Omniscience: From a Logical Point of View 9783110327090, 9783110326697

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Paul Weingartner Omniscience From a Logical Point of View

Philosophische Analyse Philosophical Analysis Herausgegeben von / Edited by Herbert Hochberg • Rafael Hüntelmann • Christian Kanzian Richard Schantz • Erwin Tegtmeier Band 23 / Volume 23

Paul Weingartner

Omniscience From a Logical Point of View

ontos verlag Frankfurt I Paris I Ebikon I Lancaster I New Brunswick

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Preface The main task of this book is to clarify the concept of omniscience and to reject attacks which are based on false or very questionable premises or on invalid argumentation. The book thereby defends the possibility to attribute omniscience to God in a consistent way. The method is to divide the main task into 12 chapters which are formulated as basic questions. Each chapter begins with arguments pro and contra. Then a detailed answer is proposed which contains a systematic discussion of the question. This is the repective main part of the chapter. These arguments pro and contra express different positions concerning the concept of omniscience and attacks against a consistent formulation of it. These problems are discussed and clarified in the commentaries to the objections at the end of the chapters. It has to be emphasized however that what is expressed in the pros and contras is not the opinion of the author. It is sometimes the opinion of other scholars as shown by quotations. The opinion of the author is expressed in the main part of the chapters and in the commentary to the objections. The last chapter 13 contains a theory of omniscience formulated as an axiom system. It is to show that theism claiming an omniscient God, who knows everything about himself and about his creation (including the universe) is possible in a consistent way. It should be observed moreover that this book is not a book about the existence of God; but about the possibility of a consistent concept of omniscience which can be attributed to a presupposed object of religion (God) which is usually understood as a most perfect being and as creator of this world (universe). This does not mean that this book is only readable for theists. Any reader interested in the topics of omniscience may study the book, accepting the assumptions which seem questionable to him only conditionally. Aknowledgements: The author wants to thank Ursula Stranzinger, Eva Stieringer and Albert Anglberger for providing the typoscript and the layout. Further thanks go to Dr. Rafael Hüntelmann of Ontos Verlag for the efficient cooperation. Salzburg, March 7, 2007

Paul Weingartner

Contents 1. Whether Everything is Ttrue What God Knows.......................................... 1 1.1 Arguments Against ................................................................................ 1 1.2 Argument Pro ........................................................................................ 3 1.3 Proposed Answer................................................................................... 3 1.31 Analysis of the Concept of Knowledge............................................. 3 1.32 Further Support: Logical and Deductive Omniscience...................... 5 1.33 Further Support: Logical and Deductive Infallibility ........................ 6 1.34 Further Support: God's knowledge of logic is not restricted to PL1.. 6 1.35 Further Support: God's knowledge comprises also the facts of the world (universe) ....................................................................................... 7 1.36 Further Support: In God there is no Belief........................................ 7 1.4 Answer to the Objections....................................................................... 9 1.41 The Divine Liar (to 1.11).................................................................. 9 1.42 True Justified Belief (to 1.12)......................................................... 13 1.43 Belief not different from Knowledge (to 1.13) ............................... 14 1.44 Necessity of Contingency (to 1.14)................................................. 15 2. Whether God Necessarily Knows Whatever He Knows ........................... 19 2.1 Arguments Against .............................................................................. 19 2.2 Argument Pro ...................................................................................... 20 2.3 Proposed Answer................................................................................. 20 2.4 Answer to the Objections..................................................................... 22 2.41 God’s Knowledge is Complete (to 2.11)......................................... 22 2.42 lKp ≠ Klp (to 2.12)...................................................................... 22 2.43 God’s knowledge and will concerning necessity (to 2.13) .............. 23 3. Whether God Knows Something at Some Time........................................ 25 3.1 Arguments Pro..................................................................................... 25 3.2 Argument Contra ................................................................................. 25 3.3 Proposed Answer................................................................................. 25 3.31 Knowing at some time and knowing that something happens at some time ........................................................................................................ 26 3.32 Analysis of Time ............................................................................ 26 3.321 Time of this world (universe) .................................................... 26 3.322 Time as a Chronological Order.................................................. 29 3.323 Time as Biological and Psychological Time.............................. 33 3.4 Answer to the Objections..................................................................... 35 4. Whether God Knows All Past and Present Events .................................... 37

4.1 Arguments Against ..............................................................................37 4.2 Arguments Pro .....................................................................................37 4.3 Proposed Answer .................................................................................38 4.4 Answer to the Objections .....................................................................39 5. Whether God's Knowledge Exceeds His Power ........................................41 5.1 Arguments Against ..............................................................................41 5.2 Arguments Pro .....................................................................................42 5.3 Proposed Answer .................................................................................42 5.31 Definition of Omnipotence (God's power) ......................................43 5.32 God's knowledge exceeds his power ...............................................47 5.4 Answer to the Objections .....................................................................50 6. Whether God Causes Everything What He Knows ...................................53 6.1 Arguments Pro .....................................................................................54 6.2 Arguments Contra................................................................................55 6.3 Proposed Answer .................................................................................55 6.31 The knowledge need not to be a sufficient condition for causing something ...............................................................................................55 6.32 The statement "God causes everything what he knows" leads to absurd consequences...............................................................................56 6.33 The thesis "God causes everything what he knows" excludes cooperation and learning processes in creatures......................................56 6.34 If God causes everything what he knows, then he is normative and volitive inconsistent ................................................................................57 6.35 If God causes everything what he knows, then he causes everything ................................................................................................................58 6.36 The thesis of the allcausing God and transitivity.............................59 6.4 Answer to the Objections .....................................................................61 6.41 God’s knowledge – a necessary cause (ad 6.11)..............................61 6.42 God’s knowledge – not a sufficient cause (ad 6.12) ........................61 6.43 Omniscience and Freedom (ad 6.13)...............................................62 7. Whether God Knows Singular Truths?......................................................67 7.1 Arguments Contra................................................................................67 7.2 Arguments Pro .....................................................................................68 7.3 Proposed Answer .................................................................................69 7.4 Answer to the Objections .....................................................................73 7.41 Discursive Knowledge (to 7.11)......................................................73 7.42 Irrelevant truths (to 7.12) ................................................................73 7.43 God knows A-propositions? (to 7.13) .............................................74

8. Whether God's Knowledge of Singular, Contingent Truths Implies the Mutability of God ......................................................................................... 79 8.1 Arguments Pro..................................................................................... 79 8.2 Arguments Contra ............................................................................... 79 8.3 Proposed Answer................................................................................. 80 8.31 The underlying principle................................................................. 80 8.32 Principle KCH is not generally valid .............................................. 80 8.33 God does not need to change his knowledge .................................. 81 8.4 Answer to the Objections..................................................................... 83 9. Whether God Knows What Is Not............................................................. 85 9.1 Arguments Against .............................................................................. 85 9.2 Arguments Pro..................................................................................... 85 9.3 Proposed Answer................................................................................. 86 9.31 God's knowledge extends also to that what is not in the sense of what is either impossible or incompatible with laws of nature or accidentally not, but possible. .................................................................................... 86 9.32 God’s knowledge extends to things that are not actual ................... 89 9.4 Answer to the Objections..................................................................... 90 9.41 What is not can be interpreted in two ways (to 9.11) ...................... 90 9.42 Truly negated (to 9.12) ................................................................... 91 9.43 Does “God cannot know something false” imply that he is not omniscient? (to 9.13) .............................................................................. 91 9.44 Does God know counterfactuals (9.14)? ......................................... 92 10. Whether Knowledge or Truth Can Change the Status of a State of Affairs ..................................................................................................................... 97 10.1 Arguments Pro................................................................................... 97 10.2 Arguments Contra.............................................................................. 98 10.3 Proposed Answer............................................................................... 98 10.31 Different Kinds of States of Affairs .............................................. 99 10.32 The necessary status cannot be changed by truth or knowledge.. 102 10.33 Can the status "contingent" be changed by truth or knowledge? . 104 10.4 Answer to the Objections................................................................. 112 10.41 Only closed proposition can be true (ad 10.11)........................... 112 10.42 Truth does not destroy contingency (ad 10.12) ........................... 112 10.43 The reason for truth is the obtaining fact, not the other way round (ad 10.13) ............................................................................................. 112 10.44 God’s knowledge does not change the ontological status of a state of affairs (ad 10.14).............................................................................. 113 11. Whether God Knows Future States of Affairs....................................... 115

11.1 Arguments Contra ............................................................................115 11.2 Arguments Pro .................................................................................117 11.3 Proposed Answer .............................................................................117 11.31 God knows the future states of affairs of the universe and of the creatures belonging to it by knowing their causes.................................117 11.32 God knows his power and the power of the creatures including man ..............................................................................................................120 11.33 God might have a possibility to know future states of affairs in their actual states...........................................................................................122 11.4 Answer to the Objections .................................................................127 11.41 “Present” or “actual event” is ambiguous (to 11.11) ...................127 11.42 Future events ≠ determinate events (to 11.12).............................128 11.43 “Foreknows” is inadequate for God (to 11.13)............................128 11.44 “Is a fact” is ambiguous (to 11.14) ..............................................129 11.45 Does foreknowledge destroy free will decisions? (to 11.15) .......129 11.46 Is knowledge of contingent future propositions inconsistent? (to 11.16) ...................................................................................................131 11.47 Free actions ≠ non-causal actions (to 11.2) .................................133 12. Whether God Knows Everything That is True ......................................135 12.1 Arguments Contra ............................................................................135 12.2 Arguments Pro .................................................................................136 12.3 Proposed Answer .............................................................................136 12.32 God's knowledge about himself ..................................................138 12.33 God's knowledge about his creation ............................................140 12.34 God's knowledge about Logic and Mathematics .........................141 12.341 Leibniz's idea of human knowledge concerning Logic and Mathematics ......................................................................................142 12.342 The limitations discovered in the 20th century. .......................144 12.4 Answer to the Objections .................................................................150 13. A Theory of Omniscience .....................................................................153 13.1 Introduction......................................................................................153 13.2 Theory of Omniscience ....................................................................154 13.21 Definitions of Omniscience and of Omnipotence........................155 13.22 God’s Knowledge .......................................................................160 13.23 God’s Knowledge of the Universe ..............................................161 13.24 God’s Knowledge and Will.........................................................165 13.25 God’s Knowledge and Will in Relation to Moral Evil ................166 13.26 God knows his activities .............................................................170

Literature ................................................................................................... 173 Subject Index ............................................................................................. 183 Name Index ............................................................................................... 187

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1. Whether Everything is True What God Knows The above question "whether everything is true what God knows" expressed in other words reads: Does it hold that if God knows something (say that some state of affairs obtains) then this (that some state of affairs obtains) is true. If we translate this question into the language of Epistemic Logic then it can be expressed more precisely thus: If g (God) knows that p (is the case) then p is true. Here 'p' stands for a proposition representing states of affairs. Symbolically: gKp → Tr(p)

or:

gKp → p

This question can also be expressed by asking whether God is infallible. Because some person may be called infallible if it cannot happen that this person knows something which would not be the case.

1.1 Arguments Against 1.11 If everything is true what God knows, then he has to believe all and only truths. But as Grim says there can be no such being. For suppose there is, and consider a sentence we might term the Divine Liar: God believes that (8) is false

(8)

"On the supposition that (8) is true, it is true that God believes that (8) is false. But we are supposing here that (8) is true, and thus we are forced to conclude that God holds a false belief. On such a supposition he cannot then qualify as omniscient. On the supposition that (8) is false, it is not the case that God believes that (8) is false. But our supposition here is that (8) is false, and thus there must be a truth – that (8) is false – that God does not believe and hence does not know. Here again he fails to qualify as omniscient. If (8) is either true or false, then, God is not omniscient. But, of course, God is not alone in this respect: a similar argument will hold for any being

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proposed as omniscient. It appears that there simply can be no omniscient being."1 This argument is also applicable if 'believes' is replaced by 'knows'. Thus on the supposition that (8') (God knows that (8') is false) is true, God knows a proposition which is false. But if everything is true what God knows, then he has to know all and only truths. Therefore it does not seem to hold that everything is true what God knows. 1.12 The thesis "everything is true what God knows" seems to presuppose (as a necessary condition) a concept of knowledge which is defined as true justified belief. But as Gettier has shown there are some cases where all the three conditions: truth, justification, and belief are satisfied though one cannot speak of knowledge such that this definition is not satisfied. Therefore the thesis "everything is true what God knows" does not seem to hold. 1.13 If everything is true what God knows, then the presupposed concept of knowledge seems to imply true belief. But in God there is no belief. Therefore it does not seem to hold that everything is true what God knows. 1.14 It cannot hold that everything is true what God knows. This can be shown by the following indirect proof: 1. Assumption to the contrary: Everything is true what God knows. Symbolically: gKp → p. Now this premise of infallibility must necessarily hold for God such that we can assume the stronger premise: 2. Necessarily: Everything is true what God knows. Symbolically: l(gKp → p) 3. Instantiation: We substitute for 'p': the world exists, such that we get: Necessarily: if God knows that the world exists then the world exists. Symbolically: l(gK that the world exists → the world exists). 4. By a distribution law of Modal Logic the necessity operator 'l' can be distributed on the parts of the implication: lgKp → lp or: If it is necessary that God knows that the world exists then it is necessary that the world exists. Symbolically: lgK that the world exists → l the world exists. 5. But we can generally assume – since God's knowing belongs to God's essence and actuality – that he necessarily knows whatever he knows. Symbolically: gKp → lgKp. 1

Grim (1991, IUN) p. 8.

3

Thus we have w.r.t. this instance: Necessarily: God knows that the world exists. And from 4. and 5. it follows (on the assumption that God knows that the world exists, gKp): 6. Therefore: Necessarily the world exists. 7. But that the world exists is contingent and not necessary.2 8. Therefore it does not seem to hold that everything is true what God knows.

1.2 Argument Pro Denying that everything is true what God knows would mean to deny that gKp → p. But this means that God would know that p is the case although p is not the case, i.e. for some p: gKp and ¬p. But this seems to be impossible for a perfect being. Therefore it seems to hold: Everything is true what God knows, i.e. gKp → p.

1.3 Proposed Answer Everything what God knows is true. Or: If God knows that p (is the case) then p is true. Symbolically: gKp → p. That this holds can be substantiated as follows: 1.31 Analysis of the Concept of Knowledge It is supported by an analysis of the concept of knowledge itself. The argument is this: If a strong concept of knowledge – in the sense that knowing that p implies the truth of p – is applicable already to men's knowledge, all the more it must be applicable to God's knowledge. But as will be shown subsequently a strong concept of knowledge (in the above sense) can be applied to men's knowledge. The concept of men's knowledge is usually understood in such a way that it implies that what is known is true. Or: If person a knows that p (is the case) then p (is the case), where 'p' stands for any meaningful statement or proposition representing a state of affairs. Symbolically this is expressed as:

2

Premise 2. and assumption 5. have been used (together with 4.) by Charles Hartshorne to derive conclusion 6. The Argument 1.14 originates in Thomas Aquinas (STh) I, 14, 13 objection 2. The answer of Thomas Aquinas is however different from ours given below 1.44.

4

KT

aKp → p

Therefore the traditional definition of knowledge contains truth in the definiens: knowledge is true, justified belief.3 Since aKp → p is logically equivalent to ¬p → ¬aKp what is expressed by it can also be stated in the following way: If it is not the case that p (is true) then p is not known. That is, there is no knowledge of something false. Though there might be knowledge that some proposition is false. Because that some proposition is false can be true and in this sense there can be knowledge of it. The principle KT can be further defended by investigating its negation or asking: what is the claim of somebody who denies that aKp → p? It is the claim that the negation of it is true. Its negation is: aKp ∧ ¬p; that is "person a knows that p (is the case) and (but) ¬p (i.e. p is not the case)". This is a situation which we would view as impossible; i.e. for the usual understanding of the concept of knowledge it is impossible that someone is said to know something (say that the sun is shining) when this is false (when the sun is not shining). From this consideration it follows w.r.t. the usual understanding of knowledge that the above principle KT is true. There is also a discussion of this principle in Epistemic Logic. Some have claimed that such a strong concept of knowlege is not defensible. But it can easily be shown by a lot of examples that such a concept of knowledge is defensible:4 There would be a widespread or maybe even complete agreement under scientists that we know in the strong sense of the above principle KT simple propositions of finite number theory, simple theorems of logic (of Propositional Logic and of First Order Predicate Logic), simple facts of sense perception, simple facts of our own inner experience (that we feel joy or anger or satisfaction … etc.) the results of very well corroborated experimental tests in different sciences … etc. This agreement is a fact, though most of the scientists today are very careful with using the concept of knowledge or knowing. That means they are very much aware of the distinction between stronger concepts such as "know" and weaker ones such as "believing", "asserting", "assuming" and "conjecturing". But still there is agreement that w.r.t. special areas a strong concept of knowledge is defensible. And since a strong concept of knowledge in the sense that truth is 3

Concerning the question whether there are exceptions to this definition see the answer to the objection 1.12. 4 Also Chisholm (1966, ThK) ch. 2 and 3 and Hintikka (1962, KaB) p. 43ff. defended such a strong concept of knowledge.

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a necessary condition for knowledge (KT) is applicable to humans, all the more it must be applicable to God. 1.32 Further Support: Logical and Deductive Omniscience Observe however that agreement w.r.t. a strong concept of knowledge in the above sense does not imply agreement with strong further axioms about human knowledge such as "logical omniscience" and "deductive omniscience": Logical Omniscience (LO): "Logical Omniscience" has someone who knows all the truths of logic (say all the theorems of First Order Predicate Calculus with Identity (PL1)). Deductive Omniscience (DO): "Deductive Omniscience" has someone who knows all the valid inferences of logic (PL1). Can humans possess LO? To answer that question we have to distinguish an implicit knowledge of the theorems of PL1 and an explicit one. Since PL1 is a complete system, by knowing a complete set of axioms + derivation rules, we may say that one can have an implicit knowledge of the theorems of PL1. And in this implicit sense humans can have LO w.r.t. PL1. But humans cannot have an explicit knowledge of all the theorems of PL1 simply because they are infinite in number. Can humans possess DO? A similar answer can be given here too. Since PL1 is a complete system, by knowing a complete set of rules either in addition to a complete set of axioms or without using axioms (if the system is built up as a system of derivation rules) we may say that one can have an implicit knowledge of the derivation rules of PL1. On the other hand humans cannot have an explicit knowledge of all the derivation rules which are valid rules in PL1.5 For God however it would be imperfect to have LO or DO only in the implicit sense like men. Therefore we have to say that he must have both LO and DO in the implicit and in the explicit sense.

5

There are systems of Epistemic Logic which (claim to) describe human knowledge and have LO and DO as their consequences. This is the case for example with the system of Hintikka in his (1962, KaB). Such systems can be accepted however only as describing an ideal(istic) concept of knowledge or as describing a kind of implicit knowledge in the above sense. For a criticism of LO, DO (and LI, DI, see below) as properties of human knowledge cf. Weingartner (1982, CRC). There a system for the concepts knowledge, belief and assumption is proposed which does not have these idealistic properties.

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1.33 Further Support: Logical and Deductive Infallibility A strong concept of knowledge which satisfies KT does not however imply logical infallibility (LI) or deductive infallibility (DI). Logical infallibility has someone who never commits an error w.r.t. logical theorems used. And deductive infallibility has someone who never commits an error w.r.t. logical deduction (i.e. the validity of inferences) used. Observe that although LI follows from LO and DI from DO the opposite does not hold. Thus these pairs of conditions (LO and LI; DO and DI) are not equivalent, since the ability not to commit an error may be restricted just to those truths or inferences which are in fact investigated. It will be easily understood that also LI and DI are not human properties. It has been an old experience of mankind that to err is human. There is no logician or mathematician who would not commit some error sometimes. Therefore LI and DI do not hold for humans.6 Of God on the other hand we cannot say that he could commit an error of logic which would violate LI or DI. His perfection must easily comprehend LI and DI. According to Thomas Aquinas already angels have the ability of LI and DI, they are able to understand all logical consequences of something known without any discursive process: "But if from the knowledge of a known principle they were straightway to perceive as known all its consequent conclusions, then there would be no discursive process at all. Such is the condition of the angels … Therefore they are called intellectual beings …".7 If Thomas Aquinas is right with this description of the angels then already the angels cannot commit logical errors and thus it is impossible that God who has created them could commit an error in matters of logic. 1.34 Further Support: God's knowledge of logic is not restricted to PL1 When defining LO and DO, and LI and DI above we have referred to PL1. But a number of truths of logic were skipped that way: (1) Metalogical theorems about PL1 (for example that it is consistent, complete, not decidable … etc.). (2) Theorems of Higher Order Logics + their metalogical therems about Higher Order Logic. 6

LI and DI are nevertheless consequences of some systems of Epistemic Logic, like that of Hintikka (1962, KaB) and Lenzen (1980, GWW). Cf. note 5 above. 7 Thomas Aquinas (STh) I, 58, 3.

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(3) Theorems of logics which are weaker than Classical Logic (like Minimal Logic, Intuitionistic Logic, Intermediate Logics) + their metalogical theorems about these weaker logics. All these theorems and metatheorems are infinite in number. Men (logicians) can have at most some implicit knowledge of these theorems and metatheorems. According to Thomas Aquinas already the angels may possess LO, DO and LI, DI unrestrictedly (i.e. without restriction to PL1). All the more God who has created these intellectual beings must possess LO, DO and LI, DI unrestrictedly. From what has been said in 1.32 – 1.34 it follows that the following principle holds for God: If something is logically true (in the broad sense indicated by the theorems of PL1 and by the conditions (1) – (3) above) then God knows that it is so. Or: If p is a truth of Logic (p ∈ L) then God knows that p. Symbolically: LK

p ∈ L → gKp

1.35 Further Support: God's knowledge comprises also the facts of the world (universe) If we assume that God created the world then his knowledge of the world must be perfect. Thus he must know all the facts about his creation. And it is impossible that he commits errors about facts of the world. This is of course compatible with the fact that he created the living organisms as learning organisms, i.e. as organisms which improve and develope via trial and error.8 1.36 Further Support: In God there is no Belief Concerning the relation of belief to knowledge we have to distinguish (at least) two different kinds of belief, a stronger and a weaker one: the stronger will be called knowledge-exclusive belief (abbreviated as G-belief) and the weaker will be called knowledge-inclusive belief (abbreviated as B-belief). The former (G-belief) is charaterized by the condition, that if someone believes something, then he does not know it and if he knows it, he does not (or need not) believe it. Whereas the second (B-belief) is characterized by the condition, that if someone knows something, he also believes it, but if he does not believe it, he also does not know it. 8

The necessary learning process is emphasized strongly by many contemporary biologists. Cf. Dobzhansky (1937, GOS), Maynard-Smith (1982, ETG).

8

These conditions can be expressed more precisely as follows: aGp → (¬aKp ∧ ¬aK¬p) aKp → aBp aGp → aBp

aKp → ¬aGp aK¬p → ¬aGp ¬aBp → ¬aKp

Examples for G-belief: Before the proof of the independence of the Continuum Hypothesis (from the axioms of set theory) was given, v. Neumann believed (but he didn't know), that the Continuum Hypothesis is independent. After Gödel proved the first part, i.e. that the General Continuum Hypothesis (GCH) can be conistently added to the axioms of Neumann-Bernays-Gödel-Set Theory (even if very strong axioms of infinity are used), v. Neumann wrote: "Two surmised theorems of set theory, or rather two principles, the so-called 'Principle of Choice' and the so-called 'Continuum-Hypothesis' resisted for about 50 years all attempts of demonstration. Gödel proved, that neither of the two can be disproved with mathematical means. For one of them we know that it cannot be proved either, for the other the same seems likely, although it does not seem likely, that a lesser man than Gödel will be able to prove this."9 But after the proof of the second part – that also the negation of GCH can be consistently added to the axioms of Set Theory (it holds for both systems, that of Zermelo-Fraenkel and that of Neumann-Bernays-Gödel) – was given by Paul Cohen in 1963, von Neumann didn't any more believe it, but knew that GCH was independent (from the axioms of Set Theory). In general we can say that scientific belief (belief in scientific hypotheses) – be it in mathematics or in natural science – is always G-belief: one does not yet have knowledge in the strong sense of KT. Examples for B-belief: No special examples for B-belief are necessary since B-belief may be interpreted in the following way: To B-believe that something (p) is the case means just to think that p is true (valid), to hold that p is true (valid), to strongly assume that p is true (valid) etc. Thus if someone knows that chromosomes duplicate, then he also B-believes it and also if someone G-believes that GCH is independent (from the axioms of Set Theory), then he also B-believes it. Religious belief – like scientific belief – is always knowledge-exclusive, i.e. is always first of all G-belief. Since if one believes religiously – for instance that Christ came for the salvation of mankind or that there will be some kind 9

v. Neumann (1969, TbG).

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of conscious life after death – one does not know it (and knows that one does not know it). And this holds for all religious beliefs even if not necessarily for all the statements of the creed of some special religion. Since the statements of the creed might not be all logically independent of one another such that some believer may infer one proposition of the creed from some others. And in this case he knows that one is a consequence of the other. Such inferences may be also done by theological argumentation. Still the propositions so derived are not known but believed, as known consequences of others which are believed.10 Now God does have neither B-belief nor G-belief. Since B-belief is a weaker consequence of knowledge, if he possesses knowledge he does not possess Bbelief, except in an inclusive way in the sense that if he knows something he inclusively also thinks that this is true. But "thinking that it is true" cannot be interpreted in a weaker sense than knowing – as it can be interpreted in man if someone thinks that something is true but does not yet know it. Moreover God cannot have G-belief either. Since G-belief is knowledge-exclusive this would mean that God lacks some knowledge w.r.t. a certain area and has only belief there. Some have claimed this for those future contingencies which are depending on men's free decisions. This difficult question will be treated in chapter 11. below. But the main point here is that God cannot commit any error; since this would be incompatible with his perfection. Thus independently in what sense his relation to future contingencies is expressed, he could never have a belief which is false.

1.4 Answer to the Objections 1.41 The Divine Liar (to 1.11) Grim thinks that the assumption: everything what God knows (believes) is true – gKp → p – leads to a contradiction by constructing a Liar sentence and therefore cannot be true. To be more accurately Grim needs for the contradiction also the opposite implication: everything which is true God knows (p → gKp). This latter principle will be discussed in ch. 12. Although it belongs to the concept of omniscience this question will have to be analyzed in detail, because of its difficult subquestions: Does God know all past events, all future contingencies, all truth about the world, about himself 10

For similarities and differences between scientifc and religious belief see Weingartner (1994, SRB).

