∀zF(x, z) ∧ ∀x´(EXMAE(y, x´) ∧ ∀zF(x´, z) ⊃ PPR(x´, x)). It is easily demonstrable that for each y the universally predicable property λo(o = y ∧ A(y)) is the universal existence-essence of y.

7.5

The (Transworld) Identity of Individuals and Logical I-Essences – and Regarding the Question What Individuals There Are

What has been said so far logically implies that the General Principle of the Identity of Indiscernibles is a rather trivial truth: ∀z∀z´(∀x(PR(x) ⊃ (EXM(z, x) ≡ EXM(z´, x))) ⊃ z = z´). Proof: Assume ∀x(PR(x) ⊃ (EXM(z, x) ≡ EXM(z´, x))). Therefore [since PR(λo(z = o)), according to P30]: EXM(z, λo(z = o)) ≡ EXM(z´, λo(z = o)). The predicative fit of λo(z = o) is doubtlessly universal, for any z (cf. what has been said in Section 7.4 concerning the property λo(o = a0)). Hence we have by the restricted prin-

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ciple of property-exemplification [because of F(λo(z = o), z) and F(λo(z = o), z´)]: EXM(z, λo(z = o)) ≡ z = z, EXM(z´, λo(z = o)) ≡ z = z´. Hence by combining the equivalences: z = z ≡ z = z´. And therefore: z = z´.

The General Principle of the Identity of Indiscernibles does not provide a substantial answer to the question of sufficient identity-conditions for states of affairs, properties, and individuals. When being asked, “Under which conditions are states of affairs, properties, individuals identical?,” it is pointless to point to the Indiscernibility Principle (as I shall more briefly say instead of “General Principle of the Identity of Indiscernibles”), for that principle is quite trivial (for, in effect, it says that z and z´ are identical if they are identical), whereas the question asked is not. Substantial answers regarding the questions of sufficient identity-conditions for states of affairs and properties have already been provided (P3 and P29). But what is a substantial answer to the question of what is a sufficient identitycondition for (all) individuals? In fact, there is no such answer. Prima facie, the following seems a good idea: The Identity Principle for Individuals ∀y∀y´(I(y) ∧ I(y´) ∧ ∀x(ESSI1(x, y) ≡ ESSI1(x, y´)) ⊃ y = y´). According to this principle, individuals are identical if they have the same logical I-essence (or in other words: the same 1-I-essence; see definition D31). But this is no news: it is already provable. And if ∀x(ESSI1(x, y) ≡ ESSI1(x, y´)) is replaced by what is provably equivalent to it: λIo(y = o) = λIo(y´ = o), then the above principle seems just as trivial as the Indiscernibility Principle. This is a just impression. But it should be noted that the Identity Principle for Individuals has a quite different character than the Indiscernibility Principle. This can be brought out vividly by considering two indiscernibility principles that appear to be less trivial than the Indiscernibility Principle. First, the Indiscernibility Principle Fortified: ∀z∀z´(∃w[MC(w) ∧ ∀x(PR(x) ⊃ (EXM(z, x, w) ≡ EXM(z´, x, w)))] ⊃ z = z´). Proof: Assume MC(w) ∧ ∀x(PR(x) ⊃ (EXM(z, x, w) ≡ EXM(z´, x, w))). We obtain: EXM(z, λo(z = o), w) ≡ EXM(z´, λo(z = o), w), that is (according to D35, D16): P([λo(z = o), z], w) ≡ P([λo(z = o), z´], w). Hence according to P30: P(that (z =

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z), w) ≡ P(that (z = z´), w). Because of z = z, we have according to P32: t* = that (z = z), and therefore [because of S(w) – which is a consequence of the assumption MC(w) – and the theorem ∀q(S(q) ⊃ P(t*, q))]: P(that (z = z), w). Hence: P(that (z = z´), w). But then we must also have: k* ≠ that (z = z´). Otherwise we would get P(k*, w), and therefore – because of the theorem ∀q(S(q) ⊃ P(q, k*)) and P1 – ∀q(S(q) ⊃ P(q, w)), and hence – because we also have S(w) – T(w), according to D5. But this contradicts the assumption MC(w), according to D15. Since we have: k* ≠ that (z = z´), it follows by P32: z = z´. [Note that P32 can be applied, since both “z” and “z´” are referentially stable terms, being variables.]

Second, the Indiscernibility Principle Extra Fortified: ∀z∀z´(∃w∃w´[MC(w) ∧ MC(w´) ∧ ∀x(PR(x) ⊃ (EXM(z, x, w) ≡ EXM(z´, x, w´)))] ⊃ z = z´). The Indiscernibility Principle Extra Fortified looks stronger than the Indiscernibility Principle Fortified. In fact, it is not stronger. The reason is this: if the worlds w and w´ are different, then there is bound to be a property x such that EXM(z, x, w) and ¬EXM(z´, x, w´). Which property x do I have in mind? This: λo(o = z ∧ O(w)). λo(o = z ∧ O(w)) is universally predicable [cf. the property λo(o = a0 ∧ O(w0)) considered in Section 7.4]. Hence [according to P30, P15, P32, P24, and keeping in mind S(w), because of MC(w)]: [λo(o = z ∧ O(w)), z] = that (z = z ∧ O(w)) = conj(that (z = z), that O(w)) = conj(t*, w) = w. Therefore: P([λo(o = z ∧ O(w)), z], w) [because of P2], and hence: EXM(z, λo(o = z ∧ O(w)), w) [D16, D35]. In contrast, [λo(o = z ∧ O(w)), z´] = that (z´ = z ∧ O(w)) = conj(that (z´ = z), that O(w)) = conj(that (z´ = z), w). According to P32, t* = that (z´ = z), or k* = that (z´ = z). If t* = that (z´ = z), then conj(that (z´ = z), w) is w, and therefore: [λo(o = z ∧ O(w)), z´] = w. Hence ¬P([λo(o = z ∧ O(w)), z´], w´), since ¬P(w, w´) [because of w ≠ w´ and because of the following theorem of the mereology of states of affairs: ∀w∀w´(MC(w) ∧ MC(w´) ∧ P(w, w´) ⊃ w = w´)]. Therefore: ¬EXM(z´, λo(o = z ∧ O(w)), w´). If, however, k* = that (z´ = z), then conj(that (z´ = z), w) is k*, and therefore: [λo(o = z ∧ O(w)), z´] = k*. Hence ¬P([λo(o = z ∧ O(w)), z´], w´), since ¬P(k*, w´) [because of MC(w´)]. Therefore once again: ¬EXM(z´, λo(o = z ∧ O(w)), w´). On the basis of the assumption that w and w´ are different worlds, we have now proven: EXM(z, λo(o = z ∧ O(w)), w) and ¬EXM(z´, λo(o = z ∧ O(w)), w´).

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The above considerations show that ∃w∃w´[MC(w) ∧ MC(w´) ∧ ∀x(PR(x) ⊃ (EXM(z, x, w) ≡ EXM(z´, x, w´)))] implies that the worlds w and w´ are identical. Hence the antecedent of the Indiscernibility Principle Extra Fortified provably implies the antecedent of the Indiscernibility Principle Fortified (in addition to the trivially provable converse implication), and hence the two principles turn out to be provably equivalent. But if it is now clear that, for different worlds w and w´, the properties x´ such that EXM(y´, x´, w´) are not the properties x such that EXM(y, x, w), does it not follow that the individuals y and y´ are different? – It does not follow if one and the same individual may have different properties in different possible worlds – and there certainly seem to be such individuals. In fact, for every individual y and possible world w, there is a property x such that y has x in w and in no other possible world. This is true because the following is true (provably so, along the lines exhibited above): ∀y∀w[I(y) ∧ MC(w) ⊃ EXM(y, λo(o = y ∧ O(w)), w) ∧ ¬∃w´(MC(w´) ∧ w´ ≠ w ∧ EXM(y, λo(o = y ∧ O(w)), w´))]. Thus, whatever individual y one is considering, in whatever possible worlds w and w´, if w and w´ are different, then the properties of y in w will not be the properties of y in w´; for in w, y has the property λo(o = y ∧ O(w)), which it does not have in w´; and in w´, y has the property λo(o = y ∧ O(w´)), which it does not have in w. On the basis of the above theorem, it is rendered particularly perspicuous what we have already found above: that whatever individuals y and y´ one is considering, in whatever possible worlds w and w´, if w and w´ are different, then the properties of y in w will not be the properties of y´ in w´. For in w, y has the property λo(o = y ∧ O(w)). There are two possible cases: y = y´, or y ≠ y´. In the first case, λo(o = y ∧ O(w)) is not a property of y´ in w´, since it is not a property of y in w´ and y´ = y. In the second case, λo(o = y ∧ O(w)) is not a property of y´ in w´, since [λo(o = y ∧ O(w)), y´] = k* – because of y´ ≠ y – and ¬P(k*, w´).

Although the properties of individual y in world w are inescapably different from the properties of individual y´ in a world w´ that is different from w, y and y´ can be the numerically same individual, constituting a case of so-called transworld identity (cf. Section 4.2). For the following statement is provably not true: ∀y∀y´∀w∀w´∀x(I(y) ∧ I(y´) ∧ MC(w) ∧ MC(w´) ∧ PR(x) ∧ ¬(EXM(y, x, w) ≡ EXM(y´, x, w´)) ⊃ y ≠ y´). But of

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course y and y´ can also be different individuals when they have different properties in different possible worlds. Which criterion, then, decides whether they are identical or not? – The one already presented above: the criterion contained in the Identity Principle for Individuals. Accordingly, if we have ∀x(ESSI1(x, y) ≡ ESSI1(x, y´)), then y and y´ are identical; and if we have ∃x¬(ESSI1(x, y) ≡ ESSI1(x, y´)), then y and y´ are different (the latter being already a consequence of the logic of identity). Both ∀x(ESSI1(x, y) ≡ ESSI1(x, y´)) and ∃x¬(ESSI1(x, y) ≡ ESSI1(x, y´)) can be ascertained without considering the properties y and y´ have in the actual or any other possible world; for the concept of logical I-essence is defined without bringing in the concept of world-relative exemplification, either explicitly (as in EXM(y, x, w)) or implicitly (as in EXM(y, x), which amounts to EXM(y, x, w*)). Nevertheless, the following sufficient conditions of identity for individuals y and y´ have now been seen to be provably equivalent: ∀x(PR(x) ⊃ (EXM(y, x) ≡ EXM(y´, x))),6 ∃w[MC(w) ∧ ∀x(PR(x) ⊃ (EXM(y, x, w) ≡ EXM(y´, x, w)))],7 ∃w∃w´[MC(w) ∧ MC(w´) ∧ ∀x(PR(x) ⊃ (EXM(y, x, w) ≡ EXM(y´, x, w´)))], ∀x(ESSI1(x, y) ≡ ESSI1(x, y´)). Thus, since you and I are different individuals, we differ in our properties, we differ in our logical I-essence, and in the properties we have in any possible world. But all four sufficient identity-conditions are useless for estab6

For the equivalence of this condition with the fourth one, consider, first, that ∀x(PR(x) ⊃ (EXM(y, x) ≡ EXM(y´, x))) implies ∀x(ESSI1(x, y) ≡ ESSI1(x, y´)) on the basis of the (proven) Indiscernibility Principle; and consider, second, that ∀x(ESSI1(x, y) ≡ ESSI1(x, y´)) implies ∀x(PR(x) ⊃ (EXM(y, x) ≡ EXM(y´, x))) (for individuals y and y´) on the basis of the (separately provable) Identity Principle for Individuals. 7 For the equivalence of this condition with the first one, consider, first, that ∃w[MC(w) ∧ ∀x(PR(x) ⊃ (EXM(y, x, w) ≡ EXM(y´, x, w)))] implies ∀x(PR(x) ⊃ (EXM(y, x) ≡ EXM(y´, x))) on the basis of the (separately proven) Indiscernibility Principle Fortified; and consider, second, that ∀x(PR(x) ⊃ (EXM(y, x) ≡ EXM(y´, x))) implies ∃w[MC(w) ∧ ∀x(PR(x) ⊃ (EXM(y, x, w) ≡ EXM(y´, x, w)))], because of (provable) MC(w*) and ∀x´∀y´´(EXM(y´´, x´) ≡ EXM(y´´, x´, w*)).

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lishing the identity of individuals y and y´, since they cannot be established as true of y and y´ without presupposing the very identity of y and y´. In this sense, the above-considered identity-principles that correspond to the four considered conditions are trivial and not substantial (but not, I trust, in the sense of being obvious and uninteresting). I finally touch on the question what individuals there are. In answer to this question, I am content to offer examples of individuals (often myself); I will not offer general descriptions of kinds of individuals. Likewise, I will remain largely silent on the question of how many individuals there are. Doubtlessly, there are very many. But I am not prepared to set down as a numbered principle any specification of the number of individuals that is more specific than the following assertion: P34 ∃y∃y´(I(y) ∧ I(y´) ∧ y ≠ y´). In any case, the number of individuals does not exceed the number of maximal-consistent totally defined properties of individuals, where totally defined properties of individuals are defined as follows: D41 PRIT(x) =Def PRI(x) ∧ ∀y(I(y) ⊃ F(x, y)), and maximal-consistent totally defined properties of individuals as follows: D42 PRITMC(x) =Def PRIT(x) ∧ x ≠ λI(o ≠ o) ∧ ∀z(PPR(x, z) ∧ z ≠ x ⊃ z = λI(o ≠ o)). As a consequence of D42, a maximal-consistent totally defined property of individuals is a proper intensional part only of λIo(o ≠ o). That property, in turn, is the I-restriction of the universally predicable property λo(o ≠ o). According to P30, P32, P14, and a theorem of the mereology of states of affairs, we have for any entity y: [λo(o ≠ o), y] = that (y ≠ y) = neg(that y = y) = neg(t*) = k*. Hence [λIo(o ≠ o), y] is k* for any individual y, and c* for any non-individual y. What would a totally defined property of individuals, x, have to be like – regarding its saturations – to be a proper part only of λIo(o ≠ o) (which is itself a totally defined property of individuals, as we have just seen, and of which, obviously, every such property is an intensional part)? It, x, would have to be like this: for one individual y, [x, y] is a maximal-consistent state of affairs (that is, a possible world), for every other individual y´, [x, y´] is k*. In this way the property x differs

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minimally from λIo(o ≠ o) – and this is precisely what a maximalconsistent totally defined property of individuals is: a property that differs minimally from λIo(o ≠ o). We have the following theorem (or we might have formulated it as a definition, turning D42 into a theorem): ∀x[PRITMC(x) ≡ PRIT(x) ∧ ∃y(I(y) ∧ MC([x, y]) ∧ ∀y´(I(y´) ∧ y´ ≠ y ⊃ [x, y´] = k*))]. Next question: how many maximal-consistent totally defined properties of individuals are there? Every maximal-consistent totally defined I-property (that is, property of individuals) x can be indexed by an individual Ix and a possible world Wx, where Ix is the individual y such that MC([x, y]), and Wx is the possible world w such that [x, Ix] = w. And obviously we have for all maximal-consistent totally defined I-properties x and x´: if x ≠ x´, then Ix ≠ Ix´ or Wx ≠ Wx´. Thus the maximal-consistent totally defined Iproperties x are one-to-one correlated with ordered pairs , consisting of an individual in the first place and a possible world in the second. Suppose all ordered pairs consisting of an individual in the first place and a possible world in the second are in the indicated way one-to-one correlated with maximal-consistent totally defined I-properties. This gives us the greatest possible number of such properties. Since any other supposition seems arbitrary, we postulate: ∀y∀w[I(y) ∧ MC(w) ⊃ ∃x(PRITMC(x) ∧ y = Ix ∧ w = Wx)]. But, in fact, this need not be postulated, because it is already provable: Let y be an individual, w a possible world. Consider the property λIo(o = y ∧ O(w)). Since λo(o = y ∧ O(w)) is universally predicable (just like a property λo(o = a0 ∧ O(w0)); see Section 7.4), its I-restriction, λIo(o = y ∧ O(w)), must be predicable of all individuals. Hence we have: PRIT(λIo(o = y ∧ O(w))). Moreover, we have: (i) [λIo(o = y ∧ O(w)), y] = w, and hence MC([λIo(o = y ∧ O(w)), y]) because of MC(w); (ii) ∀y´(I(y´) ∧ y´ ≠ y ⊃ [λIo(o = y ∧ O(w)), y´] = k*). [Cf. the considerations concerning λo(o = a0 ∧ O(w0)) in Section 7.4.] Therefore we obtain: PRITMC(λIo(o = y ∧ O(w))), according to the theorem immediately preceding the one to be proven. And if we let “x” stand for “λIo(o = y ∧ O(w)),” it is clear that we have: PRITMC(x) and y = Ix and w = Wx.

It follows that for every individual y there are as many corresponding maximal-consistent totally defined I-properties as there are pairs (y, w),

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where w is a possible world. If y is an arbitrary individual and w an arbitrary possible world, then the maximal-consistent totally defined I-property x corresponding one-to-one to the ordered pair (y, w) is specified by saying: (i) for every non-individual z: [x, z] = c*; (ii) for every individual y´ different from y: [x, y´] = k*; (iii) [x, y] = w. And any arbitrary maximalconsistent totally defined I-property x corresponds in the said way one-toone to an ordered individual-world pair, namely to , since we have: (i) for every non-individual z: [x, z] = c*; (ii) for every individual y´ different from Ix: [x, y´] = k*; (iii) [x, Ix] = Wx. As a consequence of these findings, the cardinal number of maximal-consistent totally defined Iproperties is given by the following equation: card(PRITMC) = card(I) × card(MC). Since card(MC) > 0, it follows that card(I) does not exceed card(PRITMC) (as was asserted below P34). Yet card(I) might be equal to card(PRITMC). This would be the case if card(MC) were 1. But the number of possible worlds has already been found to be greater than 1; see Chapter 3. The equality in question would also result if card(I) were denumerably infinite and card(MC) at most denumerably infinite. In any case, if card(I) were equal to card(PRITMC), it would be somehow possible to correlate individuals and maximal-consistent totally defined I-properties one-to-one. But this one-to-one correlation would not be an occasion for identification – that is, reduction – for the identification of individuals with maximalconsistent totally defined I-properties would contradict a fact already established in Section 7.1: no individual is a property. Moreover, the one-toone correlation in question would presumably not be an intuitively satisfactory correlation, perhaps be not even expressible. Thus there is no basis for identifying individuals with maximal-consistent totally defined Iproperties, even if the relevant cardinalities were right for this. There are, however, further perspectives on that issue if one envisages a different conception of individual than is here adhered to, as we shall see in the following section. 7.5.1

Notiones Completae and L(eibniz)-Individuals

Maximal-consistent totally defined I-properties fill the bill of a Leibnizian notion: the notion of notio completa. A notio completa is the ontological

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analog of a consistent and complete (but not necessarily true) description of an individual. If we add the definition D43 nc(y, w) =Def λIo(o = y ∧ O(w)), then we can prove: ∀y∀w[I(y) ∧ MC(w) ⊃ PRITMC(nc(y, w))]; ∀y∀y´∀w∀w´[I(y) ∧ I(y´) ∧ MC(w) ∧ MC(w´) ∧ (y ≠ y´ ∨ w ≠ w´) ⊃ nc(y, w) ≠ nc(y´, w´)]; (iii) ∀x[PRITMC(x) ⊃ ∃y∃w(I(y) ∧ MC(w) ∧ x = nc(y, w))]; (iv) ∀y∀w[I(y) ∧ MC(w) ⊃ ∀x[PRIT(x) ∧ EXM(y, x, w) ≡ PPR(x, nc(y, w))]]. (i) (ii)

This theorem states (a) that the notiones completae (of individuals y with regard to possible worlds w) are precisely the maximal-consistent totally defined I-properties [(i) and (iii)], (b) that the notio completa of an individual y with regard to a possible world w is the notio completa only of that individual with regard to that possible world [(ii)], (c) that the notio completa of an individual y with regard to a world w comprises precisely the totally defined I-properties which y exemplifies in w [(iv)]. Part (iv) of the theorem is the part of it which justifies the reading “notio completa of y with regard to w” for “nc(y, w)” in case y is an individual and w a possible world. Proof of (iv): Suppose in the first place: I(y), MC(w), PRIT(x), EXM(y, x, w). Hence PRIT(nc(y, w)) [according to (i) of the above theorem], and therefore [because of PRIT(x)]: ∀y´(F(x, y´) ≡ F(nc(y, w), y´)). Now assume F(x, y´), and consider [x, y´] and [nc(y, w), y´]. There are two cases: (1) y´ = y, (2) y´ ≠ y. In case (1): [x, y´] = [x, y], and therefore [because of EXM(y, x, w), D35, D16]: P([x, y´], w); and also: [nc(y, w), y´] = [nc(y, w), y] = [λIo(o = y ∧ O(w)), y] = that (y = y ∧ O(w)) = conj(that (y = y), that O(w)) = conj(t*, w) = w. Therefore: P([x, y´], [nc(y, w), y´]). [Comment: For all individuals y´´, [λIo(o = y ∧ O(w)), y´´] = [λo(o = y ∧ O(w)), y´´]; hence – because of I(y) – [λIo(o = y ∧ O(w)), y] = [λo(o = y ∧ O(w)), y], and [λo(o = y ∧ O(w)), y], in turn, is identical with the state of affairs that (y = y ∧ O(w)), because of P30 (λo(o = y ∧ O(w)) is – for any y and w – universally predicable). For the rest, consider P15, P32, P24, and the theorem ∀z(S(z) ⊃ conj(t*, z) = z).] In case (2): [nc(y, w), y´] = [λIo(o = y ∧ O(w)), y´] = that (y´ = y ∧ O(w)) = conj(that (y´ = y), that O(w)) = conj(k*, w) = k*. Therefore again: P([x, y´], [nc(y, w), y´]).

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[Comment: y´ is an individual because F(x, y´) was assumed and also PRIT(x); conj(that (y´ = y), that O(w)) = conj(k*, w) is a consequence of the assumption y´ ≠ y, and of P32, P24, and the assumption MC(w). For the rest, consider the theorem ∀z(S(z) ⊃ conj(k*, z) = k*).] We have now derived: PPR(x, nc(y, w)) (applying D32). Suppose in the second place: I(y), MC(w), PPR(x, nc(y, w)). Hence PRIT(nc(y, w)) [according to (i) of the above theorem], and therefore: PRIT(x), because of PPR(x, nc(y, w)) and D32. In turn, because of PRIT(x) and I(y) we have: F(x, y). Hence P([x, y], [nc(y, w), y]), because of PPR(x, nc(y, w)) and D32. Hence P([x, y], w), because [nc(y, w), y] = w, as we have already seen. Therefore: EXM(y, x, w), according to D16 and D35.

We have seen in the previous section that there is no apparent one-toone correlation between individuals and notiones completae (or maximalconsistent totally defined I-properties). There is, however, such a correlation between what I call “L-individuals” and notiones completae. What are L-individuals? First of all, L-individuals are not individuals (in the sense used in the present theory), but entities which Gottfried Wilhelm Leibniz (or a fictive philosopher rather similar to him) called “individuals.” But Lindividuals and individuals are closely related: L-individuals are, so to speak, individuals with a world-localization added to them. Thus: (i) for each L-individual m, one can speak of the world of m, w(m) – which is some maximal-consistent state of affairs, and of the kernel of m, k(m) – which is a certain individual (in the sense used in the present theory); (ii) each L-individual is distinguished from every other L-individual by its world or by its kernel: there are no two L-individuals which are such that their worlds are identical and also their kernels; (iii) for each individual y and world w there is an L-individual m which is such that y = k(m) and w = w(m). There can be no more than one such L-individual (if there were two, then it would not be the case that each L-individual is distinguished from every other L-individual by its world or by its kernel); the Lindividual determined by y and w will be designated by “li(y, w).” If we introduce the concept of L-exemplification, in short: L-EXM(m, x), the relationship between L-individuals and individuals as regards exemplification can be compendiously characterized as follows: For all L-individuals m and properties x: L-EXM(m, x) ≡ EXM(k(m), x, w(m)). For all individuals y, worlds w, and properties x: EXM(y, x, w) ≡ L-EXM(li(y, w), x).

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I am not going to introduce, into the present theory, L-individuals as a separate basic category of entities (besides states of affairs, individuals, and properties). But this is no reason not to speak here about L-individuals (using the predicate “L-I(x)”), for they can be identified with (i.e., reduced to) entities that are already available in the present theory. L-individuals could be identified with ordered pairs that consist of an individual in the first place and a possible world in the second. But the order of the two components is irrelevant, and hence L-individuals could also be identified, more simply, with pair-sets whose two elements are, respectively, a possible world and an individual. If they are reduced in this manner, then Lindividuals will turn out to be properties, since the pair-sets L-individuals are taken to be according to the envisaged reduction are (in the present theory) nothing else than the (universally predicable) properties λo(o = y ∨ o = w), for each individual y and possible world w. L-individuals can also be identified with maximal-consistent totally defined I-properties, i.e., with notiones completae – which reduction of them is less simple, but also more natural and more interesting (also from the historical point of view) than the two (possible) reductions considered in the previous paragraph. The identification of L-indviduals with notiones completae, then, is effected by the following definition: D44 L-I(m)=Def PRITMC(m). In order to avoid confusion, the following theorem – which is obvious in view of the theorem ∀m(PR(m) ⊃ ¬I(m)), proven in Section 7.1 (below D19), and in view of D44, D42, D41, D19, D18 – should always be kept in mind: ∀m(L-I(m) ⊃ ¬I(m)) – “L-individuals are not individuals.” Note that this assertion is not an artifact of the identification of L-individuals with maximal-consistent totally defined I-properties. Though it is provable on the basis of that identification, it is true in any case, with or without that identification.

Fitting the above reductive definition of L-individuals (i.e., D44), the functional terms k(m), w(m) and li(y, w), already used above, can be defined as follows:

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D45 (i) k(m) =Def ιy´(I(y´) ∧ MC([m, y´])); (ii) w(m) =Def ιw´(MC(w´) ∧ [m, k(m)] =w´); (iii) li(y, w) =Def nc(y, w). Note that the functional terms defined by (i) and (ii) of D45 will have their intended readings only if m is an L-individual (i.e., if L-I(m), with L-I(m) =Def PRITMC(m)), and that the functional term defined by (iii) of D45 will have its intended reading only if y is an individual (i.e., I(y)) and w a possible world(i.e., MC(w)). k(m) and w(m), for L-individuals m, are of course no other notions than the previously used Ix and Wx, for maximal-consistent totally defined I-properties x.