10

… etc. These questions will be dealt with in the subsequent chapters. In any case Grim's main point is that the Liar can be applied to God's knowledge (and belief) as it is shown in 1.11 To the quoted argument of Grim we shall say three things: (1) It is certainly not necessary that God would have such an ambiguous belief (or knowledge) as expressed in a Liar sentence. (2) There are many different solutions of this and other Liar sentences which show that contradictions coming up with Liar sentences are not unavoidable. (3) One can give an explicit solution of Grim's Divine Liar. Therefore the conclusion drawn by Grim (that there cannot be an omniscient being) is not proved. ad (1) The special construction of the Liar sentence in 1.11 uses self reference, like similar constructions; one of the shortest is "(8) is false …… (8)". In both (in this and Grim’s) constructions '(8)' is used in two different meanings: On the one hand '(8)' means a particular sentence (not mentioned) which is false; on the other hand '(8)' means the sentence: "(8) is false" or in Grim's example it means the sentence: "God believes that (8) is false". But that equivocations and explicit ambiguities lead to false and sometimes contradictory consequences is an old experience; already Aristotle emphasized that equivocation is the main source of fallacies. And why should we assume that a most perfect being should have such ambiguous beliefs? ad (2) There are many different solutions for Liar sentences known today. One is that of Tarski which is based on the distinction between object language and metalanguage. This distinction unmaskes the ambiguity (the two different meanings of '(8)') shown above. The oldest solution seems to be that of Paulus Venetus who used an extension of Tarski's truth condition.11 There are non-Tarskian proposals like that of Kripke or Hintikka which can solve Liar paradoxes.12 A simple proposal which is based on an extension of Tarski's truth condition was made elsewhere by myself.13 Thereby Tarski's truth condition "s is true iff p" (where 's' is the metalinguistic name of a sentence and 'p' is the translation of the respective sentence into the metalanguage) is extended by two explicit components or conditions which are implicit in Tarski's formal apparatus of his essay on truth.14 The first condition (abbreviated as MC) is that (under normal conditions) an indicative sentence s is interpreted in such a way that s means its content (its proposition p); thus 'snow is white' means that snow is white. The second is that (under normal conditions) an indicative sentence s is interpreted in such a way that s 11

For details cf. Weingartner (2000, BQT) ch. 7.361. Kripke (1975, OTT), Hintikka (1996, PMR) ch. 6 and 7. 13 Cf. Weingartner(2000, BQT) p.129ff. and Weingartner(2006, SDT) 14 Tarski (1935, WBF), translated in Tarski (1956, LSM), pp. 152–278. 12

11

says of itself (of s) that it is so as it says (i.e. that s is true); thus 'Caesar crossed the Rubicon' is interpreted in such a way that it says of itself that it is so as it says (or that this is true). This condition is called also the positive (or serious) usage of language (abbreviated as PS). PS may be violated in the ironical (unserious) way of talking (sometimes marked by a smile, but more hidden in written texts). It is also violated in lies for the liar himself (who is aware of the lie) but purports (nonviolation of) PS to the addressee. If we extend now Tarski's truth condition by MC (for short: s means that p) and PS (for short: s says that s is true) we receive the following truth condition TMP*: TMP*

If s means that p then: s is true iff p and s says that s is true

Instead of TMP* one can have a truth condition which places PS also in the antecedent: If MC and PS then: s is true iff p and PS (TMP+). Or one can even drop PS altogether, though in this case an important component is missing. Nevertheless it can be shown that with all three extended truth conditions many known versions of simple and complicated Liars (as cyclic ones, strengthened Liars, Liar equivalences etc.) can be solved.15 ad (3) Solution of the simple and the Divine Liar (a) Solution of the simple Liar The simple Liar is often stated thus: (s) is false

(s)

But this is a rather unclear way of expressing the Liar since the function of '(s)' on the side is not precise. Therefore we shall express the short Liar sentence thus: L

s means that s is false.

Moreover if we add also PS we shall express the full Liar sentence L* thus: L*

15

s means that s is false and s says that s is true.

For details see Weingartner (2000, BQT) pp. 121–140. The name ‘TMP*’ is used accordingly in my book (2000, BQT) ch. 7.

12

Using L as the first premise we apply (with the respective substitution for p) the extended truth condition TMP* as the second premise: TMP* is true.

If s means that s is false then: s is true iff s is false and s says that s

The solution is now the following: Since the equivalence (in the consequent of the instantiated TMP* which can be derived from L and TMP* by Modus Ponens) has the form p ↔ (¬p ∧ q) this is (by Propositional Logic) equivalent to ¬p ∧ ¬q. That is, the solution reads thus: s is false and not: s says (of itself) that s is true (PS). If we use the full Liar L* we have to use TMP+ as truth condition, since TMP+ contains PS in the antecedent. The conclusion (solution) is then the same as in the case of the short Liar. Similar solutions can be received for many versions of Liar sentences including very complicated and sophisticated ones. In all these cases no contradiction follows. (b) Solution of the Divine Liar Instead of: "God believes that (8) is false …… (8)" we use the more precise form DL (replacing '(8)' by 's') in order to avoid the mentioned ambiguities: DL

s means that God believes that s is false.

By adding PS we receive the full Divine Liar sentence DL* DL*

s means that God believes that s is false and s says that s is true.

DL (DL*) is the first premise. The second premise will be again the instantiated extended truth condition: TMP*

If s means that God believes that s is false then: s is true iff God believes that s is false and s says that s is true.

The solution is now as follows: As said above it must hold that God believes (knows) that s is false iff s is false. Therefore ‘God believes that s is false’ can be replaced by ‘s is false’. Thus the conclusion (solution) is: s is false and not: s says that s is true

13

When using the full Divine Liar (DL*) and applying TMP* then the solution is the same. In both interpretations DL and DL* of the Divine Liar the solutions are as follows: The positive (serious) usage of language is violated (by applying a misleading or cheating usage: non-PS) and s is false. But no contradiction follows. Since there are other solutions of Liar sentences there will be also other solutions of the Divine Liar. The solution given above and possible other solutions show quite clearly that Grim's claim that omniscience is impossible because of the (alleged unsolvability of the) Divine Liar is wrong.16 Therefore the conclusion in objection 1.11 is not proved. On a more general point of view it seems rather important to be very careful not to project to the attributes of God all (or a part of) the many paradoxes, inconsistencies and confusions man produces because of his imperfect and restricted mind. 1.42 True Justified Belief (to 1.12) To this objection one can say two things: (1) The definition of knowledge as true justified belief is a reasonable definition of human knowledge for many applications (though not for all; see (2) below). But it is not a reasonable definition for God's knowledge. This can be seen as follows: First because there is no belief in God as has been substantiated above (section 1.36). Secondly there is no need for justification in God's knowledge because he does not know certain propositions because of others in the sense for example that some are more evident or transparent than others or function as an explanation for others. Further justifying propositions with the help of others implies that the knowledge is discursive which is also not the case with God's knowledge. To this point Thomas Aquinas says: "In our knowledge there is a twofold discursion; one is according to succession only, as when we have actually understood anything, we turn ourselves to understand something else; while the other mode of discursion is according to causality, as when through principles we arrive at the knowledge of conclusions."17 16

See Simmons (1993, AAO) who also stresses that there are many ways out of Liar and other paradoxes and criticizes Grim's attacks of omniscience from some other aspects. 17 Thomas Aquinas (STh) I, 14, 7. Cf. (SCG) I, 57.

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Thomas Aquinas then justifies that both kinds of discursion cannot hold for God. From these considerations it follows that the definition of knowledge as true justified belief is not adequate (applicable) to God. But since the definition of knowledge as true justified belief does not apply to God's knowledge the conclusion in the objection 1.12 is not proved by this argument. (2) Independently of that one may ask whether the traditional definition (which goes back to Plato) of knowledge as true justified belief when applied to human knowledge is always satisfied. The answer to this question is this: Although this definition is satisfied in many cases of both common and scientific knowledge it is not satisfied in all cases. That there are exceptions has been shown by Gettier in a paper of 196318 which has been discussed widely since. But Gettier uses two very artificial examples as counterexamples and the first one does not seem to be a genuine counterexample at all. However it is very easy to give some real (non artificial) counterexamples from the history of science.19 In fact every well justified scientific conjecture which is correct is true justified belief although no scientist would call it knowledge before the conjecture is proved or experimentally confirmed. One example, v. Neumann's conjecture concerning the independence of the Continuum Hypothesis, has been given above (see section 1.36). Another is Fermat's famous conjecture which has been proved by Wiles in 1994 or Poincaré’s conjecture which has been proved by Perelman in 2006, again other examples are Einstein's three famous conjectures (or predictions) of his General Theory of Relativity in 1915 about the perihelion of Mercury, the deviation of light due to gravitational fields, the red-shift of the light from distant stars. They have been well confirmed ever since 1919. This shows that the concept of human knowledge cannot be completely comprised by the definition of knowledge as true justified belief, although this definition has a wide field of application also in the sciences. 1.43 Belief not different from Knowledge (to 1.13) The concept of the knowledge of God does not imply true belief in the sense that there could be belief in him which would differ from his knowledge. This 18

Gettier (1963, JTB). For a discussion of Gettiers counterexamples and for more details on genuine counterexamples see Weingartner (1996, NGP). 19

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was shown in section 1.36. Therefore the first premise of 1.13 is not correct whenever belief (in God) means something different than knowledge (in God); but without such a difference the second premise (also interpreting belief as different from knowledge) would have no point. Therefore, since the first premise is not true the conclusion in 1.13 is not proved. 1.44 Necessity of Contingency (to 1.14) In order to discuss objection 1.14 in a precise way we ask two questions: Is the argument (inference) valid, i.e. does the conclusion follow logically from the premises? And secondly: Are all premises true? Only if both questions can be answered positively the conclusion is proved by this argument. As to the first question one can see easily that the answer is positive, i.e. the argument is logically valid. Concerning the second question we shall go over the particular premises: Premise 1 and 2 can be accepted since premise 1 was defended in section 1.3. with different reasons. Premise 2 is a strengthening of premise 1 which can also be accepted on the ground that God's knowledge belongs to his essence. Premise 4 is a theorem of Modal Logic which is valid in very many different Modal Logics such that it can be generally accepted. Premise 5 can be defended independently such that it can be accepted too. It will be defended in ch. 2. below. Premise 7 can also be accepted as true. Since 6 and 8 are conclusions, the only premise left is number 3 which seems to be harmless because it is a substitution instance of premise 2 which is accepted. But on a closer look 3 is only a half truth. This can be seen as follows: Men's knowledge is often incomplete because it selects one aspect or property and does not mention another at the same time. This may have different reasons. Sometimes the reason is just that not all aspects or properties are comprehensible for us at the same time; may be because they are too many or because of deeper reasons like in the case of quantum mechanical objects of which not all properties are available to us by sharp measurement at the same time. Or the reason may be simply ignorance, like in the case that we know some properties of a new elementary particle but not the others. Now all these different kinds of incompleteness and additional other ones are impossible for God's knowledge. Since he has complete knowledge of everything what he has created. Therefore he does know not only that a thing exists but in one act of knowledge also how it exists. Whereas with us it often happens that we know only that something exists because we discover some

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causal effect on other things without any knowledge of what it exactly is and how (with which properties etc.) it exists. From this consideration it is clear that premise 3 is insufficient and incorrect w.r.t. God's knowledge. It is impossible for God's knowledge that he knows that the world (which he created) exists without knowing that the world exists contingently because he knows this in one act of knowing (which moreover includes all other properties of the world). Therefore we have to revise premise 3 by making a more complete instantiation 3* of premise 2: 3*. Necessarily: if God knows that the world exists contingently then the world exists contingently. Symbolically: l(gK that the world exists contingently → the world exists contingently). 3* can be obtained also in the following way. We instantiate premise 2 by l(gKCont(p) → Cont(p)) where 'Cont(p)' means 'contingently p'. For 'p' we can then substitute 'the world exists' which leads directly to 3*. Leaving all other premises unchanged we conclude then as the new conclusion 6*: 6*. Necessarily: the world exists contingently. Contingency can be defined in different ways. Usually one distinguishes the following two kinds which have been used implicitly already by Aristotle:20 Cont(p) ↔ m¬p ↔ ¬lp Cont(p) ↔ (m p ∧ m ¬p) ↔ (¬l¬p ∧ ¬l p) But with both definitions of contingency we obtain the revised conclusion 6*. Moreover it should be mentioned that if Cont(p) is interpreted as ¬lp there is an axiom, which leads from the modal system T (Feys or v. Wright) to S5; it has the following form: ¬lp → l¬lp. Thus this axiom leads directly to the conclusion 6* when ‘p’ is instantiated by ‘the world exists’. The reply to objection 1.14 is therefore this: Premise 3 is incorrect w.r.t. the knowledge of God because it mentions only one part of the truth. But knowing only one part of some truth is impossible for God's knowledge. Therefore the conclusions 6 and 8 are not proved by this argument. If 20

Cf. Hintikka (1973, TaN) ch. 2, especially p. 34.

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however we correct premise 3 by 3* then the right conclusion 6* can be derived from these premises.

18

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2. Whether God Necessarily Knows Whatever He Knows This question can also be expressed as follows: Does it hold, that if God knows something, then he necessarily knows it? Or: Does it hold for everything God knows that he necessarily knows it? Symbolically: ∀p(gKp → lgKp)?

2.1 Arguments Against 2.11 In chapter 1. it was shown that whatever God knows is true. This certainly holds necessarily. Symbolically: l(gKp → p). Now according to an axiom of Modal Logic (which is valid in very many different Modal Logics) l can be distributed to the parts of the implication such that we get: lgKp → lp; which means: if God necessarily knows that p, then necessarily p. Thus if it holds that whatever God knows he necessarily knows, then we can replace 'gKp' by 'lgKp'. Thus by modus ponens it follows that p is necessary, symbolically lp, for every proposition p of which God has knowledge. Consequently there are only necessary facts (expressed by 'lp') or God knows only the necessary facts. Both consequences are absurd. Therefore it is not correct to say that God necessarily knows whatever he knows. 2.12 God's knowledge includes knowledge about himself and about his creation (about the universe). Since God is a necessary being he necessarily knows everything what he knows about himself. But since the world (universe) is not a necessary, but a contingent being, it seems that what he knows about the universe, he does not necessarily know. Therefore it does not seem to hold generally that God necessarily knows whatever he knows. 2.13 God does not necessarily will everything what he wills. Though he necessarily wills his own goodness, he wills things apart from himself (for example the creation of the world) not necessarily, but freely. "Accordingly, as to things willed by God, we must observe that he wills something of absolute necessity: but this is not true of

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all that he wills. For the divine will has a necessary relation to the divine goodness, since that is its proper object. Hence God wills His own goodness necessarily, even as we will our own happiness necessarily... But God wills things apart from himself insofar as they are ordered to his own goodness as their end... Hence, since the Goodness of God is perfect and can exist without other things inasmuch as no perfection can accrue to him from them, it follows that his willing things apart from himself is not absolutely necessary."21 But both God's will and God's knowledge belong to his nature. Therefore it seems also to hold for God's knowledge that he does not necessarily know whatever he knows.

2.2 Argument Pro Although the universe (God’s creation) is contingent God’s knowledge about his creation need not to be contingent too. And since his knowledge belongs to his essence it must be necessary. Therefore it seems to hold that whatever God knows he necessarily knows.

2.3 Proposed Answer God necessarily knows whatever he knows. Symbolically: ∀p(gKp → lgKp). That this is true can be seen by the following indirect proof: (1) Assume that this is not so, i.e. that for some p, God knows that p but not necessarily knows that p; i.e. symbolically: ∃p(gKp ∧ ¬lgKp). (2) From this it follows by modal logic that ∃p(gKp ∧ m¬gKp), i.e. that for some p, God knows that p, but possibly he does not know that p. (3) But this is impossible (i.e. (2) must be false) as can be seen by the following consideration: (a) Since God knows that p (gKp), p must be true (according to the result of Question 1) because otherwise it would not be the case that God knows that p (and then m¬gKp would of course be true). (b) If 'possible' (m) is interpreted with the help of time, then (2) would say that God knows that p and for at least some (period of) time he does not know that p. But this interpretation is impossible for two reasons: First, 21

Thomas Aquinas (STh) I, 19,3.

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because God is outside time, as we assume here. Since we assume in accordance with the Theory of Relativity that time is the time of our universe. And since God created the universe, he created time by creating moving and changing material objects. Second because this would mean that his knowing (and thinking) would be only partially actual and partially potential or like a habitus. But we assume here that there is no potency or habitus in God.22 In contradistinction to that, for mankind's knowledge both is true: that something is known at some time but not at another (for example earlier) time and that knowledge is habitual and not always actual. (c) If 'possible' (m) is interpreted w.r.t. the domain of knowledge, then (2) would say that there is a domain (i.e. for some propositions p), where God has knowledge but contingent knowledge in the sense that he possibly does not know. Now this happens frequently with man: First his knowledge is restricted to a special domain and second within this domain he often has only contingent knowledge; thus he knows a proof for a mathematical theorem, but possibly not, i.e. he may have failed to do the proof. Or he succeeded to design a new experiment with a new result, but he may easily have failed to design it...etc. But for God this kind of contingency is impossible. First his knowledge is not restricted to a special domain since it includes knowledge about himself and about his creation and about things (even universes) which he could have created but did not create. Second, if he is a necessary being, as we assume here, then he must necessarily know whatever he knows of himself; i.e. it is impossible that he would know something about himself what he possibly does not know. But this seems to be equally true w.r.t. his creation; if he has created and designed the universe, he must know it in a most complete way and it is impossible that he would know something of the universe what he possibly does not (would fail to) know. The same holds w.r.t. those things (universes) which he could have created but did not create, because his knowledge cannot be narrower than his power (cf. chapter 5 below). Since the considerations in (3)(a) - (c) show that (2) is false and since (2) follows from (1), the assumption (1) must be false. Therefore the negation of it must be true and so it must be true that God necessarily knows whatever he knows. 22

This is also defended by Thomas Aquinas (SCG) I, 56. There are some recent proposals for habitual or dispositional knowledge of God (cf. Hunt (1995, DOS)). However, the premises on which their arguments are based do not seem convincing.

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2.4 Answer to the Objections 2.41 God’s Knowledge is Complete (to 2.11) The answer to this objection is the same as that given in 1.44 (to objection 1.14 of chapter 1): God's knowledge is always complete. That is if 'p' represents a necessary truth, then God knows that and if it represents a contingent truth, then God will also know that. Thus it is impossible for him to know that p is the case without knowing how and in what sense it is the case. Therefore if necessarily God knows that p is the case, but it is the case not necessarily but contingently, it follows that necessarily: p is the case not necessarily but contingently. Or, in other words, the correct conclusion is: necessarily, p is contingent. And so the conclusion drawn in 2.11 that there are only necessary facts or that God knows only necessary facts does not follow. 2.42 lKp ≠ Klp (to 2.12) Though the argument in 2.12 is logically valid, the third premise of it is the problematic one: First it should be clear that the necessity of knowing should not be confused with the necessity of what is known. Second there is no law with the help of which we could conclude the second from the first or the first from the second. Thus a contingent fact (say a prediction of an eclipse) can be proved logically and mathematically from a dynamical law (differential equation) plus some other contingent facts (initial conditions: constellation of sun, earth and moon). Now the thinking (knowing) involved in the proof process (step by step) can be necessary knowledge though what is proved (predicted) is a contingent fact; whereas the premises are partially contingent (the initial conditions) partially physically (naturally) necessary (the law). On the other hand a difficult mathematical theorem (a necessary fact) may be necessarily known by some mathematicians who know the detailed proof of it but may be only contingently known by others who are learning how to prove it. Therefore in general we will agree that a scientist who invents or succeeds to make a rigorous proof – no matter whether the conclusion is a necessary or a contingent fact – has necessary knowledge w.r.t. the proof process (its detailed steps and its logical interrelations). If we now assume instead of the scientist a perfect being who has created the universe with all its variety and multiplicity it will not be difficult to assume that he can necessarily know everything about the contingent facts of his creation. Therefore nothing hinders that contingent facts of this world are

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necessarily known by God without being known to be the case necessarily. And therefore the conclusion of objection 2.12 is not proved. 2.43 God’s knowledge and will concerning necessity (to 2.13) There is a difference concerning God's knowledge and God's will w.r.t. the question of this chapter. Though God necessarily knows whatever he knows it does not hold that he necessarily wills whatever he wills. Because he necessarily wills his own existence, his own goodness and whatever follows from his essence. But since his creation does not follow necessarily from his essence23, he freely and not necessarily wills his creation. In this sense Thomas Aquinas concludes his article "Whether whatever God wills he wills necessarily?" as follows: "it follows that God knows necessarily whatever he knows, but does not will necessarily whatever he wills."24 Considering the argument 2.13 in more detail the conclusion (of 2.13) is based on the premise "both God's will and God's knowledge belong to his nature." From this the argument infers by analogy that everything what holds for God's will, also holds for God's knowledge. To see that this is not correct, one has to consider two different principles where necessity is involved: In one of them (1) necessity is applied to God's will and God's knowledge; in the other (2) necessity is applied to what God wills or to what God knows: (1a) If God wills that p occurs, then necessarily he wills that p occurs. (1b) If God knows that p occurs, then necessarily he knows that p occurs. Symbolically: (1a) ∀p(gWp → lgWp) (1b) ∀p(gKp → lgKp) Of these (1b) is true as was shown in the answer by an indirect argument. But (1a) is not generally true, since it fails in all the cases where God wills (freely) something apart from himself. (2a) If God wills that p occurs, then God wills that p occurs necessarily. (2b) If God knows that p occurs, then God knows that p occurs necessarily. Symbolically: (2a) ∀p(gWp → gWlp) (2b) ∀p(gKp → gKlp)

23

At least not according to the Christian Doctrine. Though this is so according to the doctrine of emanation ("necessary overflow") of Plotinus. 24 (STh) I, 19,3, ad 6.

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W.r.t. these second principles, (2a) and (2b), the will of God would behave in a similar way as his knowledge: But both (2a) and (2b) are not generally true. Because for some states of affairs only (for those belonging to God's nature) he wills and knows that they necessarily occur. And this kind of necessity is a strong and unconditional necessity. For other states of affairs which obey laws of nature (which may be said to be physically or naturally necessary) he wills and knows that they occur naturally or physically necessary. For still other states of affairs which do not occur necessarily w.r.t. both types of necessity above, he wills and knows that they occur contingently. Thus of the four principles above only (1b) is generally true. The three others only hold for some states of affairs and not for others, thus not for all. Instead of (1a), (2a), (2b) the following principles hold (where 'Nlp' means 'naturally (physically) necessary p'). (1a') ∃p(gWp ∧ lgWp) ∃p(gWp ∧ ¬lgWp) (2a') ∃p(gWp ∧ gWlp) ∃p(gWp ∧ gW¬lp) (2a'') ∃p(gWp ∧ gWNlp) ∃p(gWp ∧ gW¬Nlp) (2b') ∃p(gKp ∧ gKlp) ∃p(gKp ∧ gK¬lp) (2b'') ∃p(gKp ∧ gKNlp) ∃p(gKp ∧ gK¬Nlp)

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3. Whether God Knows Something at Some Time This question can also be expressed as follows: Does it hold for something that God knows that he knows it at some time?

3.1 Arguments Pro 3.11 Since God is eternal, what he knows about himself he does not know at some time. But since the events of this world are at some time, what he knows about the events of this world he seems to know at some time. Therefore there seems to be something that he knows at some time. 3.12 If everything is true that God knows, then God cannot err about past events. But to have knowledge without error about past events implies to know at what time they happened. Now as it was shown in chapter 1., everything is true that God knows. Therefore God knows something (past events) at some time. 3.13 Analogously to 3.12 we can argue about present events, because to know about present events also implies to know at what time they happened. Therefore God knows something (present events) at some time.

3.2 Argument Contra Any point of time belongs to (lies on) some time scale. Every time scale distuingishes between earlier and later or between past and future. But eternity has no past or future or earlier and later. Therefore, since God himself and his activity like his knowledge is eternal it cannot be at some time.

3.3 Proposed Answer What God knows he does not know at some time. This answer can be substantiated as follows. First (3.31) it becomes clear with the help of a distinction. Second (3.32) it can be shown by an analysis of time.

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3.31 Knowing at some time and knowing that something happens at some time To know that something happens at some time does not mean to know this at some (the same) time. More accurately: To know that p is the case at time t is not the same as (and does not imply) to know at time t that p is the case. Symbolically: aKpt is not the same and does not imply: aKt p. Nor does it imply: aKt pt. This can easily be seen from the fact that in the first case (aKpt) the time operator is attributed to the event (or to the proposition describing the event), whereas in the second case it is attributed to the knowledge (or to the action of knowing). From this it is plain that if God knows that some event happens at time t, it does not follow from this that his action of knowing also happens at some time. In other words it would be a logical fallacy to infer aKt p (God knows at t that p) from aKpt (God knows that p occurs at t). Thus it is very well compatible that God knows at what time (of this world) certain events occur with the thesis that his action of knowing does not occur at a certain time. However it should be added that concerning human knowledge it is correct that knowing (in the sense of actually knowing, not in the sense of dispositionally knowing) that p is the case at t1 implies that the action of knowing occurs at a certain time t2. If pt1 is a contingent event (fact) of the external world and mediation of senses and brain processes are necessary for recognition, then t1 will not be simultaneous with t2 but earlier, since every causal propagation needs time (according to the Special Theory of Relativity). Whether it could be simultaneous if the event and the action of knowing are reflecting mental processes, like in cases of introspection, is an open question. 3.32 Analysis of Time An analysis of time shows that time can be understood in a threefold way: 3.321 as time of this world (universe), 3.322 as a chronological order in the logical or mathematical sense, 3.323 as biological or psychological time. 3.321 Time of this world (universe) Time as time of this world has the following characteristics which show clearly that God cannot be subject of this time: (i) It belongs to this world, it is bound to this world; i.e. there cannot be time "outside" this world or independent of this world. In this sense

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already Aristotle, who had also a relativistic concept of place25, defines time as the measure of change w.r.t. earlier and later26. That means that time is bound to change and movement in this world in such a way that change or movement are necessary conditions of time. This is also in accordance with the Theory of General Relativity according to which time is a component of spacetime and spacetime is dependent on the matter distributed in the universe (i.e. it is curved spacetime). (ii) The time of this world is not Newton's absolute time, but is relative; i.e. every physical system (say a planet or a planetary system or a cluster of stars) which is in movement w.r.t. another has its own time. Assuming inertial systems in the universe (valid only very locally) universal time would still be definable via Einstein Synchronisation. Newton thought that there is absolute time and that absolute time "flows equably": "Absolute, true, and mathematical time, of itself, and from its own nature, flows equably without relation to anything external..."27 But in fact as we know from the Theory of General Relativity, there is no such time in our universe. The time of our universe is according to General Relativity relative, it flows unequably, and it flows with relation to the distribution of matter and the fields, forces and boundaries produced by it. This does not rule out of course that locally time flows (approximately) equably. Assuming the Cosmological Principle one could defend Friedmann-universes which allow a universal time. This assumption is however very questionable; since if there are rotating subsystems in the universe, then no standard synchronization or universal time can be established. (iii) According to the Theory of General Relativity the space of the universe is closed (even if the universe is expanding); that is, there are closed spatial coordinates or closed space-like geodesics. On the other hand, we usually assume that the time coordinate is not closed; i.e. we assume that there are no closed time-like geodesics. This assumption has been called the chronology condition of spacetime.28 However, even if the time coordinate would be closed, as long as the period would be much greater (for several orders of magnitude) than the life time of the universe so far, it would not make any difference for all practical (experimental) purposes. 25 26 27 28

Cf. Jammer (1954, CSP), ch. 1. Aristotle (Phys) IV, 220a25. Cf. Mittelstaedt/Weingartner (2005, LNt), ch. 6.3.2. Newton (Princ) I, Scholium. Cf. Mittelstaedt (2008, CTP) Cf. Hawking, Ellis (1973, LSS), p. 189.