In view of the reductive definition D44, we can define the fitting concept of exemplification for L-individuals (the one already made use of above): D46 L-EXM(m, x) =Def PPR(x, m). If we call m subjectum and x praedicatum, then D46 mirrors the well-known Leibnizian conception of the truth of singular predications: praedicatum inest subjecto.

Given definitions D44, D45 and D46, the above principles that correlate the world-relative exemplification by individuals with the Lexemplification by L-individuals can be proven, thus completing the reductive process. It is an interesting question to what extent the conception of individuals of Leibniz’s 20th-century successor, David Lewis, is captured by the conception of L-individuals. Regarding this, see Section 8.1.

7.6

The Actuality of Properties

When is a property actual? – The actuality of properties is governed by the following principle of two parts: P35 (i) ∀x(PR(x) ∧ ∃y(EXM(y, x) ∧ A(y)) ⊃ A(x)); (ii) ∀x(PR(x) ∧ ¬∃yEXM(y, x) ⊃ ¬A(x)). These principles clearly show that the actuality of properties is based on the actuality of states of affairs. For in view of D20 and D6, “EXM(y, x)” can be replaced in them by “S([x, y]) ∧ A([x, y]).” Moreover, “A(y)” in the first principle can be replaced by “A([λoA(o), y]).” This latter possibility of replacement is based on the general theorem ∀y(A(y) ≡ A([λoA(o), y])), which, in fact, implies that – not only the

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actuality of properties but – the actuality of anything is, in a clear sense, based on the actuality of states of affairs. Since λoA(o) is universally predicable, [λoA(o), y] is always – for every y – a state of affairs (according to D17). And because of this universal predicability of λoA(o), it follows according to the restricted principle of property exemplification (a consequence of P30 in Section 7.4): ∀y(EXM(y, λoA(o)) ≡ A(y)), that is (because of D20, D6): ∀y(S([λoA(o), y]) ∧ A([λoA(o), y]) ≡ A(y)), that is (because of ∀yS([λoA(o), y])): ∀y(A(y) ≡ A([λoA(o), y])).

According to P35, a property which is exemplified by something actual is itself actual, and a property which is not exemplified by anything is not actual. This is plausible enough. But there are two obvious questions: Why is it that the condition of being exemplified by something actual is only assumed as sufficient, and not also as necessary for the actuality of a property? Why is it that the condition of being exemplified (by something) is only assumed as necessary, and not also as sufficient for the actuality of a property? The simple answer to these questions is this: there are counterexamples. Consider the property being at t0 thought of by U.M., and suppose that, although I think of something at t0, all that I think of at t0 is not actual (this has certainly happened at some moment in time, call it “t0”). Does this make λo(o is thought of at t0 by U.M.) a property that is not actual? Certainly not: λo(o is thought of at t0 by U.M.) is already an actual property on the mere basis of its being exemplified; the actuality of what it is exemplified by is not needed for its actuality. Consider, in turn, the property being different from, but genetically identical with U.M. This property is plausibly exemplified by something: by a non-actual individual that is different from me, but genetically identical to me. Does this make λo(o ≠ U.M. ∧ o is genetically identical with U.M.) a property that is actual? Certainly not: though exemplified, λo(o ≠ U.M. ∧ o is genetically identical with U.M.) is still not actual, the reason being that nothing that exemplifies this property is actual. We can now address the question of actualism for properties. The thesis of actualism for properties consists in the assertion that every property is actual. This assertion is false, for we have just seen that there is a counterexample to it: λo(o ≠ U.M. ∧ o is genetically identical with U.M.) is a property that is not actual. One might rashly object that this result depends on the assumption that there are non-actual individuals (see the consideration in the previous paragraph). Not so. For if there were no nonactual individuals, then λo(o ≠ U.M. ∧ o is genetically identical with U.M.)

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would not be exemplified by anything at all, and therefore this property would again turn out to be non-actual – according to P35, (ii). Note that the – true – counterfactual conditional in the preceding statement is highly interesting. What is its basis? It is clearly the following – obtaining – state of affairs: that there is no actual individual that exemplifies λo(o ≠ U.M. ∧ o is genetically identical with U.M.). Compare what a traditional analysis of counterfactual conditionals – along the lines of Lewis or Stalnaker – would make of the counterfactual under consideration.

If, on the one hand, one bases non-actuality judgments for properties solely on P35, (ii), and, on the other hand, is ready to countenance exemplification by non-actual entities, then the number of properties of which one can be certain – solely on the basis of P35, (ii) – that they are nonactual dwindles drastically: “The property of being a centaur is a nonactual property because it is not exemplified by anything. – No, wait a second! Might it not be exemplified by something non-actual?” But the non-actuality of a property is not only implied by its not being exemplified by anything; in some cases it is already implied by the weaker condition of its not being exemplified by anything actual. So it is in the case of λo(o ≠ U.M. ∧ o is genetically identical with U.M.), and so it is in the case of the property of being a centaur. It is, however, not so in the case of λo(o is thought of at t0 by U. M.), as we have seen. In view of this situation, it is helpful to distinguish between properties appropriate for 1-actuality and properties appropriate for 2-actuality: D47 PRA1(x) =Def PR(x) ∧ (A(x) ≡ ∃yEXM(y, x)). D48 PRA2(x) =Def PR(x) ∧ (A(x) ≡ ∃y(A(y) ∧ EXM(y, x))). λo(o is thought of at t0 by U.M.) is a property appropriate for 1-actuality which is not a property appropriate for 2-actuality (as we have seen), and λo(o ≠ U.M. ∧ o is genetically identical with U.M.) is a property appropriate for 2-actuality which is – plausibly – not a property appropriate for 1actuality (as we have seen). Is there a property that is both a property appropriate for 1-actuality and 2-actuality? There is indeed: λoA(o) – the property of being actual – is such a property, and so is every property λo(A(o) ∧ B[o]); in fact, every property that is exemplified only by actual entities (if at all) is both a property appropriate for 1-actuality and a prop-

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erty appropriate for 2-actuality. If actualism for individuals were true (concerning this issue, see below), then every property that is exemplified – if at all – only by individuals (and therefore a fortiori every property of individuals, i.e., every property that is predicable only of individuals) would be both a property appropriate for 1-actuality and a property appropriate for 2actuality. But is there a property that is neither a property appropriate for 1actuality nor a property appropriate for 2-actuality? Of such a property x, neither A(x) ≡ ∃yEXM(y, x) nor A(x) ≡ ∃y(A(y) ∧ EXM(y, x)) is true. Hence: (1) A(x) is true and ∃yEXM(y, x) is not, or (2) ∃yEXM(y, x) is true and A(x) is not; and also: (1´) A(x) is true and ∃y(A(y) ∧ EXM(y, x)) is not, or (2´) ∃y(A(y) ∧ EXM(y, x)) is true and A(x) is not. But (1) contradicts P35, (ii), and (2´) contradicts P35, (i). This leaves us with (2) and (1´); but they contradict each other. – These considerations demonstrate that the following assertion is true: ∀x(PR(x) ⊃ PRA1(x) ∨ PRA2(x)). And thus the question this paragraph began with has been answered negatively. Actualism for properties is just about as implausible (not to say: absurd) as actualism for states of affairs. There can be no reasonable doubt about the non-actuality of λo(o ≠ o) or the non-actuality of λo(A(o) ∧ o is a centaur); for if the former property were actual, then something would not be identical with itself, and if the latter property were actual, then there would be at least one actual centaur. Actualism for individuals, however, is a different matter. But note: if one believes of any property exemplified only by individuals that it is a property appropriate for 2-actuality, but not a property appropriate for 1-actuality – and it is rather tempting to believe this of the property λo(o human being), for example –, then one has already opted against actualism for individuals. This is seen as follows: Suppose that x is a property exemplified only by individuals, and that it is, moreover, a property appropriate for 2-actuality, but not a property appropriate for 1-actuality. Hence: (1) A(x) ≡ ∃y(A(y) ∧ EXM(y, x)), and (2) ¬(A(x) ≡ ∃yEXM(y, x)). From (2) we obtain: either (3) A(x) ∧ ¬∃yEXM(y, x), or (4) ∃yEXM(y, x) ∧ ¬A(x). (3) is logically incompatible with (1). Hence what remains is (4), and from (4) we obtain – because of ¬A(x) (contained in (4)) and (1) – ¬∃y(A(y) ∧ EXM(y, x)), and therefore because of ∃yEXM(y, x) (also contained in (4)): ∃y(¬A(y) ∧ EXM(y, x)), and there-

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fore finally (because x is supposed to be exemplified only by individuals): ∃y(I(y) ∧ ¬A(y) ∧ EXM(y, x)). The denial of actualism for individuals is implied by this.

7.7

The Actuality of Individuals

For some reason it has been utterly unbelievable to very many people – and great philosophers among them – that there should be individuals that are not actual, or in other words: individuals that do not exist. This can hardly have anything to do with the meaning of the word “individual,” for this category-term is as neutral between actuality (existence) and nonactuality (non-existence) as are the category-terms “state of affairs” (in contrast to “fact”) and “property.” Nor can it have anything to do with confusing the actuality-meaning of the word “exist” with the mere-beingmeaning of that word (see Section 1.4); for this confusion – though it certainly does occur – does not discriminate between states of affairs, properties, and individuals; therefore, once the confusion is replaced by clarity, and once one has safeguarded oneself against it by stipulating that “to exist” is, in the present theoretical context, always to be taken in the sense of “to be actual” (see Section 1.4), actualism for individuals should seem in the same degree plausible, or in the same degree implausible, as actualism for states of affairs, or for properties. But actualism for individuals is certainly not on the same level of plausibility, or implausibility, as the latter actualisms: actualism for individuals seems vastly more plausible than either actualism for states of affairs or actualism for properties, which latter actualisms, really, do not seem plausible at all. Why is this? I presume that it has something to do with the following epistemological fact. Consider states of affairs and properties of which it is both possible that they be actual and possible that they be not actual. In cognition (i.e., as intentional objects), these states of affairs and properties have palpable content, no matter whether they are actual or not, and no matter whether they are taken to be actual or not. But it seems to be quite otherwise with individuals. U.M. is an individual of which it is both possible that he be actual and possible that he be not actual. Suppose U.M. were not actual. What, then, would U.M. be? Hard to say, since under the supposition that U.M. is not actual, U.M. appears – in cognition, as intentional object – to be voided of all content. Under such circumstances, it becomes easy to believe that in the non-actual individual’s place there is just nothing at all.

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To express the matter differently: “If a state of affairs is something, then it is actual” and “If a property is something, then it is actual” are statements that – as a matter of the facts of cognition – appear to be clearly false; not so the statement “If an individual is something, then it is actual.”

But the cognitive “annihilation” of individuals that are hypothesized to be non-actual is an illusion. After all, there is such a thing as the universal logical essence of an individual y, λo(o = y), and such a thing as its logical I-essence, λIo(o = y) [that is, ires(λo(o = y)), according to D37; for the definition of ires(z), see D36]. Consider the latter essence. The logical Iessence of a contingent individual (for example, U.M.) is not affected by the individual’s actuality or non-actuality; it is exemplified by that individual, and by nothing else, under all circumstances, whatever they may be. And the logical I-essence of an individual y is certainly not “nothing.” We have seen in Section 7.5.1 that λIo(o = y ∧ O(w)) contains precisely the totally defined I-properties the individual y has in the possible world w (see part (iv) of the theorem below D43), and consider now that, for any individual y and world w, λIo(o = y ∧ O(w)) is the conjunction of the property λIo(o = y) and the property λIo(o = o ∧ O(w)). This must be so because, for all individuals y´, [λIo(o = y ∧ O(w)), y´] = conj([λIo(o = y), y´], [λIo(o = o ∧ O(w)), y´]).8 Thus: if λIo(o = y) meets – so to speak – a possible world w, then it is completely determined which totally defined I-properties y has in w, and not only that: then it is also completely determined which properties (simpliciter) y has in w. Clearly, λIo(o = y) is not “nothing” – even if y were non-actual. That the complete fixing of the totally defined I-properties y has in w implies the complete fixing of the properties y has in w is seen as follows (and we will make a significant addition to the property-generating postulates P31 and P33 stated in Section 7.4). If it is completely determined (the latter word meaning here no more than “fixed”) which I-properties an individual y has in a world w, then it is also completely determined which properties y has in w. For suppose, contrapositively, that there is no complete determination regarding the properties of y in w: that it is not determined whether y has, for example, the property x in w. Hence x must fit at least one individual; for otherwise it would be determined that y does not have property x in w, that is

For individuals y´, [λIo(o = o ∧ O(w)), y´] = [λo(o = o ∧ O(w)), y´], and therefore, according to P30, [λIo(o = o ∧ O(w)), y´] = that (y´ = y´ ∧ O(w)) (since λo(o = o ∧ O(w)) certainly fits all individuals), and hence (following the usual steps) [λIo(o = o ∧ O(w)), y´] = w. 8

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(according to D35, D16): it would be determined that P([x, y], w) is not the case, because [x, y] would be c* (according to P25), and c*, being not a state of affairs (see Section 7.1), is not an intensional part of any state of affairs (according to P0). Consider, then, ires(x) – the I-restriction of x (see D36; the existence of a unique Irestriction for each property that fits at least one individual is guaranteed by P33 and P29). ires(x) is an I-property such that [ires(x), y] = [x, y]. According to the supposition, and the definitions D35 and D16, it is not determined whether P([x, y], w). Hence it is not determined whether P([ires(x), y], w). Hence it is not determined whether y has the I-property ires(x) in w. Hence it is not completely determined which Iproperties y has in w. However, the result that we want to establish does not (yet) follow, namely: that if it is completely determined which totally defined I-properties an individual y has in a world w, that then it is also completely determined which properties y has in w. The reason is this: some I-properties may not be totally defined, may not fit every individual. While it seems somewhat problematic to postulate that every I-property fits every individual (does the property λo(o smiles at t0) – plausibly an I-property – fit a hydrogen atom?), it does seem entirely safe to postulate that every I-property has an Icompletion:

P36 ∀x1(PRI(x1) ⊃ ∃x2[∀y(¬I(y) ∨ [x1, y] ≠ c* ⊃ [x2, y] = [x1, y]) ∧ ∀y(I(y) ∧ [x1, y] = c* ⊃ [x2, y] = k*)]). An I-completion x2 of an I-property x1 is a property that is, at least, largely identical with x1. Its falling short of full identity with x1 would be due to certain individuals y for which [x1, y] = c*; with regard to such individuals we have: [x2, y] = k* (and, of course, k* ≠ c*). Clearly, an I-completion of an I-property is a totally defined Iproperty. It is easily seen that there can be at most one I-completion of any I-property, and we define:

D49 icom(x1) =Def ιx2[∀y(¬I(y) ∨ [x1, y] ≠ c* ⊃ [x2, y] = [x1, y]) ∧ ∀y(I(y) ∧ [x1, y] = c* ⊃ [x2, y] = k*)]. We can now show that if it is completely determined which totally defined Iproperties an individual y has in a world w, then it is also completely determined which I-properties y has in w – and therefore also (as we have already seen) which properties y has in w. For suppose, contrapositively, that there is no complete determination regarding the I-properties of y in w: that it is not determined whether y has, for example, the I-property x in w. Hence [x, y] ≠ c* (for otherwise it would be determined that y does not have x in w; see the considerations above). Consider icom(x), which is a totally defined I-property. Because of [x, y] ≠ c*, we have: [icom(x), y] = [x, y], and therefore, since it is not determined whether P([x, y], w) (which is what “y has x in w,” EXM(y, x, w), amounts to according to D35, D16), it must also be not determined whether P([icom(x), y], w). Hence it is not determined whether x has the

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totally defined I-property icom(x) in w, and hence it is not completely determined which totally defined I-properties y has in w.

But what, now, is a positive argument for the thesis that there are non-actual individuals (in addition to the mere removal of apparent obstacles for assuming this thesis)? In response to this question, it must first be noted that with regard to individuals the situation is rather dissimilar to the situation with regard to properties and states of affairs. We can exhibit – i.e., name in a self-explaining manner – a non-actual states of affairs: that U.M. is 165 cm tall in July 2003, and a non-actual property: λo(o = U.M. and o is 165 cm tall in July 2003); but we cannot exhibit a non-actual individual. This is certainly another epistemological fact that contributes considerably to the widespread incredulity with regard to the thesis that there are non-actual individuals. But is not Pegasus a non-actual individual? The problem with this “example” is that Pegasus (Chiron, Leopold Bloom, etc.) is not an individual; indeed, it is doubtful whether Pegasus is non-artificially anything at all. If it is non-artificially something (which is the case if “Pegasus” has a non-artificial referent, and not the case if “Pegasus” has only an artificial referent, namely, c*), then Pegasus (Chiron, Leopold Bloom, etc.) is a Meinongian entity. A Meinongian entity is in some ways similar to an individual, and if one so wishes (I do), one can call Meinongian entities “M(einong)-individuals.” But one will have to keep in mind that Mindividuals are not individuals, just like L(eibniz)-individuals are not individuals (see Section 7.5.1); M-individuals and L-individuals are called “individuals” only by way of analogy, not in the proper sense of the word. I claim that the sense of the (unmodified) word “individual” (and of the corresponding formal predicate I(x)) in which that word is used throughout this book is the proper sense of the word. For what this means regarding the transworld identity of individuals, see Section 4.2, and Section 7.5 above.

M-individuals can be said to M-have (or M-exemplify) properties – to have properties in the special way a Meinongian entity has properties – which are also had (that is, exemplified in the sense of D20) by individuals. For example, the M-individual Leopold Bloom M-has the property of being a human being and the property of being actual; however, he does not have either one of these properties (whereas U.M. has both of them). The M-exemplification (M-having) of properties by M-individuals differs from the exemplification of properties by individuals in two particu-

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larly striking respects. Consider the following M-individuals: the individual which is round at t0 and square at t0 and Leopold Bloom. No individual exemplifies both the property of being round at t0 and square at t0. But the first of our two M-individuals M-exemplifies the property of being an individual, the property of being round at t0, and the property of being square at t0; this makes it a property-inconsistent object with respect to Mexemplification. In turn, every individual exemplifies either the property λo(o is a living individual at noon, September 1, 1914) or the property λo(o is an individual that is not alive at noon, September 1, 1914). But Leopold Bloom M-exemplifies neither the one property nor the other. This makes him a property-incomplete object with respect to Mexemplification. The typical M-individuals are, with respect to M-exemplification, property-inconsistent or property-incomplete objects, and all such objects are ipso facto non-actual (or as the Meinongians prefer to say: nonexistent). Therefore, since there are lots of apparently non-artificially referring and self-explanatory names for such M-individuals, it is easy to apparently exhibit M-individuals that are not actual. But individuals are not, with respect to exemplification, property-inconsistent or propertyincomplete objects; they are not M-individuals at all. As far as I can see, it is impossible to exhibit – to name in a self-explaining manner – an individual that is not actual. Indeed, if it were otherwise, it would be difficult to understand why there are any actualists regarding individuals at all. Nevertheless, I advance two arguments for there being non-actual individuals (or more precisely: for there being at least one non-actual individual), noting from the start that they are only plausibility-arguments: they make their conclusion (more) plausible, they certainly do not establish it beyond reasonable doubt. The first argument is the argument from ontological symmetry. It cannot be reasonably denied (I, in particular, cannot reasonably deny it) that I am actual but might not have been actual. Thus, there is certainly an individual that is actual but might not have been actual. Ontological symmetry requires that there is also an individual that is not actual but might have been actual. Hence there is at least one non-actual individual. Consider carefully the costs of denying that there is an individual that is not actual but might have been actual. If it is denied, then it is accepted that every non-actual individual is non-actual necessarily, or in other words: that every individual for which it is possible to be actual is actual. This, however, is a principle of plenitude that seems false.

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The second argument, too, appeals to considerations of ontological symmetry. Suppose all individuals are actual, that is: all individuals are actual in the actual world, w*. Consider then a possible world w´, different from w*. It would be a rather remarkable ontological asymmetry if some individual were not actual in w´, while every individual is indeed actual in w*. And this asymmetry would certainly make one wonder endlessly: why is it that all individuals are actual? Is it just a metaphysical accident that all individuals are actual (i.e., actual in w*)? Is it due to divine ordinance that all individuals are actual? But, under the assumption of asymmetry, neither this nor that nor any other account would make the fact properly understandable. The only good explanation of the supposed fact that all individuals are actual in w* is presumably this: in every possible world, all individuals are actual, that is: ∀w[MC(w) ⊃ ∀y(I(y) ⊃ EXM(y, λoA(o), w))] – and this is an explanation which excludes the very assumption of asymmetry that, if made, brings about a thirst for explanation that cannot be properly satisfied. However, the following is certainly true: there is an individual y (I, for example) that is not actual in some possible world, that is: ∃y(I(y) ∧ ∃w[MC(w) ∧ ¬EXM(y, λoA(o), w)]). This, obviously, contradicts the proffered explanation of the supposed fact that all individuals are actual. And thus, we are confronted with a dilemma: either to accept the actuality of all individuals as a fact that is not well explicable, or to deny that it is a fact. There is a highly popular attempt to escape from this dilemma. It consists in interpreting the only good explanation of the supposed fact that all individuals are actual in the following way: “in every possible world, all individuals are actual” just means as much as “in every possible world, all individuals in the domain of that world are actual,” that is (in terms of the present theory): ∀w[MC(w) ⊃ ∀y(I(y) ∧ D(y, w) ⊃ EXM(y, λoA(o), w))], where D(y, w) is to be read as “y belongs to the domain of w.” Then the contradiction pointed out above vanishes, and with the contradiction the dilemma, although it remains as certain as before that there is an individual that is not actual in some possible world: All that results from ∀w[MC(w) ⊃ ∀y(I(y) ∧ D(y, w) ⊃ EXM(y, λoA(o), w))] and ∃y(I(y) ∧ ∃w[MC(w) ∧ ¬EXM(y, λoA(o), w)]) is ∃y(I(y) ∧ ∃w(MC(w) ∧ ¬D(y, w))) – that is, the apparently entirely acceptable consequence that there is an individual that is not in the domain of every world.

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Interpreting all-quantification in modal formulas as quantification over all individuals, the second argument against actualism for individuals can be translated into modal formulas in the following way9: 1) 2) 3) 4) 5)

∀yA(y) ∀yA(y) ⊃ ∀yA(y) ∀yA(y) ∃y¬A(y) ¬∀yA(y)

actualism for individuals required if 1) is to be well explicable logically from 1) and 2) undeniable assumption logically from 4), but contradicting 3)

Interpreting all-quantification in modal formulas as quantification over all individuals in the domain of the relevant world of reference, this argument is blocked, since 5) does no longer follow logically from 4). If, moreover, “A(y)” is interpreted as true of an individual y in a world w iff y is in the domain of w, then 1), 2) and 3) are made logically true. Both steps of interpretation are standard in quantified modal logic. See the modeltheoretic interpretation of existence (= actuality) and the “actualist treatment of the objectual quantifiers” in model-theoretic modal semantics, as described by Graeme Forbes in his Languages of Possibility, pp. 4-5. This treatment of the objectual quantifiers goes back to Saul Kripke; see Hughes/Cresswell, An Introduction to Modal Logic, pp. 178-182, where it is also shown (in effect) that ∀yA(y) ⊃ ∀yA(y) – the converse of an instance of the Barcan formula – fails to be logically true in the (standard) Kripke-semantics of modal predicate logic. By noticing that ∃y¬A(y) ⊃ ¬∀yA(y) is a contrapositive of ∀yA(y) ⊃ ∀yA(y) (and therefore logically equivalent to it in every classical approach), it becomes apparent that it is precisely the invalidity of this formula according to the “actualist [or Kripkean] treatment of the objectual quantifiers” which makes the step from 4) to 5) above come out as not logically valid – if, of course, one subscribes to that treatment. If not, then the logical truth of ∀yA(y) ⊃ ∀yA(y), and hence also of its contrapositive ∃y¬A(y) ⊃ ¬∀yA(y), is simply the effect of adding the very weak modal system T to standard elementary predicate logic; see An Introduction to Modal Logic, p. 143.10 9

An ordinary-language formulation of the argument can already be found in my paper “Modalität und Existenz,” pp. 51-52. 10 That acceptance of the converse of the Barcan formula might be motivated by belief in non-existent (non-actual) but possible things, and apparently also the inverse relationship of grounding, is noted by Michael Fara and Timothy Williamson in their recent paper “Counterparts and Actuality,” pp. 5-6, where they also display some sym-

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Thus it seems that one can easily reconcile necessary actualism for individuals – the only good explanation of actualism for individuals – with the undeniable fact that some individual is not necessarily actual. But this maneuver is merely verbal. For the question of actualism for individuals is not whether all individuals in the domain of w* are actual in w*, that is, (simpliciter) actual; the question of actualism for individuals is whether all individuals are actual in w*. And therefore the only good explanation of actualism for individuals, if actualism for individuals is true, will not be that in every possible world w all individuals in the domain of w are actual in w, but that in every possible world w all individuals are actual in w. Suppose there is an individual y that is not in the domain of w*. Since only those individuals can be actual in a world that are in the domain of that world, it follows that y is not actual in w*, and therefore not actual. Hence actualism for individuals is false under the supposition that some individual is not in the domain of w* (a supposition, by the way, that is shared by many metaphysicians of modality) – although it may still be true that in every possible world w all individuals in the domain of w are actual in w. Indeed, this can easily be made trivially true by stipulating that “y is actual in w” is synonymous with “y is in the domain of w.” But its truth is simply irrelevant for the issue of actualism for individuals. Acknowledging that there is no escape from the above dilemma, it may, in reaction to it, still seem better – to some – to accept actualism for individuals as a fact that has no good explanation, rather than to reject that actualism. Divine ordinance would be an explanation of actualism for individuals in the absence of the necessity of that actualism – especially if it turned out that w* is the only possible world in which all individuals are actual. Actualism for individuals would then appear to be a mark of perfection, a part of w0’s11 being the best of all possible worlds, which is for that reason chosen by God to be the actual world, w*. But hardly any philosopher these days would be ready to count this explanation as a good explanation of actualism for individuals.

pathy with “CBF” for a motive other than belief in non-existent but possible things. Opposing David Lewis’s counterpartist way of doing modal theory, they recommend “orthodox Kripke semantics” (ibid., p. 26). Orthodox Kripke semantics, however, will not yield the converse of the Barcan formula as a logical truth (as Fara and Williamson are well aware). 11 “w0” rigidly designates the world that “w*” designates non-rigidly; see Section 4.3.