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(iv) Time as time of this world is measured in units. These units are taken from two kinds of physical processes (changes) which have a certain regularity: periodic processes where the state of the system repeats itself after a finite period of time and monotone processes. Examples for the first are day, year, pendulum, crystal clock, atomic clock (which gives the most exact measurement of time units so far). Examples for the second is the uniform movement of a body on a straight line (equal distances in equal times). The main point to recognise concerning units of time is that they are conventional to some extend – as was observed by Poincaré and Mach – and more importantly that they are relative and not absolute. Thus the most exact atomic clock has to be referred to sea level because of its dependence on gravitation. In addition fast clock transport changes the units of time (see (v)) below. (v) Like there is no absolute or universal unit of time, there is also no absolute or universal concept of simultaneity, except for inertial systems. Thus there can be dilatation of time depending on the velocity of the physical systems (parts of the world) relative to each other. In a similar way the units of distance and of mass are not absolute, but there may be contraction of length and increasing of mass depending on movement. In general: There are no freely movable measuring rods or clocks which are rigid, i.e. resistant against change (movement).29 From this analysis of "real" time, i.e. of the time of our universe, it is clear that this time cannot be attributed to God or to God's knowledge. This can be seen by considering properties (i) to (v) of the time (of this universe): ad (i): If time is a property of this universe and God has created this universe, then he has created time by creating a changing and developing universe. This is also the view of the great Christian philosophers, like Thomas Aquinas: " The phrase about things being created in the beginning of time means that the heavens and earth were created together with time; it does not suggest that the beginning of time was the measure of creation."30 But if this is true, then it makes no sense to attribute this (created) time to God or to his actions. ad (ii): Since every system of the universe (earth, stars etc.) has its own time (depending on the different movements of the systems) and the whole universe could have at most an average of this relative times, but cannot have a universal time, it seems the more absurd to attribute one such time (which one?) to God. 29

For details see any textbook on Special Relativity. For Space Time invariance of laws of nature cf. Mittelstaedt/Weingartner (2005, LNt), ch. 6. 30 Thomas Aquinas (STh) I, 46,1 ad 4.

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ad (iii): Independently of whether the time coordinates (time-like geodesics) are closed or not, they imply the distinction of earlier and later or of past and future (even if only locally in case the time-like geodesics were closed). But if God is eternal, past and future cannot be attributed to him (cf. 3.3221 below). ad (iv): If the time of this world can be attributed to God (or to God's actions), then also the time units or time standards (measured by regular processes of this world). Now as we know from the Theory of Special Relativity there is no universal time and no universal or rigid time unit, i.e. time (and its units) are dependent on movement such that there may be time dilatation. But it would be rather absurd to attribute such a relativity of time units to God. Therefore it would be absurd to attribute the time of this world to God. ad (v): An analogous consideration is concerned with simultaneity: If the time of this world can be attributed to God (or to God's actions), then also simultaneity. However, it is clear from the Theory of General Relativity that there is no universal concept of simultaneity in a universe with gravitation and rotation; thus it would be rather absurd to attribute to God (or to his actions) such a relative concept of simultaneity. Therefore it would be absurd to attribute the time of this world to God. 3.322 Time as a Chronological Order (i) Time as a chronological order (with a binary relation earlier than or later than) in the logical or mathematical sense is usually described by at least four properties which are formulated with the help of axioms: Irreflexivity, Transitivity, Asymmetry, i.e. Partial Ordering and Density. Time in this sense is discussed in Tense Logics.31 It has several additional characteristics which we list under (ii) - (iv) below: (ii) This kind of chronological time is not relative in the way the time of our universe is; i.e. it is assumed to "flow equably" and not to be dependent on movement (velocity) or on matter and its fields. In this sense it is a kind of "absolute" or "conceptual" time. It is also assumed that this kind of time has a unique (or universal) concept of simultaneity and conceptual clocks which are rigid and which are not dependent on gravitation or fast transport. (iii) This kind of chronological time may be applied to real processes in order to describe the successive series of human actions or other processes in nature and technology of everyday life. If the domain is 31

One of the first to do this with the means of Symbolic Logic was A.N. Prior. Cf. Prior (1957, TMd). For later development cf. Van Benthem (1991, LgT).

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restricted in such a way, the aspects of Special and General Relativity have no impact because of the restricted "locality". Thus for time tables of buses and trains and even of aeroplanes ideal or rigid clocks are sufficiently accurate. But nevertheless for any time measurement one has to use physical or biological clocks (even if they are understood as purported ideal and rigid clocks) which refer to the revolution of the earth or to some other approximately constant (and usually periodical) process. Otherwise chronological or conceptual time is not applicable at all. (iv) Although chronological time is not bound to the change in our universe it is still bound to some kind of local change. This change might be change in our (human) environment or change of our actions which involve physical, physiological (for example brain waves) and mental changes or change in a purely idealised mental sense. Thus also chronological time has change as its necessary condition. (v) Also chronological or conceptual time involves earlier and later or past and future. The distinction of past and future is necessary for any kind of time and is also implied by any kind of change (recall 2.321(i) and ad (iv)). That time involves past and future (earlier and later) is evident from both time (understood) as time of our universe and time (understood) as chronological time: Every dynamical law describes the time development of a physical system in such a way that the state S1(t1) of the system at time t1 (in the past) corresponds to a solution of the differential equation and the state S2(t2) of this system at t2 (in the future) corresponds to another solution of the differential equation. In this case state S2(t2) of the future can be predicted with the help of the dynamical law and state S1(t1) of the past. In a similar way other laws, besides dynamical also statistical laws, describe the time development of systems in the universe, even if in the case of statistical laws the predictions concern a huge ensemble and not states of singular objects. Also time in the sense of chronological order involves past and future. This is clear from its underlying axioms (for example partial ordering) and from its application to processes be they processes of nature, of technology or of human (inclusive mental) activity. From the above analysis of chronological time it is also evident that this time cannot be attributed to God or to God's knowledge. This can be seen as follows: According to (i) time as chronological order is characterised by four axioms. Now the axiom of partial ordering implies the distinction between earlier and

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later or past and future; since it says that event A is earlier (later) than event B or B is earlier (later) than A or A and B are simultaneous. But as it was said above, since God is eternal and a reasonable concept of eternity does not involve past and future, chronological time cannot be attributed to God. According to (iv) chronological time implies some kind of change: even if it is not bound to the change of (in) our universe, it is measured by clocks which are changing in physical or biological systems. Now every change implies the distinction of past and future (v). But since God is eternal, there cannot be past and future in him or in his knowledge. This is so, since any reasonable concept of eternity is distinguished from infinite time in that it lacks past and future, whereas on the other hand every concept of time (finite or infinite) involves past and future or earlier and later. Therefore also chronological time cannot be attributed to God. 3.3221 Eternity – no succession, no past and future That a reasonable concept of eternity does not involve change nor succession, nor the distinction between earlier and later or between past and future or between beginning and end was defended since Augustin and Boethius: "Tempus autem quoniam mutabilitate transcurrit, aeternitate immutabili non potest esse coaeternum."32 "Eternity, then, is the total and perfect possession of life without end, a state which becomes clearer if compared with the world of time; for whatever lives in time lives in the here and now, and advances from past to future."33 Thomas Aquinas agrees with Boethius that eternity is simultaneously whole. Moreover he points out that it would not be sufficient to say that eternity has neither beginning nor end (whereas time has). Since some assume that movement (in the universe) goes on forever. Under such an assumption time could not be the measure of the whole movement, because an infinity is not measurable, but it could measure finite parts, i.e. periods, revolutions, which have a beginning and end: "Because granted that time always was and always will be, according to the idea of those who think the movement of the heavens goes on forever, there would yet remain a difference between eternity and time as Boethius says (De Consol. V), arising from the fact that eternity is simultaneously whole;

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Augustine (Civ) XII, 16. Boethius (Cons) V, 6.

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which cannot be applied to time."34 "Thus eternity is known from two sources: first because what is eternal is interminable – that it has no beginning nor end (that is, no term either way); secondly, because eternity has no succession, being simultaneously whole."35 3.3222 Universe – finite in time Concerning the question whether the universe is finite or infinite in time, Thomas Aquinas defended that our universe has a finite age. But he gave reasons that this fact cannot be proved rigorously from our knowledge about the universe where a rigorous proof is understood as a derivation from laws (in this case laws of nature). In his quarrel with Bonaventura at the University of Paris he defended the view that the beginning in time of the world (universe) cannot be proved from universal principles (laws) of (about) this world. Because universal principles which have their foundation in the essence of things (or nature) abstract from hic (place) et nunc (point of time). That is, laws of nature are spacetime invariant and therefore we cannot extract a certain point of time (beginning of the universe) or a singularity from a law: "That the world has not always existed cannot be demonstratively proved, but is held by faith alone. ... The reason is this: the world considered in itself offers no grounds for demonstrating that it was once all new. For the principle for demonstrating an object is its definition. Now the specific nature of each and every object abstracts for the here and now, which is why universals are described as being everywhere and always. Hence it cannot be demonstrated that man or the heavens or stone did not always exist."36 In this connection I want to mention that the question whether it can be demonstrated that the world has always existed or that it has a beginning in (with) time – answered differently by competing theories of the universe – is a question about the completeness of the laws of nature. Or at least of that laws we know. A system of laws L about a certain part P of reality is complete if and only if every truth about P is provably (derivable) from L. Thomas Aquinas' standpoint was that the universal laws of nature (about this world) are not complete with respect to all questions (all truths) about this world. It is not just our insufficient knowledge of the laws of nature what he 34 35 36

Thomas Aquinas (STh) I, 10,4. Ibid. 10,1. Thomas Aquinas (STh) I, 46,2.

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has in mind, but the true laws itself are incomplete according to him with respect to some special questions. That means that there are some statements about this world which cannot be decided with the help of the laws about this world. Or in more modern terms: the laws of nature are incomplete with respect to some important initial conditions. This problem plays an important role in the Big Bang Theory of the cosmological evolution in respect to (at least) the "first three minutes".37 3.323 Time as Biological and Psychological Time Periodic or oscillating processes can be observed on all levels of living organisms. The frequency spectrum extends from milliseconds to hours, days, weeks, months and years. The timer or "biological clock" for these periods or frequencies is localised in the organisms (even in proteins of tissues and organs) though connected or partially synchronised with periods from external factors; here light and temperature are the most important external timers w.r.t. the unit of approximately a day (circadiane); where the deviations in areas far from equator are adapted by the organisms. The organisms however do not follow passively these environmental periods, but have developed (during evolution) their own biorhythm, which is with higher organisms very often a circadiane rhythm. An example is the daily movement of the leaves of plants. Another example for such a biorhythm or "biological clock" in mammals is the production of the hormone melatonine in the pineal gland during the dark period. Moreover these "biological clocks" are not only dependent on environmental factors but also on the specific DNA.38 From these facts it is clear that like in the physical world also in the biological world there is no absolute time; and what the Theory of Special Relativity tells for the physical world – that every huge physical system (say planetary system or system of stars) has its own time – holds here too, analogously: every species of organisms has its own time. And both the units of time and the uniformity of flow are relativised to internal periodic processes of the 37

Cf. Weinberg (1977, FTM). That we cannot decide concerning such initial conditions (like the question whether the universe has a certain age) was true until very recently indeed when such things as the cosmic background radiation have been discovered (by Penzias and Wilson in 1965) which is a rather strong support for the finite age of the universe – even if we could not say it is an absolute proof in the sense of demonstration from laws. Since there are also consistent theories of cosmology without a Big Bang. 38 Cf. Baumgartner (1994, ZBZ), especially chapter 2 and 3. For more information see Moore-Ede/Sulzmann (1981, ITO) and Treismann et al. (1990, ICl).

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organism and external periodic processes with an approximate synchronisation between them. A further sign for the relativity is the different lifetime of different organisms. Some bacteria live 20 minutes before they split, flies live one day, mice live 100 days, humans live up to about 100 years, and sequoias live 4000 years. Does time flow equably for all these different living creatures? Humans know that their experience of how fast time flows is different when waiting and different when engaged in an interesting and exciting activity. Of a two-week holiday the first week is usually experienced as passing more slowly than the second. The same seems to hold for earlier and later periods of life. This is so despite the fact that humans use clocks and watches which show them (locally) objective units of time or time intervals. The successive character of psychological time is experienced very clearly when we consider to mobilise means in order to reach a goal. The general underlying structure of such an experienced succession is the axiom of partial ordering even if the intervals may be sometimes experienced as stretched or contracted. This shows that time in human experience (psychological time) is such that both its units and the uniformity of its flow is relativised to the kind of experience, to the kind of age, to the kind of culture etc. Moreover it is plain that, in contrast to biological and psychological time, physical time is clearly more objective, even if it is also relative w.r.t. units, to simultaneity and flow. From this analysis of biological and psychological time it will be clear that such a time cannot be attributed to God or to God's knowledge. This can be seen from the following reasons: (i) As the considerations above show biological and psychological time are still much more depending on local and specific differences – like change of the length of the day or DNA of the respective species – than physical time of this world. But physical time of this world (universe) cannot be attributed to God as has been shown above (3.321). Therefore all the less biological or psychological time can be attributed to God. (ii) From all that we know about biological and psychological time – and Chronobiology and Psychology know a lot more than what has been touched above – it seems much more reasonable to assume that God has designed the time schedule and the "biological and psychological clocks" when creating these organisms than to think that we should apply these relativised kinds of time units and time flows to God himself or to his knowledge.

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3.4 Answer to the Objections 3.41 (to objection 3.11) The second premise of this argument is a fallacy. The mistake involved in this fallacy is already clarified in 3.31. From knowing that some event p happens at some time (Kpt), one cannot conclude that the action of knowing happens at some (the same or other) time (Kt p). This is a confusion of two different time indices, where one belongs to the event and the other to the action of knowing. Now although for humans it is factually true that if they know that something, p, happens at some time t, then they also know at some time t, that p(is the case) – this does not hold in general. Especially it does not hold for God: Though he knows that a certain event p of this world happens at time t of this world, pt (i.e. some specific time measured by some kind of clock) and also that pt1 happens earlier than event qt2 (in this world), it does not follow from that that his knowledge is at a certain time. Therefore the second premise is false and the conclusion in 3.11 is not proved. 3.42 (to objections 3.12 and 3.13) It is correct to say that to know about past (present) events implies to know at what time they happened (happen). But to know at what time event p happens, does not imply to know this at some (this) time, as has been clarified in 3.31 and 3.41 above. Thus if God knows that the past event (present event) happens at t (of this world) it does not follow that he knows this also at t (of this world) or at some other time of this world.

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4. Whether God Knows All Past and Present Events This question can also be expressed in the following way: Is it correct to say that if event e occurred in the past or event e occurs at present, then God knows that e occurred in the past or at present? Or symbolically, where pt ≤ 0 means that event e occurred at t < 0 (past) or t = 0 (present): ∀p(pt ≤ 0 → gKpt ≤ 0)?

4.1 Arguments Against 4.11 That God knows all past and present events means that if an event occurred (occurs) in the past or at present, then God knows it. Thus the occurrence of the respective event is the reason or cause for God's knowledge of it. But this is impossible, since God has created the world in which all past and present events occur. Therefore it does not seem to be right to say that God knows all past and present events. 4.12 If the knowledge of God is the cause of things (events) and if God knows all past and present events, then God is the cause of all past and present events. Now according to Thomas Aquinas "the knowledge of God is the cause of things".39 Thus God seems to be the cause of all past and present events. But this is impossible since under the past and present events there are free immoral actions (sins) of men which cannot be caused by God. Therefore it cannot be true that God knows all past and present events.

4.2 Arguments Pro God's knowledge must be much more perfect and much more complete than man's knowledge. And on the assumption that God has created this universe, his knowledge about its (past and present) events must be most perfect and most complete. Now to know the past and present events might be possible in principle for man even if it is not possible in fact because of the huge number of past and present events (of this universe) and also because of the many hidden parameters not known so far. 39

Thomas Aquinas (STh) I, 14,8.

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So much the more God must know all past and present events.

4.3 Proposed Answer God knows all past and present events. This can be supported by the following two indirect arguments: (1) Suppose there would be an event (in the past or at present) such that God does not know that it occurred. Since the events with which we are concerned are events of this world (universe), the time at which they occur is also the time of this universe (cf. ch. 3.32). Thus the events and their occurrence at some time belong to this world (universe). To be ignorant of one of the events of this universe would mean that God has insufficient or incomplete knowledge of his own creation. And since we assume (see introduction) that God has really created this world – not out of anything and he has not merely given a structure to something already there – it is impossible that he is ignorant of some part of his creation. Therefore God knows all past and present events (of this world, universe). Observe that "real creation not out of anything" does not yet determine further properties. It does not presuppose that creation is finished at a certain time and is perfectly compatible with a long evolutionary process in which also degrees of freedom and chance have their place. It is only incompatible with an evolution out of not anything (out of "nothing") by chance. Because in this case the concept of chance seems to be problematic, if not inconsistent, since any definition of chance presupposes things, objects or states (which again presuppose things or objects) which are already there. This can be seen by an analysis of the concept of chance. A wider definition of chance w.r.t. events is this: event e2 happens by chance after (or w.r.t.) event e1 iff there is no dynamical (deterministic) law such that we could explain and predict e2 with the help of e1 plus the respective law. A more narrow definition is: e2 happens by chance after (or w.r.t.) events preceding e2 iff there are neither dynamical nor statistical laws such that e2 could be explained or predicted with the help of such laws and preceding events. A different definition is that of a chance sequence (i.e. a sequence of events by chance) due to Chaitin: A chance sequence is a sequence of objects (numbers, letters, things, ... in the simplest case a 0-1-sequence) for which there is no description or code or computer program which is shorter than the sequence itself. All three definitions of chance or more accurately of "event by chance" presuppose objects or events or states (composed of objects) where these objects are real objects, not merely conceptual objects like numbers. Thus

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if there are no objects of any sort, there cannot be states or events and consequently chance cannot be defined. Also an event or state cannot develop by chance out of nothing (out of not anything) since there is only chance for an event w.r.t. earlier events. Since real creation (not out of anything) does not presuppose objects, events or states which are already there, real creation cannot occur by chance. (2) If there is an event (in the past or at present) of this world of which God would not know, then God would not know his own power. Because God can know his own power only if he knows to what his power extends. But his power extends first to all facts of his creation (of the world) w.r.t. past and present and second, moreover to possible facts which are compatible with his Essence and with God's plan and commands concerning his creation.40 Therefore being ignorant of some event (in the past or at present) of this world would mean to have insufficient (and not perfect and not complete) knowledge of the extension of his power concerning the world. But this is impossible for God as a perfect being. Since a perfect being must know his own power. Therefore we have to assume that there is no event of this world (in the past or at present) of which God would not have knowledge. And consequently: God knows all past and present events (of his creation).41

4.4 Answer to the Objections 4.41 (to 4.11) The correct meaning of the statement "God knows all past and present events" is an if-then statement (implication) in the sense: If event e occurred in the past or if event e occurs at present, then God knows that e occurred in the past or that e occurs at present. Symbolically: ∀p(pt ≤ 0 → gKpt ≤ 0). And we may add: If event e occurred in the past or event e occurs at present, then God also knows how e occurred in the past and how e occurs at present. But an if-then statement (implication) does not necessarily represent a causal relation. Although some causal relations are formulated as implications; but also in this case the meaning of an implication (given by two valued truth 40

Cf. Weingartner (2003, EDK), ch. 7.451 and 7.452 and Appendix Def. 4, and this book ch. 11 and chs. 13.22 and 13.23. 41 If in addition to this world (universe) other parts of creation (like that of immaterial spirits or angels) are included, the extension of God's power will also include this part of creation.

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tables or by rules in Classical Logic or by other matrices or rules in Weaker Logics) is not sufficient to describe a causal relation; it can be at best a necessary condition for such a relation. Thus the second premise of the argument in 4.11 is false and therefore the conclusion is not proved. 4.42 (to 4.12) The knowledge of God is not a sufficient cause of things (events) of his creation but only a necessary one like the knowledge of the designer is a necessary condition for the things he designs. Thus Thomas Aquinas says in the same article cited in 4.12: "His knowledge must be the cause of things, insofar as his will is joined to it."42 Therefore God can have knowledge of free immoral actions of men without causing them. Since concerning immoral actions he neither wills that they occur nor wills that they do not occur but permits them to occur. Because if he would will that they do not occur, they would not occur, since his will is always fulfilled. Therefore he keeps back (or keeps off) his will w.r.t. free actions of men since he is not an allwilling and not an allcausing God because he has given the ability of causing to his creatures, and that of freely causing, to some of them.43

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(STh) I, 14,8. For a detailed discussion of the wrong thesis of an allwilling and allcausing God – which is not a thesis of the great religions Judaism, Christianity and Islam anyway – and for its connection with religious fatalism see Weingartner (2003, EDK), ch. 6.4. 43

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5. Whether God's Knowledge Exceeds His Power God's power extends to all those states of affairs which he can bring about (or can make to occur) but does not need to actually bring about (does not need to make occurring). Thus this question can also be formulated in this way: Do the states of affairs which are known by God exceed the states of affairs which he can bring about (can make) including those he actually brings about (actually makes occurring).

5.1 Arguments Against 5.11 God's power (or omnipotence) exceeds the facts; that is he can bring about (cause) states of affairs which are not realised for instance that a further species of animals occur under the living species. But what is not the case (that this further species lives) cannot be known by him, otherwise he would know something which is false, which is impossible (under the supposition that he is omniscient). Therefore God's power exceeds his knowledge. 5.12 Whatever God knows is true (see ch. 1.3); and whatever is true corresponds to a fact. Whatever comes under God's power (i.e. whatever God can cause or can will) must be consistent with God's essence. But there are states of affairs that are consistent with his essence and which come under God's power (like creating another world) although they are not realised facts; and thus they cannot be known by him. Therefore God's power exceeds his knowledge. 5.13 There seem to be states of affairs (events) which God can (could) will to bring about (can cause) but which he (in fact) does not will to bring about: "So nothing prevents there being something in the divine power which he does not will."44 But under those events which God does not will to bring about are those which nobody else brings about. And of these it cannot be known that they are. Therefore God's knowledge does not exceed God's power. 44

Thomas Aquinas (STh) I, 25,5 ad 1.

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5.14 Whatever God knows he knows necessarily (ch. 2). Among those states of affairs which can be willed by God some are willed necessarily (all those concerning his essence) and some others are either willed not necessarily (those concerning the world) or can be willed not necessarily (those which could be realised but are not). Therefore what God can will (God's power) exceeds God's knowledge.

5.2 Arguments Pro 5.21 Whenever an event (human action) is a moral evil, then God does not will it and God cannot will (and cannot cause) it. This is so for two reasons: first because everything that God can will (can cause) and everything that he wills (or causes) is good. Secondly because God has created man with free will which allows him (man) also to commit morally bad actions. And thus it would be inconsistent to create man with free will and give him moral commands on the one hand and cause moral evil on the other. But the events which are moral evils committed by men are known by God. Therefore God's knowledge exceeds God's power.

5.3 Proposed Answer God's knowledge exceeds God's power. That this is so can be substantiated by showing that the states of affairs (events) which are known by God exceed the states of affairs which he can bring about (can cause or can will). God’s knowledge extends to four great domains: His knowledge about himself (about his essence), his knowledge about logic and mathematics, his knowledge about his creation and his knowledge about all possibilities consistent with his essence and with logic and mathematics. Now his power concerns only his creation and the possibilities consistent with his essence and with logic and mathematics; it does neither concern himself (his essence) nor logic and mathematics. Therefore God’s knowledge exceeds God’s power. Moreover it can be shown that also concerning the domain of creation (and creatures) and that of possibilities God’s knowledge exceeds his power. This will become evident from a definition of omnipotence. We shall therefore first give a definition of omnipotence and then consider the parts of its definiens w.r.t. the question whether God's knowledge exceeds his power.

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5.31 Definition of Omnipotence (God's power) Def. 1: God is omnipotent iff (1) Whatever God wills is realised and (2) God can cause (can will, can make) every state of affairs (events) which (a) is self consistent and (b) is compatible with God's essence and (c) is conditionally compatible with God's providence and (d) is compatible with God's commands. In this definition condition (a) follows from condition (b); such that one could dispense with (a). However, for a detailed discussion it seems better to list all the four conditions. A short elaboration of these conditions is as follows: (1) states that God's will is always fulfilled, i.e. if God wills that a certain state of affairs (a certain event) p occurs, then p occurs. Or in other words: it is not the case (in fact: it cannot be the case) that God wills something, say that event e occurs, but event e does not occur. This frequently happens with the will of men that their desires are not realised, but it cannot happen to an omnipotent (or almighty) God. It could be objected that God wills that man does not sin, although man sins. But in this case we have to assume that man's action is a free will decision, otherwise it could not be a sin. And concerning sins that occur, God neither wills that they occur nor wills that they do not occur (otherwise they would not occur); because he is neither an allwilling nor an allcausing God and can keep off his will from such free actions. Therefore we have to say that concerning morally relevant actions or concerning what he wills according to his commands (towards man) God wills that man should will and should act in a morally good way. But from "God wills that man should will (act) morally good" it does not follow that "God wills (directly) that man wills (acts) morally good". And although it follows that an action which is in accordance with what God wills that man should will and do is morally good, it does not follow that the respective action is actually committed by man, since it is still a free will decision.45 ad (2): But God's omnipotence is not restricted to what he wills or to what he causes; i.e. it is not restricted to the effects of his will and cause: "Whence it follows that His effect is always less than this power."46 Therefore in part (2) of the definiens four necessary (together sufficient) conditions are 45

For further details see Weingartner (2003, EDK), chs. 6.4, 7.45; Appendix, definitions D4, D7, D11, D12. Cf. below ch. 13, D13 and D15 and 13.25. 46 Thomas Aquinas (STh) I, 25,2 ad 2.

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listed for God's power in the sense of what God can cause (or can will or can make or can bring about). Although he does not need to, in fact, cause it (or will it, or make it, or bring it about). Before I shall comment on the four conditions it is necessary to say something about the relation of causing and willing in God. According to the definition D5 of ch. 13 below “God causes that p” (gCp) is defined as “p belongs to the theorems about (God’s) creation and God wills that p” (gWp). Thus it holds that whatever God causes he wills. But the opposite does not hold since God (necessarily or by his own nature) wills (and can will) his own existence and his goodness but he does not (and cannot) cause it. Also as a first cause he is only related to his creation and creatures but not to himself. The socalled “causa sui” is an invention of Modern Times (cf. Spinoza, Ethics, Def.1) but in Aristotle and in the Middle Ages the causal relation is always irreflexive. Similarly it holds: Whatever God can cause he can will; or: whatever he cannot will (or does not will) he cannot cause (or does not cause). Therefore concerning his creation, what he causes and what he wills coincide and since his power is also concerned with his creation what he can cause and what he can will coincide too. The first condition says that what God can cause (can will, can make, can bring about) must be self-consistent. Inconsistent states of affairs (events) cannot be caused and cannot be done by a perfect person. In other words God's power is bound to consistency. But observe that this is only a sign of perfection since only imperfect beings like humans can have inconsistent thoughts. The second condition says that what God can cause (can will, can make, can bring about) has to be compatible with God's essence. We may express this also by saying that it has to be compatible with all what God necessarily knows and necessarily wills about himself. In this sense it is impossible that God could cancel his own existence or his own goodness. The third condition says that God can cause (can will, can make, can bring about) only states of affairs (events) which are conditionally compatible with his providence. God's providence is his plan concerning creation. It does not concern himself (his essence) and it does not concern the laws of logic or of mathematics. We may give a definition of providence as follows: Def. 2: A state of affairs or an event p belongs to God's providence iff (1) God knows that p and (2) p belongs to the theorems about creation (3) God leads p towards its goal or God subordinates p under some goal or higher good or wills conditionally that p in order to satisfy a higher good.

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(4) God permits that p. Thus what God can cause (can will, can make, can bring about) has to be compatible with God's knowledge (Def. 2 (1)). Observe that also events which possibly occur but do not in fact occur are included in God's providence via his knowledge: he knows that (and how) they are possible. God’s providence is concerned with creation (Def. 2.(2)). Thus God’s power or omnipotence must be compatible with what happens through his creation, i.e. with all the past and present events of creation (cf. ch. 4) and (conditionally) with all the future events which can be in his plan and knowledge (cf. chs. 10 and 11). Condition (3) says that every state of affairs belonging to God’s providence is one which is led by him towards its goal or is subordinated under some goal or higher good or is willed conditionally to satisfy a higher good. Thus he leads man by his commandments and relevation towards his goal (without violating man’s free will) and he leads living creatures without intellect to their goals by giving them instinct and abilities to learn and improve. The same is true for those states of affairs (events) which God wills conditionally for some goal, viz. for some higher good. For example he wills just punishment because he wills the higher good (goal) of ultimate poetic justice; or he wills that some creatures are less perfect than others because he wills the multiplicity and differentiated structure of the universe; or he wills that living organisms learn by trial and error because he wills that imperfect creatures contribute to their own development and to the development of the whole universe. According to Inwagen47 free will, natural indeterminism and the initial state of the created universe do not belong to God’s plan or providence. The reason for such a view could be that Inwagen identifies God’s plan (God’s providence) with God’s will. But although it holds that whatever God wills concerning his creation belongs also to his providence, the opposite does not hold; i.e. there are some states of affairs, like occurring moral evils, that belong to God’s knowledge and to God’s providence but not to his will since he does not will them to occur nor does he will them not to occur but he permits them to occur. And concerning his will it holds that his will is always fulfilled viz. whatever he wills is the case. So in this respect there is no place for freedom, indeterminism or chance. Now concerning man’s actions of free will we have already said that God’s commands which belong to his providence mean that God wills that man should will and act accordingly; but it does not mean that God wills (directly) that man wills and acts that way, 47

Inwagen (1995, GKM) p.54

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otherwise man would always obey God’s will without having freedom (cf. ch. 13, definition 15 about God’s will w.r.t. man). Concerning indeterminism we may say that God must have created the world in such a way that there are degrees of freedom in reality already at the level of non-living creatures in all the domains where we use statistical laws in physics (cf. ch. 7 below). And on higher levels w.r.t. living creatures they are created in such a way that they learn via trial and error which again presupposes degrees of freedom and that they got the ability of being causes and participating in the development of the world. Concerning the chance of the first initial conditions todays cosmology knows very little about the questions what kind of chance this really was. What is rather certain is that these first initial conditions which include also important constants of nature can hardly have alternative values if only a very weak version of the anthropic principle is correct.48 Thus why could not something like the anthropic principle belong to God’s plan or providence! What God permits, belongs to his providence but not necessarily to his power (Def. 2 (4)). Thus he permits that events occur which are moral evils. That means that he does not will that such events do not occur. Because otherwise moral evil would not occur since his will is always fulfilled as condition (1) of the definition of omnipotence says. And then he would prevent with his will the free actions of men which would be inconsistent with his will concerning creation, since he endowed men with free will. Moreover he does not will that moral evils occur, and he cannot will that moral evils occur, since this would be inconsistent with his essence. From this together with his permission of moral evil it follows that he keeps off his will from moral evil: neither he wills that moral evils occur, nor he wills that they do not occur. Which also means that he is neither an allwilling, nor an allcausing God.49 Coming back to the fourth condition of the definition of omnipotence (Def. 1) we observe that it is concerned with God's ethic and moral rules w.r.t. his creation. It states that God cannot will (cannot cause, make or bring about) states of affairs (events) which violate his ethical or moral rules. And this follows from the fact that his will cannot be normatively inconsistent. It would be normatively inconsistent if he would will that man should act that p is the case (honour your parents) but would prevent p or cause non-p. 48

Cf. Barrow and Tipler (1986, ACP) p.16. Mittelstaedt, Weingartner (2005, LNt) ch. 8.2.2.1 49 For a detailed discussion of the wrong thesis (or assumption) of an allwilling and allcausing God (also in connection with religious fatalism) see Weingartner (2003, EDK), ch. 6.4 and Appendix: Axiom 4 and theorems T68 and T89. Cf. ch. 13.25 below.

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5.32 God's knowledge exceeds his power This can be seen from an examination of this question in reference to the parts of the definiens of God's power (omnipotence): Condition (1) is restricted to those states of affairs (events) of which God wills (causes) that they occur. Since this range of states of affairs (events) is included in that of condition (2) – i.e. in the states of affairs (events) which God can will (or cause) – we need not to be concerned with (1). But independently of that God's knowledge exceeds God's power (omnipotence) w.r.t. condition (1), because he knows also those occurring states of affairs (events) which he does not will (which he does not cause) like moral evil. Condition (2a) of Def. 1 of omnipotence We can distinguish three cases: (α) Whatever God can cause (can will, make, bring about) must be logically consistent or self-consistent. In this case God knows that it is logically consistent. And inconsistent states of affairs cannot be caused (willed, made, brought about) and cannot be known. (β) If God can cause (will, make, bring about) that the state of affairs (event) p occurs, but does not cause it, such that p does not occur, then p is (still) possible. In this case God knows that p is possible. (γ) If God can cause (will, make, bring about) that p occurs and causes (wills, makes, brings about) that p occurs, then p obtains. In this case God knows that p obtains. Concerning condition (2a), then we may say that God's knowledge has the same extension as God's power. Condition (2b) of Def. 1 of omnipotence Whatever God can cause (will, make, bring about) must be compatible with God's essence, i.e. with everything what God necessarily knows about himself and with everything what God necessarily wills about himself. Thus for example God cannot will to change himself or he cannot diminish his own goodness. In this case it holds that everything that God knows about himself, he also wills about himself and everything God wills (and can will) about himself he also knows about himself. Therefore w.r.t. condition (2b) God's knowledge has the same extension as God's power. Condition (2c) of Def. 1 of omnipotence

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Concerning God's providence which is concerned with God's creation it holds that God's knowledge about his creation has the same extension as God's providence. This can be seen as follows. According to condition (1) of Def. 2 (providence) it holds that whatever belongs to God's providence is known by God. Does the opposite hold also, does it also hold that whatever God knows about his creation and directs to some goal (Def. 2 (3))belongs to his providence? The answer to this question is positive if providence is defined in the wide sense as in Def. 2. Providence may be defined in a narrower sense if one requires additionally that what belongs to God's providence is willed by God. In this case immoral human actions cannot come under God's providence. However, we assume here the wider concept of providence in the sense that nothing which occurs, can escape God's providence; this view of providence was also taken by Augustine and Thomas Aquinas, who defended the principle: Whatever occurs, is either willed or permitted by God.50 Thus the wide concept of providence exceeds the concept of God's power since his providence includes such facts as immoral actions, but these cannot be willed or caused by God. That the opposite implication, i.e. whatever God knows concerning his creation (condition (2)) and directs to some goal (condition(3)) belongs to his providence, is also true – provided one accepts the wide sense of providence of Def. 2 – can be seen as follows: (i) Whatever God knows is the case, is true (ch. 1.3 above); i.e. ∀p(gKp → p) (ii) Whatever is the case (is true) is (at least) permitted by God. This is the principle ∀p(p → gPp) mentioned above. (iii) From (i) and (ii) it follows that: Whatever God knows is permitted by God. The conclusion concerning providence (condition 2c of Def. 1) is therefore that God's knowledge about his creation has the same extension as God's providence. However, if we take God's knowledge in general (or in an unrestricted sense), then his knowledge exceeds his providence because his knowledge extends to both, truths about Himself and truths about creation whereas his providence extends only to his creation. Concerning the relation between God's power, providence and knowledge we have to consider the following case: If we assume God could have created another world, there are some states of affairs holding in the other world which are not in God's providence (plan) 50

For a discussion of this principle see Weingartner (2003, EDK) ch. 6.45, principle WP and Appendix theorem T15: ∀p(p → gPp) ('p' for 'permits'). From this theorem the above principle is derivable.

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which is concerned with this existing world, i.e. "conditionally compatible with God's providence” means that under the condition that God's plan (providence) for the existing universe is not made, God's power is also compatible with another plan (providence). However, under the condition that the plan is made, God cannot will something against his plan which would mean to will something against his free decision of his will, i.e. in this case his will would be inconsistent. Thus abstracting from a given providence (plan) about this universe (more generally about creation) and considering a possible plan God's power exceeds his providence (plan). Does it follow from this that God's power exceeds also his knowledge? This is not the case for the following reason: God's knowledge is concerned with what is true or what is a fact not only in the sense that it is true or that it is, but also how it is true or how it is. Now the states of affairs concerning which God's power exceeds his providence are states which are possible and not states which are actual. And as such, as possible states, they are known by God. Thus the range of his knowledge is not smaller than the range of his power, although the range of providence may be narrower if it is considered as God's freely chosen selection. Condition (2d) of Def. 1 of omnipotence First we want to show that condition 2d (of Def. 1) does neither follow from condition 2b nor from condition 2c (of Def. 1). This can be seen as follows: Assume that p is a state of affairs (an event) which occurs and which is a (human) moral evil. Then p is consistent with all states of affairs which hold of God's essence. – This is so since all states of affairs which hold of the creation or of the world are compatible with everything that holds about God's essence. The ultimate reason for that is that the creation is not a necessary outcome of God's essence, but the effect of his free decision of his will. Similarly under the above assumption p is consistent (compatible) with God's providence. – This is so because everything which is the case, every occurring event, of the world which is directed or subordinated to some goal comes under God's providence (according to Def. 2 above). But in the case of moral evil p is inconsistent (incompatible) with God's commands. Second, it can be shown that w.r.t. condition (2d) of Def. 1 (of omnipotence) God's knowledge exceeds God's power: Assume again that p is a state of affairs (event) which occurs and which is a (human) moral evil. Then God knows that p occurs, but God cannot will (and cannot cause, make, bring about) that p occurs.

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Summing up now we can say: Human moral evil which occurs is a consistent state of affairs (event) which is known by God and it is compatible with God's essence and is compatible with God's providence; but it is not compatible with God's commands. Since compatibility with God's commands is a necessary condition of every state of affairs (event) which can come under God's power (or God's omnipotence) moral evil which occurs cannot come under God's power (omnipotence). But occurring moral evil is known by God. Therefore – and in this sense – God's knowledge exceeds God's power.

5.4 Answer to the Objections 5.41 (to 5.11) Those states of affairs (events) which God can (could) cause (will, make, bring about) but are not caused by him and therefore are not realised are known by him just as those which are not realised but which could be realised by him or are possible in a special sense. Thus the second premise of the argument (5.11) should read: But what is not the case cannot be known by him as being the case, but is known by him as possible or as being in his power otherwise he would know something false… If this correction is inserted, the false conclusion does not follow any more. 5.42 (to 5.12 and to 5.13) In order to give an answer to the objection 5.12 it has to be noticed that the expression "not realised fact" in the fourth premise of this argument is ambiguous. It can mean that a state of affairs (event) p is not realised until now but will be realised in the future. Or it can mean that p is (will be) never realised. In both cases p is a state of affairs of the universe (or more generally of creation). According to this distinction the reply can be given as follows: If the respective state of affairs p is not realised until now but will be realised in the future, then – provided that p is consistent with God's commands – p comes under God's power (or God can cause or will that p) not as p in the past or at present (i.e. not as pt ≤0) but as p in the future (as pt >0); where ‘past’, ‘present’ and ‘future’ always mean ‘past, present or future relative to a reference system of this world (universe)’. Otherwise pt≤0 would be inconsistent with God's already decided providence of which pt >0 is a theorem and ¬pt≤0 is a theorem. But the respective state of affairs is in God's knowledge also as p in the future (as pt >0), i.e. God knows that pt >0 (gK pt >0). The above mentioned proviso "provided that p is consistent with God's commands" is necessary: pt>0 (p in the future) can come under God's power only if p is compatible with God's commands (towards man). Thus it cannot

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be in God's power to cause a sin (an immoral action). God cannot will and cannot cause immoral actions. But such events can be of course in his providence if he foresees that some man will commit a sin. If the respective state of affairs (event) p is (will be) never realised (in the universe or in the creation), then p is not a theorem (representing a state of affairs) in God's providence. However, since God's providence is not determined by his essence, but is his plan for the universe (creation) freely decided and selected by his will, one could ask whether some state of affairs q which is not in his providence can be in his power; for example whether God could have created another universe, one which differs in initial conditions, or also in laws or constants of nature; or more generally whether a different plan (providence) than the one chosen comes under his power (omnipotence). This question is answered positively by Thomas Aquinas: "But we showed that God does not act from natural necessity (…) whence in no way at all is the present course of events produced by God from any necessity, so that other things could not happen (…). Wherefore we must simply say that God can do other things than those He has done."51 "So nothing prevents there being something in the divine power which He does not will and which is not included in the order which He has placed in things."52 In the arguments 5.12 and 5.13 there is a further expression which is ambiguous: "they are not realised facts; and thus they cannot be known by him." If the state of affairs p is not realised, then p cannot be known, i.e. "some person knows that p" would be false. But what is known then is non-p, in the modus of "necessarily not-p" or "contingently not-p" or "factually notp". Thus in the case where the state of affairs p concerning creation is not realised, God knows that contingently not-p but possibly p. Since in this case his power extends also to possibly p, his power does not exceed his knowledge. 5.43 (to 5.14) The question whether God's knowledge exceeds his power is concerned with a comparison of the scope (extension) of those states of affairs that God knows with the scope (extension) of those states of affairs that God wills; but it is not concerned with the kind or modality of knowledge or the kind of modality of will (necessary or not necessary). Thus although (according to ch. 2) whatever God knows he necessarily knows – 51 52

Thomas Aquinas (STh) I, 25,5. Ibid. 25,5 ad 1.

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symbolically: (∀p)(gKp → lgKp) – nothing hinders that he necessarily knows that something is contingent or that something is not realised but possible; and moreover – if we abstract from his providence – he might know that he has the power (he can will, he can cause) to bring that about. From this consideration it is clear that God's knowledge concerns also what is possible and therefore his will or his power does not exceed his knowledge.

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6. Whether God Causes Everything What He Knows This question may be expressed also differently: Whether it holds that if God knows something, then he causes it. Or: Is it true that whatever God knows he causes? Symbolically the question can be expressed thus: ∀p (gKp → gCp)? Before we begin with the arguments and the answer, some preliminary remarks are needed concerning the concept of cause as used in this respect. The first remark is that the causal relation is a two place relation where one member of the relation is God and the other is the world (the universe) or events of this world. Both are not identical and the existence of the first (God) is necessary and that of the second (universe) and its events is contingent (at least w.r.t. the kind of necessity applied to God's existence). Second, concerning the properties of the causal relation what is required is that it is irreflexive and asymmetric; that means it is ruled out that x causes x such that also Spinoza's concept53 of a causa sui is ruled out. And asymmetry means that if x causes y, then it is not the case that y causes x. There are several further properties which a causal relation has to have if it is applied among things or states (events) of the universe. These are for example temporal order (the effect cannot be earlier than the cause), chronology condition (there are no closed time-like curves, i.e. the time coordinate does not make a loop), limit of causal propagation (velocity of light), objectivity (independence of reference system).54 None of them should be required for the causal relation between God and the world (universe) or between God and states or events of this world. Also transitivity is not applicable as it will be shown below. Independently of that transitivity is not generally satisfied for the causal relation expressed by laws of nature either: it is only satisfied w.r.t. dynamical laws, but not w.r.t. statistical laws; the same holds for continuity and for counterfactuality. On the other hand counterfactuality may be applied to the causal relation between God and creation since God as the first cause must be a cause in the sense of a necessary condition.

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Cf. his Ethics, Def. 1. Cf. Weingartner (2005, PCC) and Mittelstaedt,Weingartner (2005, LNt), ch. 9.

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6.1 Arguments Pro 6.11 If the knowledge of God is the cause of things, then God causes everything what he knows. Now as Thomas Aquinas says, the knowledge of God it he cause of things: "I answer that the knowledge of God is the cause of things."55 Therefore God causes everything what he knows. 6.12 Because the creatures exist, man can have knowledge of them. But for God the opposite must hold, as Augustine says: "Not because they are, does God know all creatures spiritual and temporal, but because he knows them, therefore they are."56 Therefore God causes everything what he knows. 6.13 1. If God is omniscient and exists at t1 then, under the condition that Jones acts at t2 that p (is the case), God believes at t1 that Jones acts at t2 that p (t1 < t2, i.e. t1 is assumed to be earlier than t2). 2. If God believes that p, then p (is the case, is true). 3. If God believes at t1 that Jones acts at t2 that p, then, under the condition that Jones can act at t2 that non-p, one of the following three conditions are satisfied: (i) Jones can act at t2 such that God believed at t1 that p, but p is false. (ii) Jones can act at t2 such that God did not believe as He did at t1. (iii) Jones can act at t2 such that God did not exist at t1. 4. If God is omniscient, then all three (i), (ii) and (iii) are false. 5. Therefore: If God is omniscient and exists at t1, then, under the condition that Jones acts at t2 that p, it is not the case that Jones can act at t2 that nonp.57 Thus it seems that God, believing that Jones acts at t2 that p causes John to act this way.

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Thomas Aquinas (STh) I, 14,8. Agustine (Trin) XV. 57 This is an abbreviated version of an argument by Nelson Pike in his (1970, DOV). Cf. also Craig (1991, DFH), p. 23. We express "it is in his power at t, to bring it about that p" by "he can act at t, that p" (in accordance with Pike, p. 84). This argument does not use possible worlds since Pike uses such a version only in his (1977, DFH). Although according to chapter 3 the attribution of time indices to actions of God does not make sense, it is accepted here for the sake of the argument. The conclusion of the argument is interpreted by Pike as saying that Jones' action at t2 cannot be free, but is determined by God's foreknowledge, i.e . God causes by his foreknowledge. 56

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6.2 Arguments Contra If God causes everything he knows, then, since God knows that Hitler breaks His ethical rules (commandments), it follows that God himself causes this breaking of His rules. This is absurd. Therefore the thesis that God causes everything what he knows must be false.

6.3 Proposed Answer It is not the case that everything what God knows he causes. This can be seen from the following reasons: 6.31 The knowledge need not to be a sufficient condition for causing something This can be justified as follows: In humans it is a fact that knowing an action or a process is not a sufficient condition for bringing it about or for causing it. For example a masterbuilder knows how to build a house; he had the idea in his mind and designed an exact plan. But his knowledge is not sufficient to execute the plan. In order to execute his plan he has to employ (mobilise) his will. This is so also for other human actions: knowledge is never sufficient for causing them, i.e. for their execution. However, knowledge about the action or process (at least some kind of knowledge) is a necessary condition for (consciously) causing the action or process. That is, the opposite implication does generally hold: If some human person (consciously) causes some action or process, then he has some kind of knowledge of it. From the fact that for human persons knowledge is not a sufficient condition for causing, it follows that the statement "For all persons A, everything what A knows is caused by A" is false; because it does not hold for human persons. Therefore this statement does not necessarily hold in the sense that it is universally true for all persons or belongs to the nature of persons. We may therefore conclude that it need not to hold for God. That it does not and cannot hold for God is shown by the subsequent arguments in 6.32 – 6.35.

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6.32 The statement "God causes everything what he knows" leads to absurd consequences Since we have to assume that God knows all the immoral actions which have been committed by men,58 it would follow that he causes (caused) all these immoral actions. From this, three further absurd consequences follow: (i) that such a God would be inconsistent, since immoral actions are inconsistent with his commands (specifically the Ten Commandments) given to men (see below 6.34). (ii) that such a God who causes immoral actions cannot be all good or perfect. (iii) that man cannot have free will. And if man cannot have free will, he cannot have responsibility for his actions. If man cannot have responsibility for his actions, juridical institutions like courts, punishment law, prison etc. not only make no sense, but are in fact fraud and deception. This consequence is absurd; and that it is absurd can be shown by empirical evidence. Therefore, on the assumption that God knows the immoral actions committed by humans, it cannot be true that God causes everything what he knows. 6.33 The thesis "God causes everything what he knows" excludes cooperation and learning processes in creatures This can be seen as follows. If God causes everything what he knows, then it is impossible that God wills the imperfect cooperation and imperfect contribution of imperfect creatures in the happenings of the world (universe). Because since he knows these imperfect contributions he causes them himself. This is first of all inconsistent with his perfection. Secondly, there are a number of good reasons that God wills the cooperation and contribution of imperfect creatures caused by them and not caused by God: "In another way one is said to be helped by a person through whom he carries out his work, as a master through a servant. In this way God is helped by us; inasmuch as we execute his orders, according to 1 Cor. III, 9: We are God's coadjutors. Nor is this on account of any defect in the power of God, but because he employs intermediary causes, in order that the beauty of order may be preserved in the universe; and also that he may 58

That this holds follows from what has been defended in chapter 4: that God knows all past and present events.

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communicate to creatures the dignity of causality.59 Further: If God causes everything what he knows, then there cannot be a genuine process of learning in higher organisms and in man. This can be seen as follows. Every genuine process of learning requires that the higher organism or the human person himself has at least some degrees of freedom to make trials and errors and afterwards to make decisions for improvement. Since learning without trial and error seems to be impossible. But if God, since he knows all occurring learning processes, himself causes every trial, error and improvement of the organism or human person, then we cannot speak of a genuine process of learning. We could not interpret a learning process as an instinct (caused by God) either, because an instinct leads directly and straightforwardly to the optimal action without trial and error. Since we know from the biological sciences that there are genuine learning processes in higher organisms and in human persons where trial and error play an essential role. Therefore it cannot be true that God causes everything what he knows. 6.34 If God causes everything what he knows, then he is normative and volitive inconsistent We say that a person is normative and volitive inconsistent if and only if the person either wills or orders (commands) that p occurs (should occur) but (at the same time) causes or permits that non-p occurs. An example would be a dictator who orders that people should not be imprisoned for political opposition but allows (or commands) his police to imprison them. More accurately and applied to God we may say that God is normative and volitive consistent. And this we might define as follows: Def. 1 God is normative and volitive consistent iff, if p is a theorem of God's will w.r.t. man, then God permits p. Def. 2 p is a theorem of God's will w.r.t. man iff for all humans h: (a) either God wills that h wills that p or (b) God wills that h acts with the intention to bring about p or (c) God wills that h should will that p or (d) God wills that h should act with the intention to bring about p60

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Thomas Aquinas (STh) I, 23,8 ad 2. For a formal (axiom) system in which these definitions are used cf. Weingartner (2003, EDK) Appendix, Def. 12 and 13 and ch. 13, D6, D6.1, D15 of this book. 60

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In the cases (a) and (b) man h wills that p and acts with the intention to bring about p and p is something good if we assume that both of the following conditions hold for God: (i) God's will is always fulfilled, i.e. whatever God wills is the case. (ii) Whatever God wills is good. Cases (a) and (b) hold for instance if there is a natural inclination to some good in man like to survive or to live in a community. On the other hand in cases (c) and (d) man h, since his actions are free, may not be willing that p (or may even will that non-p), although God wills that he should will that p. Similarly concerning his action with the intention to bring about that p: he might not act with such an intention or might act with an opposite intention, although God wills that h should act with the intention to bring about p. Coming back now to the thesis that God causes everything what he knows, it follows that if this thesis is true, then God causes immoral actions such that he is normative inconsistent w.r.t. conditions (c) and (d) because he has given commands (for example the Ten Commandments) to man. Since we cannot attribute to God a volitive and normative inconsistency (an inconsistency of his will), the thesis "God causes everything what he knows" must be false. 6.35 If God causes everything what he knows, then he causes everything Since God does not cause immoral actions, he does not cause everything and therefore the thesis that God causes everything what he knows must be false. The above thesis, that if God causes everything what he knows, then he causes everything, can be substantiated as follows. In chapter 4 it was defended that God knows all past and present events. Thus if p is a past or a present event (of this universe), then God knows that p. Symbolically: ∀pt ≤ t0 (p → gKp). Together with the wrong thesis: God causes everything what he knows (symbolically: ∀p (gKp → gCp)) it follows from it that God causes (caused) every past an present event. Symbolically: ∀pt ≤ t0 (p → gCp). It will be defended later that also the more general thesis holds for God's knowledge: God knows all events that occur; or God knows everything which is the case. Symbolically: ∀p (F(p) → gKp), where 'F(p)' stands for 'p is the case' or 'p obtains'. From this assumption together with the (false) thesis God causes everything what he knows, it follows that God is allcausing. This can be seen as follows: from F(p) → gKp and gKp → gCp it follows that F(p) → gCp, i.e. whatever

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is the case is caused by God. From this it follows (by contraposition, by substitution of ¬p for p and by the assumption that F(¬p) = ¬F(p)) that for every event p either God causes that p or God causes that non-p. Symbolically: ∀p (gCp ∨ gC¬p). This thesis we may call the thesis of the allcausing God. It was discussed in detail and shown to be false together with the analogous thesis of the allwilling God in another publication.61 The thesis of the allcausing God leads to the same absurd consequences which have been discussed already in ch. 6.32 above. Thus they need not be repeated here. These consequences show that – since we can defend the thesis that God knows everything which is the case – the remaining other premise must be false, i.e. the premise: God causes everything what he knows, must be false. 6.36 The thesis of the allcausing God and transitivity Observe that the (wrong) thesis of the allcausing God would also follow from a wrong application of transitivity to the causal relation between God and creation. Since if God is the first cause in a causal chain or in a causal tree and if the causal relation is transitive, then the first cause causes everything what the subsequent members cause, i.e. then the first cause causes everything (every member of the chain or tree ) in the sense of a sufficient cause. In such a case God would be allcausing. Thus we are asking the question: Does it follow from being a first cause (in a chain or net of causal relations) that the first cause causes every member (in the chain or net)? It is certainly manifest that the first cause and first member of a causal chain must cause the second member. And if there is a branching (a "tree"), then there are more (than one) second members and all these must be caused by the first cause (member). But the second members are causally related to the third member (or members) and cause them; those again cause the fourth... etc. But the second members need not cause the fourth members although they are causally related (connected) with them via the third members; and thus also the first member need not to cause the third or fourth members although it is causally connected with them via the second members. This means we have to distinguish the following two things: (1) being causally connected (related) to some member (x is causally related to y in some way) and (2) being caused by some member (x is caused by y) or y being causally dependent on x. 61

See Weingartner (2003, EDK), ch. 6.4.

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An example may illustrate the difference: The 4 grandparents are the cause of the parents and these parents are the cause of their children. But the grandparents are not the cause of their grandchildren although they are causally related (connected) with them. Assume that a grandchild (as an adult) solves some important scientific problem. We cannot say that the parents or grandparents caused this solution although both are causally connected with that grandchild; or assume that the grandchild (as an adult) committed a crime. Again, we cannot say that the parents or grandparents caused that crime (presupposing that the child had a normal education). Thus it is clear that the grandparents are not a sufficient cause (not a cause as a sufficient condition) for the grandchildren, but they are certainly a necessary cause (a cause in the sense of a necessary condition) for them. Similarly the first cause (God) is a necessary cause (a cause in the sense of a necessary condition) for all other causes and events, but not a sufficient one. From this it follows that transitivity holds for the causes as necessary conditions, but not for the causes as sufficient conditions. A similar counterexample to transitivity (of causes as sufficient condition) is given by Pearl; where state X is capable of changing the state Y and Y is capable of changing the state Z, yet X is incapable of changing Z. "That causal dependence is not transitive is clear... The question naturally arises as to why transitivity is so often conceived as an inherent property of causal dependence..."62 The main point is that X is not sufficient in order to change Z. But the earlier (ancestors) in the causal chain or net might be necessary and the first member is certainly necessary for all the others, but not sufficient. In this respect it is interesting that Thomas Aquinas in his second (causal) way to prove the existence of God required only irreflexivity (explicitely in his premises) of the causal relation which holds between things and a first element. Interestingly enough he didn’t assume transitivity. Certainly he assumed also asymmetry between God as the cause and the world as the effect. Concerning the causal relation within the world we know today: The fundamental laws of Quantum Mechanics and Relativity Theory are timereversal symmetric such that they do not designate a causal order in one direction only. However the direction of the actual movement (of stars, planets, atoms … etc.) cannot be reversed; its just that the dynamical laws don’t forbid the opposite direction. And – to give an example from another area – the neuronal connections in the brain are reciprocal (neuron A fires to 62

Cf. Pearl (2000, CMR), p. 237. The failure of transitivity is also illustrated by Pearl with the help of a figure which represents a model as a counterexample. For further counterexamples against transitivity see Galles-Pearl (1997, ACR).