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And perhaps accepting actualism for individuals as a brute fact – as a colossal ontological accident – is not positively irrational. Then, although one believes that all individuals are actual, one can nevertheless mimic nonactualism for individuals. This can be done as follows. Suppose all individuals are actual. Consider L-individuals, introduced in Section 7.5.1 and there identified with the maximal-consistent totally defined I-properties, the notiones completae. Some of these properties – for example, nc(U.M., w*) – are exemplified, some are not exemplified – for example, nc(U.M., w#), where w# is a possible world that is different from w* (and where “w#” is assumed to be a referentially stable term). That nc(U.M., w#) is not exemplified is seen as follows. Suppose ∃yEXM(y, nc(U.M., w#)). Hence according to D43, D20: O([λIo(o = U.M. ∧ O(w#)), y]). Hence also O([λo(o = U.M. ∧ O(w#)), y]). [Regarding this last step: Since λo(o = U.M. ∧ O(w#)) is a property that fits some individual, λIo(o = U.M. ∧ O(w#)), or ires(λo(o = U.M. ∧ O(w#))), is a property such that for all individuals y´: [λIo(o = U.M. ∧ O(w#)), y´] = [λo(o = U.M. ∧ O(w#)), y´], and for all non-individuals y´: [λIo(o = U.M. ∧ O(w#)), y´] = c*. See P33, D36, D37, and surrounding considerations. Now, y must be an individual; for otherwise – because of the theorem ¬O(c*) (following from ¬S(c*), according to D6) – we would get ¬O([λIo(o = U.M. ∧ O(w#)), y]), which contradicts the assumption. Therefore: [λo(o = U.M. ∧ O(w#)), y] = [λIo(o = U.M. ∧ O(w#)), y].] Hence according to P30 [since λo(o = U.M. ∧ O(w#)) is a totally defined property]: O(that (y = U.M. ∧ O(w#))). Hence according to P15: O(conj(that (y = U.M.), that O(w#))), hence according to P24 [w# being a state of affairs, and “w#” a referentially stable term]: O(conj(that (y = U.M.), w#)). Now, y = U.M.; for otherwise k* = that (y = U.M.), according to P32, and therefore [because of O(conj(that (y = U.M.), w#))]: O(conj(k*, w#)), hence O(k*) – which is provably false. Then, because of y = U.M.: t* = that (y = U.M.), according to P32, and therefore [because of O(conj(that (y = U.M.), w#))]: O(conj(t*, w#)). Hence O(w#), hence A(w#) [because of D6]. Hence according to the Actuality Principle for States of Affairs: P(w#, w*). Hence [because of MC(w#), MC(w*), and the theorem ∀w∀w´(MC(w) ∧ MC(w´) ∧ P(w´, w) ⊃ w´ = w)] w# = w* – contradicting the assumption that w# is different from w*.

For the exemplified notiones completae, it is also true that they are exemplified by at least one actual entity: because they are exemplified, they are exemplified by an individual (since they are I-properties), and therefore they are also exemplified by an actual individual (since all individuals are actual, according to supposition). Hence it follows, according to P35, (i), that the exemplified notiones completae are actual. For the non-

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exemplified notiones completae, on the other hand, it follows, according to P35, (ii), that they are not actual. Thus there are actual L-individuals and non-actual L-individuals (since the L-individuals have been identified with the notiones completae). Thus, for L-individuals, actualism is provably false, and non-actualism provably true. Note that this result is obtainable as long as there is at least one individual and at least two possible worlds.

Now, under the supposition of actualism for individuals, the actual individuals (which are the individuals under this supposition) and the actual Lindividuals are one-to-one correlated. If y is an actual individual, then nc(y, w*) is an actual L-individual. If y and y´ are different actual individuals, then nc(y, w*) and nc(y´, w*) are different actual Lindividuals. This much is already clear after Section 7.5.1 and 7.6. For obtaining that every actual L-individual corresponds via nc(y, w*) to an actual individual y, assume that x is an actual L-individual. Hence (according to D44): x is an actual maximal-consistent totally defined I-property. Hence – according to the theorem ∀x[PRITMC(x) ⊃ ∃y∃w(I(y) ∧ MC(w) ∧ x = nc(y, w))] (item (iii) below D43 in Section 7.5.1) – x = nc(y, w), for some individual y and some world w. Because of the supposition of actualism for individuals, y is an actual individual. Moreover: since x is actual (according to assumption), x is exemplified [according to P35, (ii)], and hence nc(y, w) is exemplified, that is: ∃y´O([λIo(o = y ∧ O(w)), y´]) [according to D43, D20]. But ∃y´O([λIo(o = y ∧ O(w)), y´]) has the following consequence: w = w*. [If y´ were not an individual, [λIo(o = y ∧ O(w)), y´] would be c* and O([λIo(o = y ∧ O(w)), y´]) could not be true. Hence y´ is an individual, and therefore: [λIo(o = y ∧ O(w)), y´] = [λo(o = y ∧ O(w)), y´] = that (y´ = y ∧ O(w)) = conj(that (y´ = y), that O(w)). Hence: O(conj(that (y´ = y), that O(w))). If y´ were not y, conj(that (y´ = y), that O(w)) would be k* and O(conj(that (y´ = y), that O(w))) could not be true. Hence y´ = y, and therefore: conj(that (y´ = y), that O(w)) = conj(t*, w) = w. Hence finally: [λIo(o = y ∧ O(w)), y´] = w, and ∃y´O([λIo(o = y ∧ O(w)), y´]) is seen to imply O(w). This, by the Actuality Principle for States of Affairs, implies P(w, w*), which, in view of MC(w) and MC(w*), can only be true if w = w*.] And therefore: the assumption that x is an actual L-individual has been seen to imply ∃y∃w(A(y) ∧ I(y) ∧ MC(w) ∧ w = w* ∧ x = nc(y, w)), hence: ∃y(A(y) ∧ I(y) ∧ x = nc(y, w*)) – what was to be shown.

Thus, under actualism for individuals, the non-actual L-individuals – that is: the non-actual notiones completae, the non-actual maximal-consistent totally defined I-properties – can be regarded as ersatz for non-actual indi-

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viduals (which, after all, are not only a metaphysical horror to some people, but also useful for some purposes),12 although under the supposition of actualism for individuals – a supposition that I do not share, but do not consider to be positively irrational – there aren’t really any such entities. It remains to be noted that the proposed ersatz entities for non-actual individuals – supposing actualism for individuals – are without exception mere world-specific property-variants of the actual individuals: for each actual individual, there are as many non-actual L-individuals that are its property-variants as there are non-actual possible worlds (there is no transworld identity for L-individuals); non-actual L-individuals are never entities that are more “alien” than that. It cannot be otherwise; for each of the non-actual L-individuals – each of the non-actual maximal-consistent totally defined I-properties – has an individual as its kernel whose totally defined I-properties in exactly one non-actual possible world it precisely comprises (see Section 7.5.1 for more on this): a kernel-individual which – under the supposition of actualism for individuals – must be actual.

12

Note that there is a double ersatzism involved, for matters can be regarded in the following way (inversely to the order of presentation in this book): First, non-actual Lindividuals are substituted for genuine non-actual individuals (precisely speaking: they assume the roles the non-actual individuals would have filled if there had been such things). Then, maximal-consistent totally defined I-properties – notiones completae – are substituted for genuine L-individuals (see Section 7.5.1), which means, in particular, that non-actual notiones completae are substituted for genuine non-actual Lindividuals (already substituting for genuine non-actual individuals). Regarding the substitution of ersatz entities for non-actual individuals (under the pressure of actualism), compare David Lewis, On the Plurality of Worlds. But the above ersatzist conception cannot be found among the ersatzisms Lewis considers, which is not surprising in view of his ontological framework, which is rather different from the present one (see Section 8.1).

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8

Properties, Essences, and Actualism (Again)

Modal theory has two thematic pivots: the necessity of states of affairs – some would say: propositions – on the one hand, and the necessity of properties – in their relation to individuals – on the other hand. These two thematic pivots, which can very well be subsumed under the traditional headings necessitas de dicto and necessitas de re (taking the first one of these Latin phrases not quite literally, namely: in a purely ontological sense), are closely related, and the nature of that relationship, I hope, has become clear in Chapter 7, building on Chapter 3. Its general character is that necessitas de dicto, the necessity of states of affairs, is more basic than necessitas de re, the necessity of properties. This final chapter is dedicated to addressing some of the issues specifically surrounding necessitas de re. 8.1

David Lewis on Possible Worlds, Individuals, and I-Properties

David Lewis’s position on individuals and I-properties (in other words: properties that are only defined for individuals) can be summarized in the main by the following five theses (they can easily be gleaned from Lewis’s On the Plurality of Worlds): [0] There are several possible worlds. [1] All possible worlds are individuals, but not all individuals are possible worlds. [2] Every individual is a part of exactly one possible world. [3] In an extended sense of “individual,” mereological sums consisting of individuals (as characterized by [1] and [2]) from different possible worlds could also be regarded as individuals. But this option will be ignored. [4] I-properties (or as they are also called: properties of individuals) are sets of individuals, and any set of individuals is an I-property.

8 Properties, Essences, and Actualism (Again)

Clearly, Lewis’s conceptions of possible worlds, individuals, and Iproperties are different from the conceptions of these things that are advocated in this book. But let us see how, and to what extent, Lewis’s conceptions can be represented within the present framework. Though possible world are states of affairs in the present framework, and individuals for Lewis, there is a natural one-to-one correspondence between the Lewis-worlds, in short: LI-worlds, and the worlds of the present framework. This is the case because Lewis certainly has no other structural conception of what (coarse-grained) logical possibilities there are than everybody else – whatever may be the differences between him and other metaphysicians in their ontological interpretation. The LI-worlds, therefore, can be represented by the possible worlds of the present framework, that is: by the maximal-consistent states of affairs (as conceived of in the present framework). Since states of affairs are for Lewis sets of LI-worlds, Lewis himself recognized a one-to-one correspondence between the LI-worlds and the maximal-consistent states of affairs, which in Lewis’s framework are simply the singleton sets of LI-worlds, obviously matching one-to-one the LI-worlds themselves. When speaking simpliciter of “maximal-consistent states of affairs,” I will have in mind maximal-consistent states of affairs as conceived of in the present framework, that is, the entities satisfying the predicate MC(x).

I-properties apply only to individuals – this is common ground between Lewis and the present framework. But it is evident from the above theses that, for Lewis, some I-properties apply to possible worlds. For Lewis, some I-properties have possible worlds among their elements, and they apply to – are exemplified by – their elements. In contrast, Iproperties, according to the present framework, never apply to possible worlds. According to the present framework, possible worlds are states of affairs, hence not individuals (see P27), and therefore I-properties are not exemplified by – do not apply to – possible worlds. But Lewis-I-properties, in short: LI-properties, can be put into three groups: (1) LI-properties that apply only to Lewis-individuals – in short: LI-individuals – which are not LI-worlds [i.e., LI-properties which are such that all their elements are LI-individuals that are not LI-worlds], (2) LIproperties that apply only to LI-worlds [i.e., LI-properties which are such that all their elements are LI-worlds], (3) LI-properties that apply both to a LI-world and to a LI-individual which is not a LI-world [i.e., LI-properties which have both a LI-world and a LI-individual that is not a LI-world

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among their elements]. Let us focus our attention on the first group: the LIproperties that apply only to LI-individuals which are not LI-worlds, in short: the LI*-properties. From a Lewisian point of view, these entities are the I-properties that one would consider to be the I-properties in a narrow or strict sense. Note that the extensionally empty LI-property – the empty set ∅ – belongs, according to construction, both to group (1), and therefore to the LI*-properties, and to group (2) (and, note, it is the only LI-property for which this is true). Group (2) comprises the Lewisian propositions, which, incidentally, are also the Lewisian states of affairs. Thus, for Lewis, ∅ is not only an I-property in the strict sense but also a proposition and state of affairs. I guess, one can live with that.

Just as Lewis has no other structural conception of what logical possibilities there are than everybody else, so he also has no other conception of what I-properties in a strict sense there are than everybody else (whatever may be the differences between him and other metaphysicians in their ontological interpretation). Thus, the LI*-properties can be taken to stand in a natural one-to-one correspondence to the entities of which the predicate PRIT(x) is true (just as the LI-worlds were taken to stand in a natural one-to-one correspondence to the entities of which the predicate MC(x) is true), and therefore the LI*-properties can be represented by the totally defined I-properties as conceived of in the present framework, by the “PRITs,” so to speak. Why are LI*-properties represented by the entities of which the predicate PRIT(x)1 is true, and not by the entities of which the predicate PRI(x)2 is true? Why are the LI*-properties represented by the totally defined I-properties (as conceived of in the present framework) and not simply by the I-properties (as conceived of in the present framework)? The reason is that Lewis himself does not mention the difference between properties of individuals that are defined for all individuals and properties of individuals that are not defined for all individuals. This is a clear indication that all properties of individuals are, for Lewis, defined for all individuals. And, indeed, it is not an audacious step of interpretation to ascribe to him the doctrine that properties of individuals, like all properties, are defined for all entities, but that they, qua properties of individuals, can only apply to individuals. Translated into [reduced to] the present framework, and limiting our attention to LI*-properties (Lewisian I-properties in the strict sense), this position amounts to the following: ∀x[LI*-PR(x) ≡ PR(x) ∧ ∀yF(x, y) ∧ ∀y([x, y] ≠ k* ⊃ I(y))]. But since the entities of which PR(x) ∧ ∀yF(x, y) ∧ 1 2

The definitional pedigree of this predicate is D17, D18, D19, D41. The definitional pedigree of this predicate is D17, D18, D19.

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∀y([x, y] ≠ k* ⊃ I(y)) is true correspond one-to-one to the entities of which PR(x) ∧ ∀y(I(y) ⊃F(x, y)) ∧ ∀y([x, y] ≠ c* ⊃ I(y)), or in other words: of which PRIT(x) is true, and since the latter predicate connects more easily with what has been said in Sections 7.5 and 7.5.1, LI*-properties are here represented by “PRITs.” Therefore, the Lewisian position, translated into [reduced to] the present framework, turns out to be this: ∀x[LI*-PR(x) ≡ PRIT(x)].

What about LI-individuals, i.e., Lewis-individuals? How can they be represented within the present framework? The preliminary thing to be said regarding this issue is this: Since we have limited our attention to LI*properties (disregarding other LI-properties), the consistent thing to do is to also limit our attention to LI*-individuals – the entities which, from a Lewisian point of view, one would consider to be the individuals in a narrow or strict sense. The LI*-individuals are simply the LI-individuals which are not LI-worlds. How can LI*-individuals be represented within the present framework? How are they related to the entities fulfilling the predicate I(y) – that is, to the entities conceived to be individuals according to the present framework? For the sake of brevity, let us call these latter entities “MIindividuals.” LI*-individuals are rather different from MI-individuals. This much, among other things, can be gathered from the following considerations. In Section 7.5, I have introduced the notion of a maximal-consistent totally defined I-property; the formal predicate was PRITMC(x) (see D42). In Section 7.5.1, the maximal-consistent totally defined I-properties were also called “notiones completae” (since they can be regarded as the entities that Leibniz called by that name). A maximal-consistent totally defined Iproperty or notio completa is a totally defined I-property that is a proper intensional part of only one totally defined I-property: λI(o ≠ o). This is precisely the content of D42: PRITMC(x) =Def PRIT(x) ∧ x ≠ λI(o ≠ o) ∧ ∀z(PPR(x, z) ∧ z ≠ x ⊃ z = λI(o ≠ o)). ∀x´[PRIT(x´) ⊃ PPR(x´, λI(o ≠ o))] is a theorem, and hence there is no need to include PPR(x, λI(o ≠ o)) in the definiens of D42. There is also no need to write ∀z(PPR(x, z) ∧ PRIT(z) ∧ z ≠ x ⊃ z = λI(o ≠ o)) in the definiens, since ∀x´∀z(PRIT(x´) ∧ PPR(x´, z) ⊃ PRIT(z)) is a theorem. There is, finally, no need to include PRIT(λI(o ≠ o)) in the definiens, since this is a theorem, too.

Since the PRITs correspond one-to-one to the LI*-properties (as we have already established), which are the LI*-properties to which correspond

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one-to-one the maximal-consistent PRITs? – It is easily seen: those LI*properties are the singleton sets whose sole element is a LI*-individual. A LI*-property x is a LI-intensional part of a LI*-property z iff z is a subset of x. (Nota bene: it is not the other way around!) This given, it is clear that the singleton sets whose sole element is a LI*-individual are precisely the LI*-properties which are different from the empty set (matched by the PRIT λI(o ≠ o)) and which are such that all LI*-properties of which they are proper LI-intensional parts are identical with the empty set.

But the LI*-individuals obviously correspond one-to-one to the singleton sets whose sole element is, in each case, a LI*-individual. Hence it follows that the maximal-consistent PRITs correspond one-to-one to the LI*individuals – and the correspondence is entirely natural. In contrast, there is no such natural one-to-one correspondence between maximal-consistent PRITs (or notiones completae) and MI-individuals, as we have seen (in Section 7.5). Based on the natural one-to-one correspondence between the LI*individuals and the maximal-consistent PRITs, we can represent LI*individuals in the present ontological framework by maximal-consistent PRITs, that is, by notiones completae. Hence the LI*-individuals, the central Lewis-individuals, are represented by the very same entities that the Lindividuals, the Leibniz-individuals, are represented by (see Section 7.5.1), and there is no reason no to identify the LI*-individuals with the Lindividuals. – Summing up the results that have been reached so far, we have: The Lewis-worlds [LI-worlds, LI-W(z)] are represented within the present framework by the maximal-consistent states of affairs. The Lewis-I-properties [LI-properties] that do not have Lewis-worlds as elements, that is: the LI*-properties [LI*-PR(x)], are represented within the present framework by the totally defined I-properties (i.e., by the entities satisfying PRIT(x)). The Lewis-individuals [LI-individuals] that are not Lewis-worlds, that is: the LI*-individuals [LI*-I(y)], are represented within the present framework by the maximal-consistent totally defined Iproperties (i.e., by the entities satisfying PRITMC(x)).

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Given these representations, one can go one step further and reductively define within the present ontological framework: LI-W(z) =Def MC(z), LI*-PR(x) =Def PRIT(x), LI*-I(y) =Def PRITMC(y).3 There are several tests for the adequacy of the above representations. For example, according to Lewis, an individual exemplifies an I-property simply by being an element of it, in other words: a LI-individual LIexemplifies a LI-property simply by being an element of it. How is this aspect of Lewis’s theory mirrored within the present framework? – We limit our attention to LI*-individuals and LI*-properties. Then, let y be an LI*individual and z a LI*-property. We have: PRITMC(rep(y)) and PRIT(rep(z)), where rep(y) and rep(z) are the representations of y and z within the present framework. Note that the representation of the LI*individual y within the present framework coincides with the representation of the LI*-property {y}: rep(y) = rep({y}). And further we have: y LI-exemplifies z iff y ∈ z iff {y} ⊆ z iff LI*-property z is a LIintensional part of LI*-property {y} iff PPR(rep(z), rep({y})) iff PPR(rep(z), rep(y)). Therefore: (A) y LI-exemplifies z iff PPR(rep(z), rep(y)). This result once more shows how close Lewis is to Leibniz; compare what is said about L-individuals and L-exemplification (Leibnizexemplification) in Section 7.5.1. A striking aspect of Lewis’s views regarding individuals is thesis [2] above: that each individual is a part of exactly one possible world. How is this doctrine mirrored within the present framework? – Again, we limit our attention to the LI*-individuals, which are represented within the present framework by maximal-consistent totally defined I-properties (as conceived of in the present framework). Concerning such properties, we have the following theorem, which made its appearance already in Section 7.5: 3

These definitions effect a reduction of Lewisian views regarding possible worlds, Iproperties that are not exemplified by possible worlds, and individuals that are not possible worlds, to the present ontological theory; they do not, of course, capture the meanings Lewis had in mind when he spoke of possible worlds, etc. But note that reductive definitions need not capture meanings.

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The central theorem regarding maximal-consistent PRITs ∀x[PRITMC(x) ≡ PRIT(x) ∧ ∃y(I(y) ∧ MC([x, y]) ∧ ∀y´(I(y´) ∧ y´ ≠ y ⊃ [x, y´] = k*))]. This implies: The incorporation theorem For each maximal-consistent PRIT x, there is exactly one maximalconsistent state of affairs w such that ∃y(w = [x, y]). Proof: Assume PRITMC(x). Hence it follows according to the central theorem (regarding maximal-consistent PRITs): ∃y(I(y) ∧ MC([x, y])), and therefore: ∃w(MC(w) ∧ ∃y(I(y) ∧ w = [x, y])). Assume moreover: MC(w) ∧ ∃y(I(y) ∧ w = [x, y]) and MC(w´) ∧ ∃y´(I(y´) ∧ w´ = [x, y´]). There are two cases: (1) y´= y, (2) y´ ≠ y. In case (1), [x, y] = [x, y´], and therefore: w = w´. In case (2), [x, y´] = k* according to the central theorem, and therefore MC(k*) – contradicting the theorem ¬MC(k*). Thus the second case is excluded, and we have derived from the initial assumption: ∃=1w(MC(w) ∧ ∃y(I(y) ∧ w = [x, y])). Moreover: since x is a PRIT (according to assumption), we have: ∀w[MC(w) ∧ ∃y(I(y) ∧ w = [x, y]) ≡ MC(w) ∧ ∃y(w = [x, y])], and therefore we have also derived ∃=1w(MC(w) ∧ ∃y(w = [x, y])) from the initial assumption.

Now, let y be a LI*-individual, and w a LI-world. We have: PRITMC(rep(y)) and MC(rep(w)), where rep(y) and rep(w) are the representations of y and z within the present framework. And we have: (B)

y is a part of w iff ∃y´(rep(w) = [rep(y), y´]).

Lewis’s thesis [2], restricted to individuals that are not possible worlds, can be proven true on the basis of (B) and the incorporation theorem (proven above). For [2], if restricted in the indicated manner, amounts to this: every LI*-individual is a part of exactly one (possible) LI-world, and this can be seen to be true as follows: Proof: Assume that y is a LI*-Individual. Then we have: PRITMC(rep(y)), and therefore according to the incorporation theorem: ∃=1w(MC(w) ∧ ∃y´(w = [rep(y), y´])). From this, abbreviating “ιw(MC(w) ∧ ∃y´´(w = [rep(y), y´´]))” by “w+”, we obtain: MC(w+) ∧ ∃y´(w+ = [rep(y), y´]). Because of the one-to-one correspondence between the LI-worlds and the maximal-consistent states of affairs, we also have: w+

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= rep(w´), for some LI-world w´. Therefore: MC(rep(w´)) ∧ ∃y´(rep(w´) = [rep(y), y´]), and therefore (according to (B)): y is a part of w´. Suppose now (for reductio) that y is also a part of another LI-world, w´´. Hence: MC(rep(w´´)) ∧ ∃y´(rep(w´´) = [rep(y), y´]) (by applying (B)). But since w´´ ≠ w´ (according to supposition), rep(w´´) ≠ rep(w´) [because of the one-to-one correspondence between the LI-worlds and the maximal-consistent states of affairs] – contradicting the incorporation theorem.

It should be noted that equivalence (A) and equivalence (B) – bridgeprinciples between two ontological frameworks – determine the degree of naturalness of the representation of LI*-individuals, LI*-properties, and LI-worlds by – respectively – maximal-consistent PRITs, PRITs, and maximal-consistent states of affairs (as conceived of in the present framework): MCs. That representation – the one-to-one correspondence rep – is natural in that degree (no less and no more) in which (A) and (B) are natural (non-artificial) truths. There remains an important question that was posed above, but that has not been explicitly answered so far: How are the LI*-individuals related to the MI-individuals? – The former are related to the latter simply in the following way. There is a natural one-to-one correspondence between the maximal-consistent PRITs and the pairs – ordered pairs, or alternatively: unordered pair-sets – that consist of a MI-individual and an MC (see Sections 7.5 and 7.5.1); the nature of that correspondence can be directly read off the central theorem regarding maximal-consistent PRITs (see above). Hence, in view of what has been established above, there is also a natural one-to-one correspondence between the LI*-individuals and the -pairs (worlds being MCs; this understanding is presupposed when, subsequently, I will be using the word “world” simpliciter), and this, clearly, answers the question how the LI*-individuals are related to the MI-individuals. So far, so good. But here comes David Lewis’s objection: Carnap, Kanger, Hintikka, Kripke, Montague, and others have proposed interpretations of quantified modal logic on which one thing is allowed to be in several worlds. A reader of this persuasion might suspect that he and I differ only verbally: that what I call a thing in a world is just what he would call a pair, and that what he calls the same thing in several worlds is just what I would call a class of mutual counterparts. But beware. Our difference is not just verbal, for I enjoy a generality he cannot match. The counterpart relation will not, in general, be an equivalence relation. So it will not hold just between those of his pairs with the same first term, no matter how he

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may choose to identify things between worlds. (“Counterpart Theory and Quantified Modal Logic,” p. 28.)

One wonders how it could be maintained that “[t]he counterpart relation is our substitute for identity between things in different worlds” (ibid., p. 27) if it is not even, in general, an equivalence relation; it must be a bad substitute indeed. But no matter, Lewis’s objection seems to be that LI*individuals cannot be represented by -pairs, since that representation would not lead to a correct mirroring of the formal properties of the counterpart relation between LI*-individuals. In reply, merely consider that any counterpart relation between LI*-individuals that Lewis countenanced, and any that he would have countenanced if he had thought of it, can be mirrored with all its formal properties (whatever they are) on the side of the -pairs simply by defining: (I) p´ is a counterpart# of p =Def p´ is a -pair and p is a -pair, and the LI*-individual represented by p´ is a counterpart of the LI*-individual represented by p. But, of course, one can, in defining, also go the other way around: (II) y´ is a counterpart of y =Def y´ is a LI*-individual and y is a LI*individual, and the -pair that represents y´ (i.e., rep´(y´)) is a counterpart# of the -pair that represents y (i.e., of rep´(y)). One counterpart# relation that can be used (in the manner of (II)) to define a (Lewisian) counterpart relation is mentioned by Lewis himself in the above quotation. It is the counterpart# relation that is defined as follows: (III.1) p´ is a counterpart#1 of p =Def p´ is a -pair and p is a -pair, and the first constituent of p´ is identical with the first constituent of p.

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This counterpart# relation is an equivalence relation (over -pairs), and on the basis of (II) it leads to a (Lewisian) counterpart relation that is indeed an equivalence relation (over LI*-individuals): (II.1) y´ is a counterpart1 of y =Def y´ is a LI*-individual and y is a LI*individual, and the -pair that represents y´ is a counterpart#1 of the -pair that represents y. But there is nothing objectionable in this, since we are talking about a counterpart relation (and the defined relation is surely a counterpart relation, since it fulfills all of Lewis’s formal requirements for such a relation; see “Counterpart Theory and Quantified Modal Logic,” p. 274), and not about the counterpart relation or about all counterpart relations. It might be conceded that it would be objectionable to assert that the counterpart relation is an equivalence relation, or that all counterpart relations are equivalence relations. But no such thing is asserted. “There is no way to make sense of a non-qualitative counterpart relation,” says Lewis (On the Plurality of Worlds, p. 230). But the counterpart1 relation looks suspiciously like a non-qualitative counterpart relation, and we have just made sense of it. However, it must be admitted that Lewis did not choose to avail himself of the advantages provided by the present theory of modality (or a theory more or less similar to it).

On the basis of the counterpart1 relation – but of course not on the basis of all counterpart relations, and probably not on the basis of the counterpart relation, whatever rare relation this relation might be – MIindividuals can obviously be represented by equivalence classes of LI*individuals (as Lewis envisages in the above quotation): by the classes of mutually counterpart1-related LI*-individuals. Is there anything more that is needed for our complete ontological satisfaction than that MI-individuals can be represented in the indicated manner on the Lewisian side of being 4

Lewis’s postulates P1 – P8 are provably true (on the basis of the given representation of his modal ontology) if the four predicates they contain are defined as follows: Cxy =Def x is a counterpart1 of y; Wx =Def x is a LI-world; Ixy =Def x is a LI*-individual which is such that the second constituent of the representation of x as an -pair represents y; Ax =Def x is a LI*-individual such that the second constituent of the representation of x as an -pair is w*.

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(so to speak): by basing the representation on one particular counterpart relation, the counterpart1 relation? I do not believe there is. One might require that the counterpart1 relation be specifiable without recourse to the counterpart#1 relation. It may in fact be specifiable in such a manner, although it is hard to see how. But if it turned out that the counterpart1 relation is indeed not specifiable without recourse to the counterpart#1 relation, what has this to do with ontology? What is important for ontology is that there is such a relation as the counterpart1 relation, and that it can be picked out somehow.

Yet it must be admitted that the counterpart1 relation is very peculiar from the Lewisian point of view. The reason is this: every LI*-individual has a counterpart1 in every LI-world. Before presenting the proof of this, here is some useful terminology – and terminology used below – for dealing with ordered pairs: if p is an ordered pair, then 1(p) is the first constituent (“first term,” as Lewis says) of p, and 2(p) the second. Let y, then, be an arbitrary LI*-individual, and w an arbitrary LI-world. Consider the following -pair: . [“rep´(y),” and not “rep(y),” because we are now considering the representation of LI*-individuals by -pairs, and not by maximalconsistent PRITs.] Because of the one-to-one correspondence between LI*-individuals and -pairs there is some LI*-individual y´ such that rep´(y´) = . Since y and y´ are LI*-individuals and rep´(y´) is a counterpart#1 of rep´(y) (according to definition (III.1)), it follows, according to definition (II.1), that y´ is a counterpart1 of y. Moreover, y´ is a part of w. For this, according to (B), one must show: ∃y´´(rep(w) = [rep(y´), y´´]). Now, the two representations of LI*-individuals, rep and rep´, are related in such a manner that we have: [rep(y´), 1(rep´(y´))] = 2(rep´(y´)),5 and therefore: ∃y´´(rep(w) = [rep(y´), y´´]) (since 2(rep´(y´)) = rep(w)).

That every LI*-individual has a counterpart1 in every LI-world is not prohibited by the formal requirements Lewis posited for counterpart relations. But doesn’t it mean that every LI*-individual exists at every LI-world? 5

rep represents the LI*-individuals by the maximal-consistent PRITs, and the LIworlds by the maximal-consistent states of affairs; rep´ represents the LI*-individuals by the -pairs. The two representations are related in the following way: They differ only regarding LI*-individuals; if y is a LI*-individual, then rep(y) is λIo(o = 1(rep´(y)) ∧ O(2(rep´(y)))), and rep´(y) is . As a consequence of (B) and the described relation between rep and rep´, we have for all LI*-individuals y and LI-worlds w: y is a part of w iff rep(w) = 2(rep´(y)).

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This is indeed how Lewis himself would have interpreted the situation – since to exist/to be actual at a world w means for him to have a counterpart in (or “at,” says Lewis) w; see On the Plurality of Worlds, p. 11, p. 12 – which would have led, we may be certain, to the rejection by Lewis of the counterpart1 relation as an acceptable counterpart relation. That ordinary individuals exist only contingently is one of the most deeply entrenched modal intuitions – also for Lewis (see ibid., p. 11). It is easily seen that it is not only true that every LI*-individual has a counterpart1 in every LI-world, but also true that every LI*-individual has exactly one counterpart1 in every LI-world. In “Counterparts and Actuality,” p. 25, Fara and Williamson assert that adopting the assumption “that every possible object has exactly one counterpart in every possible world” “would effectively collapse counterpart theory.” Indeed, many of the possibilities that are dear to Lewis and that are left open by Lewisian counterpart theory would be shut down in that case, but the theory itself would certainly not collapse into something that can no longer be properly called “counterpart theory.” It would merely become a different – and still non-trivial – counterpart theory, one closer to Leibnizian intentions. This remains true even if one adds to the general exactly-one-assumption the general no-merger-assumption: that no counterpart of any possible object is the counterpart of any other possible object. (Note, in this connection, that it is also provably true that no counterpart1 of any LI*individual is the counterpart1 of any other LI*-individual.)

However, matters can be somewhat improved – regarding adequacy to Lewisian views – by defining a counterpart relation that does not have the described drawback of the counterpart1 relation: of making existence (as interpreted by Lewis) a necessity for all LI*-individuals. This relation is the counterpart2 relation; it is based on a counterpart# relation – the counterpart#2 relation – which is different from the counterpart# relation that the counterpart1 relation is based on. (III.2) p´ is a counterpart#2 of p =Def p´ is a -pair and p is a -pair, and: p´ = p, or 1(p´) = 1(p) and EXM(1(p), λoA(o), 2(p))6 and EXM(1(p´), λoA(o), 2(p´)). And making use of the relation defined in (III.2), the next definition is this: This expression is read as “the first constituent of p exemplifies λoA(o) – actuality – in the second constituent of p,” or in other words: “1(p) is actual in 2(p).” For the definition of EXM(y, x, w), see D35 in Section 7.4. 6

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(II.2) y´ is a counterpart2 of y =Def y´ is a LI*-individual and y is a LI*individual, and the -pair that represents y´ is a counterpart#2 of the -pair that represents y. The counterpart2 relation is obviously symmetrical. It is also a relation that is reflexive over LI*-individuals: Assume y is a LI*-individual. According to (II.2) and (III.2): y is a counterpart2 of y iff the -pair that represents y – call it “p” – is a counterpart#2 of itself iff p = p.

Finally, the counterpart2 relation is transitive: Assume that y´´ is a counterpart2 of y´ and y´ a counterpart2 of y. Let p´´ be the -pair that represents y´´, p´ the -pair that represents y´, and p the -pair that represents y. On the basis of the assumption and (II.2), we have: p´´ is a counterpart#2 of p´ and p´ a counterpart#2 of p. According to (III.2.), there are four cases: (1) p´´ = p´ and p´ = p. Hence: p´´ = p, and therefore: p´´ is a counterpart#2 of p, and therefore: y´´ is a counterpart2 of y. (2) p´´ = p´ and 1(p´) = 1(p) and EXM(1(p), λoA(o), 2(p)) and EXM(1(p´), λoA(o), 2(p´)). Hence: 1(p´´) = 1(p) and EXM(1(p), λoA(o), 2(p)) and EXM(1(p´´), λoA(o), 2(p´´)), and therefore: p´´ is a counterpart#2 of p, and therefore: y´´ is a counterpart2 of y. (3) 1(p´´) = 1(p´) and EXM(1(p´), λoA(o), 2(p´)) and EXM(1(p´´), λoA(o), 2(p´´)) and p´ = p. Hence: 1(p´´) = 1(p) and EXM(1(p), λoA(o), 2(p)) and EXM(1(p´´), λoA(o), 2(p´´)), and therefore: p´´ is a counterpart#2 of p, and therefore: y´´ is a counterpart2 of y. (4) 1(p´´) = 1(p´) and EXM(1(p´), λoA(o), 2(p´)) and EXM(1(p´´), λoA(o), 2(p´´)) and 1(p´) = 1(p) and EXM(1(p), λoA(o), 2(p)) and EXM(1(p´), λoA(o), 2(p´)). Hence: 1(p´´) = 1(p) and EXM(1(p), λoA(o), 2(p)) and EXM(1(p´´), λoA(o), 2(p´´)), and therefore: p´´ is a counterpart#2 of p, and therefore: y´´ is a counterpart2 of y.

Thus, the counterpart2 relation – like the counterpart1 relation – is an equivalence relation over LI*-individuals, and we can also take equivalence classes with respect to the counterpart2 relation. But whereas every LI*-individual has a counterpart1 in every LI-world, we may expect that not every LI*-individual has a counterpart2 in every LI-world. It was shown above that there is, for an arbitrary LI*-individual y and an arbitrary LI-world w, a LI*-individual y´ which is such that: rep´(y´) = and [rep(y´), 1(rep´(y´))] = 2(rep´(y´)). It follows, according to

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(B), that y´ is a part of w. But it does not yet follow that y´ is a counterpart2 of y (though it does already follow that y´ is a counterpart1 of y). For obtaining that y´ is a counterpart2 of y, one needs to show in addition: either rep´(y´) = rep´(y), or: EXM(1(rep´(y)), λoA(o), 2(rep´(y))) and EXM(1(rep´(y´)), λoA(o), 2(rep´(y´))) – and showing this is in some cases, we may take it, simply not possible: because, for example, EXM(1(rep´(y´)), λoA(o), 2(rep´(y´))) is not true, while EXM(1(rep´(y)), λoA(o), 2(rep´(y))) is true (although 1(rep´(y´)) = 1(rep´(y))). Indeed, a LI*-individual y does not have any counterpart2 in any LIworld w that is different from its own world and which is such that 1(rep´(y)) does not exemplify actuality in rep(w), and we may take it that there are LI*-individuals y and LI-worlds w such that w is different from the world of y and 1(rep´(y)) does not exemplify actuality in rep(w). Suppose, for reductio, a LI*-individual y had a counterpart2 y´ in a LI-world w that is different from the LI-world of y and which is such that 1(rep´(y)) does not exemplify actuality in rep(w). According to (II.2) and (III.2), we therefore must have: rep´(y´) = rep´(y), or: 1(rep´(y´)) = 1(rep´(y)) and EXM(1(rep´(y)), λoA(o), 2(rep´(y))) and EXM(1(rep´(y´)), λoA(o), 2(rep´(y´))). Since, according to assumption, the LI-world of y [the LI-world of which y is a part] is different from w, which is the LI-world of y´, it follows that y and y´ must be different LI*-individuals, and therefore it cannot be that rep´(y´) = rep´(y). Moreover, according to assumption, we have: ¬EXM(1(rep´(y)), λoA(o), rep(w)), and therefore: ¬EXM(1(rep´(y´)), λoA(o), rep(w)) [because of 1(rep´(y´)) = 1(rep´(y))]. But rep(w) is 2(rep´(y´)), and therefore: ¬EXM(1(rep´(y´)), λoA(o), 2(rep´(y´))) – contradicting EXM(1(rep´(y´)), λoA(o), 2(rep´(y´))), which, however, is a consequence of the assumptions made (see above). What remains to be shown is this: rep(w) = 2(rep´(y´)). Since y´ is a part of w [according to supposition], we have, according to (B): ∃y´´(rep(w) = [rep(y´), y´´]). Since MC(rep(w)) and ∃=1w´(MC(w´) ∧ ∃r(w´ = [rep(y´), r])), it follows: rep(w) = ιw´(MC(w´) ∧ ∃r(w´ = [rep(y´), r])). But, according to the relation between rep and rep´ (see footnote 5), rep´(y´) = . Hence: rep(w) = 2(rep´(y´)).

However, the equivalence classes that can be taken with respect to the counterpart2 relation – exhaustively dividing the LI*-individuals among them – do not correspond one-to-one and in a natural way to the MIindividuals. Consider the actual LI*-individual u.*m.*7 and the class of its 7

The representation of the LI*-individual u.*m.* as -pair is . According to Lewis, I am the LI*-individual u.*m.* (he would not have scrupled to call that LI*-individual “U.M.”); according to myself, however, I am the MI-individual U.M.

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counterparts2: [u.*m.*]CP2. Each element y in [u.*m.*]CP2 has something in common with every other element y´ in [u.*m.*]CP2: the first constituent of the representation of y as -pair is identical with the first constituent of the representation of y´ as -pair: 1(rep´(y)) = 1(rep´(y´)); this is so because 1(rep´(y)) = U.M., for every y in [u.*m.*]CP2. But unfortunately there is a LI*-individual y*, and the class of its counterparts2 [y*]CP2 is such that the following is true: 1(rep´(y)) = U.M., for every y in [y*]CP2, and [y*]CP2 ≠ [u.*m.*]CP2. Take a maximal-consistent state of affairs w´ which is such that U.M. does not exemplify actuality in w´. (There certainly is such a maximal-consistent state of affairs.) Consider the ordered pair: . Let y* be the LI*-individual which corresponds to . y* is an element of [y*]CP2, since y* is a counterpart2 of itself. But y* is not an element of [u.*m.*]CP2, since y* is not a counterpart2 of u.*m.*: according to what we know about w´ and w*, neither is rep´(y*) [= ] identical with rep´(u.*m.*) [= ], nor do we have both EXM(1(rep´(u.*m.*)), λoA(o), 2(rep´(u.*m.*))) [≡ EXM(U.M., λoA(o), w*)] and EXM(1(rep´(y*)), λoA(o), 2(rep´(y*))) [≡ EXM(U.M., λoA(o), w´)].

Thus: while every equivalence class with respect to the counterpart2 relation determines (in the way indicated) exactly one MI-individual, and every MI-individual is determined by some such equivalence class, it is not always the case that different such equivalence classes determine different MI-individuals. There is no natural one-to-one correspondence between the equivalence classes with respect to the counterpart2 relation and the MIindividuals. But consider the set of equivalence classes with respect to the counterpart2 relation, MCP2, and the predicate “x ∈ MCP2 ∧ y ∈ MCP2 ∧ x determines [in the way indicated] the same MI-individual as y.” This predicate, in its turn, expresses an equivalence relation. The equivalence classes that can be taken with respect to this higher-order equivalence relation – exhaustively dividing the classes in MCP2 among them – do obviously correspond one-to-one and in a natural way to the MI-individuals. Therefore: based on the counterpart2 relation, we have found, in the end, a second way to represent MI-individuals in the Lewisian framework, a way that is, from the Lewisian point of view, more adequate than the first way (which was based on the counterpart1 relation). Nevertheless, there are aspects of the above-described relationship between MI-individuals and LI*-individuals that are bound to jar with Lewisian ways of thinking. These aspects have nothing to do with counter-

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part relations; they have to do with existence. If a MI-individual x does not exist in a certain possible world w – that is, I(x) ∧ ¬EXM(x, λoA(o), w) ∧ MC(w) – then there is nevertheless a LI*-individual x´, matching , that is a part of the LI-world w´, matching w. And since x´ is a part of w´, it follows, according to the Lewisian ways of thinking, that x´ exists (is actual) in w´. Lewis decidedly prefers to say “is actual” instead of “exists”; for others, it is the other way round. No matter which way of speaking is preferred, the important thing to remember in this context is that, in this book, “exist” is taken to express actuality (or, in other words, the Second Ontological Meaning of Existence; see Section 1.4). Regarding the Lewisian concepts of existence/actuality at a LI-world w´, respectively in a LI-worlds w´: That LI*-individual x´ exists (is actual) in w´ implies that x´ exists (is actual) at w´, but not conversely. For x´’s existing in w´, it is necessary and sufficient that x´ is a part of w´. For x´ existing at w´, it is necessary and sufficient that x´ has a counterpart at (synonymously: in) w´, i.e., that a counterpart of x´ is a part of w´: exists in w´. Clearly, x´ may have a counterpart at (in) w´ without being a part of w´. Equally clearly, if x´ is a part of w´, it must have a counterpart at w´ (namely itself, since the counterpart relation is reflexive over LI*-individuals). Note that in the here advocated ontological framework, no difference is made – or need be made – between existence/actuality at a world, and existence/actuality in a world.

Thus: if a MI-individual x does not exist in a certain possible world (maximal-consistent state of affairs) w, the LI*-individual corresponding to x in the LI-world that corresponds to w is nevertheless – from the point of view of Lewis – bound to exist in that LI-world. The remedy for this inadequacy of representation is to give up the interpretation of “[LI*individual] x´ exists (is actual) in [LI-world] w´” as “[LI*-individual] x´ is a part of [LI-world] w´,” and to treat the predicate “[LI*-individual] x´ exists in [LI-world] w´” just like an ordinary LI-world-relative predicate, say, like the predicate “[LI*-individual] x´ is at some time a philosopher in [LIworld] w´.” Regarding this latter predicate, it is entirely in harmony with Lewisian positions to suppose, for all LI*-individuals x´and LI-worlds w´, (C)

x´ is at some time a philosopher in w´ iff x´ is a part of w´ ∧ EXM(1(rep´(x´)), λo(o is at some time a philosopher), 2(rep´(x´))).

For example (instantiating (C)): u.*m.* is at some time a philosopher in the world of u.*m.* iff u.*m.* is a part of the world of u.*m.* ∧ EXM(1(rep´(u.*m.*)), λo(o is at some time a philosopher), 2(rep´(u.*m.*))). Since u.*m.* is a part of the

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world of u.*m.*, and 1(rep´(u.* m.*)) = U.M., and 2(rep´(u.*m.*)) = w* (see footnote 7), we obtain: u.*m.* is at some time a philosopher in the world of u.*m.* iff EXM(U.M., λo(o is at some time a philosopher), w*). Note that “x´ is a part of w´” can be equivalently replaced in (C) by “2(rep´(x´)) = rep(w´).” This is possible, since, on the basis of (B), we have, for all LI*-individuals x´ and LI-worlds w´, 2(rep´(x´)) = rep(w´) iff x´ is a part of w´. (For the relationship between rep and rep´ which is needed for proving this equivalence, see footnote 5.)

Applying (C) to the predicate “x´ exists in w´,” instead of the predicate “x´ is at some time a philosopher in w´,” we obtain, for all LI*-individuals x´and LI-worlds w´, x´ exists in w´ iff x´ is a part of w´ ∧ EXM(1(rep´(x´)), λoA(o), 2(rep´(x´))). Thus, if “x´ exists in w´” were treated analogously to “x´ is at some time a philosopher in w´,” on the basis of (C), then the above-described problem would not occur, since “EXM(1(rep´(x´)), λoA(o), 2(rep´(x´)))” would then be a necessary condition of “x´ exists in w´” in addition to “x´ is a part of w´.” But Lewis would certainly have been very loath to change his ways regarding existence/actuality. But his views regarding existence/actuality do produce difficulties. Consider the assertion (a) “w´ is a LI-world at which u.*m.* does not exist” and the assertion (b) “w´´ is a LI-world at which u.*m.* is not a philosopher.” There are two non-equivalent but equally plausible possible ways to interpret assertion (b) according to Lewisian counterpart theory: (1) no counterpart of u.*m.* is both a philosopher and a part of the LIworld w´´; (2) some counterpart of u.*m.* is not a philosopher but a part of the LI-world w´´.8 This ambiguity shows that Lewisian counterpart theory has a large lacuna regarding a matter that is of central importance. By analogy, there should also be two ways to interpret assertion (a): (1´) no counterpart of u.*m.* both exists and is a part of the LI-world w´; (2´) some counterpart of u.*m.* does not exist but is a part of the LI-world w´. But Lewis, in fact, chooses a third way to interpret (a): (L) w´ is a LIworld at which u.*m.* has no counterpart (this interpretation of (a) 8

Note that (1) and (2) would be equivalent if every LI*-individual had exactly one counterpart in each LI-world – as would be the case if the counterpart relation were taken to be the counterpart1 relation.

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emerges clearly enough from On the Plurality of Worlds, p. 11, p. 12), or in other words: no counterpart of u.*m.* is a part of the LI-world w´. Consider now the assertion (c) “w´´ is a LI-world at which u.*m.* is married but not a philosopher.” For this assertion, in contrast to assertion (b), there is only one intuitively satisfactory interpretation according to Lewisian counterpart theory: (3) some counterpart of u.*m.* is married, not a philosopher, and a part of the LI-world w´´. Consider finally the assertion (d) “w´ is a LI-world at which u.*m.* is a person, but does not exist.” Prima facie (d) seems analogous to (c), and therefore it seems it should be interpreted analogously to (c): (3´) some counterpart of u.*m.* is a person, does not exist, and is a part of the LIword w´. But this may seem wrong, for the following reason that is inherent in Lewis’s property-theory: while one can say absolutely of a LI*individual y´ that it is married (by this I mean: married at some time) but not a philosopher (by this I mean: not a philosopher at any time), or that it is a person – one is just saying that y´ is, or is not, an element of the respective LI*-property – and bring in the relativization to a LI-world w´ only by adding that y´ or a counterpart of it is a part of w´, one does not appear to be able to say absolutely (in the sense indicated) of a LI*individual y´ that it does, or does not, exist (“to exist,” remember, being taken in the sense of “to be actual”). What is evidently true, however, is that (d) implies both (e) “w´ is a LI-world at which u.*m.* is a person” and (a) “w´ is a LI-world at which u.*m.* does not exist.” But then (d) cannot be true, according to Lewis. For according to Lewis (see above), (a) means this: (L) no counterpart of u.*m.* is a part of the LI-world w´; and (e) can only be taken to mean: (4) some counterpart of u.*m.* is a person and a part of the LI-world w´. (4) obviously contradicts (L). But might it not be the case that (d) is true (w´ being appropriately chosen)? If so, then there must be something wrong with Lewis’s interpretation of existence/actuality. In particular, existence/actuality at a LI-world w´ must mean something else than having a counterpart at (in) w´, i.e., something else than having a counterpart that is a part of w´. Regarding the interpretation of existence/actuality there is a deep conceptual difference between the present framework and Lewis’s – and it is the point where the representation of the latter framework within the former reaches its limits. The LI*-properties have been represented by the totally defined I-properties as they are conceived of in the present work: by the PRITs (and this step of representation is crucial for obtaining the repre-

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sentation of the LI*-individuals by the maximal-consistent PRITs). But λIoA(o) is a PRIT. Hence there must be a LI*-property that matches it. In fact, it is easy to specify that LI*-property. If we choose the representation rep´ of LI*-individuals by -pairs, then the LI*property that matches λIoA(o) is the following set: {y is a LI*-individual: EXM(1(rep´(y)), λIoA(o), 2(rep´(y)))}. If we choose the representation rep of LI*-individuals by maximal-consistent PRITs, then the LI*-property that matches λIoA(o) is identical with the set {y is a LI*-individual: PPR(λIoA(o), rep(y))}. Both sets are provably identical. This is a consequence of a more general theorem, namely: for all LI*individuals y and PRITs f, EXM(1(rep´(y)), f, 2(rep´(y))) ≡ PPR(f, rep(y)). Since for all LI*-properties z we have PRIT(rep(z)), the theorem implies on the basis of (A): for all LI*-individuals y and LI*-properties z, y LI-exemplifies z [that is, y ∈ z] iff EXM(1(rep´(y)), rep(z), 2(rep´(y))).

But are the two sets (which really are the same set) the sets that Lewis could have called “the property of actuality for individuals* (i.e., for individuals excluding possible worlds)”? No, because for Lewis there is no such thing as the LI*-property actuality for individuals*, or in other words: for Lewis, there is no LI*-property that is identical with actuality for individuals* – for him, actuality for individuals* is not a LI*-property. Thus, for Lewis, {y is a LI*-individual: EXM(1(rep´(y)), λIoA(o), 2(rep´(y)))} and {y is a LI*-individual: PPR(λIoA(o), rep(y))}, being LI*-properties, cannot be identical with actuality for individuals*. Why not? Why not simply identify Lewisian actuality for individuals* with the LI*-property {y is a LI*-individual: EXM(1(rep´(y)), λIoA(o), 2(rep´(y)))} and assume accordingly, for all LI*-individuals y´ and LI-worlds w´, y´ is actual/exists in w´ iff y´ is a part of w´ and y´ ∈ {y is a LI*individual: EXM(1(rep´(y)), λIoA(o), 2(rep´(y)))},9 y´ is actual/exists at w´ iff a counterpart y´´ of y´ is a part of w´ and y´´ ∈ {y is a LI*-individual: EXM(1(rep´(y)), λIoA(o), 2(rep´(y)))}?

9

Compare (C) above, applied to “x´ exists in w´.”

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Lewis’s famous indexical analysis of actuality (see his “Anselm and Actuality”) is the reason why Lewisian actuality for individuals* does not follow the model of Lewisian being a person, which is a LI*-property sure enough and identical with {y is a LI*-individual: EXM(1(rep´(y)), λIo(o is a person), 2(rep´(y)))}; for this latter set, it is certainly true that, for all LI*-individuals y´ and LI-worlds w´, y´ is a person in w´ iff y´ is a part of w´ and y´ ∈ {y is a LI*individual: EXM(1(rep´(y)), λIo(o is a person), 2(rep´(y)))} [in other words: iff y´ is a part of w´ and LI-exemplifies the LI*-property being a person],10 y´ is a person at w´ iff a counterpart y´´ of y´ is a part of w´ and y´´ ∈ {y is a LI*-individual: EXM(1(rep´(y)), λIo(o is a person), 2(rep´(y)))} [in other words: iff y´ has a counterpart at (in) w´ that LI-exemplifies the LI*-property being a person]. As has already been adumbrated above, for Lewis, actuality for individuals* has no world-independent identity. For him, it makes no sense to speak of actuality for individuals* independent of an implicit world-index. Note the difference to being a person. For Lewis (as for me), being a person does have a world-independent identity; it does make sense to speak of being a person independent of an implicit world-index.