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neuron B and vice versa) and do not show an asymmetry either (except the asymmetry introduced by the time of propagation of the neuronal effect which cannot exeed the velocity of light).63

6.4 Answer to the Objections 6.41 God’s knowledge – a necessary cause (ad 6.11) Thomas Aquinas does not claim that the knowledge of God is a sufficient "cause of things". This is plain from the end of article 8 where he says: "His knowledge must be the cause of things, in so far as His will is joined to it."64 And in article 9, which deals with the question whether God has knowledge of things that are not, he makes that point even more explicit: "The knowledge of God, joined to His will is the cause of things. Hence it is not necessary that whatever God knows, is, or was, or will be; but only is this necessary as regards what He wills to be, or permits to be."65 Moreover, it is clear from the text of article 8 that the kind of cause which is meant here is the causa formalis (intelligible form) and not the causa efficiens. God's knowledge w.r.t. creatures is compared to the knowledge of the artificer w.r.t. his works of art. But the causa formalis is not sufficient as the following quotation shows: "... the intelligible form does not denote a principle of action insofar as it resides in the one who understands unless there is added to it the inclination to an effect, which inclination is through the will."66 6.42 God’s knowledge – not a sufficient cause (ad 6.12) Also the quotation of Augustine need not be interpreted in such a way that the knowledge of God would be a sufficient cause for the existence of the creatures. But it can be interpreted as saying that God's knowledge w.r.t. creatures in the sense of his plan of the creation must be principally prior and must be a necessary condition (cause) for their existence. What is pointed out by Augustine is that whereas in the case of man, things are first and then man can have knowledge of them, it is the opposite with God who does not need things in order to have knowledge. 63

For neuronal interaction see Popper-Eccles (1984, SfB), p. 228 and 241ff. For a discussion of the causality relation which is expressed by laws of nature, see Mittelstaedt/Weingartner (2005, LNt), ch. 9. 64 Thomas Aquinas (STh) I, 14,8. 65 Ibid. 14,9 ad 3. 66 Ibid. 14,8.

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6.43 Omniscience and Freedom (ad 6.13) Before we give a comment on the argument in 6.13 we shall translate it into symbolic form. The argument has four premises of which the third has a more complicated form. We state therefore the third premise in a simpler form first and give its details only after the argument. 1. (OSg ∧ E!gt1) → (jAt2p → gBt1(jAt2p)) 2. gBp → p 3. gBt1(jAt2p) → [jCAt2¬p → ((i) ∨ (ii) ∨ (iii))] 4. OSg → ¬((i) ∨ (ii) ∨ (iii)) 5. Therefore: (OSg ∧ E!gt1) → (jAt2p → ¬jCAt2¬p) The three alternatives in premise 3. are as follows: (i) jCAt2 (gBt1p ∧ ¬p) (ii) jCAt2 (¬gBt2p ∧ gBt1p) (iii) jCAt2 (¬E!gt1) Since we have turned the argument into precise form, we shall first ask whether this argument is logically true or valid. The answer to this question is yes. One can see this easily if one makes the following abbreviations in order to obtain a very simple argument structure: OSg ... A ... God is omniscient jAt2p ... B ... Jones acts at t2 that p gBt1(jAt2p) ... C ... God believes at t1 that Jones acts at t2 that p jCAt2¬p ... D ... Jones can act (has the power to act) at t2 that non-p (i) ∨ (ii) ∨ (iii) ... I E!gt1 ... E ... God exists at t1 Then the argument becomes the following simple form: 1'. (A ∧ E) → (B → C) 2'. C → (D → I) 3'. A → ¬I Therefore: (A ∧ E) → (B → ¬D) One can see immediately that this argument is valid. It is an abbreviation of the former more detailed argument without premise 2. To derive the above conclusion the second premise (2.) is not needed. But we have not yet analysed I and premise 4. For the justification of premise 4., premise 2. is needed . Before we make a comment to the premises we shall first emphasise again that according to chapter 3 the attribution of time indices to God or to God's actions does not make sense. As it was said there, time is understood here as the time of this universe which is measured differently and relatively in

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subsystems of this universe, i.e. for instance at the earth (not to speak of London-time or Los Angeles-time). There is no universal (or absolute) time which would be valid for the whole universe or for all its subsystems. Accordingly, although we can say that God, being omniscient, knows that some event A (for instance a free action of a human person) happens at a time t2 relative to a time measurement used on earth and moreover that this event A occurred later than another one B at t1 (according to the same time measurement), time is only attributed to events of this world (universe) but not to God or his knowing and willing. In this connection it is interesting to ask whether dropping all the time indices attributed to God's belief would make a change w.r.t. the validity of the above argument. The answer to this question is: No. The validity remains untouched in this case. But the premises will receive a different meaning of course. We are turning now to a discussion of the four premises. The most uncontroversial premise is premise 2: Everything what God believes is true. If we replace 'believes' by 'knows', then the respective premise has been defended already in chapter 1: Everything what God knows is true. And although it is not very reasonable to attribute beliefs to God, this can be accepted for the sake of the argument since premise 2 guarantees that all of God's beliefs are true. It might be worth mentioning that Pike presupposes here that 'belief' is understood in such a way that “person a knows that p” implies “person a believes that p”; i.e. what one knows, one also believes67 in the sense of holding it to be true. There is however another concept of belief which has a different, i.e. exclusive relation to knowledge: what one believes one does not (yet) know and what one knows one does not (need to) believe any more. This kind of belief is far more important than the former one. Since what one might call scientific belief (the belief in scientific hypotheses) and also religious belief is of this second kind.68 Premise 1. is certainly acceptable if we drop the time index t1 attributed to God's existence and believing. The gist of it is then just that God knows what happens at t2 (where t2 is a point of time of this world). Therefore it must be true to say: if Jones acts at t2 that p, then God knows (believes) that Jones acts at t2 that p. Since this is the consequent of premise 1., the whole premise must be true. If we add the time indices, then we first have to consider E!gt1: God exists at t1. Although this sentence would be false if we interpret it as saying that God's existence is bound or dependent upon time (the time of this 67 68

book.

Pike (1970, DOV) For a detailed discussion see Weingartner (1994, SRB). Cf. section 1.36 of this

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universe) it can be accepted if it just means: God certainly exists at any point of time in which this universe exists. In the same sense we may interpret the consequent. God knows at t1 that Jones acts at t2 that p. This is to mean that what God knows that it happens at some time (of this world) he knows at any point of time of this world. Interpreted in this sense the first premise can be accepted to be true. The most complicated and problematic one is premise 3. The main question is whether the antecedent is contradictory and then the premise would be true, but logically true or trivially (or emptily) true. This can be seen as follows: From God believes (at t1) that Jones acts at t2 that p, it follows that: Jones acts at t2 that p (by premise 2). But if the latter is true, Jones cannot at the same time t2 act that non-p. Even if he acts voluntarily, i.e. freely. Since it is just an impossibility to both act at t2 that p and act at t2 that non-p. It is also impossible to both act at t2 that p and have the power (or ability) at t2 to act at t2 that non-p. Such a kind of "power" nobody can have not even God, since it would mean an inconsistency. This inconsistency is symbolically expressed thus: jAt2p ∧ jCAt2¬p. The power to act that ¬p Jones could have earlier than t2 but not exactly at t2 when he acts that p. Therefore with the help of premise 2 it follows from premise 3 that the above contradiction implies (i) ∨ (ii) ∨ (iii) which is also logically true: (jAt2p ∧ jCAt2¬p) → (i) ∨ (ii) ∨ (iii). Thus it does not matter that (i) ∨ (ii) ∨ (iii) is itself contradictory. And so premise 4 is correct of course because it is logically true. Therefore we have to say: even with free (voluntary) actions it holds that if the event (action) takes place (at t2) it cannot not take place (at t2). But from this one cannot conclude that the action (at t2) is necessary or not voluntary, or not free or not contingent: That I am sitting now at t3 and writing this chapter is both contingent and voluntary (free), although I am not able (I do not have the power, I cannot) now (at t3) to act in such a way that I am not sitting or not writing; i.e. under the condition that I am sitting I cannot do something else which would imply not-sitting. This was called "conditional necessity" in the philosophical tradition. As is clear from the examples, conditional necessity is perfectly compatible with the action being contingent and voluntary (free). Therefore, if the interpretation of "conditional necessity" is applied, the conclusion of Pike's argument is even logically or trivially true because the second part is logically true, namely: If (i.e. under the condition that) Jones acts at t2 that p, it is not the case that Jones can act at t2 that non-p. Symbolically: jAt 2 pt 2 → ¬jCAt 2¬pt 2. From this consideration it follows that Jones can act (or has the power to act) that non-p only at a point of time earlier than t2 (at which he acts that p). At

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time t2 the action is being committed and cannot be altered any more. And earlier he could have decided otherwise (on the assumption that it was a free voluntary action). Although one has to observe that preliminary preparing actions committed by Jones for the action at t2 that p, can make it more and more improbable that Jones is able to act that non-p before acting at t2 that p. And if Jones would have decided (and committed himself) to act that non-p at t2 (instead of acting that p), then God would have known that he is acting that non-p at t2. Another interpretation of Pike’s argument was given by Plantinga.69 He tries to show convincingly that neither of (i) or (ii) or (iii) follows from “John acts at t2 that p and John can act at t2 that ¬p” provided the latter is interpreted as consistent. Instead the counterfactual propositions (52’), (53b) and (54’) follow. They are rather harmless and do not imply that God is not omniscient. For the question whether foreknowledge can change the ontological status of an action (be it contingent and free or necessary) see chs. 10. and 11. below.

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Plantinga (1974, GFE) p.68ff.

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7. Whether God Knows Singular Truths? Concerning terminology a singular truth is a true statement about an object which is understood as individuated in some way or about a group of objects which are united together to a single object. For example an atom, a living being, a person; but also an individual state or event or human action, where the latter are individuated by the place or the point of time or by both. Instead of saying "true statement about..." we could also say "fact about..." or "obtaining state of affairs about...". In this sense we could also speak of "singular facts". If the individuation is expressed more precisely, one uses space time coordinates, like in the sentence "the explosion of the first tower in Manhattan on Sept. 11, 2001". In general a sentence p with space time coordinates may be expressed as "pl,t" where l locates position (place) and t time. In discussions on omnipotence the so-called A-propositions or Asentences do not mention the position, but only the point of time; such Apropositions can be formulated thus: pt1, qt2, rt3... pt2, qt3... etc. Introducing the operators 'K', 'CK', 'TE', and 'TO' for 'knows that', 'can know that', 'truly expresses that' and 'truly tokens that' respectively we can form sentences like aKpt, bCK pt, cTEpt ... etc. (where 'a', 'b' and 'c' stand for persons).

7.1 Arguments Contra 7.11 The aim of the perfect scientist is not to know all the singular truths and data, but to know the axioms and laws from which they follow. In a similar way Aristotle characterises the wise man: "We suppose first, then that the wise man knows all things, as far as possible, although he has no knowledge of each of them individually."70 Now, since God is most perfect and most wise, he knows rather the axioms than the infinite singular truths and data and the latter as consequences of the former. Therefore God does not seem to have direct knowledge of singular truths and of truths about singular things. 7.12 There are many despicable and irrelevant truths under the singular truths. But as Augustine says, these are better not to know than to know: "The situation is completely different however if someone knows this, and the other one that, the one useful things, the other less useful or even derogatory things. Who would not 70

Aristotle (Met), 982a7.

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prefer – in the latter case – the one who does not know over the one who knows? There are even things of which it is better not to know them, than to know them."71 But since we should attribute to God rather what is better, it seems that he need not to know all singular truths. 7.13 Every true A-proposition is a singular truth. But as Gale says, God cannot know A-propositions. Therefore God does not know all singular truths. That God cannot know A-propositions Gale tries to prove in the following way:72 (1) A person can know a proposition only if she can truly express it. From this we can make the instantiation: (1a) A person can know an A-proposition only if she can truly express it. (2) A person can truly express an A-proposition only if she can truly token an A-sentence (where an A-sentence is a sentence expressing an Aproposition). (3) A person can truly token an A-sentence only [if she truly tokens it] at a time. (4) A person truly tokens an A-sentence at a time only if she exists in time. (5) A person can know an A-proposition only if she exists in time (from (1a) to (4)). (6) Enthymem: A timeless being does not exist in time. (7) Therefore: A timeless being cannot know an A-proposition. And consequently: Supposing that God is a timeless being he cannot know an A-proposition and therefore he does not know all singular truths.

7.2 Arguments Pro According to chapter 4 God knows all past and present events. But many true descriptions of past and present events are singular truths. And some of them have the form of A-propositions. Therefore God knows singular truths and he knows also A-propositions.

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Augustine (Ench), 17. See Gale (1993, NEG), p. 65f. This report of the argument is a reconstruction, but one that tries to do justice to Gale's intention though putting it into a more concise form (for example avoiding changes from "is expressible" to "a person can truly express" etc.) and adding enthymemic premises. 72

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7.3 Proposed Answer God knows singular truths. One reason for that is as follows: According to ch. 5 God's knowledge exceeds God's power. But God's power extends certainly to singular truths or singular facts. Examples are his creatures and his creation. Therefore also his knowledge must extend to singular truths or singular facts. A second reason can be given by a comparison with man’s knowledge of singular truths. Man knows singular truths (singular facts) via two different ways: first through experience, second with the help of laws. In the first sense, by experience he knows only that they are, in the second sense he knows why they are so and not otherwise. The second way of knowledge for singular truths is the scientific way, although it also has to include the first way (experience) in a refined and methodologically controlled way. It is roughly as follows:73 In order to explain or predict (retrodict) singular truths (facts) we use well confirmed laws (of nature) together with initial conditions which are also singular truths (facts). For example in order to explain or predict an eclipse, the initial condition is the constellation of sun, earth and moon at a certain time, which together with Newton's dynamical laws allow to derive the description of the eclipse (as a singular truth) from these laws. Thus in general it holds that singular truths can be known in the sense of being explained and predicted (retrodicted) with the help of other singular truths (used as initial conditions) plus dynamical laws. However, it has to be emphasised that the laws have to be dynamical laws. Statistical laws would not be sufficient. But since large areas of human knowledge are only accessible through statistical laws, many singular truths in these areas cannot (at least not so far) be known by man. These areas are: thermodynamics, radiation, friction, diffusion, electric transport, measurement process in Quantum Mechanics, processes of growth, in general: processes of biology, psychology and cosmology. The reason why statistical laws are not suitable for explaining and predicting singular truths (or singular facts) is the following difference between two respective characteristics of dynamical laws (D1 and D2) and statistical laws (S1 and S2): D1 The state of the physical system S at any given time ti is a definite function of its state at earlier time ti-1. A unique earlier state (corresponding to a unique solution of the differential equation) leads

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For a detailed analysis of laws of nature, especially in the two forms of dynamical and statistical laws cf. Mittelstaedt/Weingartner (2005, LNt), ch. 7.

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under the time evolution to a unique final state (again corresponding to a unique solution of the equation). D2 Condition D1 is also satisfied for every part of the physical system, especially for every individual body (object) as part of the system even if the individual objects may differ in the classical or in the quantum mechanical sense.74 S1 The state of the physical system at ti is not a definite function of an earlier state at ti-1. The same initial state may lead to different successor states (branching). S2 Statistical laws describe and predict the states for the whole physical system, but they do not describe or predict the individual (objects) of this system. It is easy to see that there is an essential difference between the conditions D1 and S1. Like D1 is necessary for dynamical laws, S1 is necessary for statistical laws. This presupposes however that we interpret S1 (and by it statistical laws) realistically (i.e. in an ontic sense). That is we assume there is real branching in reality. An epistemic interpretation according to which branching is only a sign for our lack of knowledge whereas in the underlying reality everything is determined (by hidden parameters and dynamical laws of which we are ignorant) we do not find justified.75 This can be substantiated by the fact that the above mentioned types of processes do not satisfy D1 (but satisfy S1) as is evident from all the sophisticated knowledge we possess today about these processes. Similarly, D2 and S2 differ in an important point. Statistical laws are bound to huge ensembles – they describe physical systems consisting of a huge number of objects. The greater the number of objects, the more strict is the law about the whole ensemble. Though there is indeterminacy for every individual system, there is a strict law for the whole system if the ensemble is large enough. To some extent such laws "emerge" from the "lawless" behaviour of a large number of individual systems. In this sense Wheeler

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It has to be observed however that 'physical system' can be understood in a twofold way: classically and quantum mechanically. Classically D1 holds for example for every subsystem of a planetary system (for planets and parts of planets) and thus for individual objects. Quantum mechanically there is no definite composition of subsystems (they are not composable by Boolean operations only) and objects as parts of physical systems have to have commensurable properties. 75 Cf. Weingartner (1998, SLG).

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spoke of "law without law"76. This problem was clearly understood and emphasised already by Boltzmann and Poincaré: How can the law of entropy emerge from random behaviour of individual systems? Schrödinger gave the following answer in his inaugural lecture of 192277. "In a very large number of cases of totally different types, we have now succeeded in explaining the observed regularity as completely due to the tremendously large number of molecular processes that are cooperating. The individual process may, or may not, have its own strict regularity. In the observed regularity of the mass phenomenon the individual regularity (if any) need not be considered as a factor. On the contrary, it is completely effaced by averaging millions of single processes, the average values being the only things that are observable to us. The average values manifest their own purely statistical regularity…" Summing up we have to say that man's knowledge of singular truths (singular facts) is restricted: First it is possible to attain such knowledge to a high degree of accuracy in those domains where we have dynamical laws (satisfying D1 and D2, but also condition D3, see below) and a precise description of initial conditions. Then also the individual future states and the position and other properties of the individual objects can be predicted with high accuracy. Second, it is only partially possible through direct experiment or observation since only a part of the individual objects or states on the earth or in the cosmos are accessible in this sense. Third, man's knowledge of singular truths is very restricted concerning processes which can be only described by statistical laws. This is so because if there are real degrees of freedom (of different type and perfection) in reality – on the level of the molecules in a gas, on the level of living organisms in learning processes, on the level of human freedom – then we do not have laws to explain or predict these singular processes (we can predict them only with a very low probability). Thus we do not have genuine knowledge about them or respectively about the corresponding singular truths. We have only knowledge about the average values or about the development of a huge ensemble of singular objects. This is expressed in the third condition (S3) of statistical laws below. 76

Wheeler (1983, RLL). For more details on this question see chapter 13.2 of Mittelstaedt/Weingartner (2005, LNt). 77 At the University of Zurich. This lecture was later published under the title "Was ist ein Naturgesetz?". Cf. Schrödinger (1961, WNG), p. 11.

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But also concerning dynamical laws we have to add an important proviso. Prediction of singular facts with the help of dynamical laws is only possible if the system described by the law satisfies condition D3. Otherwise the system becomes chaotic and then there is an exponential increase of loss of information about the individual objects of the system. In other words there is an exponential increase of ignorance concerning singular truths. D3 The physical system S has a certain type of stability which obeys the following condition: Very small changes in the initial states, say within a neighbourhood distance of ε lead to proportionally small (no more than in accordance of a linearly increasing function of time) changes h(ε) in the final state. This kind of stability which survives small perturbations and leads to relaxation afterwards is called perturbative stability and holds in many linear systems.78 S3 The loss of information (and consequently the difficulty of prediction) about the state of an individual object (or a small part) of the whole system increases exponentially with the complexity of the system. On the other hand: (accuracy of the) information about the average values of magnitudes (parameters) of the state of a huge number of individual objects (or particles) increases also with the complexity of the system. None of these restrictions which pertain to the specific human intellect can be attributed to God. First because there is no restriction in God like that of our senses (plus the technical instruments to extend them) w.r.t. observation; since God knows with his intellect. Secondly, there is no restriction in God, like the one we have concerning processes and events describable only by statistical laws; since under the assumption of God as creator he knows which degrees of freedom he has given to the different types of individuals and he himself has ordered that creatures contribute to their own development by a learning process via trial and error, providing for them degrees of freedom on different ontological levels. Thirdly, there is no restriction in God, like that of our ignorance w.r.t. chaotic motion; since under the assumption of God as the creator he himself has created a world in which chaotic motion can develop because of sensitive dependence on initial conditions such that we must

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For chaotic motion in the sense of Dynamical Chaos cf. Schuster (1989, DCh) for a discussion of properties of dynamical chaos cf. Weingartner (1996, UWT) and Chirikov (1996, NLH). For other kinds of chaos like Quantum Chaos cf. Casati, Chririkov (1994, QCh). For the restriction concerning predictability if D3 is not satisfied cf. Mittelstaedt/Weingartner (2005, LNt), ch. 9.4.

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assume that he knows what Prigogine called the "Laws of Chaos"79 and in fact much more than that.

7.4 Answer to the Objections 7.41 Discursive Knowledge (to 7.11) As it was described in the answer, man's knowledge is such that it proceeds from the knowledge of initial conditions (singular truths) and from the knowledge of axioms or laws, taken together to the knowledge of predictions (other singular truths). This procedure is discursive and step by step as it is appropriate to the restricted and imperfect abilities of the human mind. However, there is no need to assume such imperfect restrictions for the knowledge of God; i.e. nothing hinders that he encompasses initial conditions, laws (axioms) and predictions in one action of knowledge. Thus we do not need to assume for God's knowledge a procedure in time going from one truth to the other, nor do we need to assume for his knowledge a kind of causal connection in the sense that he would know one truth because of another like we know the conclusion because of knowing the premises.80 7.42 Irrelevant truths (to 7.12) As Augustine says, there are many (in fact infinitely many) trivial and irrelevant truths in general and consequently also among the singular truths. As examples take the substitution instances of the general law of identity x = x (where 'x' is an individual variable) or of the law of equivalence p ↔ p (where 'p' is a propositional variable for singular propositions). Moreover, there are a lot of redundant and irrelevant factual truths in the consequence class of any factual proposition: For instance, if p is a singular truth, then p ∨ q, ¬p → q, q → p, p ∨ p, p ∧ p, (p ∧ q) ∨ (p ∧ ¬q)… etc. are non-tautological (factual) consequences of p which are redundant and irrelevant. The redundancy and irrelevance can be seen by the fact that in the above consequences the variable 'q' can be replaced on one or on both occurrences by an arbitrary (other) variable (also by the negation ¬q of q) salva validitate

79 80

Prigogine (1995, GCh). Cf. Thomas Aquinas (STh) I, 14,7.

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of the inference (consequence relation); similarly the second occurrence of 'p' in p ∨ p and p ∧ p can be dropped.81 It is also right to point out that man and especially philosophers and scientists can be distracted and misled by such irrelevant truths. This is evident from the many paradoxes in different domains which have their root in redundant and irrelevant elements of the consequence class as it is defined by Classical Logic and which have been discussed in the scientific literature for decades. Such domains of paradoxes are: Theory of explanation, of confirmation, of law statements, of disposition predicates, of versimilitude, of Quantum Logic, of Epistemic Logic of Deontic Logic.82 Although Augustine's concern was not about these kinds of modern paradoxes his point was nevertheless not less important: man may be seduced or distracted by despicable and irrelevant truths in a similar way as he may be seduced by knowing immoral actions. And in this sense for man, both, irrelevant and despicable truths and immoral actions are better not known than known. However, in contradistinction to man, we cannot assume that a most perfect being is misled or distracted by knowing the errors of man or as he cannot be seduced or distracted by irrelevant and redundant truths. Like God cannot be misled by knowing the immoral actions of man, so he cannot be distracted by irrelevant and redundant truths, even if they are infinite in number. We have to assume therefore that he knows the irrelevant and redundant truths implicitly without being distracted by them. 7.43 God knows A-propositions? (to 7.13) Like in previous answers to arguments we shall ask two questions: Is the argument valid? and: Are the premises true? In order to decide this in a more precise way we shall put the argument into a symbolic form: (1) Premise aCKp → aTEp83 CK… can know (1a) aCK(A) → aTE(A) TE… can truly express (2) Premise aTE(A) → aCTOs(A) 'a'… variable for persons 81

For a precise treatment of relevance and irrelevance in this sense see Weingartner (2000, RFC) and (2000, BQT) ch. 9. 82 For a solution see Weingartner/Schurz (1986, PSS), Schurz/Weingartner (1987, VDR) and Weingartner (2000, RFC) and (2004, RSL). 83 We do not use quantifiers for propositional variables here, since all occurrences can be universalised anyway. Also 'a' can be universally quantified. The quantification for time variables however is necessary.

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(3) Premise aCTOs(A) → (∃t)aTOt s(A) A… A-proposition (3a) Enthymem 1: (∀t)(aTOts(A) → a exists at (in) t) s(A)… A-sentence CTO... can truly token TO… truly tokens (4) (∃t)(aTOt s(A)) → (∃t)(a exists at (in) t) from Enth. 1 (5) aCK(A) → (∃t) a exists at (in) t from (1a), (2), (3) and (4) (6) Enthymem 2: (∀x)(x is a timeless being → ¬(∃t) x exists at (in) t) (7) Therefore: (∀x)(x is a timeless being → ¬xCK(A)) Concerning the first question, one can easily see that the argument – in this version – is valid. Gale claims for his version that it is valid too; although this cannot be so easily checked, because his version is not in symbolised form and presupposes therefore sometimes intuitive understanding of different formulations like "is expressible" and "can truly express" (4e, 4f) or "by the tokening of" and "can truly token". Moreover, he seems to presuppose what we called Enthymem 1 and Enthymem 2. The relation of implication which is used by Gale: p only if q is interpreted as usual (in textbooks of logic) as p → q. This is in accordance with Gale's understanding which becomes clear from his commentary to the steps of the argument, i.e. q is a necessary condition for p. Further we understand by an A-proposition a contingent statement with an index of time. For example: Socrates is sitting at time t, symbolically: pt. After these preliminaries we turn now to the discussion of the premises: (1) The first question is here what it means to "truly express a proposition". Gale ensures us that not overt expression in some public language is necessary and the possibility of private language is permitted. If this is assumed, we may say three things to this premise: (i) It may be right for all human actions of knowing that they occur together with a kind of internal or mental "speaking" (a theory which goes back to Augustin)84. (ii) Although this makes sense for humans, it is completely unintelligible why this should hold for God. Why should God have, besides his action of knowing, an action of "expressing" and "tokening" (second premise)? In a "private Deitese language" with Deitese letters? This would be a rather bizarre, if not completely absurd assumption, which shows that the first premise, if applied to God, must be false.

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We leave that question open; here research in psychology has to find out whether this is generally true.