For Lewis, there are just the mutually exclusive sets m(w) – m(w) =Def {y is a LI*-individual: y is a part of w} –, one m(w) for each LI-world w, and the phrase “actual individual*”, which denotes, wherever and whenever it occurs, some set m(w) – not always the same – out of {x: ∃w(w is a LIworld ∧ x = m(w))}. There is indeed a LI*-property that is the union of the m(w)s, namely {y: y is a LI*-individual}. But for Lewis this LI*-property can certainly not be identical with actuality for individuals*.11 Nor can ac-

10

Compare (C) above. There is, however, one passage that throws doubt on this assertion: “If I had to, I too could say that everything [hence every LI*-individual] is actual, only not all that is actual is part of this world.” (“Anselm and Actuality,” p. 25.) But could he really say so if he had to? For Lewis also says: “If I were convinced that I ought to call all the worlds actual [which would be the case if, and only if, all LI*-individuals were actual

11

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tuality for individuals*, from the Lewisian point of view, be the set {y: y is a LI*-individual and y is a part of @} – or in other words: the set m(@), where “@” is a rigid name for the actual LI-world. For that LI*-property is not Lewisian actuality for individuals*, it is Lewisian actuality in @ for individuals*. If actuality for individuals* is to be anything property-like from the Lewisian point of view, then actuality for individuals* can only be the set {x: ∃w(w is a LI-world ∧ x = m(w))} – which is not a LI*property but a certain Lewisian second-order property: a Lewis-property of certain LI*-properties. 8.2

Alvin Plantinga and David Lewis on Essence and Essentiality

Regarding essence, Alvin Plantinga asserts: E is an essence of Socrates if and only if E is essential to Socrates and there is no possible world in which there exists an object distinct from Socrates that has E. (The Nature of Necessity, p. 70.) Accordingly, Socrateity is an essence of Socrates. But there are others. Each of Socrates’ world-indexed properties, as we have seen, is essential to him. Now let P be any property he and he alone has – being married to Xantippe, for example, or being the shortest Greek philosopher, or being A. E. Taylor’s favourite philosopher; and consider the world-indexed property having-P-in-α. … [F]or any property P and world W, if in W Socrates alone has P, then having-Pin-W is one of his essences. (Ibid., p. 72.)

I consider Plantinga to be referring in these passages to properties with universal fit, that is, to properties that can be predicated of everything. Taking Plantinga’s “essence” and “essential” to be tantamount to “universal logical essence” and “logically essential,” abbreviating “Socrates” by “Sortes,” and replacing “α” by “w0” (this being the designator here employed for designating the same entity rigidly that is designated non-rigidly by “w*”), Plantinga’s main claims in the two cited passages – explicitly stated or logically implied – translate into the following statements:

for Lewis] … then it would become very implausible to say that what might happen is what does happen at some or another world.” (On the Plurality of Worlds, p. 100.)

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Plant1

For every property E which is such that ∀yF(E, y): ESSU1(E, Sortes) ≡ EXME1(Sortes, E) ∧ ¬∃w∃y(y ≠ Sortes ∧ MC(w) ∧ EXM(y, E, w)).

Note that this translation involves rendering Plantinga’s “there is no possible world in which there exists an object distinct from Socrates that has E” as “there is no possible world w such that there is something distinct from Socrates that has E in w.” This is justified from the point of view of interpreting Plantinga, since “that has E” in Plantinga’s phrase must certainly be taken to be implicitly complemented by “in that world,” and since, for Plantinga, an object’s exemplifying (i.e., having) a property in a world implies its existing in that world (see Section 8.5 below). Moreover, the word “object” in Plantinga’s phrase is here taken to have its most general sense (which interpretation is at least consistent with Plantinga’s intentions, I presume), which means that “there is an object” is tantamount to “there is something.” Plant2

ESSU1(λo(o = Sortes), Sortes).

What Plantinga calls “Socrateity” is of course the property λo(o = Sortes). Plant3

For every property P which is such that ∀yF(P, y): EXM(Sortes, P) ∧ ∀y(EXM(y, P) ⊃ y = Sortes) ⊃ ESSU1(λoEXM(o, P, w0), Sortes).

Plantinga’s “y has P” is rendered by “EXM(y, P).” Accordingly, Plantinga’s “world-indexed property having-P-in-α.” is rendered by “λoEXM(o, P, w0)” (“w0” replacing “α”). Plant4

For some property P which is such that ∀yF(P, y): EXM(Sortes, P) ∧ ∀y(EXM(y, P) ⊃ y = Sortes) ∧ λoEXM(o, P, w0) ≠ λo(o = Sortes).

In the present framework, the statements Plant1 – Plant3 are provably true; Plant4, however, is provably false. In Sections 7.3 and 7.4, we have demonstrated that an entity that is rigidly designated by “a0” has exactly one universal logical essence, namely, the property λo(o = a0). Hence we have in particular (“Sortes” rigidly designating Socrates):

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(A) ESSU1(λo(o = Sortes), Sortes) ∧ ∀x(ESSU1(x, Sortes) ⊃ x = λo(o = Sortes)). Thus, obviously, Plant2 has already been shown to be true. Moreover, we can show: (B) EXME1(Sortes, λo(o = Sortes)) ∧ ¬∃w∃y(y ≠ Sortes ∧ MC(w) ∧ EXM(y, λo(o = Sortes), w)) ∧ ∀x[PR(x) ∧ ∀yF(x, y) ∧ EXME1(Sortes, x) ∧ ¬∃w∃y(y ≠ Sortes ∧ MC(w) ∧ EXM(y, x, w)) ⊃ x = λo(o = Sortes)].12 Proof: (a) According to D30, proving EXME1(Sortes, λo(o = Sortes)) is involved in proving ESSU1(λo(o = Sortes), Sortes), and the latter proving has already been done (in proving (A)). (b) Suppose, for reductio, y ≠ Sortes ∧ MC(w) ∧ EXM(y, λo(o = Sortes), w). Hence, according to D35, D16: P([λo(o = Sortes), y], w). Hence [λo(o = Sortes), y] ≠ k* (because of MC(w)). But we also have [λo(o = Sortes), y] = k*, because of y ≠ Sortes, P32, and P30. [From y ≠ Sortes, by employing P32 (“Sortes” being a referentially stable term): k* = that (y = Sortes); and [λo(o = Sortes), y] = that (y = Sortes), according to P30, since λo(o = Sortes) is a universally predicable property.] Contradiction. (c) Assume: PR(x) ∧ ∀yF(x, y), EXME1(Sortes, x), ¬∃w∃y(y ≠ Sortes ∧ MC(w) ∧ EXM(y, x, w)). Suppose for reductio: x ≠ λo(o = Sortes). Hence, according to P29 [and because of ∀yF(x, y) and ∀yF(λo(o = Sortes), y)]: [x, z] ≠ [λo(o = Sortes), z], for some z. Now, either z = Sortes, or z ≠ Sortes. In the first case: Because of EXME1(Sortes, x), we have according to D22: [x, Sortes] = t*. Moreover: [λo(o = Sortes), Sortes] = t*.13 Therefore: [x, Sortes] = [λo(o = Sortes), Sortes], and hence (given z = Sortes): [x, z] = [λo(o = Sortes), z] – contradicting the conclusion already reached. In the second case: ∀w(MC(w) ⊃ ¬EXM(z, x, w)) [because of ¬∃w∃y(y ≠ Sortes ∧ MC(w) ∧ EXM(y, x, w)) and z ≠ Sortes], hence by D35: ∀w(MC(w) ⊃ ¬O([x, z], w)). 12

In other words: being identical with Socrates is the only universally predicable logically essential property of Socrates which, in any possible world, is exemplified by nothing that is different from Socrates. [PR(λo(o = Sortes)) and ∀yF(λo(o = Sortes), y) – being provable or presupposed – need not be included in the theorem.] 13 [λo(o = Sortes), Sortes] = that (Sortes = Sortes) = t* (according to P30 and P32, because Sortes = Sortes).

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Since we have S([x, z]) [because of ∀yF(x, y), D17], this implies: [x, z] = k*.14 But, for z ≠ Sortes, [λo(o = Sortes), z] is also k*.15 Hence once again: [x, z] = [λo(o = Sortes), z] – contradicting the conclusion already reached. The proof is now complete.

Given the above two principles, (A) and (B), it is easy to derive Plant1. Plant4, in turn, is shown to be false by proving its negation: (C) For every property P which is such that ∀yF(P, y): EXM(Sortes, P) ∧ ∀y(EXM(y, P) ⊃ y = Sortes) ⊃ λoEXM(o, P, w0) = λo(o = Sortes). Proof: We will have to make use of one additional first principle – a principle that is to be presupposed for properties designated by terms of the form “λoEXM(o, P, w0).” This principle will be introduced in the manner of prefixing “[1]” to it (see below). Assume, then, PR(P), ∀yF(P, y), EXM(Sortes, P), ∀y(EXM(y, P) ⊃ y = Sortes). Suppose for reductio: λoEXM(o, P, w0) ≠ λo(o = Sortes). [1] ∀yF(P, y) ⊃ ∀yF(λoEXM(o, P, w0), y). Hence, according to P29: [λoEXM(o, P, w0), z] ≠ [λo(o = Sortes), z], for some z. Now, either z = Sortes, or z ≠ Sortes. Before going into these two cases, consider: λoEXM(o, P, w0) = λoP([P, o], w0), according to D35, D16. And therefore: [2] ∀y(P([P, y], w0) ⊃ [λoEXM(o, P, w0), y] = t*) ∧ ∀y(¬P([P, y], w0) ⊃ [λoEXM(o, P, w0), y] = k*). For assume P([P, y], w0). Then, according to P22 and P30 (which are safely applicable, since “y” and “w0” are referentially stable terms): t* = that P([P, y], w0) = [λoP([P, o], w0), y] = [λoEXM(o, P, w0), y]. And assume ¬P([P, y], w0). Then, according to P22, P30: k* = that P([P, y], w0) = [λoP([P, o], w0), y] = [λoEXM(o, P, w0), y]. Now back to the two possible cases: (i) z = Sortes, (ii) z ≠ Sortes. In the first case: Since, according to assumption, EXM(Sortes, P), we have: EXM(z, P), that is (according to D20): O([P, z]), and hence (according to D6 and the Actuality Principle for States of Affairs, stated in Section 3.3): P([P, z], w*), and therefore (because w0 = w*): P([P, z], w0). According to [2], this means: [λoEXM(o, P, w0), z] = According to the theorem ∀x[S(x) ⊃ (◊1(x) ≡ ∃y(MC(y) ∧ O(x, y)))] in Section 3.8 and the definition ◊n(x) =Def S(x) ∧ ¬P(neg(x), bn) in Section 3.1, we have because of S([x, z]) (and considering that b1 = t*): ¬P(neg([x, z]), t*) ≡ ∃y(MC(y) ∧ O([x, z], y)), and therefore, because of ∀w(MC(w) ⊃ ¬O([x, z], w)): P(neg([x, z]), t*). Hence (according to the mereology of states of affairs): neg([x, z]) = t*, and therefore: [x, z] = k*. 15 [λo(o = Sortes), z] = that (z = Sortes) = k* (according to P30 and P32, because z ≠ Sortes). 14

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t*. But, for z = Sortes, we also obtain: [λo(o = Sortes), z] = t*. Hence: [λoEXM(o, P, w0), z] = [λo(o = Sortes), z] – contradicting the conclusion already reached. In the second case: Since, according to assumption, ∀y(EXM(y, P) ⊃ y = Sortes), we have: ¬EXM(z, P), that is: ¬O([P, z]), and hence (according to D6, the Actuality Principle for States of Affairs, and S([P, z]) because of ∀yF(P, y)): ¬P([P, z], w*), and therefore: ¬P([P, z], w0). According to [2], this means: [λoEXM(o, P, w0), z] = k*. But, for z ≠ Sortes, we also obtain: [λo(o = Sortes), z] = k*. Hence once again: [λoEXM(o, P, w0), z] = [λo(o = Sortes), z] – contradicting the conclusion already reached. The proof is complete.

Given the proven truth of (C) – which is the negation of Plant4 – Plant3 is easily derivable by using Plant2. One of the four Plantingian claims considered turned out to be false. But this is not due to a deep difference in the conception of essence between the present framework and Plantinga’s. Rather, it is due to a difference in the conception of the differentiation of properties. For Plantinga, properties are fine-grained, fitting the fine-grainedness of states of affairs in his conception (see Section 4.1). Thus he believes that being married to Xantippe in α (i.e., in w0) is a different property than being Socrates. For me, properties are coarse-grained, fitting the coarse-grainedness of states of affairs in my conception. Thus I believe that being married to Xantippe in α is the same property as being Socrates. Who finds this absurd should take into consideration that being married to Xantippe in α and being Socrates are obviously, with logical necessity, co-extensional properties. On the coarse-grained conception of properties, this is generally regarded as sufficient for identifying them.

Plantinga distinguishes between essences which are haecceities – that is: properties each of which is, for some x, the property of being identical with x – and essences which are not haecceities.16 Using this terminology, the difference between the present conception of [universal logical] essence and Plantinga’s can be expressed in the following way: according to the present conception of essence, all essences are haecceities; according to Plantinga’s conception of essence, some essences are haecceities, and others are not.

16

“Among an object’s essences, there is its haecceity: the property of being that very object, or the property of being identical with that very object.” (“Self-Profile,” p. 92.)

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Turning now to the conception David Lewis has of essence and essentiality, we find him defining: An attribute that something shares with all its counterparts is an essential attribute of that thing, part of its essence. The whole of its essence is the intersection of its essential attributes, the attribute it shares with all and only its counterparts. (“Counterpart Theory and Quantified Modal Logic,” p. 35.) Essence and counterpart are interdefinable. We have just defined the essence of something as the attribute it shares with all and only its counterparts; a counterpart of something is anything having the attribute which is its essence. (Ibid.)

Lewis’s conceptions of possible worlds, individuals, and properties of individuals are rather different from the conceptions of such things within the present ontological framework. Nonetheless, his conceptions are representable within that framework; see Section 8.1 above. I am using the terminology introduced in that section to distinguish his conceptions from mine: Lewisian individuals minus Lewisian possible worlds are called “LI*-individuals”; Lewisian properties of LI*-individuals are called “LI*properties”; they are simply the sets of LI*-individuals. Lewisian (possible) worlds are called “LI-worlds.” Consider, then, the essence of a LI*-individual, say, of u.*m.* (that is, of the LI*-individual corresponding to me in the actual LI-world; according to Lewis, of course, u.*m.* does not only correspond to me, but is me; see footnote 7). According to Lewis, (1)

the essence of u.*m.* =Def {y: y is a counterpart of u.*m.*}.

Moreover, we have according to Lewis, for all LI*-properties x, (2)

x is an essential property of u.*m.* =Def ∀y(y is a counterpart of u.*m.* ⊃ y has x) (in other words: the essence of u.*m.* ⊆ x).

Now, an existence-essential LI*-property of u.*m.* is a LI*-property such that u.*m.* cannot exist (be actual) without it. In other words, for all LI*properties x, (3)

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x is an existence-essential property of u.*m.* =Def ∀w(w is a LIworld and u.*m.* exists at w ⊃ u.*m.* has x at w).

8 Properties, Essences, and Actualism (Again)

Prima facie, there seems to be a great difference between being an essential LI*-property of u.*m.* and being an existence-essential LI*-property of u.*m.* On closer inspection, however, one finds that the apparent difference is not in fact there. (A real difference, also for Lewis, is between absolute essentiality and existence-essentiality; see the end of this section.) In Section 8.1 we have seen that, according to Lewis, a LI*-individual exists at a LI-world w if, and only if, it has a counterpart at w. This Lewisian doctrine – call it “LD” for further reference – makes it possible to prove, for all LI*-properties x, that (4)

x is an essential property of u.*m.* iff x is an existence-essential property of u.*m.*,

if one assumes that no LI*-individual has more than one counterpart at a given LI-world. Lewis, of course, has explicitly rejected this assumption.17 But he would have done well to think twice before rejecting it. Suppose u.*m.* – or any other LI*-individual – has two counterparts at a LI-world w´ (that is, two counterparts that are both parts of w´), y and y´; y and y´ are bound to have different LI*-properties, say, y has the LI*-property x, whereas y´ does not have x, which means that it has the LI*-property non*x (this being simply the set of all LI*-individuals that are not elements of x). Does u.*m.*, or does he not, have x at w´? – Obviously, if for u.*m.*’s having a LI*-property f at w´ it is sufficient that some counterpart of u.*m.* at w´ has f, then u.*m.* has x at w´; but by the same token he also has non*-x at w´ – which is absurd. If, however, for u.*m.*’s having f at w´ it is necessary that all counterparts of u.*m.* at w´ have f, then, obviously, u.*m.*, though existing at w´, has neither x nor non*-x at w´ – which is just as absurd. In view of these considerations, I proceed on the assumption that no LI*-individual has more than one counterpart at a given LI-world. Then the proof of (4) is this: Let x be an arbitrary LI*-property. (i) Assume: x is an essential property of u.*m.*, that is: ∀y(y is a counterpart of u.*m.* ⊃ y has x), and assume: w is a LI17

“It would not have been plausible to postulate that nothing in any world had more than one counterpart in any other world. Suppose x4a and x4b in world w4 are twins; both resemble you closely; both resemble you far more closely than anything else in w4 does; both resemble you equally. If so, both are your counterparts.” (“Counterpart Theory and Quantified Modal Logic,” p. 29.)

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world and u.*m.* exists at w. Hence u.*m.* has a counterpart at w (according to LD). This counterpart has the LI*-property x, according to the first assumption. Therefore: u.*m.* has x at w (since u.*m.* has a counterpart at w which has x). – We have now derived: ∀w(w is a LI-world and u.*m.* exists at w ⊃ u.*m.* has x at w), that is: x is an existence-essential property of u.*m.*. (ii) Assume x is an existence-essential property of u.*m.*, that is: ∀w(w is a LI-world and u.*m.* exists at w ⊃ u.*m.* has x at w), and assume: y is a counterpart of u.*m.*. Since y is a LI*-individual (being a counterpart of u.*m.*), there is a LI-world w´ such that y is a part of w´ (see no. [2] of the Lewisian principles that were stated at the beginning of Section 8.1), and therefore: y is a counterpart of u.*m.* at w´ (since y is a counterpart of u.*m.* and a part of w´), that is: u.*m.* has a counterpart at w´. Hence (according to LD): u.*m.* exists at w´. Therefore (according to the initial assumption of (ii)): u.*m.* has x at w´. This means: u.*m.* has a counterpart at w´ which has (the LI*-property) x. But this counterpart can only be y, since y is a counterpart of u.*m.* at w´ and u.*m.* has at most one counterpart at w´. Therefore: y has x. – We have now derived: ∀y(y is a counterpart of u.*m.* ⊃ y has x), that is: x is an essential property of u.*m.*.

But (4) spells trouble for Lewis’s approach. For there seems to be a certain LI*-property, existence for LI*-individuals, or, in others words, actuality for LI*-individuals: LI*-existence, LI*-actuality. LI*-existence, if a LI*property, is an existence-essential property of u.*m.*, according to (3), for there is obviously no LI-world at which u.*m.* exists but does not have LI*-existence. But then, according to (4), LI*-existence is also an essential property of u.*m.*. On the basis of (2), however, this conclusion does not seem to be correct, given that LI*-existence is equated with LI*-actuality; for it certainly does not seem to be true that every counterpart of u.*m.* has LI*-actuality. What should be done about this? Should one call the interpretation of LI*-existence as LI*-actuality illegitimate, and say that the only legitimate interpretation of LI*-existence is to interpret it as the LI*-property being identical with some LI*-individual? This removes the problem, but at the price of making it impossible to say truly that u.*m.* might not have had LI*-existence (for u.*m.* could certainly not have been different from every LI*-individual). Moreover, from the point of view of interpreting positions of David Lewis, one ought to respect the fact that Lewis – though usually saying “is/is not actual” when he means to say that something is/is not actual – does also use “exists/does not exist” when he means to say that something is/is not actual.18 18

For example: “So it seems Humphrey does satisfy ‘necessarily x exists’ and doesn’t satisfy ‘possibly x does not exist’. That is wrong. For all his virtues, still it really will

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Should one deny the obvious assumption after all? Should one deny that there is no LI-world at which u.*m.* exists but does not have LI*existence? One might do so, but this move can only be acceptable if at the same time one denies that LI*-existence is a LI*-property. Then there is some LI-world at which u.*m.* exists but does not have LI*-existence, simply for the reason that there is some LI-world at which u.*m.* exists, whereas LI*-existence cannot be identified with any of the LI*-properties had (in the way LI*-properties are had – which is the way the “have” is meant in the obvious assumption) by u.*m.* or his counterparts, it not being identifiable with any LI*-property at all. Thus one reaches the same conclusion as the one already reached at the end of Section 8.1, which conclusion was there expressed in the following way: for Lewis, actuality for individuals* is not a LI*-property. (What was called “actuality for individuals*” in Section 8.1 is here called “LI*-existence” or “LI*-actuality.”) I turn to a somewhat different subject in connection with Lewis’s conception of essentiality. Lewis points out the following difficulty for the theory of modality: (3) We want it to come out that he [Humphrey] satisfies the modal formula ‘necessarily x is human’, since that seems to be the way to say something true, namely that he is essentially human. (4) We want it to come out that he satisfies the modal formula ‘possibly x does not exist’, since that seems to be the way to say something else true, namely that he might not have existed. (5) We want it to come out that he does not satisfy the modal formula ‘possibly x is human and x does not exist’, since that seems to be the way to say something false, namely that he might have been human without even existing. So he satisfies ‘x is human’ at all worlds and ‘x does not exist’ at some worlds; so he satisfies both of them at some worlds; yet though he satisfies both conjuncts he doesn’t satisfy their conjunction! How can that be? (On the Plurality of Worlds, p. 11.)

Lewis believes that the solution any theory of modality might offer for the difficulty described by him – for example, his own theory (i.e., counterpart theory), or the theory of “the friend of boxes and diamonds” – will be unsatisfactory to some extent (see On the Plurality of Worlds, pp. 11-12). What can be said about this matter? Given that one lets the word “exist” express actuality and accords “possible” and “necessary” their widest – respectively, narrowest – sense, one must still decide whether by the word “essential” one means absolutely not do to elevate Humphrey to the ranks of the Necessary Beings.” (On the Plurality of Worlds, pp. 10-11.)

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essential or rather existence-essential, and one must still decide whether by the word “human” one means simpliciter-human or rather human-andexistent. Moreover, one must decide whether Humphrey is a LI-individual (an individual in Lewis’s sense19) or a MI-individual (an individual in my sense). If Humphrey is taken to be a LI-individual, then one is moving within a Lewisian counterpart-theoretic framework, with the corresponding Lewisian interpretation of existence/actuality; if Humphrey is taken to be a MI-individual, then one is moving within a framework that is more or less like the one presented in this book. There are eight possible combinations of decisions (to prejudge matters as little as possible, there will be no talk of properties in what follows): (a) “human” means simpliciter-human, and “essential” absolutely essential. Humphrey is a MI-individual. (a´) “human” means simpliciter-human, and “essential” absolutely essential. Humphrey is a LI-individual. (b) “human” means simpliciter-human, and “essential” existenceessential. Humphrey is a MI-individual. (b´) “human” means simpliciter-human, and “essential” existenceessential. Humphrey is a LI-individual. (c) “human” means human-and-existent, and “essential” absolutely essential. Humphrey is a MI-individual. (c´) “human” means human-and-existent, and “essential” absolutely essential. Humphrey is a LI-individual. (d) “human” means human-and-existent, and “essential” existenceessential. Humphrey is a MI-individual. (d´) “human” means human-and-existent, and “essential” existenceessential. Humphrey is a LI-individual. In case (a): Humphrey satisfies the modal formulas “necessarily x is human” [because k* = that H. is not (simpliciter-)human], “possibly x does not exist” [because k* ≠ that H. does not exist] and “possibly x is human and x does not exist” [because k* ≠ that H. is human and does not exist]. Therefore: Humphrey does not exist essentially, but is essentially human.

19

If Humphrey is a LI-individual, then he is also a LI*-individual, since he is certainly not a LI-world.

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In case (a´): Humphrey does not satisfy “necessarily x is human” [because there are LI-worlds at which H. has no human counterpart] and does not satisfy “possibly x is human and x does not exist” [because there is no LI-world at which H. is human but has no counterpart]. But Humphrey satisfies “possibly x does not exist” [because there are LI-worlds at which H. has no counterpart]. Therefore: Humphrey neither exists essentially, nor is essentially human. In case (b): As in case (a), Humphrey satisfies the modal formulas “necessarily x is human,” “possibly x does not exist,” and “possibly x is human and x does not exist.” But now Humphrey exists essentially [that is: existence-essentially, because k* = that H. exists and does not exist] in addition to being essentially [that is: existence-essentially] human [because we have: k* = that H. exists and is not human, which identity is entailed by the following identity: k* = that H. is not human]. In case (b´): As in case (a´), Humphrey satisfies “possibly x does not exist,” but does neither satisfy “necessarily x is human” nor “possibly x is human and x does not exist.” But now Humphrey exists essentially [that is: existence-existentially, because there is no possible world at which he exists – has a counterpart – and does not exist at that world] and is also essentially [that is: existence-essentially] human [because at every world at which H. has a counterpart – exists –, he has a human counterpart]. In case (c): Humphrey satisfies “possibly x does not exist,” but does not satisfy “necessarily x is human” and does not satisfy “possibly x is human and x does not exist.” It follows that Humphrey neither exists essentially, nor is essentially human. In case (c´): Humphrey satisfies “possibly x does not exist,” but does not satisfy “necessarily x is human” [simply because there is some possible world at which H. does not exist, which implies that there is some possible world at which H. is not human-and-existent] and does not satisfy “possibly x is human and x does not exist.” It follows that Humphrey neither exists essentially, nor is essentially human. In case (d): As in case (c), Humphrey satisfies “possibly x does not exist,” but does not satisfy “necessarily x is human” and does not satisfy “possibly x is human and x does not exist.” But now Humphrey exists essentially [i.e., existence-essentially] and is essentially [i.e., existenceessentially] human.20 20

Suppose Humphrey were not existence-essentially human-and-existent. Hence it could be the case that he exists but is not human-and-existent. But that seems clearly

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In case (d´): As in case (c´), Humphrey satisfies “possibly x does not exist,” but does not satisfy “necessarily x is human” and does not satisfy “possibly x is human and x does not exist.” But now Humphrey exists essentially [i.e., existence-essentially] and is essentially [i.e., existenceessentially] human [because there is no world at which H. exists but is not human-and-existent21]. The problem that Lewis points out in the above quotation is based on the irreconcilability of certain modal intuitions, which each have an equally strong claim on our allegiance as long as conceptual clarity has not been established. The problem disappears as soon as conceptual clarity has been established; then we no longer have all of these intuitions, but only those that are compatible with one another, and hence we are no longer bothered by their irreconcilability. (a) – (d´) are eight possible ways to obtain the desired conceptual clarity. Is there a best way among them? I favor solution (a). It accords best with the way we usually speak, and I do not believe that there is anything unsatisfactory about it at all. The objection that, according to (a), it is possible for Humphrey to be human without even existing, is sufficiently answered by pointing out that “human” is taken to mean simpliciter-human. Like other predicates, “human” can certainly be taken to have an existence-neutral sense. What is the best solution from a Lewisian, counterpartist point of view? (b´) or (d´), I believe (no saying which of the two is better), which both require, however, that essentiality be identified with existenceessentiality (i.e., x essentially Fs if, and only if, x Fs at every world at which x has a counterpart), and not with absolute essentiality (i.e., x essentially Fs if, and only if, x Fs at every world). Solutions (b´) and (d´) do not quite so well accord with the way we usually speak as does solution (a). But the objection that, according to both (b´) and (d´), Humphrey exists essentially (though not a “Necessary Being”; cf. footnote 18) is sufficiently answered by pointing out that “essential” is taken to mean existenceessential. false. It is clearly false, in view of what has been said regarding (a), if human-andexistent can be analyzed as simpliciter-human and existent. 21 Suppose there were a world at which Humphrey exists but is not human-andexistent. Hence he would have a counterpart at that world – that is, he would have a counterpart which exists at that world – but no counterpart of him at that world would be human-and-existent. This can hardly be the case, given that every counterpart of Humphrey is simpliciter-human (which is an assumption obvious in itself and certainly accepted by Lewis).

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8.3

Negative, Disjunctive, and Empty Properties

Once negative and disjunctive states of affairs are accepted – and they have been accepted in this book, and their acceptance has been defended (see Chapter 3 and Section 4.4) – it is a matter of course to allow that there are also negative and disjunctive properties. The predicate “x´ is a [replaceable by the; see below] negation of property x” is defined as follows: D50 NEGPR(x´, x) =Def PR(x´) ∧ PR(x) ∧ ∀y(F(x´, y) ≡ F(x, y)) ∧ ∀y(F(x´, y) ⊃ [x´, y] = neg([x, y])). And the predicate “x´ is a [replaceable by the; see below] disjunction of properties x and z” is defined as follows: D51 DISJPR(x´, x, z) =Def PR(x´) ∧ PR(x) ∧ PR(z) ∧ ∀y(F(x´, y) ≡ F(x, y)) ∧ ∀y(F(x´, y) ≡ F(z, y)) ∧ ∀y(F(x´, y) ⊃ [x´, y] = disj([x, y], [z, y])). The following four principles are easily provable on the basis of D50, D51, P29, etc.: ∀x∀x´∀x´´(NEGPR(x´, x) ∧ NEGPR(x´´, x) ⊃ x´ = x´´) ∀x∀z∀x´∀x´´(DISJPR(x´, x, z) ∧ DISJPR(x´´, x, z) ⊃ x´ = x´´) ∀x∀x´(NEGPR(x´, x) ⊃ NEGPR(x, x´)) ∀y(F(λoB[o], y) ≡ F(λo¬B[o], y)) ⊃ NEGPR(λoB[o], λo¬B[o]). I see no reason against postulating ∀x∀z(PR(x) ∧ PR(z) ∧ ∀y(F(x, y) ≡ F(z, y)) ⊃ ∃x´DISJPR(x´, x, z)). Regarding ∀x(PR(x) ⊃ ∃x´NEGPR(x´, x)), however, it should be noted that if it is assumed, then every universally predicable property will have a universally predicable negation. Since sets are special universally predicable properties (see Section 7.2) and since λo(o ≠ o) is certainly a set (namely, the empty set), it will follow that its negation is also a set, with everything being an element of it. But set-theorists are quite unanimous in assuming that there is no

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such set (though some of them will call λo(o = o) – or in other words: {y: y = y} – a class22). In any case, for showing that there are negative and disjunctive properties, one need not assume the two strong principles just mentioned. Even without them, it can easily be shown that both ∃x(PR(x) ∧ ∃x´NEGPR(x´, x)) and ∃x∃z(PR(x) ∧ PR(z) ∧ ∃x´DISJPR(x´, x, z)) are true. Consider λIo(o is self-identical) and λIo(o is self-different). Both entities are properties and fit the same entities, namely, all and only individuals, since λo(o is selfidentical) – the property of which λIo(o is self-identical) is the I-restriction – and λo(o is self-different) – the property of which λIo(o is self-different) is the I-restriction – certainly fit all individuals (whether or not they fit all entities). Moreover, we have ∀y(F(λIo(o is self-different), y) ⊃ [λIo(o is self-different), y] = neg([λIo(o is selfidentical), y])): Assume F(λIo(o is self-different), y), hence I(y), hence (1) [λIo(o is self-different), y] = [λo(o is self-different), y] and (1´) [λIo(o is selfidentical), y] = [λo(o is self-identical), y]. (For the justification of these steps, see Section 7.4, P33, D36, D37.) According to P30 (since λo(o is self-identical) and λo(o is self-different) – fitting all individuals – both fit y), (2) [λo(o is self-identical), y] = that y is self-identical, and (2´) [λo(o is selfdifferent), y] = that y is self-different. And we also have: (3) that y is self-different = neg(that y is self-identical) [because: that y is self-different = that (y ≠ y) = that ¬(y = y) = neg(that (y = y)) = neg(that y is self-identical); the third identity in this sequence follows according to P14]. Therefore: [λIo(o is self-different), y] =(1) [λo(o is self-different), y] =(2´) that y is selfdifferent =(3) neg(that y is self-identical) =(2) neg([λo(o is self-identical), y]) =(1´) neg([λIo(o is self-identical), y]). Therefore: [λIo(o is self-different), y] = neg([λIo(o is self-identical), y]) – which is what was to be deduced. Applying D50, we have now shown: NEGPR(λIo(o is self-different), λIo(o is selfidentical)), and therefore we have also shown: ∃x(PR(x) ∧ ∃x´NEGPR(x´, x)).

22

I prefer to regard the words “set” and “class” as synonyms, which, in itself, is a purely terminological preference. For the distinction that those set-theorists have in mind who distinguish between sets and classes could nevertheless be expressed in the following way: classes/sets that are elements of classes/sets versus classes/sets that are not elements of classes/sets. (One might call the former normal sets/classes, and the latter supersets/superclasses.) Note, however, that because of ∀x∀zF(λo(o = x), z), we can derive in the present ontological theory: ∀y∃x(SET(x) ∧ y ∈ x).

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For showing ∃x∃z(PR(x) ∧ PR(z) ∧ ∃x´DISJPR(x´, x, z)), consider λIo(o is selfidentical), λIo(o is self-different), and λIo(o is an individual) [= λIo(o is self-identical or self-different)].

Thus: since a negative property is the negation of some property x, it follows that there is at least one negative property, and since a disjunctive property is the disjunction of some properties x and z, it follows that there is at least one disjunctive property. But it might be objected that a negative property is not simply the negation of some property or other, but the negation of a positive property. – Whatever “positive property” may mean (and if our aim is to define “negative property” on the basis of “positive property,” we had better not say that a positive property is a non-negative one), we may be sure, I believe, that λIo(o is self-identical) is a positive property. Therefore the above argument also shows that there is a negative property in the envisaged stronger sense of “negative property.”

Some philosophers – notably David Armstrong (see A World of States of Affairs, and Universals and Scientific Realism) – have adopted the position that there are no negative and disjunctive properties. They presumably would be unimpressed by the above argument. Accordingly, they must have a rather narrow conception of what properties there are. To adduce another example, the Armstrongian position implies the denial that λIo(o is identical with U.M.) and λIo(o is different from U.M.) are both properties – for if it is allowed that they are both properties (fitting all and only individuals), then λIo(o is different from U.M.) is the negation of λIo(o is identical with U.M.) (and vice versa), and hence it must be allowed that there is a negative property after all. Moreover, the Armstrongian position implies also the denial that λIo(o is sexed), λIo(o is male), and λIo(o is female) are, all three of them, properties – for if it is allowed that these three items are properties (as is only natural), then λIo(o is sexed) is the disjunction of λIo(o is male) and λIo(o is female), and hence it must be allowed that there is a disjunctive property after all. Besides believing that there are no negative and disjunctive properties, Armstrongians also believe that there are no empty – non-exemplified – properties. In other words, according to the Armstrongians, all properties are exemplified. This is not yet actualism for properties, because most properties require for actuality not only that they be exemplified, but also that they be exemplified by something actual (see Section 7.6). However, one can be sure that the Armstrongians are not going to honor the distinc-

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tion between a property’s being exemplified and its being exemplified by something actual, and that they will, in consequence, hold that the former condition as much as the latter entails the actuality of the property (but compare P35). Even if P30 is left out of the picture, λo(U.M. is thinking about o at t1) – being thought about by U.M. at t1 – is certainly a property if I am thinking at t1, say, about going to bed. For then λo(U.M. is thinking about o at t1) is exemplified – is it not? – and therefore it is a property (the exemplification of λoB[o] entailing the propertyhood of λoB[o] uncontroversially, I should say). If to this is added the denial of non-exemplified – empty – properties, then it follows that λo(U.M. is thinking about o at t1) is not a property if I am not thinking about anything at t1; for in that case λo(U.M. is thinking about o at t1) does not appear to be exemplified. Therefore, whether λo(U.M. is thinking about o at t1) is a property or not depends on my thinking. This consequence seems strange, to say the least. Armstrongians will reject the example. For them, λo(U.M. is thinking about o at t1) is neither scientific nor natural nor fundamental enough to be even a candidate for propertyhood. But take any item λoB[o] that also Armstrongians would acknowledge to be a property. Is it a matter of λoB[o]’s being exemplified that it is a property? Suppose that λoB[o] – exemplified and a property – is like many properties: it is neither necessarily exemplified nor necessarily not exemplified. On the other hand, it appears to be indubitable that λoB[o] is either necessarily a property or necessarily not a property. Hence it cannot be true that both, (1), λoB[o]’s being exemplified necessarily implies its propertyhood and that, (2), λoB[o]’s propertyhood necessarily implies its being exemplified. Since it is indubitable that λoB[o]’s being exemplified necessarily implies its propertyhood, it follows that λoB[o]’s propertyhood does not necessarily imply its being exemplified. Will the Armstrongians relish this conclusion? I suppose not. Which assumption that leads to this conclusion will they, therefore, give up? I suppose they will give up the assumption that λoB[o] is necessarily a property or necessarily not a property, and will arrive at the result that λoB[o] is as contingently a property as it is contingently exemplified. But this result, too, is quite unpalatable. Here is a puzzle for Armstrongians: Suppose, for the sake of the argument, that λo(U.M. is thinking about o at t1) is a candidate for propertyhood, and suppose that all I am thinking about at t1 is the non-obtaining (non-existing) state of affairs that nothing exists. Is λo(U.M. is thinking about o at t1) – or is it not – a property? Suppose, alter-

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natively, that all that I am thinking about at t1 is this: λo(U.M. is thinking about o at t1). Once more: is λo(U.M. is thinking about o at t1) – or is it not – a property? Suppose, finally, that all that I am thinking about at t1 is λo(o is actual and a unicorn). Once again: is λo(U.M. is thinking about o at t1) – or is it not – a property?

I do not believe that the Armstrongian position is at all plausible. The best way to make sense of it is this: the Armstrongians have in mind a highly discriminatory notion of propertyhood. This notion analytically excludes negative, disjunctive, and empty properties. The analytic exclusion of empty – non-exemplified – properties leads inexorably to the result that some properties are properties only contingently. If there is an analytic equivalence between being a property and being an exemplified property, contingency of exemplification implies contingency of propertyhood.

But in view of the fact that ostensible intentional reference to negative, disjunctive, and empty properties is constantly occurring – not only outside of science but also within it –, adopting such a discriminatory notion of propertyhood seems to me insufficiently motivated, to say the least. 8.4

Haecceitism? Anti-Haecceitism?

David Lewis gives the following characterization of (purely) qualitative differences: Suppose we had a mighty language that lacked for nothing in the way of qualitative predicates, and lacked for nothing in its resources for complex infinitary constructions, but was entirely devoid of proper names for things; then the qualitative differences would be those that could be captured by descriptions in this mighty language. (On the Plurality of Worlds, p. 221.)

Haecceitism, according to Lewis (ibid.), is the doctrine that some possible worlds, w and w´, have no qualitative differences between them, but yet differ “in what they represent de re concerning some individual.” Antihaecceitism is the negation of this doctrine. Lewis regards himself as an anti-haecceitist (ibid.). He also says: [G]enuine modal realism with overlap of worlds is congenial to haecceitistic differences. There is no reason why proponents of this view – if there were any – should not accept haecceitism. (Ibid., p. 228.)

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Well, I consider myself to be a proponent of “genuine modal realism with overlap of worlds,” since possible worlds are in my eyes maximalconsistent states of affairs, and therefore entities real enough for realism, and since different possible worlds qua states of affairs certainly overlap by having individuals as shared constituents. (See Section 4.2, where I also answer Lewis’s famous objection against overlapping worlds, “the problem of accidental intrinsics.”). Is the present theory of modality haecceitistic? It needs to be pointed out that Lewis’s distinction between haecceitism and anti-haecceitism is geared to possible worlds in his sense. Lewis takes possible worlds to be spatiotemporally extended individuals, and the present or absent “qualitative differences” between them are taken to be differences in the qualities of spatiotemporally extended individuals. But the distinction is also applicable, in an appropriately analogous sense, if we are talking about possible worlds qua states of affairs. Are at least two such worlds different in what they represent de re concerning some individual while being, nonetheless, qualitatively identical, that is: while having exactly the same states of affairs that are expressible in a (purely) qualitative superlanguage (a language like Lewis’s “mighty language”) as intensionals parts? Let w and w´ be two qualitatively identical worlds (in the sense just described) and suppose the state of affairs that ∃yF(y) is an intensional part of w, where F(y) is a predicate of the qualitative superlanguage which can only be true of individuals. Hence that ∃yF(y) must also be an intensional part of w´. Since w is a maximal-consistent state of affairs, that ∃yF(y) is an intensional part of w only if, for some individual y, that F(y) is an intensional part of w. Suppose w is a maximal-consistent state of affairs, that ∃yF(y) is an intensional part of w, but there is no individual y such that that F(y) is an intensional part of w. Hence, since w is a maximal-consistent state of affairs, neg(that F(y)) is an intensional part of w, for every individual y; and therefore (according to P14): that ¬F(y) is an intensional part of w, for every individual y. Moreover, since F(y) is a predicate that can only be true of individuals, that ¬F(y) is an intensional part of w also for every non-individual y.23 Hence: for every y, that ¬F(y) is an intensional part of w. Hence Suppose, for some non-individual y´´, that ¬F(y´´) is not an intensional part of w. Hence neg(that ¬F(y´´)) is an intensional part of w (due to w being a maximalconsistent state of affairs), hence that ¬¬F(y´´) is an intensional part of w (because of P14), and hence that F(y´´) is an intensional part of w (because of EQU*, generalized 23

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CONJx∃y(x = that ¬F(y)) is an intensional part of w, and therefore, according to P16, that ∀y¬F(y) is an intensional part of w. Since ∃yF(y) is defined as ¬∀y¬F(y)24 and since according to supposition that ∃yF(y) is an intensional part of w, we also have: that ¬∀y¬F(y) is an intensional part of w; hence (according to P14) neg(that ∀y¬F(y)) is an intensional part of w. But this, finally, contradicts the consistency (the being different from k*) of the maximal-consistent state of affairs w.

Hence: for some individual y, that F(y) is an intensional part of w. By the same reasoning: for some individual y´, that F(y´) is an intensional part of w´. Suppose now that some individual y´ which is such that the state of affairs that F(y´) is an intensional part of w´ is not identical with any y which is such that the state of affairs that F(y) is an intensional part of w: ∃y´(I(y´) ∧ P(that F(y´), w´) ∧ ¬P(that F(y´), w)). Does this supposition contradict the assumed qualitative identity of w´ and w? If not, then w and w´ are different in what they represent de re concerning individual y´, although they are qualitatively identical worlds. And, indeed, it is the case that the envisaged situation does not contradict the assumed qualitative identity of w´ and w. Let [F]w be the set of all individuals y such that that F(y) is an intensional part of w, and let [F]w´ be the set of all individuals y´ such that that F(y´) is an intensional part of w´. According to the situation envisaged above, the two sets are different. Yet the difference will not show up between w and w´ with respect to the states of affairs that are expressible in the qualitative superlanguage if the following condition is fulfilled: For all predicates P[y] (complex or not) of the qualitative superlanguage, there are exactly as many entities for which F(y) ∧ P[y] is true at w as there are entities for which F(y) ∧ P[y] is true at w´, and there are exactly as many entities for which ¬F(y) ∧ P[y] is true at w as there are entities for which ¬F(y) ∧ P[y] is true at w´. And this condition is fulfilled if w and w´ are, as assumed, qualitatively identical worlds. Is haecceitism a bad thing? According to Lewis,

in the way P23 has been generalized; see the introductory remarks to Chapter 7). This can only mean that the predicate F(y) is true at w of a non-individual y´´ – contradicting the assumption that it can only be true of individuals. 24 See the definitions below P16 in Section 3.4.

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we should reject haecceitism not for any very direct reason, but rather because its intuitive advantage over the cheap substitute [that Lewis has on offer] … costs us far more trouble than it’s worth. (On the Plurality of Worlds, p. 228.)

There is not only no direct reason against haecceitism, but its intuitive advantage costs us no trouble at all if we decide to go with the present theory of modality. Therefore: within the present theory of modality, haecceitism – what haecceitism amounts to within it – is not a bad thing. 8.5

Graeme Forbes and Alvin Plantinga on Property Actualism and Quantifier Actualism

In his Languages of Possibility, p. 45, Graeme Forbes defines “property actualism to be the doctrine that necessarily for any object x, it is not possible for x to possess an unstructured property in a situation in which it also fails to exist.” Besides property actualism, there is also strict property actualism: “if [object] x does not exist at [world] w, x possesses no properties, structured or unstructured, at w.” (Ibid., p. 45.) Forbes calls this strict position also “strict predicate actualism” (ibid., p. 46) and notes that it “has been urged most strongly” by Alvin Plantinga (for instance, in Plantinga’s “Replies” in Tomberlin/van Inwagen, Alvin Plantinga). Avoiding modal operators (as is suggested by Forbes’ own formulation of strict property actualism) property actualism and strict property actualism can be put as follows: PA

For all possible worlds w, objects y, and properties x: if y does not exist at w and x is unstructured, then y does not have x [i.e., does not exemplify x] at w.

SPA For all possible worlds w, objects y, and properties x: if y does not exist at w, then y does not have x at w. Forbes claims that David Lewis “is immediately committed to predicate actualism” (Languages of Possibility, p. 72). How so? Very easily (more easily than Forbes explains matters): according to Lewis, a (possible) individual (= part of some possible world) y has a property of individuals x at a world w only if y has a counterpart at w, and according to Lewis, y’s existing at w is tantamount to y’s having a counterpart at w (see Section 8.1 above). Therefore: if an individual y does not exist at a world w, then y does not have any property of individuals x at w. This is SPA if the objects are identified with the individuals and the properties with the properties of individuals

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(otherwise, Lewis’s predicate actualism is only a logical consequence of SPA). Note that, from a Lewisian point of view, SPA is not stronger than PA, since according to Lewis all properties can be considered to be unstructured: to be simply the sets of their instances, this being for Lewis the primary way of regarding them; see On the Plurality of Worlds, p. 50, pp. 55-56. The consequences of Lewis’s predicate actualism are somewhat strange. According to Lewis, if an individual y does not exist at a world w, it doesn’t even have at w the property of being an individual – while, absolutely speaking, it does of course have the property of being an individual (simply by being an element of the set of all possible individuals; see Section 8.1).

Quantifier actualism, on the other hand, is nowhere explicitly defined in Forbes’ Languages of Possibility. But from the following remarks we can gather what Forbes intends by the designation “quantifier actualism”: [Q]uantifier actualism, which he [Plantinga] aptly expresses as the doctrine that ‘there is no property that is entailed by but does not entail existence’ (Languages of Possibility, p. 68, referring to Plantinga, “Replies,” p. 316; Plantinga himself is simply using the word “actualism,” and not the expression “quantifier actualism,” at the place where he gives the characterization cited by Forbes. Compare with it the somewhat different – apparently weaker – characterization of actualism by Plantinga that can be gathered from the quotation on p. 29 of this book.)

The thesis of quantifier actualism, therefore, can be put in the following way: QA For all properties x: if, for every possible world w, every object that exists at w has x at w, then, for every possible world w´, every object that has x at w´ exists at w´. It may not be immediately evident why this thesis is named “quantifier actualism” by Forbes. Forbes gives no explanation. I herewith present an explanation of the name which seems likely to me. An actualist understanding of objectual quantification in modal predicate logic is perspicuously expressed in the following way: (∀yB[y] ≡ ∀y(A(y) ⊃ B[y])) (the matching schema for “∃” is an easy consequence of the schema for “∀” on the basis of ∃y =Def ¬∀y¬). But (∀yB[y] ≡ ∀y(A(y) ⊃ B[y])) is logically equivalent to ∀yA(y),25 and, significantly, it is also logically equivalent to ∀y(A(y) (∀yB[y] ≡ ∀y(A(y) ⊃ B[y])) → (∀yA(y) ≡ ∀y(A(y) ⊃ A(y))) → ∀yA(y); ∀yA(y) → (∀yB[y] ≡ ∀y(A(y) ⊃ B[y])). (The deductions make use only of schemainstantiation and of absolutely basic principles of modal predicate logic.) 25

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⊃ B[y]) ⊃ ∀y(B[y] ⊃ A(y))26 – which is nothing else than the way in which one would express QA if one expressed it with objectual quantifiers, but without explicit quantification over properties and possible worlds, taking “to exist” in the sense of “to be actual.” Thus, what is called “quantifier actualism” by Forbes is, basically, just the adoption of an actualist understanding of objectual quantification in modal predicate logic. The designation “quantifier actualism,” though (now) understandable, is a somewhat misleading designation for this position, as, by the way, is the designation “property actualism” for the positions PA and SPA.

If PA, SPA, and QA are to have anything to do with actualism, then “to exist” in them must be understood to mean as much as “to be actual,” and “to be actual” can replace “to exist” at all places of its occurrence in PA, SPA, and QA without changing the meaning. It must be pointed out that Plantinga has explicitly repudiated this interpretation of “to exist” in theses of actualism. For him, “to exist” does not mean as much as “to be actual” in a thesis of actualism (for further explanation of this prima facie astonishing position, see Section 2.1). But if “to exist” is not taken to mean as much as “to be actual,” then its only remaining possible ontological interpretation (i.e., interpretation of it that takes it to express a property) is to let it mean as much as “to be selfidentical” (see Section 1.4). This interpretation, however, renders at once pointless the entire discussion regarding actualism. For any of the proposed theses of actualism, when formulated (like PA, SPA, and QA) by using the word “exist,” must be uncontroversially true if “to exist” is taken to mean as much as “to be self-identical” in them. In order to preserve the ontological interest of theses of actualism, I therefore ignore Plantinga’s interpretation of them, and treat his actualistic statements as if “to exist” meant as much as “to be actual” in them.

But – the interpretation urged in the statement preceding the above note in place – what do PA, SPA, and QA have to do with any actualism properly speaking, that is: with any claim that all entities of a certain kind are actual? There should be a substantial connection with some such claim; for otherwise the designation “actualism” for them would be out of place. Well, according to the present theory, the three principles have the following consequences. Consider λo(o is an object), which is a property according to P30. It is safe to assume that this property fits – is predicable (∀yB[y] ≡ ∀y(A(y) ⊃ B[y])) → (∀y(B[y] ⊃ A(y)) ≡ ∀y(A(y) ⊃ (B[y] ⊃ A(y)))) → ∀y(B[y] ⊃ A(y)) → ∀y(A(y) ⊃ B[y]) ⊃ ∀y(B[y] ⊃ A(y)); ∀y(A(y) ⊃ B[y]) ⊃ ∀y(B[y] ⊃ A(y)) → ∀y(A(y) ⊃ y = y) ⊃ ∀y(y = y ⊃ A(y)) → ∀yA(y) → (∀yB[y] ≡ ∀y(A(y) ⊃ B[y])). (The deductions make use only of schemainstantiation and of absolutely basic principles of modal predicate logic.) 26

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of – every object (at least if “object” is not taken to be a synonym for “entity,” which does seem to be in accordance with Forbes’ understanding of the term “object”27). It is also safe to assume that this property is unstructured. For where is its structure? Note that it is problematic to conclude that a property is structured from the fact that its name is structured. “λo(o is an object)” is certainly structured, but that doesn’t mean that λo(o is an object) is also structured. For Forbes, λx◊(∃y)(x = y) – or in the present notation: λo◊∃y(o = y) – is a structured property (see Languages of Possibility, p. 68), and the basis of this judgment seems to be that the property’s name, “λx◊(∃y)(x = y),” is structured. But such reasoning is questionable. For the property λx◊(∃y)(x = y) is just the property of being (in the widest sense), and that property does not seem to be structured. (Forbes himself admits a “certain uneasiness” regarding the distinction between unstructured and structured properties; ibid., p. 46.)

Applying PA or SPA to λo(o is an object), we have: PA1/SPA1 For all possible worlds w, objects y: if y is not actual at w, then y does not exemplify λo(o is an object) at w. Or in other words: For all possible worlds w, objects y: if y exemplifies λo(o is an object) at w, then y is actual at w. And applying QA to λo(o is an object), we have: QA1

If, for every possible world w, every object that is actual at w exemplifies λo(o is an object) at w, then, for every possible world w´, every object that exemplifies λo(o is an object) at w´ is actual at w´.