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(iii) A further wrong assumption in premises (1) and (2) connected with that of (ii) is that every person (therefore also God) could know only by forming propositions. But God need not to know by forming propositions or by expressing propositions; although God may know all the true propositions which are known by man he need not to know them by forming propositions, but may know them by some sort of direct insight. Thus Thomas Aquinas says: "God knows all enunciations that can be formed... He knows enunciable things not after the manner of enunciable things, as in his intellect there were composition or division of enunciations; for he knows each thing by simple intelligence."85 (2) A similar comment can be made to the second premise (it corresponds to Gale's 4d). It might be right for human actions of knowing. But applied to God, both antecedent and consequent will be false and therefore the whole premise (2) will be true, but trivially or emptily true. (3) Premise (3) (corresponding to Gale's 4g) is important since it hides a difference between two time-indices. This can be shown as follows: Asentences can be represented by propositional variables to which time indices are attached: 'pt' represents the A-proposition pt. The index 't' indicates the time when the event described by the proposition occurs, for example: Socrates sits at time t. On the other hand the action of knowing this A-proposition (or of tokening the respective A-sentence) occurs also at a certain time, say t1, if humans are knowing or tokening. Thus it holds for all humans that if a human person a knows that pt (an event occurs at time t) symbolically: aKpt – then the action of knowing also occurs at a certain time, say t1. Symbolically: ∀a∈H, ∀t(aKpt → (∃t1)aKt1pt) where t1 is usually not identical with t.86 But that this principle which holds for humans, should also hold for God is rather inconceivable and completely unjustified; i.e. the statement that God knows that p occurs at time t (where t is a point of time related to a time measurement of this world) is perfectly 85

Thomas Aquinas (STh) I, 14,14. This was also pointed out by Leftow (1990, TAO), p. 309. 86 This is certainly so if the event, the occurrence of which we know is external, so that we need our senses as mediators, since every causal propagation needs time (as we know from the Theory of Special Relativity) such that t1 must be later than t. The same holds if brain processes are needed as mediators. The only case when t1 could be simultaneous with t is one of a kind of knowing as purely mental introspection of the own mental actions or processes. But since this is also hardly possible (for humans) without any brain processes there will be no factual case where t and t1 are strictly simultaneous (even if the time interval between t and t1 can be very short).

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acceptable; but from this it does not follow that his knowledge is (or has to be) at a certain time. This need not be repeated here because it has been discussed in detail already in chapter 3. Specifically w.r.t. premise 3 the fallacious switch, which transfers the time index from the event (described by an A-proposition) to the action (of tokening), can be seen be combining premises 1a, 2 and 3, which imply: aCK(A) → (∃t)aTOts(A). This proposition is of course not generally true, since a can be any person and it is specifically false for God. (4) Enthymem 1 can be certainly accepted for humans. And then steps (4) and (5) (cf. 7.43) are only consequencies. Applied to God, the antecedent of enthymem 1 is false and therefore enthymem 1 (applied to God) is (trivially) true. (5) Enthymem 2 can be accepted with a clarification. If enthymem 2, or its instance: "if God is a timeless being, then there is no time such that God exists at (or in) that time" is interpreted in such a way that 'time' means the time of this world (universe, recall ch. 3) and 'God exists' means the respective objective proposition without reference of man's belief in God's existence, then enthymem 2 can be accepted. Now believers in God's existence have their beliefs at a certain time of this world. One might say therefore that a believer a believes at time t that God exists now (at the time when they believe it). But this is an odd way of speaking. Although it is perfectly correct to say "a believes at time t that God exists" and that in this case both this statement and its content (that God exists) is true, it would be incorrect or at least misleading to add the word 'now' (after 'exists') especially if we assume that God is timeless or "outside" the time of this world. Coming back to the whole argument now, we have shown that premises (1) and (3) must be false in application to God. And since only God can be (understood as) a timeless being, the conclusion (7) is not proved by that argument.

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8. Whether God's Knowledge of Singular, Contingent Truths Implies the Mutability of God 8.1 Arguments Pro 8.11 If God knows singular truths, then he always knows what time it is. But as Kretzmann says: "A being that always knows what time it is, is subject to change."87, i.e. is mutable. But as has been shown in ch. 7 God knows singular truths. Therefore God is mutable. 8.12 Singular contingent truths describe facts that change. But to know the changing of anything is to know first that p (is the case) and then that not-p (is the case, for some contingent p). Now a knowing person who knows first one proposition (that p) and then another one (that not-p) is a knowing person who changes.88 Thus since God knows the changing of everything, he changes himself and is therefore mutable.

8.2 Arguments Contra If a technician constructs a technical object (apparatus), then he knows that this object has a certain lifetime. Thus he knows that the object will last, say two years on the average, and will be out of order afterwards; which is a statistical kind of knowledge concerning singular contingent facts (i.e. knowledge of the probability of some singular contingent event, like that of getting out of order). Now although this technical object changes from functioning to getting out of order, it would be wrong to say that the knowledge of the technician changes. On the contrary the technician knows (and can predict) both the average time of functioning and the average point of time of getting out of order without changing his mind or knowledge with respect to that apparatus he constructed. But what holds for the technician, 87 88

Kretzmann (1966, OSI), p. 410. Cf. ibid. p. 411.

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will hold all the more for God. If God knows the changes of the objects of his own creation, it does not follow from this that his knowledge changes or that he his mutable.

8.3 Proposed Answer God's knowledge of singular contingent truths does not imply that he is mutable. This can be seen as follows: First, by showing that the underlying principle for this claim is not even universally true for man's knowledge. Second, by showing that it is not true for God's knowledge. 8.31 The underlying principle The underlying principle for the claim that God's knowledge of singular (and contingent) truths implies his mutability is the following one: KCH

If a person x knows changing facts, then the knowledge of x changes.

This or a similar form of this principle is also used in the arguments 8.11 and 8.12. Moreover this principle is also defended by Stump and Kretzmann89. 8.32 Principle KCH is not generally valid However, it can be shown that the principle KCH is not generally valid for man's knowledge. All the more we may be suspicious to apply the principle to God's knowledge. The principle KCH is not valid in all cases where we have knowledge of singular (contingent) truths with the help of dynamical laws; in the sense described already in ch. 7.3 by condition D1 and D2. In these cases predictability in a strong sense is also possible. To see that the principle KCH is not valid here, we have to look at the status of knowledge of a scientist who explains and predicts singular (contingent) facts with the help of dynamical laws. If he knows for example the initial conditions of the planetary system (sun plus some planets), i.e. its state S1 at t1. Then he also knows all the successor states Sn at tn (by calculating them with the help of the differential equation). These successor states are different at different times and they are 89

Cf. Stump-Kretzmann (1981, ETE), p. 455: "Thus a being that always knows what time it is, knows first that it is now t1 (and not t2), then that it is now t2 (and not t1) and so on; and in that way such a being's knowledge is constantly changing. And if a being's knowledge is changing in such a way that it no longer knows what it once knew, then that being itself is also changing."

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describable by different singular contingent truths. Now although the states of the system change with the time development, the knowledge of the scientist does not change. He can know all the changes of the successor states by knowing the law (differential equation) plus the initial state together with the calculation procedure. But he does not change his knowledge, neither concerning the law, nor concerning the initial conditions, nor concerning the calculation procedure. And although he may derive (calculate) the different successor states after one another, he knows them implicitly already by knowing the law and the initial conditions. To put it more generally: To know the consequences by (or after) knowing the premises and the derivation rules does not mean to change knowledge. Otherwise every logician and mathematician (who does not even deal with contingent truths) would permanently change his knowledge just by making proofs for different theorems. The only thing which can be called a change here is that the scientist or logician (mathematician) has to proceed successively since he cannot explicitly comprehend all the consequences (theorems) in one action of knowledge. But this does not mean to change his knowledge concerning the respective domain, but only that he comes to a more complete knowledge of that domain piece by piece. 8.33 God does not need to change his knowledge That the principle KCH is not valid if applied to God's knowledge can be seen as follows: (i) First God's knowledge is certainly more perfect than man's knowledge and therefore trivially, at least as perfect as man's knowledge. But since the principle KCH is not valid for man's knowledge, it is certainly not valid for God's knowledge, too. (ii) It is generally assumed that God's knowledge is not discursive or successive; if this assumption is correct, then God knows in one action of knowledge that a system (say a planetary system) is in an (initial) state S1 at t1 and that it will be (by its respective laws) in states S2, S3, S4... Sn at t2, t3, t4... tn in the future (where the time indices t1... tn refer to a reference system of this world, here to the planetary system). He therefore does not change his knowledge although he knows all the different successor states occurring at successive times of the planetary system (as a reference system of this world). (iii) Concerning principle KCH one has to keep in mind the distinction already introduced in ch. 3: (a) Acting (knowing) in such a way that the

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action of knowing occurs at a certain time (symbolically: aKt p). (b) Acting (knowing) in such a way that what is known occurs at a certain time (symbolically: aKpt). Concerning man's knowledge both, his action of knowing and that what is known (if the latter is a fact of this world) occurs at a certain (not necessarily the same) time. Concerning God's knowledge about the world (his creation) only (b) is the case, i.e. he knows that certain events of this world happen at a certain time of (a reference system of) this world. (iv) If the distinction above (iii) is kept in mind, then the difficulties which some people have with some special forms of singular (contingent) truths will disappear: Some see a difficulty with the statement "x knows what time it is". They think that either God cannot know what time it is or if God would know it, he must be mutable.90 The big confusion in such a claim is that the phrase "what time it is" is completely ambiguous. To resolve this confusion we have to point out that 'time' can only be understood as time of this world or as chronological order or as biological (psychological) time (recall ch. 3.32). In the first case what time it is, depends on a reference system. Every schoolboy today knows that New York time is different from Moscow time. And on a larger scale the time in the solar system is different from one in a system of stars which move with different velocity compared to the solar system... etc. But why should God not know what time it is in New York (at a certain moment), i.e. what time would be shown there by accurate clocks. This is even easy for man to find out, why should this be difficult for God. In the second case what time it is depends on a 'place' in a chronological order or time scale. Since there is no empirical measuring unit required by the axioms of chronology, points of time refer to real numbers (if 'time' is understood as continuous). But although chronological time cannot be stretched or contracted (as it is the case with time of this world, i.e. with physical time), there can only be the distinction between earlier and later, when we assume that the chronology condition is satisfied, i.e. when there are no closed time like curves, that is no loops. But why should God not know that one event is earlier than another one according to chronological order? It would be rather ridiculous to assume such ignorance for a perfect being. In the third case, what time it is, depends on some biological or psychological clock of an organism. Such clocks are adjusted to either inner biological processes or to external ones, like the period of day and night (where the period of day and night does certainly have a causal influence on those inner 90

Cf. Kretzmann (1966, OSI).

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biological processes). Accordingly an organism can know some point of time relative to such a clock, say when it is awaking in the morning. But, again, why should God not know when a certain animal or person is awaking in the (at some) morning (relative to some place on the earth) and that the animal or person is aware (or knows) at this time that it is awakening. On the assumption that God is the creator of the universe, the claim that God cannot know what time it is (in the sense of the three cases above) seems particularly strange. Since then he has created time by creating a changing world such that time is dependent on and relative to the kind of change happening in a certain part of the universe. And this holds also for the time on different parts and in different individuals (including human persons) on the earth. Summing up: In all these cases of knowing what time it is, God knows that an event of this world occurs at a certain time relative to a reference system of this world. If he is omniscient, he knows all these reference systems and the respective points of time for occurring events. But since he does not know successively first one event and then a second one, but comprehends the time development of physical and biological systems in a non-discursive (non-successive) action, he need not and does not change his knowledge.91

8.4 Answer to the Objections 8.41 (to 8.11) As it was said in the answer (8.33) the phrase "what time it is" is highly ambiguous. Although God knows "what time it is" relative to a certain reference system on earth or in the universe by knowing what events occur at such a relative time it does not follow from this that God's action of knowing would occur at a certain time. But only if his activity of knowing would occur at a certain time (or would have a time development), his knowledge would be changing and he would be mutable. Since his knowing activity does not occur at a certain time, and since it does not have a time development, his knowledge need not to change and God need not to be mutable. Therefore the conclusion in argument 8.11 is not proved. 8.42 (8.12) Although God knows the changing of facts and although he knows that p occurs at t1 and not-p occurs at t2, he does not know first that p and later that not-p. Since he knows the time development of a system or the succession of the different events not at the times they occur, but not at a time 91

See also the discussion in Castaneda (1967, OIR) and Kutschera (1990, VGI) p. 56f. and p. 337.

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at all. Therefore his knowledge need not to change. If an astronomer knows the orbit of a planet and consequently knows that it is at position P1 at t1 and at position P2 (P1≠P2) at t2, we do not say that the astronomer changes his knowledge (he would have to change his knowledge with every new position of the planet). On the contrary, he comprehends the whole time development of the planet knowing his orbit and therefore also the successive positions. So much the more God encompasses with his knowledge all the time developments (relative to different reference systems) of his creatures small or large on the earth or in the universe without participating himself in some time development. Therefore his encompassing knowledge need not to change.

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9. Whether God Knows What Is Not. 9.1 Arguments Against 9.11 Instead of saying "something is not (the case)" we can say that it is false. Thus Aristotle says: "Again, 'being' and 'is' mean that a statement is true, 'not being' that it is not true but false."92 But what is false cannot be known. Therefore God cannot know what is not. 9.12 What is known is true. What is true is satisfied by a model. What is not cannot be satisfied by a model. Therefore what is not cannot be known. Therefore God cannot know what is not. 9.13 It is not the case that God can fail. But if God cannot fail, then – as Leftow says – God cannot know "that being a failure oneself feels like this": "For if God cannot fail, God cannot have the kind of experience 'this' picks out, and so in a sense cannot even understand the proposition that 'being a failure oneself feels like this'."93 Therefore God cannot know something which is not (the case) of himself and consequently God does not know everything what is not. 9.14 What is known is true and what is true corresponds to a fact. But what is expressed by a counterfactual does not correspond to a fact. Since if p and q are facts the respective counterfactual reads “if p were to occur (presupposing that it does not occur) then q would occur (presupposing that it does not occur)”; but on the assumption, both p and q occur. Therefore what is expressed by a counterfactual cannot be known. And consequently there are some things that are not, i.e. those expressed by counterfactuals, which are not known by God.

9.2 Arguments Pro Everything what is the case is either willed by God or permitted by God that it is the case. And everything what is not (the case) is either willed by God that it is not or permitted by God that it is not. But everything what is either

92 93

Aristotle (Met) 1017a22. Leftow (1990, TAO), p. 313.

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willed by God or permitted by God is known by God. Therefore everything what is not is known by God.

9.3 Proposed Answer 9.31 God's knowledge extends also to that what is not in the sense of what is either impossible or incompatible with laws of nature or accidentally not, but possible. Since, what is not, is impossible by laws of logic or incompatible with laws of nature or not occurring accidentally because of certain initial conditions. Now God knows certainly what is impossible by laws of logic. And under the assumption that he is the creator of the universe, he ruled its evolution by laws and initial conditions, knowing what will occur and what will not occur according to them. Therefore God knows what is not. A more detailed answer is as follows: Concerning what is not it is necessary to distinguish different meanings. First it can refer to some state of affairs which does not obtain. Secondly it can refer to a thing which does not exist. Concerning both cases we have to distinguish that what is logically impossible from that what is – although logically possible – factually not the case. (1) States of affairs which are logically impossible, cannot obtain and things which are logically impossible, cannot exist; i.e. that an earthquake occurs in January 2004 in Persia and does not occur in January 2004 in Persia is impossible to obtain and a round square cannot exist. Impossible states of affairs and impossible things violate the principle of non-contradiction. This principle can be expressed in different versions with different strength.94 The weakest principle of non-contradiction is valid in all classical and many valued systems of logic, except paraconsistent logics95 and quasi-truth-functional logics. It is this principle: NW At most one member of the pair p, ¬p can be true (or can have a designated value). This most tolerant principle of non-contradiction was also already defended by Aristotle.96 Observe that it does not exclude many valued logic since if both p and ¬p would receive the value 94

Cf. Rescher (1969, MVL), p. 144ff. Rescher distinguishes there six versions of the principle. 95 For Paraconsistent Logics see Batens et al. (2000, FLP). 96 Aristotle (Met) 1011b14 and 1062a22.

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indefinite or undefined (as for example in Kleene's system of three valued logic), it is still satisfied. As a realist I do not think that on the level of unconscious nature or reality there can be a violation of NW. Violations of NW can only happen in human thinking and in organisms which have a learning process with trial and error. Therefore paraconsistent logics, which allow violations of NW, can be understood as models of some processes of human thinking or of some processes of trial and error of some organisms having consciousness, but not of the structure of unconscious reality. To sum up, we may say that what is not in the sense of being logically impossible is that what does not satisfy the principle NW of non-contradition. (2) States of affairs which are logically possible although factually not the case, can be of a twofold kind: Those which are not the case at present, but were the case sometimes in the past or will be the case at some time in the future (2a); and those which never obtain although they are logically possible (2b). An analogous distinction can be made concerning things (individuals); those which do not exist at present but did in the past (for example species who became extinct) or will exist in the future from those which never exist. (2a) Concerning those states of affairs which do not obtain at present but obtain either in the past or in the future97, a part of them can be predicted 97

According to Aristotle it holds for all genuine possibilities (potencies) in the sense of contingencies: First that they are actualised at some point of time (in the past, present or future). In other words this opinion is expressed as follows: If the state of affairs p is possible, then there is some time t in the past or future such that p obtains at t (mp → (∃t)pt). Cf. Aristotle (Met) 1047b2, 1047b35 – 1048a1. Secondly it holds according to Aristotle that genuine possibilities (in the sense of contingencies; they would be expressed more completely by: mp ∧ m¬p) are not actualized at some (other) time. In other words, this opinion is expressed as follows: If the state of affairs p is possible (in the sense of a contingent potency), then there is some time t such that p does not obtain at t (mp → (∃t)¬pt), (Met) 1050b10-16. For the first part of the interpretation see also Hintikka (1973, TaN), p.94 and 189. States of affairs which obtain for all times seem not to be possible, in the sense of being contingent, for Aristotle. There are however such examples in modern cosmology: The numerical amount of mass (energy) of the whole universe is not determined by laws as we understand them; the law of the conservation of energy says only that this magnitude is constant, but not that it has a certain numerical value. Now this numerical value is (according to the law) the same for all times. But it is still a contingent fact relative to the laws of nature. For a discussion see Mittelstaedt, Weingartner (2005, LNt) chs. 8.2.2.3 and 8.2.3.2. Cf. 10.31(17) below.

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or retrodicted (even by man, viz. scientists) if there are dynamical laws obeying conditions D1-D3 (recall ch. 7.3), which describe them. Another part may be such that the single events (or single systems or particles) obtain randomly and may be therefore calculated with a certain probability, while only a huge ensemble of single events (single systems or particles) behave law-like, such that the respective macrostate can be predicted according to statistical laws (cf. S1-S3 of ch. 7.3). (2b) Those states of affairs which do not obtain at any time are threefold: (2bi) they may be ruled out or may be incompatible with dynamical laws, (2bii) they may be ruled out or may be incompatible with statistical laws, and (2biii) they may not occur accidentally because of certain initial conditions. An example for (2bi) would be a stone which is not attracted by the earth, such that it would violate the law of gravitation, one for (2bii) a perpetuum mobile of the second kind, which would violate the law of entropy. An example for (2biii) would be a microstate of the universe which would never obtain. The latter can be explained as follows: A litre of air at temperature 0°C (273K) and atmospheric pressure (sea-level) contains about 2,7⋅1022 molecules. This system of molecules can be in a huge number of different possible microstates. The number is about 105⋅1022. Every such state can realise the macrostate "litre of air under the conditions mentioned". Already here we can ask the question whether all these possible microstates will ever be realised (and in what time). But let us extrapolate now the example to the whole universe. What is the number of possible microstates of all the molecules in the whole universe (including dark matter). Can all these possible microstates ever be realised in the lifetime of the universe, if this lifetime is finite? The answer to this question is very probably: No. Thus there are some microstates of the universe which will never be realised.98 This consideration also shows that the laws of nature, as we understand them, are valid not only in our universe, but also in all those others which differ from ours only in some microstates which are not realised in ours but in others.99 Summing up, we may say that states of affairs which are logically possible but do not occur, are ruled out by dynamical or statistical laws either for not occurring at a certain point of time or never; or they do not occur accidentally, either not at a certain time or never because of certain initial conditions like 98

For a detailed discussion cf. Mittelstaedt/Weingartner (2005, LNt), ch. 7.2.3.4.3(2b). 99 For a more detailed argument with additional reasons see Mittelstaedt/Weingartner (2005, LNt), ch. 8.1.6. and Weingartner (1996, UWT) ch.7.

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the limited lifetime of the universe. A similar consideration can be made for things (individuals) which do not exist. Coming back now to the question whether God knows what is not, the answer is as follows: Concerning (1) it would be rather absurd to claim of a perfect being that he would not know what is impossible according to the laws of logic, in particular what is ruled out by a most tolerant principle of non-contradiction; or in other words, why should a most perfect being not know what states of affairs cannot obtain because they are logically impossible? Thus we have to say that God knows what is not in the sense of being logically impossible. This can be substantiated further by two reasons: (i) Man has the ability to know the most basic laws of logic like the principle NW or similar simple ones. And such a knowledge seems to be rather common such that it is available without research, at one's disposal whenever needed and (under normal conditions)100 without error. (ii) According to Thomas Aquinas angels are infallible w.r.t. to logical reasoning.101 Since already man and in a more perfect way angels have this kind of knowledge, all the more God as their creator must have it.102 Concerning (2a) a similar argument as above is suitable: Already man knows, at least to a considerable extend, what is incompatible with laws of nature in form of dynamical and statistical laws. Therefore all the more God will know what is not the case (at what time of some reference system of the universe) according to laws of nature and certain initial conditions. An analogous argument holds for (2b) which differs from (2a) only in that the incompatibility with the laws or initial conditions is not restricted to some period of time. Summing up we can say that God knows also what is not, be it in the sense of what is not according to laws of logic or according to laws of nature or according to certain initial conditions. 9.32 God’s knowledge extends to things that are not actual There is an argument of Thomas Aquinas in support of the thesis that God's knowledge extends to what is not. This argument complements ours and is 100 101 102

Abnormal conditions would be brain defects or similar diseases. Thomas Aquinas (STh) I,58,3. For further supporting arguments recall chs. 1.32-1.34.

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based on the idea that things (or states of affairs) which are not actual, are in the power either of God or of a creature. But God knows his power and the power of his creatures. Therefore he has knowledge also of things that are not. Thomas Aquinas presupposes that all these "things" (states of affairs) are not logically impossible: "Now it is possible that things that are not absolutely, should be in a certain sense. For things absolutely are which are actual; whereas things which are not actual are in the power either of God Himself or of a creature, whether in active power, or passive; whether in power of thought or of imagination, or of any other manner of meaning whatsoever. Whatever therefore can be made, or thought, or said by the creature, as also whatever He Himself can do, all are known to God, although they are not actual. And in so far it can be said that He has knowledge even of things that are not."103

9.4 Answer to the Objections 9.41 What is not can be interpreted in two ways (to 9.11) First, as that state of affairs which is (truly) negated and second, as the negated state of affairs. According to the first interpretation, that which is (truly) negated can be replaced – as Aristotle says – by that which is false.104 Thus the sentence "it is not the case that the universe is spatially infinite" can also be expressed by saying: "it is false that the universe is spatially infinite". And in this case what is not (the case) is that the universe is spatially infinite. This, because it is false, cannot be known. Therefore what is not according to the first interpretation, i.e. that state of affairs which is truly negated, cannot be known and also God cannot know it; but just because something which is false, cannot be known, because what is known has to be true (recall ch. 1.31 and the principle KT). However, according to the second interpretation what is not means the negated state of affairs and this can be known. In our example the negated state of affairs is what is expressed by the true sentence "it is not the case that the universe is spatially infinite" and this fact can be known and it is known 103

(STh) I,14,9. The underlying principle here is Tarski's Truth Condition: The sentence ‘p’ is true if, and only if, p. Or: The sentence ‘p’ is false if, and only if, ¬p. For a detailed discussion of Tarski's Truth Condition, see Weingartner (2000, BQT), ch. 7. 104

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by God. Since what is not has to be interpreted according to the second interpretation in question 9. and in the argument 9.11 it follows that the conclusion in argument 9.11 is not proved. 9.42 Truly negated (to 9.12) The second and the third premise of argument 9.12 are only true if what is not is interpreted according to the first interpretation (as that which is truly negated). But if it is interpreted according to the second interpretation – i.e. in the sense of a "negative fact"105 – it can be known. 9.43 Does “God cannot know something false” imply that he is not omniscient? (to 9.13) It is correct to say that God cannot know something which is false (to say) of him. And this because in general what is false cannot be known (recall ch. 1.31). And since it would be false to say of God that he has human feelings as being a failure, this – that he has such feelings – cannot be known (because it is false). On the other hand God knows that he cannot have such imperfect feelings. Thus the expression what is not in the conclusion is only correct if it is interpreted in the first sense as that what is false. This in fact cannot be known. But God knows what is not in the sense that he knows all the negative facts, i.e. he knows what is not according to the second interpretation, as has been explained in the answer to the other two objections above. Therefore, interpreted in the second sense, the conclusion of 9.13 is not correct. Independently of what has been said as a commentary to the argument 9.13, we want to make a comment to a further claim of Leftow in the article cited above: After the quotation in 9.13 Leftow continues: "So it seems that God's very perfection, by entailing that he cannot fail, entails that he cannot be propositionally omniscient – that there are knowable truths God cannot know."106 From the fact that God cannot have feelings like he himself being a failure and from its true consequence that God cannot know that he has feelings like he himself being a failure, Leftow concludes that God is not omniscient. The structure of this fallacy is as follows: First one looks for some false proposition (for example that God feels he himself would be a failure, or 105

Concerning negative facts like that there is no perpetuum mobile or that the diagonal in the square is not rational cf. Weingartner (2000, BQT), ch. 8. 106 Leftow (1990, TAO), p. 313f.

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some other). Then one concludes correctly from this that God cannot know this false proposition.107 From this one concludes fallaciously that God is not omniscient. We may ask what can be the reason for such a fallacy. In order to find that out we put the argument into symbolic form: (1) (2) (3) (4) (5)

False(p) Assumption: p is false False(p) → (¬∃x)xKp (¬∃x)xKp There is no person x (x runs over persons) such that x knows that p. Form (1) and (2) with M.P. ¬gKp God does not know that p. Form (3) by instantiation. (∃p)¬gKp There is some proposition p such that God does not know that p.

Proposition (5) is of course true and the derivation of (5) is rather trivial. (5) is of course true, since in all cases where p is false, God does not know that p (take 'p' to stand for '2 + 2 = 5' or for 'there is a perpetuum mobile'). Now Leftow seems to conclude from (5) that God is not omniscient. Such a fallacious conclusion could arise from some strange (or better: contradictory) idea of omniscience which is this: x is omniscient means that for any proposition p, God knows that p. This definition is of course contradictory since we may instantiate for p: (q ∧ ¬q). The solution here is just that God can know only propositions that are true, which is complemented by the answer in chapter one that whatever God knows is true. And under the assumtion that p is false God cannot know that p. Thus it is correct to say that there are some propositions which are not known (as true) by God, namely the false ones; but God of course knows that they are false108 9.44 Does God know counterfactuals (9.14)? We ask two questions first: Is the argument valid and are the premises true? Concerning the first, we see immediately that the answer is Yes; when we consider the first and second premise and the conclusion (forgetting now the justification of the second premise beginning with: "Since ..."). The argument

107

Observe that this has nothing to do with God, because any strong concept of knowledge implies that what is known, is true (recall ch. 1.31). 108 There might be some further wrong ideas underlying Leftow's fallacy, but we do not want to speculate about them.