27

It is unclear whether the word “object” is used by Plantinga as a logical equivalent of the rather technical term “entity,” or has a more restricted sense for him than this latter term. He does use the word “object” in a very comprehensive way; see “SelfProfile,” p. 88. In the present context of discussing actualism, it is, however, to the advantage of both Plantinga and Forbes to interpret “object” in a more restricted sense than the word “entity” has – for the present purposes, it does not matter in which sense precisely, as long as all individuals turn out to be objects.

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But since it is true that every object exemplifies λo(o is an object) at every possible world,28 we obtain both from PA1/SPA1 and QA1: For all possible worlds w, objects y: y is actual at w. Thus necessary actualism for objects (and therefore also actualism for objects simpliciter) is seen to be implied by (so-called) property actualism, strict property actualism, and (so-called) quantifier actualism. Therefore, these three actualisms cannot be more plausible, in the end, than necessary actualism for objects – which does not seem plausible at all. (Is not your brain an object that is not actual at some possible world?) Necessary actualism for objects must not be confused with the following thesis, which is weaker and much more plausible than necessary actualism for objects: For all possible worlds w, objects y: if y belongs to the domain of w, then y is actual at w. The matter is treated in detail in Section 7.7.

But Forbes has offered an argument for property actualism, and Plantinga even an argument for strict property actualism (which Plantinga calls “serious actualism”; see “Replies,” p. 316). Let us look at these arguments and see what is wrong with them. Judging from Plantinga’s description of what he calls “actualism,” according to which it is the doctrine that “there is no property that is entailed by but does not entail existence” (“Replies,” p. 316), we may take it that Plantinga’s “actualism” can be formulated as the thesis QA above, which can be rendered as follows in the terminology of the present theory:

28

This need not be assumed on the strength of intuition. Positing an additional first principle, it can also be proven. Suppose y is an object, w a possible world, and assume for reductio: ¬EXM(y, λo(o is an object), w). Hence (according to D35, D16): ¬P([λo(o is an object), y], w). Since λo(o is an object) fits y [because λo(o is an object) fits every object – see above – and because y is supposed to be an object], we have (according to P30): [λo(o is an object), y] = that (y is an object). Therefore: ¬P(that (y is an object), w). But “y´ is an object” is an ontological predicate, and therefore we have in analogy to P22, P32: ∀y´(y´ is an object ⊃ t* = that (y´ is an object)) ∧ ∀y´(¬(y´ is an object) ⊃ k* = that (y´ is an object)). Hence: t* = that (y is an object) [because y is supposed to be an object; “y” – being a variable – is a referentially stable term and can be safely substituted into a “that”-context]. Therefore finally: ¬P(t*, w) – which contradicts the theorem ∀z(S(z) ⊃ P(t*, z)) in view of S(w) [S(w) being a consequence of the supposition that w is a possible world].

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QA* ∀x[PR(x) ∧ ∀w∀y(MC(w) ∧ Obj(y) ∧ EXM(y, λoA(o), w) ⊃ EXM(y, x, w)) ⊃ ∀w∀y(MC(w) ∧ Obj(y) ∧ EXM(y, x, w) ⊃ EXM(y, λoA(o), w))]. Plantinga claims (“Replies,” p. 319) that “serious actualism” (= strict property actualism) is a consequence of “actualism” (= quantifier actualism) and offers a somewhat unperspicuous deduction (making use of counterfactual conditionals regarding possible worlds, mixing the possible-worldsidiom with the idiom of necessity and impossibility). I prefer to ask: how does the above statement, QA*, imply SPA*

∀x[PR(x) ⊃ ∀w∀y(MC(w) ∧ Obj(y) ∧ EXM(y, x, w) ⊃ EXM(y, λoA(o), w))],

which is a (contrapositive) rendering of SPA – Plantinga’s “serious actualism” – in the terminology of the present theory? QA* implies SPA* in the following manner: The reverse implication – of QA* by SPA* – is clear enough: ∀x[PR(x) ⊃ ∀w∀y(MC(w) ∧ Obj(y) ∧ EXM(y, x, w) ⊃ EXM(y, λoA(o), w))] logically implies (in a trivial manner) ∀x[PR(x) ∧ ∀w∀y(MC(w) ∧ Obj(y) ∧ EXM(y, λoA(o), w) ⊃ EXM(y, x, w)) ⊃ ∀w∀y(MC(w) ∧ Obj(y) ∧ EXM(y, x, w) ⊃ EXM(y, λoA(o), w))]. Now consider the property λoObj(o). It is provably true (see footnote 28): ∀w∀y(MC(w) ∧ Obj(y) ∧ EXM(y, λoA(o), w) ⊃ EXM(y, λoObj(o), w)). Hence it follows on the basis of QA*: ∀w∀y(MC(w) ∧ Obj(y) ∧ EXM(y, λoObj(o), w) ⊃ EXM(y, λoA(o), w)), and therefore (again on the basis of what is proven in footnote 28): ∀w∀y(MC(w) ∧ Obj(y) ⊃ EXM(y, λoA(o), w)), hence (purely on the basis of logic): ∀x[PR(x) ⊃ ∀w∀y(MC(w) ∧ Obj(y) ∧ EXM(y, x, w) ⊃ EXM(y, λoA(o), w))]. Thus QA* implies SPA*. [The implication of EXM(y, λoObj(o), w) by Obj(y) [presupposing MC(w)], used above, is seen to be true in the following manner (as regards the main deductive steps): Obj(y) → P(that Obj(y), w) → P([λoObj(o), y], w) → EXM(y, λoObj(o), w). For details, see footnote 28.]

Though Plantinga’s “actualism” implies his “serious actualism” (and vice versa), though, in other words, quantifier actualism implies strict property actualism (and vice versa), one can hardly see in this fact an argument for strict property actualism. And in fact, Plantinga does not really want to show that “serious actualism” is true, but only that everyone who – like him – believes that “actualism” is true must, in reason, also believe

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that “serious actualism” is true (see “Replies,” p. 318). For everyone who is not already a quantifier actualist (hence also: for everyone who is not already an actualist regarding objects, since we have seen above that quantifier actualism implies actualism for objects), Plantinga’s argument for strict property actualism – if really taken as an argument for this position – is just question-begging. Strict property actualism – SPA* – not only implies necessary actualism for objects (see above), but, obviously, is implied by it, that is, by NAO*

∀w∀y(MC(w) ∧ Obj(y) ⊃ EXM(y, λoA(o), w)),

which is the way necessary actualism for objects is formulated in the terminology of the present theory. Necessary actualism for objects does not appear to be true (as was already indicated above). Hence strict property actualism does not appear to be true either. Consider also a thesis that is stronger than strict property actualism; it is obtained from SPA by replacing the word “objects” in that statement by the word “entities,” or by omitting “ ∧ Obj(y)” from SPA*; it is the assertion that, necessarily, any exemplification of any property implies the existence – that is, the actuality – of the exemplifier. But the fact that some exemplification of some property does not imply the actuality of the exemplifier is precisely the reason why the actuality-principle for properties, P35, was formulated in the following manner at the beginning of Section 7.6: (i) ∀x(PR(x) ∧ ∃y(EXM(y, x) ∧ A(y)) ⊃ A(x)), (ii) ∀x(PR(x) ∧ ¬∃yEXM(y, x) ⊃ ¬A(x)), and not simply, and more strongly, like this: ∀x(PR(x) ⊃ (∃yEXM(y, x) ≡ A(x))). The latter formulation logically implies the former, but not vice versa. However, the former formulation would be equivalent to the latter if ∀x∀y(PR(x) ∧ EXM(y, x) ⊃ A(y)) could be presupposed. But ∀x∀y(PR(x) ∧ EXM(y, x) ⊃ A(y)) certainly cannot be presupposed, because it is false. Consequently, ∀x[PR(x) ⊃ ∀w∀y(MC(w) ∧ EXM(y, x, w) ⊃ EXM(y, λoA(o), w))] (the modification of SPA* envisaged above) is also false, that is: not necessarily any exemplification of any property im-

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plies the actuality – the existence, in the sense here relevant – of the exemplifier. For [1], ∀x[PR(x) ⊃ ∀w∀y(MC(w) ∧ EXM(y, x, w) ⊃ EXM(y, λoA(o), w))], implies [2], ∀x∀y(PR(x) ∧ EXM(y, x) ⊃ A(y)) (and hence the falsity of [2] entails the falsity of [1]): Since we have MC(w*) [according to the mereology of states of affairs], we obtain from [1]: ∀x∀y(PR(x) ∧ EXM(y, x, w*) ⊃ EXM(y, λoA(o), w*)). In this, EXM(y, x, w*) reduces to (a) P([x, y], w*), and EXM(y, λoA(o), w*) to (b) P([λoA(o), y], w*), according to D35, D16. Regarding (a): P([x, y], w*) is equivalent to EXM(y, x): Assume P([x, y], w*). Hence S([x, y]) [according to P0]. Hence A([x, y]) [following from P([x, y], w*) and S([x, y]) according to the Actuality Principle for States of Affairs: ∀x(S(x) ⊃ (A(x) ≡ P(x, w*))); see Section 3.3]. Hence O([x, y]) [from S([x, y]) and A([x, y]), according to D6]. Hence EXM(y, x) [according to D20]. Conversely, assume EXM(y, x). Hence [according to D20, D6] S([x, y]) and A([x, y]). Hence [according to the Actuality Principle for States of Affairs] P([x, y], w*). Regarding (b): P([λoA(o), y], w*) is equivalent to A(y): We have just seen that P([λoA(o), y], w*) is equivalent to EXM(y, λoA(o)). And EXM(y, λoA(o)) is equivalent to A(y), according to the restricted principle of property-exemplification: ∀y(F(λoA[o], y) ⊃ (EXM(y, λoA[o]) ≡ A[y])) (see Section 7.4), because λoA(o) is universally predicable (fits every entity predicatively). The derivation of [2] – ∀x∀y(PR(x) ∧ EXM(y, x) ⊃ A(y)) – from [1] – ∀x[PR(x) ⊃ ∀w∀y(MC(w) ∧ EXM(y, x, w) ⊃ EXM(y, λoA(o), w))] – is now complete.

Why is ∀x∀y(PR(x) ∧ EXM(y, x) ⊃ A(y)) false? Because being at t0 thought of by U.M. (cf. Section 7.6) is a property that is exemplified by the non-actual state of affairs that nothing is at t0 thought of by U.M. That state of affairs is non-actual (according to D6, P10) because, in fact, U.M. thinks of something at t0 (t0 being appropriately chosen): the state of affairs that nothing is at t0 thought of by U.M. – which, in turn, means that the property being at t0 thought of by U.M. is exemplified by that state of affairs (according to the restricted principle of property-exemplification, since the property we are talking about certainly fits the state of affairs we are talking about). It is true that strict property actualism – SPA* – might still be true although ∀x∀y(PR(x) ∧ EXM(y, x) ⊃ A(y)) and (therefore) ∀x[PR(x) ⊃ ∀w∀y(MC(w) ∧ EXM(y, x, w) ⊃ EXM(y, λoA(o), w))] are false. But SPA* can only be upheld if the non-actual state of affairs that nothing is at t0 thought of by U.M. is not an object. However, it does seem to be an object, does it not?

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But I do not believe it rational to hold on to strict property actualism. To repeat: strict property actualism implies necessary actualism for objects, the thesis NAO*; but NAO* is false – evidently so, I should say. Let us now consider Forbes’ argument for property actualism, which (compare PA with SPA) is a logically weaker position than strict property actualism. The argument goes as follows (see Languages of Possibility, p. 47): (1) [Object] α does not exist at [world] w, yet possesses the unstructured property P at w [assumption for reductio]. (2) “α is P” is true at w [from (1)]. (3) There obtains at w an atomic fact to which the proposition that α is P corresponds [from (2)]. (4) is an atomic fact that obtains at w [from (3)]. (5) α is a constituent of [invoked principle; Forbes’ principle 3]. (6) [For all worlds w,] An atomic fact cannot obtain at w if one of its constituents does not exist at w [invoked principle; Forbes’ principle 4]. (7) α exists at w [from (4), (5), and (6)] – contradicting (1). Discussion: The step from (1) to (2) is unproblematic, and so is the step from (2) to (3) (although, certainly, not everybody will agree). The step from (3) to (4) is based on a substantial assumption: The proposition that α is P corresponds to exactly one atomic fact: the ordered pair . I will not here reject this assumption, but will accept it for the sake of the argument, although I do not in fact believe that, for all objects α and unstructured properties of objects P predicable of α, the atomic fact that corresponds to the proposition that α is P is the ordered pair . I do not draw in question the presently operative notion of atomic fact. But, first, I prefer to avoid speaking of a fact – which, in the proper sense of the word, is an obtaining state of affairs – when, in fact, all that is generally meant is a state of affairs (obtaining or not). Second, I do not believe that the atomic state of affairs that corresponds to the proposition that α is P is, in every case of an object α and an unstructured property P predicable of α, the ordered pair . Rather, that atomic state of affairs is the state of affairs [P, α]. (Since P fits α predicatively, [P, α] must be a state

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of affairs – according to D17.) On the basis of what has been said in Chapter 7, it should be clear that [P, α] cannot always be identified with . But for a highly restricted set of objects α´ and properties P´ predicable of α´, there is presumably a natural one-to-one correspondence between entities [P´, α´] and entities .

With the above substantial assumption in place, (4) is a straightforward consequence of (3). (5), in turn, is entirely unproblematic, and (7) does indeed follow logically from (4), (5), and (6). Therefore, the above argument would be correct if (6) were correct. But (6) is not correct. Consider the property λoObj(o) – the property of being an object; if this property is not unstructured, I do not know what an “unstructured property” is. Consider an object α´, and consider the fact ; this fact is certainly atomic, since λoObj(o) is an unstructured property. Suppose α´ does not exist at some world w´ (cf. (1) in Forbes’ argument); for example, I myself am certainly such an object. Hence is an atomic fact that obtains at w´, although one of the constituents of that fact – namely, α´ – does not exist at w´. Therefore, (6) stands refuted. Forbes does not address this objection to his argument for property actualism. What might be held against the objection? One cannot consider the supposition that α´ is an object that does not exist at some world w´ to be implausible; that supposition is entirely plausible. One cannot deny that is an atomic fact (an atomic state of affairs, properly speaking). All that one can do is to deny that obtains at w´ – to deny it precisely on the strength of α´’s not existing at w´. This move is tantamount to assuming that objecthood implies existence with logical necessity. But this is doubtful, since those that believe in non-existent objects may be mistaken, but they do not seem to make a mistake that belongs to the same order of magnitude as believing in married bachelors. As an introduction to his general argument for property actualism, Forbes asserts that there are many unstructured analytically existence[actuality]-entailing properties (Languages of Possibility, pp. 46-47). But there do not seem to be as many such properties as he thinks there are, even leaving aside the fact that it does not seem to be correct that all unstructured properties are analytically existence-entailing properties. Forbes’ (supposedly uncontroversial) examples of unstructured analytically existenceentailing properties (ibid., p. 47) are being material – λo(o is material) – and being spatiotemporal – λo(o is spatiotemporal). But it certainly seems to be possible that an object be material and spatiotemporal and non-existent. Think of Lewis’s non-actual possible worlds, and closer to home: if the sun were not actual, it would still be mate-

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rial and spatiotemporal, would it not? Of course, the property of being material and existent – λo(o is material and o is actual) – and the property of being spatiotemporal and existent – λo(o is spatiotemporal and o is actual) – are analytically existenceentailing properties (trivially so); but they do not seem to be unstructured. Analytically existence-entailing properties are also being spatiotemporally located and being made of matter; but they, too, do not seem to be unstructured. Forbes apparently believes – see ibid. – that the two last-mentioned properties, being spatiotemporally located and being made of matter, are the same properties as the two aforementioned ones, being spatiotemporal and being material. But this is far from clear. However, there does seem to be a clear case of an unstructured analytically existence-entailing property: it is the property of being conscious, as Descartes discovered. (Note the contrast: “If the sun did not exist, it would still be material” seems true or at least not clearly false; but “If I did not exist, I would still be conscious” is clearly false.)

Moreover, one can, in fact, strengthen the above objection and formulate it for the property λo(o is an entity) instead of the property λoObj(o). The outcome is: is an atomic fact that obtains at w´, although one of the constituents of that fact – α´ – does not exist at w´. Thus the refutation of (6) is repeated, only in a somewhat different key. In reaction, one might claim that being an entity implies existence with logical necessity. If this is true, then, given that α´ does not exist at w´, it follows inexorably that does not obtain at w´ – the confutation of the refutation of (6). And is it not obviously true that being an entity implies existence with logical necessity? That depends on what is meant by “existence” and “to exist.” As was pointed out in Section 1.4, there are two ontological meanings of existence. Only one of these two meanings is relevant here, where we are talking about actualisms: it is the meaning of “existence” according to which that word is synonymous with “actuality,” and therefore the verb “to exist” synonymous with “to be actual.” Given this interpretation of the claim in question, it is not at all obvious that being an entity implies existence with logical necessity, for it is not at all obvious that being an entity implies actuality with logical necessity. On the contrary, there are forceful apparent counterexamples to the latter claim: the state of affairs that U.M. is 200 cm tall at t0 is certainly an entity, but it is not an actual entity (since it is not an obtaining state of affairs). In desperation, one might claim, in order to save (6) from refutation, that λo(o is an entity) is not a property: this would keep from being a fact (i.e., state of affairs), and a fortiori from being an atomic fact that obtains at w´. The rationale for such a claim might be that

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if λo(o is an entity) were a property, it would apply to absolutely everything – and hence be prone to generate antinomies. But one can well hold that λo(o is an entity) is a property without holding automatically that is applies to absolutely everything. Indeed, according to the present theory, λo(o is an entity) is a property, but it applies to everything only if it fits everything predicatively, and that it fits everything predicatively is far from being obviously true. But nevertheless one can always find an α´ that does not exist at some possible world w´, and which is such that λo(o is an entity) fits it and hence applies to it. Thus there is no obstacle at all to holding that is a w´-obtaining fact, and there is also no obstacle to holding that it is an atomic such fact, since λo(o is an entity) is unstructured if any property is unstructured. One cannot conclude from the fact that λo(o is an entity) is, with logical necessity, co-extensional with λo∃y(o = y) that λo(o is an entity) is a structured property. For, for every property P, one can find a property λoB[o], where B[x] is a complex predicate, such that P is, with logical necessity, co-extensional with λoB[o].

Thus, Forbes’ argument for property actualism does not withstand scrutiny. In fact, it loses all semblance of plausibility as soon as one replaces in it “does not exist at w” and “exists at w” by, respectively, “is not actual at w” and “is actual at w” – which is a replacement that is entirely sanctioned by the purpose of the argument: to show that property actualism is true. For property actualism – not to be confused with actualism for properties (see Section 7.6) – would not have anything to do with actualism, and the designation “property actualism” would be a misnomer for it, if “to exist” were not synonymous with “to be actual” in the formulation of property actualism (i.e., in PA). In consequence, “to exist” must also be synonymous with “to be actual” in Forbes’ argument for property actualism; in particular, “to exist” and “to be actual” must be synonymous in (6) and can replace each other there without change of meaning. Consider, then, a contingent fact p. Hence there is at least one world w such that the atomic fact that p is possible obtains at w, while p – which is a constituent of that fact – does not obtain at w, that is: is not actual at w, that is: does not exist at w. Therefore (once again): (6) is false. One might object that that p is possible is not an atomic fact. But, from the Forbesian point of view, it can certainly be made to look like an atomic fact. Is not, from the Forbesian point of view, identical with the fact that p is possible, and is not λo(o is possible) an unstructured property? What

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else can be needed, from the Forbesian point of view, for the state of affairs that p is possible being an atomic fact? – I leave it at that. Forbes himself sanctions the equivalence of “obtains” and “exists” for facts (see Languages of Possibility, p. 47). He observes (ibid.) that there is, in contrast, no equivalence between “true” and “exists” for propositions. But the matter is more intricate. The truth of propositions and the obtaining of facts (or states of affairs, as one should really say) can be regarded as being entirely parallel to each other as soon as one distinguishes the two ontological meanings of existence (see Section 1.4). If “x exists” merely means as much as “x = x” (or “∃y(x = y)”; it does not matter logically), then there is as little equivalence between “obtains” and “exists” as there is between “true” and “exists”: then there are facts that do not obtain29 but nevertheless exist (in the sense of being self-identical, identical with something), just as there are propositions that are not true but nevertheless exist (in the indicated sense). If, however, “x exists” means as much as “x is actual,” then there is as much equivalence between “true” and “exists” regarding propositions as there is between “obtains” and “exists” regarding facts, provided it is accepted that truth is the actuality of propositions, just like obtaining is the actuality of facts (or states of affairs).

29

Widespread ontological usage does not acknowledge that “facts that do not obtain” is a self-contradictory phrase. Most ontologists see no problem in regarding the following statement as true: “It is a non-obtaining fact that George W. Bush is twenty years old in 2003.” This means that the word “fact” has, for many people, adopted the meaning of the (perhaps somewhat awkward) expression “state of affairs,” so that the two expressions are now synonyms for them. The described usage does not contribute to clarity, but there it is.

360

Bibliography Adams, E. W.: The Logic of Conditionals, Inquiry 8 (1965), 166-197. Adams, E. W.: Probability and the Logic of Conditionals, in Aspects of Inductive Logic, edited by J. Hintikka and P. Suppes, Amsterdam: North-Holland 1966, 265-316. Antipopper, E.: The Logic of Scientific Non-Discovery, Regensburg: Scepsis 2002. Armstrong, D. M.: Universals and Scientific Realism, 2 volumes, Cambridge: Cambridge University Press 1978. Armstrong, D. M.: A World of States of Affairs, Cambridge: Cambridge University Press 1997. Bealer, G.: Modal Epistemology and the Rationalist Renaissance, in Conceivability and Possibility, edited by T. S. Gendler and J. Hawthorne, Oxford: Clarendon Press 2002, 71-125. Bonevac, D.: Deduction. Introductory Symbolic Logic, 2nd edition, Oxford: Blackwell 2003. Chalmers, D. J.: The Conscious Mind, paperback edition, New York/Oxford: Oxford University Press 1997. Chalmers, D. J.: Does Conceivability Entail Possibility?, in Conceivability and Possibility, edited by T. S. Gendler and J. Hawthorne, Oxford: Clarendon Press 2002, 145-200. Chihara, Ch. S.: The Worlds of Possibility. Modal Realism and the Semantics of Modal Logic, paperback edition, Oxford: Clarendon Press 2001. Ernst, B.: Das verzauberte Auge, part III of the volume Unmögliche Welten, Cologne: Taschen 2002. Fara, M./Williamson, T.: Counterparts and Actuality, Mind 114 (2005), 1-30. Forbes, G.: Languages of Possibility, Oxford/New York: Blackwell 1989. Gendler, T. S./Hawthorne, J. (eds.): Conceivability and Possibility, Oxford: Clarendon Press 2002. Hughes, G. E./Cresswell, M. J.: An Introduction to Modal Logic, paperback edition, London: Methuen 31974. Husserl, E.: Logische Untersuchungen [Logical Investigations], vol. II, part 1, Tübingen: Niemeyer 1968. Jackson, F.: On Assertion and Indicative Conditionals, in Conditionals, edited by F. Jackson, Oxford: Oxford University Press 1991, 111-135. Kratzer, A.: Conditional Necessity and Possibility, in Semantics from Different Points of View, edited by R. Bäuerle, U. Egli, and A. von Stechow, Berlin/Heidelberg/New York: Springer 1979, 117-147. Kratzer, A.: Partition and Revision: The Semantics of Counterfactuals, Journal of Philosophical Logic 10 (1981), 201-216.

Bibliography

Leibniz, G. W.: Der Briefwechsel mit Antoine Arnauld [Letters to Antoine Arnauld], French and German, edited and translated by R. Finster, Hamburg: Meiner 1997. Lewis, D.: Counterfactuals, Oxford: Blackwell 1973. Lewis, D.: Anselm and Actuality, in Philosophical Papers, vol. I, New York/Oxford: Oxford University Press 1983, 10-25. Lewis, D.: Counterpart Theory and Quantified Modal Logic (with postscripts), in Philosophical Papers, vol. I, New York/Oxford: Oxford University Press 1983, 26-46. Lewis, D.: On the Plurality of Worlds, Oxford: Blackwell 1986. Lewis, D.: Counterfactuals and Comparative Possibility, in Philosophical Papers, vol. II, New York/Oxford: Oxford University Press 1986, 3-31. Lewis, D.: Probabilities of Conditionals and Conditional Probabilities (with postscript), in Philosophical Papers, vol. II, New York/Oxford: Oxford University Press 1986, 133-156. Lewis, D.: Causation (with postscripts), in Philosophical Papers, vol. II, New York/Oxford: Oxford University Press 1986, 159-213. Lewis, D.: Events, in Philosophical Papers, vol. II, New York/Oxford: Oxford University Press 1986, 241-269. Lewis, D.: Ordering semantics and premise semantics for counterfactuals, in Papers in Philosophical Logic, Cambridge: Cambridge University Press 1998, 7796. Meixner, U.: Ontologically Minimal Logical Semantics, Notre Dame Journal of Formal Logic 36 (1995), 279-298. Meixner, U.: Ereignis und Substanz. Die Metaphysik von Realität und Realisation [Event and Substance. The Metaphysics of Reality and Realization], Paderborn: Schöningh 1997. Meixner, U.: Axiomatic Formal Ontology, Dordrecht: Kluwer 1997. Meixner, U.: Existenz und Modalität [Existence and Modality], in Neue Ontologie und Metaphysik [New Ontology and Metaphysics], edited by R. Hüntelmann and E. Tegtmeier, St. Augustin: Academia 2000, 43-67. Meixner, U.: Theorie der Kausalität [The Theory of Causality], Paderborn: mentis 2001. Meixner, U.: The Two Sides of Being. A Reassessment of Psycho-Physical Dualism, Paderborn: mentis 2004. Meixner, U.: Causation in a New Old Key, Studia Logica 76 (2004), 343-383. Meixner, U.: Laws of Nature – A Skeptical View, in Nature’s Principles, edited by J. Faye et al., Dordrecht: Kluwer 2005, 245-254. Plantinga, A.: The Nature of Necessity, Oxford: Clarendon 1974. Plantinga, A.: Self-Profile, in Alvin Plantinga, edited by J. E. Tomberlin and P. van Inwagen, Dordrecht/Boston/Lancaster: Reidel 1985, 3-97. Plantinga, A.: Replies, in Alvin Plantinga, edited by J. E. Tomberlin and P. van Inwagen, Dordrecht/Boston/Lancaster: Reidel 1985, 313-396.