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has then the following simple (valid) structure: K → T and T → F; C → ¬F; Therefore: C → ¬K. Concerning the truth of the premises, there is no problem with the first premise, since both of its parts can be accepted as true.109 The second premise however is mistaken. This can be seen as follows: Assuming two statements p and q, there are two related counterfactuals: C1 If p were to occur, then q would occur. C2 If p were not to occur, then q would not occur. Examples: According to Boyle-Mariotte's law, it holds for an isolated system (keeping temperature constant) of an ideal gas:110 If the pressure were increased (p), the volume would decrease (q). And: If the pressure were not increased (¬p), the volume would not have decreased (¬q). According to Kepler's and Newton's laws it holds: If the constellation among sun, earth and moon were such and such at t1 (p) there would be an eclipse at t2 (q). And: If the constellation … were not such and such at t1 (¬p), there would not be an eclipse at t2 (¬q). Assuming now that both, p and q, are true, it is easy to see that also the respective first counterfactuals (of the form C1) are true. In any case the above instances are well confirmed physical laws. But this is equally so if both p and q are false. The same holds for the respective second counterfactuals (of the form C2). They are also true in both cases, i.e. if p and q are true and if p and q are false.111 This consideration shows that the second premise claiming that "what is expressed by a counterfactual does not 109

That true sentences or statements correspond to facts has been explained and formulated in different ways. It seems that there are more than one consistent ways to do that. A more complicated problem in this respect are the so-called "negative facts" (for example, facts like that the diagonal of the square is not rational or that there is no perpetuum mobile). For one solution of this problem and for the related one of "negative properties" see Weingartner (2000, BQT), ch. 8. 110 In fact it holds only for restricted domains and is replaced for the general case by state equations of thermodynamics. But this does not hinder to use Boyle-Mariotte’s law as an example in the above context. 111 For the case that the antecedent (p) (or both p and q) is (are) true, there are axioms in Lewis' theory of counterfactuals stating that, if p is true, the counterfactual p l→q reduces to p→q and that p∧q implies: p l→ q. Cf. Lewis (1975, CCP), p. 24. Although Lewis' theory of counterfactuals can be applied to the above examples, we do not adhere, in general, to Lewis' theory as an interpretation of the causal relation. Since it can be shown that this theory has only restricted domains of application for a part of the causal relations expressed by dynamical deterministic laws and no or only wrong application for causal relations expressed by statistical laws. See Hausman (1998, CAs), ch. 6 and Mittelstaedt/Weingartner (2005, LNt), ch. 9.2.

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correspond to a fact" is wrong: The above counterfactuals of the examples express well confirmed and true physical laws which correspond to facts. After having shown that the second premise of the argument (in 9.14) is false and therefore the conclusion is not proved, we may return to the main question of this argument: whether God knows counterfactuals. This question has to be divided into three parts: The first part (1) deals with counterfactuals which express a law-like connection; the second (2) with counterfactuals concerning single events in which persons with free will are involved; (3) the third with a historical remark. The question whether God knows counterfactuals is in fact a sub-question of question 9: whether God knows what is not. And that God knows what is not was already substantiated in the answer to question 9. That the question whether God knows counterfactuals is a sub-question to question 9 (whether God knows what is not) can be seen as follows: In a counterfactual of the form C2 it is presupposed that p and q both occur. And the question answered by C2 is what would happen if p were not to occur. (1) In order to know that in the sense of 9.31 (1) above, it is necessary first to know whether the non-occurrence of p is compatible with the laws of logic or with the laws of nature. But this kind of knowledge is available, at least in many cases, already for man. Secondly it is necessary to know the logical and empirical (according to laws of nature) consequences of the non-occurrence of p. But also this kind of knowledge is available to a great extend already for man by knowing laws of logic and laws of nature plus initial and boundary conditions (where by 'laws of nature' both, dynamical and statistical laws, are meant, cf. ch. 7). An analogous consideration can be made concerning counterfactuals of the form of C1 in which it is presupposed that both p and q do not obtain. Now if already man has access to counterfactual knowledge based on laws – even if not completely – we have to say that all the more God, as a perfect being and as a creator of the universe and its laws (including man), has a knowledge of counterfactuals.112 Although from this it does not follow that his knowledge means knowing by forming counterfactuals, as he need not to know by forming propositions. (2) If p and q in C1 and C2 refer to single events in which persons with free will are involved, then even underlying laws are not sufficient to explain the chosen event. Therefore man cannot predict such events (except with very low probability). 112

24.

For a reference in the Bible that God knows counterfactuals see Matthew 11, 20-

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However, we have to say that God can know such events in a multiple sense: that and how they are possible; that, if they would occur, then certain other events would occur and if they would not occur, certain others would not occur. Justifying reasons for that are as follows: (a) Assuming God as the creator, he knows what is in men's (also in a particular man's) power to bring about. (b) Assuming God as the creator, he knows the logical and empirical consequences of a man's particular decision and action. (c) Assuming God as the creator, he knows the logical and empirical consequences of the omission (non-occurrence) of a man's particular decision or action. From these considerations it is plain that God knows the facts expressed in true counterfactuals. (3) Historically, God's knowledge of counterfactuals was already discussed in the Middle Ages. According to Molina (1535-1600), knowing all true counterfactuals is one of three special types of knowledge possessed by God, which he calls middle knowledge: (i) God knows all possible states of affairs and all its combinations and complexities; this is what Molina calls natural knowledge. (ii) Further God knows all contingent facts (past and present); this is his free knowledge. (iii) God knows counterfactuals; this is what Molina calls middle knowledge. This means for Molina that God knows also especially what each particular human person with free will would do were this person to be placed in this or that or ... infinitely many situations and circumstances.113

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For an extensive historical elaboration see Craig (1988, PDF), p. 169ff. and (1991, DFH), ch. XIII. The claim that Molina was the first to point that out it is hardly tenable. At least at the time of Thomas Aquinas such questions were frequently discussed and similar views have been defended. Cf. Thomas Aquinas (STh) I, qu. 14, articles 9, 12 and 13.

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10. Whether Knowledge or Truth Can Change the Status of a State of Affairs 10.1 Arguments Pro 10.11 If the proposition "A wins the election" turns from false at t1 into true at t2, then the status of the corresponding state of affairs changes from nonfactual to factual. Therefore truth can change the status of a state of affairs. 10.12 If the proposition p, to which a state of affairs corresponds even if this state of affairs is contingent, is true (or has been true for some time or forever), then this state of affairs turns into a necessary state of affairs, because it then cannot not obtain. In this sense Aristotle says: "Hence, if in the whole of time the state of things was such that one or the other was true, it was necessary for this to happen, and for the state of things always to be such that everything that happens happens of necessity. For what anyone has truly said would be the case cannot not happen."114 Therefore truth can change the status of a state of affairs.115 10.13 For genuine knowledge we assume that what is known must be true such that the principle aKp → p is a necessary condition for genuine knowledge. Now if it is known at t1 < t0 (t0 = present) that it will be the case that pt2>t0, then pt2>t0 (the state of affairs p at t2 > t0) must obtain and cannot not obtain; otherwise it would not be correct to say it was known so to happen. Thus all contingent states of affairs, truly known before they obtain, must obtain necessarily. Therefore knowledge can change the status of a state of affairs. 10.14 According to the answer of question 1, whatever God knows is true. Now suppose that God knows some singular future contingent event like: Socrates is sitting at time t > t0 (t0 = present). Then either it is possible that Socrates is not sitting at t > t0 or it is not possible. If it is not possible, then it 114

Aristotle (Int), ch. 9, 19a1-5. Instead of "status of a state of affairs" one could say also "ontological status of a state of affairs" in order to keep it off from a more epistemological status. 115

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is impossible for Socrates not to sit at t > t0. Hence for Socrates to sit is necessary. On the other hand, if it is possible that Socrates is not sitting at t > t0, then God's knowledge would be erroneous.116 Thus on the assumption that God's knowledge cannot be erroneous, the fact that Socrates is sitting at t > t0 must be necessary, although we have assumed it to be a contingent event (state of affairs). Therefore knowledge can change the status of a state of affairs.

10.2 Arguments Contra 10.21 Every proposition which represents a state of affairs is either logically true or false, or not logically true or false. If it is logically true or false, it represents a state of affairs which is logically determined. In this case the status of the state of affairs is its logical determination. If it is not logically true or false, it represents a state of affairs which is not logically determined. But truth or falsity does not change the logical determination of the respective state of affairs. And thus truth or falsity does not change the status of this state of affairs. Similarly truth or falsity does not change the logical indetermination of a state of affairs. Similarly the fact that someone knows a proposition cannot change the status of logical determination or logical indetermination of its respective state of affairs. Therefore truth or falsity or knowledge does not change the status of a state of affairs. 10.22 If the proposition p is a law of nature, then it represents a state of affairs with the status of natural necessity. But the fact that p is known by some physicists cannot change the status of natural necessity of the respective state of affairs. Therefore knowledge cannot change the states of affairs which correspond to laws of nature. But a similar argument can be constructed for other cases of the state of affairs. Therefore knowledge cannot change the status of a state of affairs.

10.3 Proposed Answer Neither truth nor knowledge can change the status of a state of affairs which is expressed by a statement or proposition. 116

Cf. the similar argument in Thomas Aquinas (Ver) 2, 12, objection 2.

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In order to see that, we have to show first (10.31) that the status of a state of affairs which is expressed by a statement or a proposition can be of different kind. Secondly (10.32), it will be shown that in the case where the status of a state of affairs is a necessary one or a conditionally necessary one, truth or knowledge cannot change this status. In this case the corresponding propositions are necessary or conditionally necessary. A special case are states of affairs which obtained in the past (up to the present) which are later unpreventable and in this special sense "necessary". Thirdly (10.33), the question whether truth or knowledge can change the status of a state of affairs will be analysed concerning contingent future states of affairs. 10.31 Different Kinds of States of Affairs The status of a state of affairs can be of different kind. The following list may not be complete, but it is sufficiently exhaustive for our problem. (1) p is logically necessary in the sense of a law or of a theorem of logic117: p = pLg Corresponding status: Logical Necessity (2) p is mathematically necessary in the sense of a law or a theorem of mathematics: p = pM Corresponding status: Mathematical Necessity (3) p is naturally (or empirically) necessary in the sense of dynamical laws of nature: p = pN Corresponding status: Natural Necessity (dynamical) (4) p is naturally (or empirically) necessary in the sense of statistical laws of nature: p = pS Corresponding status: Natural Necessity (statistical)118 (5) p is conditionally naturally necessary in the sense that it follows from dynamical laws + initial conditions: p = pNI Corresponding status: Conditional Natural Necessity (dynamical) (6) p is conditionally naturally necessary in the sense that it follows from statistical laws + initial conditions: p = pSI Corresponding status: Conditional Natural Necessity (statistical) Observe however that since statistical laws do not apply to the single state of affairs (instantiation) in the same sense as to the whole ensemble (for example a thermodynamical macrostate) only the macrostate can be

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Usually what is meant here are the theorems of First Order Predicate Logic with Identity. 118 Examples for dynamical laws are Newton's second law of motion, Maxwell's equations or the Schrödinger equation. Examples for statistical laws are the law of radioactive decay or the law of entropy. For details on dynamical and statistical laws, their properties and their differences see Mittelstaedt/Weingartner (2005, LNt), ch. 7.

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predicted or retrodicted with certainty i.e. with natural necessity (cf. (7) below). (7) p is probabilistically contingent in the sense that it is a single state which obeys statistical laws + initial conditions. p = pPC. Since statistical laws allow degrees of freedom for the single state (for example a thermodynamical microstate) they can be predicted or retrodicted only with a certain probability whereas a macrostate can be predicted or retrodicted with natural necessity (cf. (6) above). Corresponding status: Probabilistic contingency. (8) p is unpreventably necessary in the sense that p represents a past or present state of affairs: p = pt ≤ t 0 Corresponding status: Unpreventable or Past Necessity (9) p is contingent in the sense that it is not ruled and not ruled out by any (known) laws. That means also that p is compatible with all (known) laws: p = pC Corresponding status: Contingency Possible Combinations: It is plain that each of pLg, pM, pN and pS cannot be combined with any other proposition representing (another) status. The reason is that their corresponding status is timeless. More accurately: The laws of logic and of mathematics have nothing to do with time, they may be called atemporal. Whereas all laws of nature describe processes of nature which change in space and time but are themselves understood as time (translation) invariant; i.e. as not changing through time.119 On the other hand the corresponding states of affairs of pNI, pNS, pt ≤ t0 and pt > t0 are obtaining at a certain point of time. pC may be conjoined to the latter two, but has to be treated also separately since there may be contingent states of affairs which never happen. Therefore the remaining combinations are those with past (or present) states of affairs and those with future states of affairs and finally pC separately:

119

There is the difficult question whether the laws of nature are strictly time translation invariant, because of the question whether fundamental constants entering these laws are really constant in time. But the most exact measurements did not establish a convincing deviation from constancy for constants like α, G, h or c so far. For details and further references cf. Mittelstadt/Weingartner (2005, LNt), ch. 8. The expression 'time translation invariant' should protect against confusion with 'time reversal invariant', which holds only for dynamical laws, but not for statistical ones.

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(10) pt ≤ t0 ∧ pNI represent states of affairs which are conditionally necessary (derivable from dynamical laws + initial states of affairs) and obtain at present or in the past. (11) pt ≤ t0 ∧ pNS represent states of affairs which are conditionally necessary (derivable from statistical laws + initial states of affairs) and obtain at present or in the past. (12) pt ≤ t0 ∧ pC represent states of affairs which are contingent (not ruled by laws) and obtain at present or in the past. Observe that (10), (11) and (12) have in common that their states of affairs obtain at present or in the past and are in this sense unpreventably necessary (8).But they are still different concerning their status: (10), because of the component pNI, is conditionally necessary in a stronger sense than (11), since pNS is weaker because of the statistical law. And (12) represents the weakest necessity coming only from the fact that it is a present or past event (state of affairs). Therefore the status of (12), pt ≤ t0 ∧ pC, could also be called Past Contingency. (13) pt>t0 ∧ pNI represent states of affairs correctly predictable with the help of dynamical laws + earlier states of affairs. Its status can be called (Conditional) Future Necessity (Dynamical). (14) pt>t0 ∧ pSI represent states of affairs statistically predictable with the help of statistical laws + earlier initial conditions. Such states of affairs can only be predicted with certainty if they are like thermodynamical macrostates. Corresponding status: (Conditional) Future Necessity (Statistical). (15) pt>t0 ∧ pPC represent single states of affairs statistically predictable with the help of statistical laws and initial conditions. Such states of affairs can only be predicted with a certain degree of probability. Corresponding status: Future Probabilistic Contingency. (16) p is future contingent in the sense that p is contingent (not ruled and not ruled out by any laws, cf. (9)) and represents a state of affairs which will obtain in the future: p = (pC ∧ pt>t0) or for short: p = pCt>t0. Observe that pCt>t0 is not predictable since it is not ruled by either dynamical or statistical laws. An example is a human free will decision. Its status can be called Future Contingency. (17) p is omnitemporal contingent in the sense that p is contingent (cf. (9)) and represents a state of affairs which, either always or never, obtains, i.e. which either obtains all the time or obtains neither in the past, nor at present, nor will obtain in the future: p = (∀t)pCt or p = ¬(∃t)pCt Its status

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may be called Omnitemporal Contingency. An example is the realised numerical value of the mass (energy) of the whole universe. According to the recent understanding, this value is not determined by laws of nature. An example for a contingent state of affairs, which never obtains is a numerical value which is different from the one realised. According to the law of conservation of energy, such a value is constant through time. Those omnitemporal contingent states of affairs, which are realised, could also be expressed as a combination of (12) and (16). As it was said above, (11) and (12) agree in the sense that their states of affairs are unpreventably necessary. Analogously, (13), (14) and (15) agree in the sense that their states of affairs are future. But they are also different concerning their status: The status of (13) and (14) is Future Necessity and its respective state of affairs is predictable with the help of dynamical or statistical laws. The status of (15) is Future Probabilistic Contingency, its respective state of affairs can only be predicted with a certain probability. On the other hand the respective state of affairs of (16) – status: Future Contingency – cannot be predicted at all because of the absence of both dynamical and statistical laws. Thus we have two cases of Future Contingency: (15) and (16). A special case is the omnitemporal contingency of (17). 10.32 The necessary status cannot be changed by truth or knowledge (a) The status is necessary in the strongest sense if p = pLg or p = pM (logical or mathematical necessity). If all states of affairs not possessing this status are called not necessary, i.e. contingent (in this sense), then all laws of nature (even dynamical or deterministic ones) would have to be called contingent. Also the existence of the world (universe) is contingent according to this terminology. This is important in connection with the Christian doctrine that the creation of the universe is not a consequence of God's essence, i.e. neither something what he necessarily wills, nor something what he wills that it happens necessarily (see below 10.33, contingent 1). Now it is easy to understand that neither truth, nor knowledge can change the status of logical or mathematical necessity. A first reason for that is that in the case of logical necessity a logically valid proposition like p → p or ¬(p ∧ ¬p) or (p ∧ (p → q)) → q is atemporal or valid in a timeless way. Since a change requires a different state of affairs at a different time, there cannot be such a change of the state

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of affairs corresponding to such logically valid propositions; i.e. if they are true, they are true atemporally or eternally. The same holds for mathematically valid propositions like 2 + 3 = 5 or x ⋅ y = y ⋅ x. A second reason is that truth (the truth predicate or the truth operator) is itself atemporal. States, states of affairs and events obtain at a certain time (of this world). And therefore propositions, if they represent states, states of affairs or events, may have a time index like pt, qt1, pt>t0... etc. But it makes no sense to attach a time index to the truth predicate or to the truth operator. Thus it is correct to say: "It is true that World War II ended in 1945", but not: "It is true in 1945 (or at some other time) that World War II ended in 1945". Since the states of affairs corresponding to logically or mathematically necessary propositions are atemporal, too, no time or change can be involved concerning the status of logical or mathematical necessity and the truth of its representing propositions. Concerning human knowledge, the operator "knows that" or "the person a knows that" may have a time index. But although some human person may know at a certain time that a logically or mathematically valid proposition is true, just by knowing that, the status (logical or mathematical necessity) or the respective state of affairs cannot be changed. Or to say it with an example: The status (mathematical necessity) of Fermat's theorem, i.e. that the equation xn + yn = zn has no solutions for n > 2, is not dependent on the fact (and cannot be changed by it) that it was correctly conjectured by Fermat, but was known only in 1994, when Wiles has offered (and revised and complemented) its proof. On the other hand, as is clear from chapter 3, it does not make sense to attribute a time to God’s knowledge if, first, by ‘time’ we understand always the time of our universe and, second, we assume God to be outside time (if he has created a changing universe with time). (b) The status is necessary in a strong sense if p = pN or p = pS. Since laws of nature are time translation invariant, pN (dynamical laws) and pS (statistical laws) hold also in a timeless way; or in this case we may also say they hold for all times (if time is the time belonging to the universe). Therefore the reasons why the status (natural necessity) of the states of affairs represented by such propositions cannot be changed by truth or knowledge, are similar as those given above for logically and mathematically necessary propositions. (c) The status is necessary in a weaker sense if p = pNI or p = ptt0 ∧ pPC) or p = pCt>t0 or p = (∀t)pCt or p = ¬(∃t)pCt. It is

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only this third sense of contingency which is meant in the question 10.33 and which will be discussed in this paragraph. (a) ad (15): Future Probabilistic Contingency If we mix a litre of cold and a litre of hot water, we will get two litres of lukewarm water. The probability for the state (of affairs) of lukewarm water after a certain time of mixing will have a certain very high value, close to 1, say: r. We can predict therefore the state of lukewarm water with the very high probability r. This prediction, made earlier (say at t1), than the occurring state (lukewarm water) at t2 (t2 > t1), is a true prediction (statement), i.e. it is true that a certain time interval after mixing the state of lukewarm water occurs with probability r. The state "lukewarm water" is a macrostate in the sense of thermodynamics. It occurs with statistical future necessity. On the other hand the occurring microstate which realises the macrostate is still Probabilistic Contingent since there is a huge number of possible microstates which can realise the macrostate lukewarm water. That just a particular microstate (i.e. a particular distribution of the atoms or molecules) out of the huge number would occur at t2 has an extremely low probability due to the huge number of possible microstates.120 Now this very weak (low) kind of probabilistic contingency which is the ontological status of these microstates is just a fact and will not be changed when the respective proposition describing this fact is (called) true. Therefore truth does not change the status of probabilistic contingency. The same holds for knowledge: The fact that somebody predicts correctly at t1 that a certain state (of affairs) – a particular single microstate – will occur at t2 with the small probability ∆r, does not and cannot change the status of the predicted fact (that it is probabilistic contingent). The same result can be obtained if we take a state of affairs which happens with another very low probability. An example is the so-called tunnel effect in Quantum Mechanics: a particle passes through a potential barrier with kinetic energy which is lower than that of the height of the barrier. Assume that the very low probability for such an effect is ∆s. Also then a true prediction for such a state of affairs does not change the probability value ∆s of the event and consequently does not change its status of Future Probabilistic Contingency. The same holds for knowledge as is clear if we speak of predictions. (b) ad (16): Future Contingency 120

The number of microstates which can realise the same macrostate was used by Boltzmann to define entropy.

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Here we take a state of affairs which cannot be predicted, not even with some probability via underlying statistical laws. An example is a human free will decision. The historically famous example is Aristotle's sea battle.121 There are numerous interpretations of chapter 9 of Aristotle's De Interpretatione. Summarising them, they can be grouped into two main interpretations Int 1 and Int 2. Int 1 is defended by the Epicureans, Boethius and in the 20th century by Anscombe, Lukasiewicz and Prior.122 Int 1 seems to be based first on De Int. 19a4: "What anyone has truly said would be the case, cannot not happen." In other words: If p is true, it cannot not happen that p. Now "it cannot not happen" is interpreted by "impossibly not" or "necessarily", such that we get the principle for future contingent propositions: If pt>t0. is true, then necessarily: pt>t0. Int 1 is further based on De Int. 19a39: "... yet not already true or false." Int 1 says that Aristotle held that the principle of bivalence (every proposition is either true or false) or the principle of excluded middle (every sentence of the form 'p or not-p' is true) are not valid for future contingent propositions. Int 2 is defended by the medieval commentators of Aristotle, among them Thomas Aquinas123, and in the 20th century by Hintikka and Rescher124. According to Int 2, Aristotle did not give up the principle of bivalence or excluded middle in De Int ch. 9. In order to understand this better, one has first to realise the structure of ch. 9. Already according to the medieval commentators, notably Thomas Aquinas in his commentary (1962, AIN), but also observed by Hintikka – Aristotle's text has three main parts: the first part 18a28 – 19a5 contains arguments pro necessity and determinism together with its unlikely consequences (18a28 – 18b25) and then together with its impossible (absurd) consequences (18b26 – 19a5). Inbetween there is a short chapter where Aristotle says that bivalence has to be accepted (18b17 – 25). The second part contains arguments against necessity and determinism (19a6 – 22). Only the third part (19a23- 19b4) contains Aristotle's answer. In this he says that p ∨ ¬p is necessarily true, i.e. l(p ∨ ¬p) – also if ‘p’ is a future contingent proposition – but the necessity operator must not be distributed on the parts. That is, what is 121

Aristotle (Int), ch. 9. Anscombe (1956, ASB); Lukasiewicz (1958, ASy), p. 155f.; Prior (1957, TMd), p. 86. 123 Thomas Aquinas (1962, AIN). 124 Hintikka (1964, FSF); Rescher (1968, TNT); cf. also Weingartner (1964, VFW), where the same view is defended. For an English version of it see Weingartner (2000, BQT), ch. 4. 122

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rejected for future contingent propositions is: lp ∨ l¬p. If the latter is false, its negation has to be true, i.e. ¬lp ∧ ¬l¬p or in other words mp ∧ m¬p must be true for future contingent propositions. The phrase "yet not already true or false" (19a39) is interpreted by Thomas Aquinas125 as yet not already determinably true or determinably false. This rather simple and consistent interpretation will be also adopted here. But for a more systematic solution we shall apply again the two principles which have been applied and confirmed already in the foregoing chapters: P1 Truth is atemporal. P2 Truth or knowledge cannot change the (ontological) status of a future contingent state of affairs which is represented by a future contingent proposition. P1: As has been said above 10.32(a), the truth predicate or the truth operator cannot have a time index; only states of affairs, events or states are in space and time, where time is understood as time of this world.126 Thus from the fact that state of affairs (event) p occurs at time t (pt) one can conclude that it is true that pt, i.e. Tr(pt), but not that it would be true at a certain time, say t or t1 > t that pt. Because a closed proposition p at t(pt) containing no variables, is always true or never. And further if the state of affairs p will occur at t > t0 (t0 = present) in the future, i.e. pt > t0, then it is true that pt > t0, i.e. Tr(pt> t0). The danger here lies in committing a fallacy as follows: Since truth is atemporal, it is also omnitemporal. Thus from Tr(pt) we conclude: omnitemporal Tr(pt). And from this one may wrongly conclude Tr[∀t(pt)], i.e. it is true that p occurs at all times or that p is omnitemporally necessary. The mistake here consists clearly of smuggling in a universal quantification over times taken wrongly from a property of truth, i.e. from its atemporal character. Since 'omnitemporal' is described by ∀t... and since t can be only attributed to states of affairs, events and states, it is better not to attribute omnitemporal to truth at all, but use only the attribute atemporal.127 125

Thomas Aquinas (1962, AIN), p. 123. See chapter 3.32. For more on time see Mittelstaedt/Weingartner (2005, LNt), chs. 6 and 7.235. 127 Applying P1 and consequently attributing time indices only to propositions representing states of affairs (events, states) can simplify considerably certain theses of Tense Logic. For example instead of saying: "If p is true, then it will be true n time units hence that it was true n time units ago that p" we may say: "If it is true that p at t0 (t0 = present), then it is true that p at (t0 + t – t)"; where +t means t time units in the future and – t means t time units in the past. 126

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P2: If p = pCt>t0 (cf. (16) above), then the (ontological) status represented by p is Future Contingency. That means p is neither ruled, nor ruled out by any laws and its corresponding state of affairs will obtain in the future. If p is not ruled by laws, then p is not necessary (in the sense of law-necessity) and if p is not ruled out by laws or compatible with the laws, then p is not impossible, hence possible. In order to make this a little bit more precise we use the terminology introduced above: We define law-necessity (lL) by saying that lLp holds just in case that p is one of the laws (pLg logical, pM mathematical, pN dynamical law of nature, pS statistical law of nature) or ruled by one of them, i.e. a consequence of them together with initial conditions (pNI or pSI). lLp iff p is one of pLg, pM, pN, pS, pNI, pSI.128 If p = pCt>t0, then p is not any one of those listed above; therefore if p = pCt>t0, then ¬lLp. Further, if p is future contingent (p = pCt>t0), then p is also not ruled out by any laws, i.e. p is compatible with any of the laws; therefore if p = pCt>t0, then ◊Lp. From this it follows that if p = pCt>t0, then mLp and ¬lLp: p = pCt>t0 → (mLp ∧ ¬lLp) or p = pCt>t0 → (mLpt>t0 ∧ ¬lLpt>t0) The important thing to observe now is this: Applying the truth predicate or (for reasons of simplicity), the truth operator 'Tr' to future contingent propositions does not change their contingency status since it cannot change the ontological status (of contingency) of the corresponding state of affairs. Because the obtaining of the state of affairs is the reason for the proposition being true and not otherwise. Tr(pCt>t0) → Tr(mLpt>t0 ∧ ¬lLpt>t0) Here one can also distribute Tr to the parts of the consequent: Tr(pCt>t0) → [Tr(mLpt>t0) ∧ Tr(¬lLpt>t0)] Furthermore it seems to hold: Tr(pCt>t0) → Tr(pt>t0 ∧ ¬lLpt>t0) Observe that also (p ∧ ¬lp) expresses the contingent status, although in this case we have a contingent fact which may occur at t0 (present) or even at t < t0 (past). If so no problem arises. But in the case of future

For the interrelations of necessity (l) and possibility (m) and the usual laws of Modal Logic we assume the modest system T (of Feys or v. Wright) and the usual definitions: lp ↔ ¬m¬ p, ¬l p ↔ m¬p etc. 128