362

Bibliography

Rosenberg, J. F.: Was epistemische Externalisten vergessen [What Epistemic Externalists Forget], in Argument und Analyse, edited by C. U. Moulines and K.-G. Niebergall, Paderborn: mentis 2002, 13-31. Stalnaker, R.: Indicative Conditionals, in Conditionals, edited by F. Jackson, Oxford: Oxford University Press 1991, 136-154. Stalnaker, R.: What Is It Like to Be a Zombie?, in Conceivability and Possibility, edited by T. S. Gendler and J. Hawthorne, Oxford: Clarendon Press 2002, 385-400. van Fraassen, B. C.: The Only Necessity Is Verbal Necessity, The Journal of Philosophy 74 (1977), 71-85. van Fraassen, B. C.: Essence and Existence, in Studies in Ontology, edited by N. Rescher, American Philosophical Quarterly Monograph No. 12, Oxford: Blackwell 1978, 1-25. van Inwagen, P.: Modal epistemology, in: P. van Inwagen, Ontology, Identity, and Modality. Essays in Metaphysics, Cambridge: Cambridge University Press 2001, 243-258. Wittgenstein, L.: Tractatus Logico-Philosophicus, German and English, translated by C. K. Ogden, London: Routledge and Kegan Paul 81960. Yablo, S.: Is Conceivability a Guide to Possibility?, Philosophy and Phenomenological Research 53 (1993), 1-39.

363

Index Definitions, Principles, Rules actuality principle for properties 294, 354 Actuality Principle for States of Affairs 75, 78 all-instantiation 98-99, 102, 267 analysis (I) and (II) of counterfactuals 160, 165, 168-169, 172 Central Principle of Essence 268-270 central theorem 228, 241 central theorem regarding maximalconsistent PRITs 289, 315 conditions for bases of acceptable utterances of conditionals 170, 173 D* [that A] 79-81, 90 D1 [CONJxA[x]] 67 D2 [t*] 67 D3 [k*] 67 D4 [M(z)] 68 D5 [T(z)] 68 D6 [O(x)] 69 D7 [w*] 69 D8 [conj(x, y)] 69 D9 [QA(x)] 69 D10 [EL(x)] 70 D11 [neg(x)] 71 D12 [DISJxA[x]] 71 D13 [disj(x, y)] 71 D14 [QC(x)] 72 D15 [MC(x)] 72 D16 [O(x, y)] 94 D17 [F(x, y)] 252 D18 [PR(x)] 252 D19 [PRI(x)] 252 D20 [EXM(y, x)] 252 D21 [PRE1(x)] 255 D22 [EXME1(y, x)] 255 D23 [SET(x)] 256

D24 [y ∈ x] 256 D25 [PRIE1(x)] 256 D26 [SETI(x)] 256 D27 [y ∈I x] 257 D28 [PREn(x)] 258 D29 [EXMEn(y, x)] 258 D30 [ESSUn(x, y)] 259 D31 [ESSIn(x, y)] 260 D32 [PPR(x´, x)] 260 D33 [exmo(x)] 266 D34 [λEoA[o]] 266 D35 [EXM(y, x, w)] 270 D36 [ires(x)] 276 D37 [λIoA[o]] 276 D38 [EXMAEn(y, x)] 280 D39 [EXMAE(y, x)] 281 D40 [ESSUA(x, y)] 283 D41 [PRIT(x)] 288 D42 [PRITMC(x)] 288 D43 [nc(y, w)] 291 D44 [L-I(m)] 293 D45 [k(m), w(m), li(y, w)] 292, 294 D46 [L-EXM(m, x)] 294 D47 [PRA1(x)] 296 D48 [PRA2(x)] 296 D49 [icom(x1)] 300 D50 [NEGPR(x´, x)] 341 D51 [DISJPR(x´, x, z)] 341 definitions of modal expressions 63 EQU 80-81 EQU* 81, 91, 98, 250 existential-instantiation 267 Identity Principle for Individuals 284, 287 incorporation theorem 315 P0 66-67 P1 66-67 P2 66-67 P3 66-68, 261, 284

Index

P4 66-67, 249 P5 66, 69, 70 P6 67, 71, 249 P7 67, 69, 78 P8 67, 78, 89, 130 P9 67, 69, 78, 88-89 P10 67, 69, 136, 249 P11 74, 77 P12 75, 78 P13 79, 119, 249 P14 79, 249 P15 79, 250 P16 79, 250 P17 83 P18 83, 162 P18b 162, 267 P19 83, 102-103, 152 P20 83 P21 83 P22 90 P23 91, 250 P24 96, 156 P25 251 P26 251 P27 251 P28 251 P29 254, 284 P30 264 P31 266 P32 268 P33 276 P34 288 P35 294, 354 P36 300 principles for “that” 79, 83, 96 principle of assertability of indicative conditionals 214-215 principle of extensionality 256 principle of predicability 253-254 principle of property-exemplification, unrestricted/restricted 263-264 principles of actuality for states of affairs 79, 126, 246 principles of intensional parthood for states of affairs (P0 –P6) 76, 79

366

principles of the identity of indiscernibles 283-286 provability-rules 81, 91-92 step-down rule 92, 101 substitution of identicals 81-82, 98 truth-conditions for utterances of conditionals 177

Persons Adams, Ernest 210-211 Anselm of Canterbury 55 Antipopper, Ernst 11-12 Armstrong, David 128, 133-147, 343344 Arnauld, Antoine 35-36, 38 Bealer, George 40, 46-48 Bonevac, Daniel 180 Cantor, Georg 157, 202 Carnap, Rudolf 316 Castañeda, Hector-Neri 28 Chalmers, David 40, 44, 47-51, 54, 152-154 Chihara, Charles 36-39, 157-158 Cohen, Paul 199, 203 Cresswell, M. J. 304 Descartes, René 55, 358 Fara, Michael 304-305, 320 Frege, Gottlob 60, 116 Forbes, Graeme 304, 348-352, 356-360 Gödel, Kurt 199, 203 Goodman, Nelson 187, 218 Hintikka, Jaako 316 Hume, David 1, 3, 13, 52, 87, 192-194, 242 Hughes, G. E. 304 Husserl, Edmund 109, 138 Jackson, Frank 209 Kanger, Stig 316 Kant, Immanuel 201 Kratzer, Angelika 215

Index

Kripke, Saul 46-49, 304-305, 316 Leibniz, Gottfried Wilhelm 35-36, 3840, 42, 88-89, 96, 290, 292, 294, 312-313, 320 Lewis, David 3, 36, 38, 105-109, 112116, 118, 122-126, 133, 135, 160, 174, 183, 187, 199, 209-211, 215216, 218, 221-227, 229, 236-242, 281, 294, 296, 305, 308-320, 322329, 334-338, 340, 345-349 Meinong, Alexius 17, 28, 32-33, 301302 Montague, Richard 316 Moore, George Edward 32 Plantinga, Alvin 27-35, 37, 54, 107108, 113, 116-117, 156-157, 329330, 333, 348-354 Putnam, Hilary 47-48 Reutersvärd, Oscar 44 Rosenberg, Jay 50 Russell, Bertrand 28, 32, 60 Stalnaker, Robert 40, 45, 164, 174, 188, 207-208, 226, 232, 236-237, 240, 296 van Fraassen, Bas 13, 147-150, 243245 van Inwagen, Peter 52-59, 61 Williamson, Timothy 304-305, 320 Wittgenstein, Ludwig 1, 69, 117, 136 Yablo, Stephen 57-58

Subjects absolute actuality 124-128 absolute essentiality 335, 337-340 abstract entities 116-118 abstraction of properties [λoA[o]] 262 abstractor 254, 262-263 abstract possibilities 118 accessibility-function 223 accessibility relation 148 actual existence 22-24 actualism 23-25, 30-31, 33, 37, 350, 358

actualism for objects 352, 354 actualism for/with respect to individuals 24-25, 33, 37, 297-298, 302-308 actualism for/with respect to properties 24, 295, 297-298, 343, 359 actualism for/with respect to states of affairs 23, 25, 33, 75, 297-298 actuality [A(x)] 23-24, 27, 29-35, 40, 79f, 136, 279, 281, 298, 304, 324326, 350, 358-360 actuality-essence see existence-essence actuality-essential see existenceessential actuality for LI*-individuals 327-329, 336-337 actuality of individuals 281, 298 actuality of possible worlds 123-125, 199 actuality of properties 294-297, 343344 1-actuality of properties 296-297 2-actuality of properties 296-297 actuality of propositions 360 actuality of states of affairs 69, 74-76, 79, 122, 133-135, 156, 294, 360 alethic modalities see ontic modalities alethic necessity see ontic necessity alethic possibility see ontic possibility all-quantification in modal formulas 304 analytic necessity see logical necessity a posteriori necessity 45, 47-48 assertability of conditionals 209-212, 214-215, 218, 221 assertorial completeness 120 atomic fact 356-357, 359 atomicity see simplicity atomic state(s) of affairs 69-70, 134, 140-147, 356 1 b 65, 83, 153-154, 162, 194 b2 65, 83, 102-103, 149, 153-154, 194, 267 3 b 153-154 b10 162, 194, 267 Barcan formula 304 bases of modal assertions 194

367

Index

bases of necessity 64-66, 83, 86-88, 92, 96, 152-153, 159-161, 193-194, 258 bases of utterances of conditionals 166, 170-173, 175, 186, 208, 225 Bases-Theory of Conditionals 159160, 187-188, 192, 195, 208, 214215, 217, 219, 222, 224-226, 228, 231, 233-236, 238-239, 243, 249 Bases-Theory of Modality 160, 193, 195, 243 Bases-Theory of Necessity 92-93, 147, 149-150, 249 basic mereology of states of affairs 66, 67, 73-74 basis-function 217 basis of causal necessity 193-194 basis of factuality see b10 basis of logical necessity see b1 basis of metaphysical necessity see b3 basis of nomological necessity see b2 being 23-24, 32, 298 being a fact see obtaining being an entity 358-359 Boolean algebra 73 Boolean mereology 73 Boolean principles 72 c* 251-252 causal necessity 192-193 causation 191-192 centered mereology see Boolean mereology centeredness of comparative closeness/similarity of worlds 230, 233-234, 237-240 chance 39 change of basis in an inference 179180, 185 change of basis in a sequence of assertions 223, 225-226 coarse-grained conception of properties 261, 333 coarse-grained conception of states of affairs 68, 108, 261, 333 cognitive function of sentences 110111 comparative closeness of worlds see comparative similarity of worlds

368

comparative similarity of worlds 226, 230, 232-234, 237-240 conceivability 40-43, 49-51 conceptual necessity see logical necessity conceptual skepticism 52, 55 conceptual truth of principles 76-78, 89, 91, 102-103, 162 concrete entities 117-118 concrete possibilities 118 conditional necessity see relational necessity conditionals 66, 93, 159-161, 226 conjunction of properties 270, 299 conjunction of states of affairs [CONJxA[x], conj(x, y)] 64, 67, 69, 96, 157-158 consistency 204 consistency of part of the mereology of states of affairs 79-80, 100 consistent necessities 84, 88 constituent(s) 112, 114-115, 143-144, 346, 356-357 constitution of possibilities 140-141, 143-147 context-dependence of conditionals 160-161 contingency 39 contingency of states of affairs 133, 139-140 contingent necessities 85-87 convention and necessity 244 converse of the Barcan formula 304305 conviction 93 cotenability, objective/subjective 173, 218-221 counter-counterfactual 169-170 counterfactuals see counterfactual conditionals counterfactual conditional(s) 160, 165168, 172, 185, 190, 216, 221-231, 233-237, 242, 296 counterpart(s) 334-335 counterpartism 114, 305, 340 counterpart relation(s) 113, 316-323, 325

Index

counterpart theory 318, 320, 325-326, 337-338 definitions of modalities 63-65 disjunction of properties 341-343 disjunction of states of affairs [DISJxA[x], disj(x, y)] 71, 96, 133 disjunctive property/-ties 341-343, 345 disjunctive state(s) of affairs 133, 138139, 341 dispositions 3 doxastic necessity 93, 244 doxastic possibility 210 dualism 153, 259 elemental states of affairs [EL(x)] 6971, 73, 75-76, 82, 145-146 elementhood [y ∈ x] 256 empiricism 13, 87 empty property 343-345 epistemic necessity 1, 93, 244 epistemic possibility 44-47, 49 epistemological holism regarding possibility 58 epistemological problem of causal necessity 192 epistemological problem of ontic modalities 3, 9-13, 61, 194 epistemological skepticism 52, 55-57, 60 ersatz for non-actual individuals 307308 ersatz worlds 116, 118 essence(s) 259, 262, 329, 333-334 essential exemplification see essential property of essentiality simpliciter 255, 281-282 essential LI*-property of 334-335 essential property/-ties 255-258, 279 essential property of 255, 258, 262, 278-279, 282-283, 330-331 everyday necessity 1 exemplification (of a property) [EXM(y, x)] 252-253, 292, 301, 343-344 existence 16, 18, 20-24, 27-34, 37, 115, 133-136, 281, 298, 304, 320, 324-326, 350, 357-360

existence at a world 319-320, 324, 327, 335 existence-entailing property/-ties 357358 existence-essence 283 existence-essentiality 281-282, 335, 338-340 existence-essential LI*-property of 334-335 existence-essential property of 281,282 existence in a world 324-325, 327 existence of propositions see actuality of propositions existence of states of affairs see actuality of states of affairs experiencing 196-197 explanatory conditionals 191-192 extensional modification of a property 266-267, 278 extensional property/-ties 255, 267-268 extent of predicative fit 260, 275, 277, 300 fact(s) 22, 117, 133, 135-136, 360 factmaking 136-137 factuality 85, 87-88 faithfulness of necessities to truth 84, 86, 88, 92, 102, 147, 168 Fermat’s Last Theorem 53 fine-grained conception of properties 333 fine-grained conception of states of affairs 68, 108, 157, 333 first-order states of affairs 140, 143146 force of a necessity see strength of a necessity freedom 39 free logic 20, 33, 136 general actualism 23, 33 general determinism 88 haecceitism 345-348 haecceity/-ties 333 having a property see exemplification higher-order contingency 129-130, 132 higher-order states of affairs (in two senses) 128-129, 132, 145

369

Index

I-completion of an I-property 300 identity of individuals 284, 286-288 identity of properties 254-255, 284 identity of states of affairs 67, 284 imaginability 41, 43-44, 49-52, 57, 196-197 imagining see imaginability incompatibility conditionals, trivial/non-trivial 174, 219, 235, 242 indeterminateness of predicative fit see vagueness of predicative fit indexical conception of actuality 122125, 328 indicative conditional(s) 159-160, 172, 185, 190, 207-209, 216, 226 individual(s) [I(x)] 2-3, 24-26, 37, 73, 107, 112, 116, 122, 251-252, 260, 262, 288-290, 292-293, 298-302, 309-310, 312, 334, 351 individual objects see individuals individual-world pair(s) 289-290, 293, 316-318 inner necessity see intrinsic necessity instantiation of an inference-form 178 instantiation of a property see exemplification intension(s) 46, 49, 68, 80-81 intensional contexts 81, 99, 102 intensional entities 2 intensionalistic theory 2-3 intensional parthood 129, 144 intensional parthood between properties [PPR(x´, x)] 260-262 intensional parthood between states of affairs [P(x, y)] 64, 67-68, 79, 90, 118-119, 158, 194-196, 199, 222, 243 intentional objects 138-139, 298 intentionality 138-139 intentional reference 345 intrinsic necessity 1, 244 intrinsic possibility 54 I-property see property of individuals I-restriction 275-278 k* 67 knowing a proposition 58-60 knowledge 93

370

knowledge of a mere possibility 12-16, 19, 24, 26, 61, 197 knowledge of necessity 13, 26, 61 knowledge of the non-actual see knowledge of the non-existent knowledge of the non-existent 3, 14, 24-26 Kripke-semantics of modal predicate logic 304-305 language and states of affairs 196, 198, 246-247 laws of nature 83, 86, 102-103, 149150, 153-154, 186, 192, 194, 244, 273 LD [Lewisian doctrine of existence] 335 Leibnizian conception of modality 96 Lewis-individual(s) 310-312, 338 Lewis-I-properties 310-311, 313 Lewis-world(s) 310-313, 315-316, 319, 334-335 L(eibniz)-exemplification 292, 294, 314 LI-exemplification 314 LI*-existence see actuality for LI*individuals LI-individuals see Lewis-individuals LI*-individuals 312-313, 315-319, 322-324, 327, 334-335 LI-intensional parthood between properties 313-314 Limit Assumption for comparative similarity of worlds 240-242 L(eibniz)-individual(s) 292-294, 301, 306-308, 313-314 LI-properties see Lewis-I-properties LI*-properties 311, 313, 316, 326-327, 334, 336 LI-worlds see Lewis-worlds logical empiricism 1 logical essence, universal/I-restricted 259-261, 268-270, 272, 279, 284, 287, 299, 329-331 logical essentialism 272 logical essentiality 255, 257-258, 267, 278-279, 281-283, 329-331 logical general determinism 88-89

Index

logical implication(s) 164, 195 logical incommensurability 176 logically contingent theory 200-201 logical modalities 199 logical necessity 1-2, 44, 55, 65, 83, 85, 87-90, 147, 151-155, 192-193, 195, 199, 205, 242-247, 255, 357358 logical possibility 12, 44-47, 49, 52-56, 58, 153-156, 196-197, 199, 204-205, 310 logical provability 81, 91-92, 101-102 logical space 114-115 logical truth 151-152 logic of conditionals 161, 178-185, 190, 231-232 material atoms 70 material implication(s) 161-164, 207209, 216 material mereologies 70 maximal-consistent PRITs see entry after next entry maximal-consistent states of affairs [MC(x)] 34, 72, 75, 88-89, 94-95, 106, 112-114, 116-119, 122, 145, 288, 310, 315-316, 346-347 maximal-consistent totally defined properties of individuals [PRITMC(x)] 288-293, 306-308, 312-313, 315-316, 327 MCs see maximal-consistent states of affairs meanings of sentences 68 Meinongian entities 301 mereology of sets 70 mereology of states of affairs 67, 70, 74, 151, 242, 251 metalinguistic theory of conditionals 173, 215-217, 219, 221 metaphysical essentiality 255 metaphysical necessity 1, 44, 152-155, 255 metaphysical possibility 44-47, 54, 152-155 methods for gaining modal knowledge 197-199 M(einong)-exemplification 301-302

might-conditionals 190-191 MI-individual(s) 312-313, 316-317, 322-323-324, 338 mindability 41-42 M(einong)-individuals 301-302 minimal state of affairs [M(x)] 68 modal constructivism 36 modalism 38 modal knowledge 194-197 modal logic 2, 30, 38-39, 92, 304 modal omniscience 55, 57 modal predicate logic 349-350 modal predicates 63-64 modal properties 3 modal propositional logic 92 modal realism 3, 35-39, 112, 118, 345346 modal sentence-connectives 63-64 modal skepticism 52, 55-58, 60 model-theoretic semantics 2, 38, 66, 151, 304 model theory 204 natural necessity see nomological necessity necessitas de dicto/de re 309 necessities in nature 243-244 necessity/-ties in respect of facts of soand-so kind 224-225 necessity of states of affairs and necessity of properties 309 negation of a property 341-343 negation of a state of affairs [neg(x)] 64, 71, 145 negative existentials 33 negative property/-ties 341-343, 345 negative state(s) of affairs 133, 138139, 145, 341 nomological accessibility 148-149 nomological essence, universal/Irestricted 259-261, 272-273 nomological essentiality 258 nomological implication 164 nomological necessity 1, 44, 65-66, 83, 85-86, 89, 101-102, 147-152, 154155, 244, 255 nomological possibility 153-156

371

Index

non-actual entities 3, 25-26, 31-32, 3639, 75-76, 295, 297, 301-302, 307308, 357 non-actual individuals 301-302, 307308 non-actualism 29, 306-307 non-actuality 23-25, 302 non-actual properties see non-actual entities non-actual states of affairs see nonactual entities non-being 23 non-compositum 143 non-existence 17, 19-21, 23-25, 27-28, 134-135, 302 non-existent entities see non-actual entities non-material mereologies 70, 73 notiones completae 290-293, 306-307, 312-313 number of individuals 288, 290 number of maximal-consistent PRITs 290 number of named properties 265 number of possible worlds 290 number of states of affairs 71, 75-76, 82-83, 89, 196 numerical existence 22-24 object(s) 351 objecthood 357 obtaining [O(x)] 22-23, 69, 94, 133136, 156, 196, 360 obtaining in [O(x, y)] 94-96 ontic modalities 3, 9-10, 68, 155, 191,199 ontic necessity 1-2, 87, 242, 244 ontic possibility 40-47, 49-51, 54, 155, 210 ontological category/-ries 112, 260261, 293 ontological identification see ontological reduction ontological problem of ontic modalities 9-10 ontological reduction 290, 293-294, 314 onto-nomological theory 2, 63

372

onto-nomological theory of modality 160 onto-nomological theory of necessity 93 onto-nomological theory of possibility 93 overlap of worlds 346 part(s) 112, 115, 144, 260 part(s) of a world in Lewis’s sense 309, 315 physicalism 30-31, 45, 152, 155, 259 positive property/-ties 343 positive state(s) of affairs 145 possibilia see possible beings possibilism 29 possibilities as combinations of simples 133, 140-143, 146-147 possibility 40, 54, 56, 58, 118 possibility arguments 56 possible being(s) 36, 38-39, 134-135 possible individual(s) 31 possible object(s) 32 possible properties 31 possibles see possible beings possible world(s) 2, 19, 31, 34-38, 64, 72, 75, 94, 106, 108-109, 112-119, 122-123, 125, 130, 145, 153, 242, 246, 288-290, 309-310, 334, 345347 possible worlds qua spatiotemporally extended individuals 116, 346 possible worlds qua states of affairs 116, 346 possible-worlds-semantics 38, 46, 148151, 216 potentialities 3 predicability (of x regarding y) see predicative fit predicate logic 1, 17-18, 33, 136 predicative fit (of x to y) [F(x, y)] 251254, 300 predicative saturability (of x by y) see predicative fit predicative saturation (of x by y) [[x, y]] 251-252, 260, 270, 272 Principle of Conditional Excluded Middle 232-233, 236

Index

PRITs see totally defined properties of individuals problem of accidental intrinsics 114, 346 proper necessities 85-87, 164 property/-ties [PR(x)] 2-3, 24-25, 29, 251-254, 261-262, 267-268, 290, 293, 295, 298-299, 359 property actualism, simpliciter/strict 348, 350-354, 356-357, 359 propertyhood 344-345 property-incomplete object(s) 302 property-inconsistent object(s) 302 property of being actual 279 property/-ties of individuals [PRI(x)] 252, 257, 277, 288, 299-300, 309311, 334 proposition(s) 55, 58-59, 105, 107-111, 116-117, 138-139, 311, 356, 360 quantifier actualism 349-354 quasi-atomic state of affairs [QA(x)] 69 quasi-complete state of affairs [QC(x)] 72 range of predicability see extent of predicative fit realism about abstract entities 116 realism about ontic necessities 244 realism about possible worlds 115-116 reductive theory of modality 132, 242244 referential instability see referential unstableness referential stability 120, 258, 267 referential unstableness 82, 97-99, 102103, 162 relational necessity 93, 159, 161-162, 244 relations 254 relative exemplification [EXM(y, x, w)] 270, 286-287, 294 relevantly connecting basis 212-213, 220 rigid designation see referential stability S5 92, 147-149, 151 S5-necessity 147

saturation-function see predicative saturation saturator 251, 254, 260, 263 self-exemplification 253-254 self-identity 21, 29-30, 32, 253-254 self-predicability 253-254 self-saturation 275 semantical problem of ontic modalities 9-10, 13, 63 set(s) 106, 256, 267-268, 341-342 sets of individuals, in the typetheoretical sense 256-257 sets of possible worlds 105-109, 153, 222 set-theoretical representation of states of affairs 222 set theory 2, 199-202 simple conditional(s) 159, 165-166, 168, 171-172, 185, 190, 216 simplicity 143-144 singleton sets 107 Slingshot Argument 80 somethingness 30, 32-33, 136 space-times 114-115, 199 spatiotemporal totalities see spacetimes state(s) of affairs [S(x)] 2-3, 22-26, 3132, 35, 37, 55, 58, 80, 83, 105-110, 112, 114, 116-117, 119, 122, 133, 136-139, 153, 158, 196-197, 204205, 242, 246, 251-252, 260, 262, 293, 298-299, 311, 360 state of affairs being about an individual 14, 114-115 strength of a necessity 88 strict conditional(s) 221-224 strict implication(s) 161-164, 207, 226 structure of a property 351, 357, 359 subjective probability and conditionals 211-212 sum of properties 270 sum of states of affairs 157 supercontingency see higher-order contingency supercontingent state(s) of affairs 129130,132 synthetic a priori theory 201

373

Index

t* 67, 97, 162 “that”-contexts see intensional contexts the (self-)contradictory state of affairs 68 the maximal state of affairs 67-68, 88 the minimal state of affairs 67-68 thereisness see somethingness the tautological state of affairs 68 the total state of affairs see the maximal state of affairs the (actual) world 69, 72, 75, 88-89, 94, 108, 117, 146 totally defined property/-ties of individuals [PRIT(x)] 288, 311, 313, 316, 326 total state of affairs [T(x)] 68 transworld identity of individuals 113, 286, 301, 308, 317, 346 truth according to possible worlds 130131 truth-functional propositional logic 72, 92 truthmaker see truthmaking truthmaking 133, 136-138 two-dimensionalism in semantics 47, 49 understandability 41-43, 59 universally fitting property/-ties see universal predicability universal predicability 254, 268, 270, 272, 274, 279-280 universes of states of affairs 73-74, 7980, 82-83, 89 vagueness of predicative fit 277 verbal necessity 243-246 w* 67, 69, 97, 102, 120, 162, 267 w0 120 world-localization 292 world-relative exemplification see relative exemplification

374

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