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contingencies, if we distribute now Tr, we can see immediately a danger for interpretation: Tr(pCt>t0) → [Tr(pt>t0) ∧ Tr(¬lLpt>t0)] The first part of the consequent: Tr(pt>t0) suggests now that pt>t0 is determinate true. But this wrong suggestion comes from the fact that Tr(pt>t0) is a half-truth; this is so because it is now separated from ¬lLpt>t0, which shows its contingency, because it lost the mark C (for contingent) on the letter 'p'. In order not to forget the status of contingency after distribution, the last two formulas should be better replaced by the two following ones: Tr(pCt>t0) → Tr(pCt>t0 ∧ ¬lLpt>t0) and Tr(pCt>t0) → [Tr(pCt>t0) ∧ Tr(¬lLpt>t0)] What has been said concerning truth, can be analogously said about knowledge. We may therefore replace 'Tr' by 'K' (standing for 'knows that'): K(pCt>t0) → K(mLpt>t0 ∧ ¬lLpt>t0) K(pCt>t0) → K(pt>t0 ∧ ¬lLpt>t0) If one distributes the operator 'K' to the parts of the conjunction, then there is the same danger for a misinterpretation as above: K(pt>t0) may be interpreted wrongly as saying that it is known that pt>t0 is determinate true. But as above, K(pt>t0) is a half-truth and therefore misleading, because it is separated from the part which shows the contingency and does not have the mark 'C' for 'contingent'. Therefore the last formula has to be replaced by K(pCt>t0) → [K(pCt>t0) ∧ K(¬lLpt>t0)]. But if the knowledge is human knowledge, we might attribute time indices to the action of knowing. In this case one may say: The person a knows at time t0 (present) that p will be the case at t>t0: aKt0(pt>t0). But also in this case knowledge cannot change the status of the state of affairs corresponding to the (future) proposition pt>t0. This can be seen as follows: Knowing something before its happening can be knowledge with the help of laws (of nature) in which case the future state of affairs has a certain type of necessity. On the other hand if – as it is assumed here – it is knowledge not with the help of laws (of nature), but correct conjectural knowledge, the prediction pt>t0 will be true but contingent, i.e. not necessary and therefore correctly represented by pCt>t0. Also here the danger for misunderstanding consists in only mentioning a part, i.e. aKt0(pt>t0) without the mark 'C', instead of mentioning the whole which is: aKt0(pCt>t0 ∧ ¬lLpt>t0). Here that what is known represents the status of the

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respective state of affairs: Future Contingency. And the obtaining state of affairs with this ontological status is the reason for the corresponding proposition being true and not the other way round. From these considerations it is clear that neither truth nor knowledge can change the status of Future Contingency of the respective states of affairs. Applying all this now to the problem of the sea battle or to that of a (future) free will decision, it should be clear that "it is true that there will be a sea battle tomorrow" or "it is true that there will be this free will decision next week" are incomplete predictions in the following sense: To make them complete we have to add, i.e. to explicitly mention, the contingency of the future event. Thus we may formulate the two examples above as follows: "It is true, though not necessarily, that there will be..." or: "It is true, but not determined by any law, that there will be...". Similarly for knowledge: "It is known now, but not with the help of laws, that there will be..." or: "It is known now that there will be this contingent (not necessary) effect". (c) ad (17) Omnitemporal Contingency There were views in Aristotle and in Diodorus and the Megarian School to define that what is necessary (possible) as that which is true at all (at some) times: ltp ↔ (∀t)pt mtp ↔ (∃t)pt A somewhat weaker definition is by defining the necessary as that which is the case now and ever after and the possible as that which is the case either now or at some future time. But already the medieval commentators, especially Thomas Aquinas and Cajetanus pointed out that there cannot be an equivalence, but only an implication. More accurately: Provided that the above equivalences are not understood as definitions of a new kind of a time dependent weaker necessity and stronger possibility, but are understood as necessity of laws and compatibility with laws, then there are no equivalences: "For something is not necessary because it always will be, but rather, it always will be because it is necessary; this holds for the possible as well as the impossible."129 In order to show what is meant in the above quotation necessity (necessarily: p) has to be understood here solely as necessity of laws, i.e. as pLg or pM or pN or pS, which will be abbreviated as lL*p, where mL*p ↔ ¬lL*¬p. Then the three principles adopted in the quotation are the following ones (where the second and third follow from the first): 129

Thomas Aquinas (1962, AIN), p.113.

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lL*p → (∀t)pt lL*¬p → (∀t)¬pt (∃t)pt → mL*p From the last principle one recognises immediately that there can be cases where something (some states of affairs) is possible without being realised at some time such that the opposite implication and the equivalence does not hold. There are examples in modern cosmology: the numerical value of the amount of energy (mass) of the whole universe is an omnitemporal contingency. The law of conservation of energy (in a closed system) says that this amount (numerical value) must be constant. But the law does not say how large this numerical value is. The law says if the value is M, then M is constant through time. But the fact that it is just M, is an omnitemporal contingency because it is not ruled by laws. And since this fact is not ruled by laws, it is not necessary in the sense of lL*. Now a value M', which differs a little bit from M is possible, because compatible with the laws, but it will never be realised. Another example concerns rational beings like man somewhere in the universe. As we know so far, other rational beings like men are not determined, neither ruled nor ruled out by the laws of nature. Thus it is possible that they might exist; but on the other hand their existence may never be realised during the lifetime of the universe, i.e. for all times. In this case their existence is possible, but is not realised at any time. The principle lL*p → (∀t)pt shows already that the implication goes only in one direction, i.e from the fact that some states of affairs are omnitemporal, one must not conclude that they are necessary. And thus if it is true that (∀t)pt it does not follow from this that p is necessary. On the contrary, according to the above example p (saying that the mass of the whole universe equals M) is true and holds omnitemporally, but contingently such that the whole truth is expressed thus: (∀t)pt ∧ ¬lL*p This shows again that truth cannot change the status of a state of affairs which is in this case Omnitemporal Contingency. Summarising ch. 10 we may say that it was shown first that truth and knowledge do not change the (ontological) status of necessity of states of affairs corresponding to laws of logic, of mathematics and of laws of nature. Then it was shown that truth and knowledge do not change the status of a conditional and of a weaker kind of necessity of states of affairs which are past (pt>t0) or which can be deterministically predicated (pNI, pSI). Finally it was shown that also the status of contingency (be it Future Contingency or Omnitemporal Contingency) cannot be changed by truth or

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knowledge. This result amounts also to a simple solution of the problem of the "sea battle" and analogous questions.

10.4 Answer to the Objections 10.41 Only closed proposition can be true (ad 10.11) The expression "A wins the election" is not a closed proposition (sentence), but in fact a propositional function or a sentential function, i.e. it contains some variables or some indefinite parts or lacks some parts. As such it cannot be true or false. We may construct a closed sentence out of it by adding concrete space and time indices and assuming that 'A' refers to a real person: A wins the election at time t and place x. If this proposition – that a certain event occurs at time t and place x – is true, then it is always true and not once false and once true. Moreover, truth is not the reason for the respective event or state of affairs to obtain. On the other hand the obtaining state of affairs is the reason for the closed proposition being true. Therefore truth cannot change the status of a state of affairs. 10.42 Truth does not destroy contingency (ad 10.12) The answer to Aristotle's problem of the sea battle has been given in detail in ch. 10.33. The main point is first that truth is atemporal and therefore any closed proposition pt is either always true or never. And second, the status of the state of affairs which corresponds to the true proposition pt – if it is contingent – cannot change into necessary; because the contingent state of affairs is the reason for this respective proposition being true. 10.43 The reason for truth is the obtaining fact, not the other way round (ad 10.13) If it is correctly known at t1 < t0 that pt2 > t0, then – since genuine human knowledge needs justification – it can be known either with the help of laws or without. If it is known with the help of dynamical or statistical laws, then the known prediction pt2 > t0 is conditionally necessary (recall 10.31 (5), (6), (13), (14)). If it is a single state and is known with the help of statistical laws, then the prediction pt2 > t0 is probabilistically contingent (recall 10.31 (7), (15)). If it cannot be known with the help of laws, then pt2>t0 is a future contingent statement, which is correctly conjectured (possibly by giving other reasons which are not law-like) (recall 10.31 (9), (16)). But the fact that it is

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correctly conjectured cannot change its contingency, since the reason for the conjecture being true is the obtaining of the contingent future state of affairs and this – the obtaining of the contingent future state of affairs represented by pt2>t0 – is what is known; if otherwise it would not be knowledge, but error. 10.44 God’s knowledge does not change the ontological status of a state of affairs (ad 10.14) The ontological status of the state of affairs that Socrates is sitting at some future time t > t0 or also of the state of affairs that Socrates is not sitting at t > t0 is (in both cases) contingent. But God's knowledge does not change the (ontological) status of a state of affairs. This can be substantiated by three reasons as follows: (1) God's knowledge can be best replaced by truth: gKp ↔ Tr(p). As it was shown in 10.33, truth does not change the status of contingency: (Tr(p) ∧ pC) → Tr(pC). Therefore God's knowledge does not change the status of contingency, i.e. if God knows that p and if p is contingent, then God knows that the contingent proposition p is true: (gKp ∧ pC) → gK[Tr(pC)]. Applied to the example the first justification is this: Let 'p' be the proposition that Socrates is sitting at t > t0. According to ch. 1 whatever God knows is true. Thus if God knows that p, then p is true (Tr(p)) and God knows that p is true (gK[Tr(p)]). But if p is true, (Tr(p)) and the status of p is contingent (St(p) = pC), then the contingent proposition pC is true (Tr(pC)). Now we can assume that God knows of every state of affairs (and of its representing proposition) whether it is contingent or not. Thus if the status of p is contingent, then God knows this: gK[St(p) = pC]. Therefore God knows that the contingent proposition pC is true gK[Tr(pC)]. Symbolically: 1. gKs (where 's' is the proposition that Socrates is sitting at t > t0) 2. gKs → gK[Tr(s)] 3. (Tr(s) ∧ St(s) = sC) → Tr(sC) 4. gK[St(s) = sC] 5. gK[Tr(s)] ∧ gK[St(s) = sC] from 1./2. and 4. 6. gK[Tr(s) ∧ St(s) = sC] Distribution (∧) 5. C 7. gK[Tr(s )] from 3. and 6. by the principle of Epistemic Logic: [(p → q) ∧ Kp] → Kq. (2) The second reason is this: God's knowledge does not change the contingent status even if he knows with necessity whatever he knows (cf. ch. 2). Thus

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if he knows that p and p is future contingent (p = pCt>t0), then he necessarily knows that pCt>t0. And since the contingency of pt>t0 can be expressed by not-necessarily pt>t0 (¬lpt>t0), God necessarily knows that not-necessarily pt>t0: lgK[¬lpt>t0]. From this it is seen that even God's necessary kind of knowledge does not change the contingent status: what he necessarily knows, is the contingency of the future state of affairs expressed by pCt>t0, i.e.: ¬lpt>t0. Moreover if we add to lgK[¬lpt>t0] as a second premise: Necessarily: whatever God knows is the case (l(gKp → p)), then the conclusion is: Necessarily: (not-necessarily pt>t0); i.e. l¬lpt>t0. Independently, this can be received by a theorem of the modal system S5 from ¬lpt>t0. This theorem is also used as an additional axiom leading from the weaker modal system T (Feys and v. Wright) to S5: ¬lp → l¬lp.130 (3) A third reason is this: Every obtaining state of affairs (every fact) is either willed by God or permitted by God and therefore its respective ontological status is also either willed or permitted by God. But if the status of any state of affairs is either willed by God or permitted by God, then God's knowledge cannot change this status. Otherwise his knowledge would be inconsistent with his will which is impossible.

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For details of this kind of argumentation recall ch. 1, objection 1.14 and the answer to it (1.44).

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11. Whether God Knows Future States of Affairs 11.1 Arguments Contra 11.11 If an event E is a future state of affairs and is not determined by dynamical laws, then the ontological status of E is not yet actual and still open. But what is not yet actual, can only be possible or possibly not. Therefore it cannot be known as (actually) occurring or known as (actually) not occurring and consequently God cannot know it in this sense. Therefore God cannot know future states of affairs as obtaining or not obtaining if they are not yet actual. 11.12 If God knows future states of affairs, then he knows them either in their causes or in their actuality. But since they are not yet actual, he cannot know them in their actuality. However, to know them in their causes would mean that they are determined, which is not the case of those states of affairs which are not ruled by dynamical laws, like free human actions. Therefore God does not know future states of affairs like free human actions. 11.13 If God knows the future in the sense that he foreknows all human actions, then they cannot happen otherwise then he foreknew. For example, if he knows that Judas will be a traitor, it is impossible for him not to become a traitor, that is, it is necessary for Judas to betray. Thus the actions of men follow by necessity from the foreknowledge of God;131 and consequently there are no free human actions and man is not responsible for them such that court and criminal law (and many other institutions) are in vain. But this seems to be absurd. Therefore God does not foreknow all human actions. 11.14 If a proposition is true, then it represents a fact (a state of affairs that obtains). If a proposition is known, then it is true. If a proposition is future

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Cf. the argument by Lorenzo Devalla in his Dialogue on Free Will, reprinted in Dworkin (1970, DFW), p. 111-118, p. 111. Cf. further objection 13 of Thomas Aquinas (Ver) 24.

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contingent, then it does not (yet) represent a fact (a state of affairs that obtains). Therefore: If a proposition is future contingent, then it is not known. And consequently it is not known by God.132 11.15 (1) Assume that God knows the future state of affairs that person a acts (in such a way) that p (is the case) at time t3 > t0 (where t3 is in the future relative to t0 at present). (2) According to ch. 2 it holds that whatever God knows, he necessarily knows. (3) Moreover, it holds that whatever God knows is (or will be) the case (is true) – according to ch. 1. (4) Therefore if God knows that person a acts (in such a way) that pt3 > t0, then it is necessary that person a acts that pt3 > t0. (5) If it is necessary that person a acts that pt3 > t0, then person a does not act freely that pt3 > t0. (6) Hence: If God knows that a acts that pt3 > t0, then a does not act freely that pt3 > t0. (7) Therefore: If person a does act freely that pt3 > t0, then God does not know (the future state of affairs) that person a acts freely (in such a way) that pt3 > 133 t0. 11.16 (1) Assume that p is a contingent future state of affairs. (2) Then it holds: possible p at t2 (mpt2) and possible not-p at t2 (m¬pt2), where t2 > t0, i.e. t2 is in the future relative to the present point of time t0. (3) Consequently it is possible that God knows that pt2 and it is possible that God knows that not-pt2. (4) Now by the law of excluded middle either God knows that pt2 or God does not know that pt2. (5) (a) According to ch. 2 it holds that whatever God knows, he necessarily knows. And we might add (b): Whatever God does not know he necessarily does not know. (6) Therefore, (from (4) and (5)) it follows that either necessarily: God knows that pt2 or necessarily: God does not know that pt2. But (6) contradicts (3) 132

Cf. Thomas Aquinas (Ver) 2,12, objection 9. An argument with a similar structure is discussed by Linda Zagzebski (1997, FHF), p. 291f. The difference is that there, time indexes (of the past) are attributed to God's knowing and so necessity is introduced as necessity per accidens as William of Ockham called the necessity of the past. Here the view is defended that time indexes cannot be applied to God's actions and they cannot be applied to truth (or a truth operator) either because God and truth are timeless. Time can only be applied to events or states which are in spacetime of this world (creation). However, necessity can be introduced here via premise (2) according to ch. 2. This leads to the same difficulty of God's knowledge (of future state of affairs) and human freedom as in the case of the argument discussed by Linda Zagzebski. 133

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which seems to show that God’s knowledge of future states of affairs is inconsistent.134

11.2 Arguments Pro If one knows all the causes of the contingent future events, then one knows the obtaining of these contingent future events. Now God knows all the causes of every event and consequently also those of every contingent future event. Therefore God knows the obtaining of every contingent future event.135

11.3 Proposed Answer Using a distinction of Thomas Aquinas, we can say that there are two ways how God knows future states of affairs: he may know them "in their causes" and he may know them in their actuality.136 He can know the future states in their causes because he knows the causes of the events (states) in the universe and because he knows his power and the power of every creature, especially also that of man. He can know the future states in their actuality as he is outside time. Therefore we may distinguish three cases: First (11.31), he knows the future states of affairs of the universe and of all creatures belonging to it by knowing its causes. Second (11.32), he knows them because he knows his power and the power of creatures especially that of man. Third (11.33), because he might have a possibility to know everything in its actual state if he is outside time. 11.31 God knows the future states of affairs of the universe and of the creatures belonging to it by knowing their causes 11.311 Future states of affairs ruled by laws of nature. Concerning the future states of affairs of the universe we might distinguish different cases in accordance with chapter 10 above. If the future states of 134

An argument with similar but more complicated structure is discussed by Linda Zagzebski(1997, FHF) p. 297. But the respective inconsistency can also be seen by the simplified argument above. Again the necessity introduced here is that defended in ch. 2 for God’s knowledge, whereas the necessity in Zagzebski(1997, FHF) is introduced as necessity of the past. Moreover we replaced ‘belief’ by ‘knowledge’ because we defended in ch. 1 that there is no belief in God but just knowledge in the strict sense (i.e.: ∀p(gKp → p)). 135 Cf. Thomas Aquinas, (Ver) 2,12. To the contrary 6. 136 Cf. Thomas Aquinas (STh) I, 14,13.

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affairs are ruled by dynamical or by statistical laws, they are predictable (even for humans) with the help of laws and initial conditions (cases (13) and (14) of 10.31, status: future necessity). Humans however, although they can know in principle such cases (pt>t0 ∧ pNI) and (pt>t0 ∧ pSI), they will know only a very small part of all the future states of affairs which are ruled by dynamical and statistical laws. But if we assume that God has created the universe with its laws and with the cosmological initial conditions, he must know all those future states of affairs. The same holds for future states of affairs predictable with the help of statistical laws (macrostates). In the case of dynamical laws, an earlier state will be called the cause of a later state which follows from the earlier one with the help of the law (differential equation). In the case of statistical laws, an earlier microstate will be called the cause of later microstates, which result statistically in a macrostate, even if not every individual element of them is thereby determined (case: pt>t0 ∧ pSI, status: conditional natural necessity (statistical)). In both cases also the law may be viewed as a "cause" in a different sense. In the tradition coming from Aristotle it was termed "causa formalis". Since a true law of nature describes a certain structure of reality, this structure may be viewed as a cause, which together with the initial state (as a cause, termed "causa efficiens") leads to the final state as the effect. That God knows the future states of affairs "in their causes" means then that he knows the laws of nature, the different states of the universe, plus its elements down to the singular causes, and its boundaries (like constants of nature). Concerning such causes even man can find out a lot of things: As an analysis of the causal relations shows, there is a pluralism137 of causal relations and especially also of those which are represented by laws of nature: causal relations which are represented by dynamical laws of Classical Mechanics, Electrodynamics and the Theory of Relativity, by statistical laws of Thermodynamics and Radiation, by dynamical laws underlying Dynamical Chaos, by dynamical and statistical laws of Quantum Theory. Such an analysis shows further that these causal relations have a number of important properties which can be subdivided into: (a) Logical properties like irreflexivity, asymmetry, transitivity, where the relation can be one-one, one-many or many-one. Here transitivity is not generally satisfied, but only w.r.t. dynamical laws. 137

Cf. Weingartner (2005, PCC).

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(b) Intrinsic properties like completeness of causes, robustness and necessity. Here completeness is not satisfied and robustness is not generally satisfied (for example not in chaotic motion). In causal relations which are represented by laws of nature, the necessity of the causal relation is just the necessity of the respective law. This does not mean however that every cause is a necessary condition, as claimed by the so-called Counterfactual Theory of Causality. In the case of statistical laws, the causes are usually neither necessary conditions, nor sufficient conditions. (c) Spatio-Temporal properties like continuity, no closed time-like curves138, temporal order, finite limitation of causal propagation, objectivity of causal ordering. Here continuity is not generally satisfied.139 In an analogous way an analysis could be given for Single Event Causality. In cases of single event causality, we do not assume underlying laws of nature. And this mainly for two reasons: First, because they may be completely hidden such that we do not have any knowledge of them. Secondly, because they may only partially exist since the event is a free decision of human will (see 11.312 below). The analysis of causal relations shows the huge multiplicity and richness of the structural variety of our universe. And the part of it which can be understood by man with the help of laws – though a considerable part – is still rather small in comparison to the many new things discovered and the many new questions which are still unanswered. We may say like Einstein that when one question is answered by science, ten new problems turn up which nobody has ever dreamed of before. Thus if we say that man can know the future states of affairs by knowing their causes and by knowing the causal relations represented by laws of nature, the more we have to say that the creator of the universe will know the future states of affairs "in their causes". 11.312 Future states of affairs only partially ruled by laws of nature. The more problematic cases however are those where the states of affairs are only partially ruled by laws of nature, i.e. cases where a single future event (state of affairs) is a free human action and decision. But also in these cases 138

For more about closed time-like curves, for causality in General Relativity and for the question of singularities in our universe see Hawking, Ellis (1973, LSS). 139 For a detailed analysis of the causal relations represented by laws of nature see Mittelstaedt, Weingartner (2005, LNt), ch. 9, Ruse (2001, CDC).

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God can know the free future actions and decisions of man w.r.t. their causes. The reasons for this are the following: (1) First, we have to remember from chapter 10 that no kind of knowledge can change the ontological status of a state of affairs. Thus if H is a free human action (a free and contingent decision of the human will) the fact that this event is (or can be) known by any other person (including God), does not change this free decision of the will and his contingent status. (2) In order to show that God can know also free future actions or decisions of the human will, we have to show that with respect to all important conditions for such actions: (a) A free action or decision of the human will (FADW) is one without compulsion from outside. On the assumption that God has created this world (including man)140, he will know all possibilities of compulsion hindering FADW. (b) FADW implies deliberation and planning with the help of reason. But for God the reason which a human person is taking into consideration may be even more transparent to him than to the person himself. (c) For some of the FADW ethical and moral reasons like the commandments are considered as reasons or motives. Also concerning these reasons God can know them better than the individual human person himself. (d) FADW implies indeterminacy concerning different aspects of the action or decision. For example, although man is usually not free to choose health or not health, man is largely free in choosing the means, especially if several medicaments are available and helpful. But we have to assume that God knows all counterfactuals of the type if A would occur, man would choose B, and if C would occur, man would choose D… etc. The knowledge of all possible counterfactuals by God was defended by Molina who called it middle knowledge. This point was already elaborated and defended in ch. 9.44 above. 11.32 God knows his power and the power of the creatures including man In ch. 5 it was shown that God's knowledge exceeds his power especially because of the fact that he knows also the free immoral actions of man which 140

We do not enter here the question on how God has created the world (universe). There is, however, no difficulty that God has created the universe as a universe in evolution. Cf. Weingartner (2000, EVS).

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do not come under his power even if he permits them; as states of affairs which he knows and permits, they come under his providence. But God knows also the power of his creatures. In this sense he knows all the abilities, not only of man in general, but also of every particular person. (a) FADW implies the person's ability of autonomous actions and self determination w.r.t. the person's decision. But also here we have to assume that God has a detailed knowledge of the abilities of man in general and of every particular human person. (b) FADW implies responsibility of the person's actions in connection with the evidence that the decision was in his power and that he caused the action.141 In other words: free actions and decisions of the human will are not without causes. The freely acting and deciding human person knows that he himself is the cause of those actions and decisions. But we have to assume that the power and ability to cause free decisions and actions in man is known to God far better than to the person himself. (c) For some FADWs it holds that there are inner actions going on – sometimes not without inner conflict – in preparation for a free will decision. In such cases the respective person might even himself not foreknow what he is going to decide. But from this it does not follow that a close friend of him could not know him better and foreknow what he will decide. So much the more we have to assume that God will be able to know his future decision. (d) For some events, which are or have been future events for men, it holds that they are both in God's power and willed by God. In such a case it is easy to understand that they are known by God. An example would be the birth of Christ which is predicted by Jesaja (7,14): "The virgin will give birth to a child…" Summing up chapter 11.31 and 11.32 we may say that God can know future states of affairs "in their causes", both, if the causal connection is based on laws of nature, and if it is based on his power or on the power and inner autonomous self determination of man in actions or decisions of free will (FADW). However, there is still a remaining difficulty with those future states of affairs – especially FADWs – for which no causes (motives, plans, considerations, intentions… etc.) exist so far. One simple reason for that could be that the person in question is not yet born. From such considerations it seems to 141

This condition was stressed already by Aristotle in his Metaphysics (982b25).

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follow that knowing the future states of affairs "in their causes" – though a good explanation for many cases – will not be applicable to all cases of knowing contingent future states of affairs. This is the reason why Thomas Aquinas proposed a further explanation for God's foreknowledge which will be treated in the next chapter 11.33. 11.33 God might have a possibility to know future states of affairs in their actual states The explanation Thomas Aquinas gives for this is contained in the following quotations: "In evidence of this, we must consider that a contingent thing can be considered in two ways; first, in itself, in so far as it is now in act: and in this sense it is not considered as future, but as present; neither is it considered as contingent (as having reference) to one of two terms, but as determined to one; and on account of this it can be infallibly the object of certain knowledge… In another way a contingent thing can be considered as it is in its cause; and in this way it is considered as future and as a contingent thing not yet determined to one… and in this sense a contingent thing is not subject to any certain knowledge. Hence, whoever knows a contingent effect in its cause only, has merely a conjectural knowledge of it. Now God knows all contingent things not only as they are in their causes, but also as each one of them is actually in itself. And although contingent things can become actual successively, nevertheless God knows contingent things not successively, as they are in their own being, as we do; but simultaneously. The reason is because His knowledge is measured by eternity, as is also His being; and eternity being simultaneously whole comprises all time, as said above."142 "…but the relation of the divine knowledge to anything whatsoever is like that of present to present. This may be understood by the following example. If someone were to see many people walking successively down a road during a given period of time, in each part of that time he would see as present some of those who walk past, so that in the whole period of his watching he would see as present all of those who walked past 142

Thomas Aquinas (STh) I,14,13. With the phrase "above" he refers to (STh) I,10,2.

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him. Yet he would not simultaneously see them all as present, because the time of his seeing is not completely simultaneous. However, if all his seeing could exist at once, he would simultaneously see all the passers-by as present, even though they themselves would not all pass as simultaneously present."143 "Just as he who goes along the road, does not see those who come after him; whereas he who sees the whole road from a height, sees at once all travelling by the way."144 "Therefore, since the vision of divine knowledge is measured by eternity, which is all simultaneous and yet includes the whole of time without being absent from any part of it, it follows that God sees whatever happens in time, not as future, but as present. For what is seen by God is, indeed, future to some other thing which it follows in time; to the divine vision, however, which is not in time but outside time, it is not future but present."145 11.331 God does not know the future states of affairs as future. (1) God neither knows the past events (states of affairs) as past, nor does he know the future events (states of affairs) as future. This can be substantiated as follows: (1a) If x knows the occurrence of an event e as past (for him, for x), then there must be a time interval between the point of time of his action of knowing (t1) and the occurrence of e (the past event) (t2), where t1 is later than t2 (t1 > t2). (1b) If x knows the occurrence of an event e as future (for him, for x), then there must be a time interval between the point of time of his action of knowing (t1) and the occurrence of e (the future event) (t3), where t1 is earlier than t3 (t1 < t3). (2) It follows from (1a) and (1b): If there is a time interval between the point of time of the action of knowing of x and the occurring event, then the action of knowing of x occurs at a certain point of time. (3) But it was shown in ch. 3 that it is not the case that God knows something at some time. Or in other words: No time index can be attributed to God's action of knowing; although God can know that some event happens at some 143 144 145

Thomas Aquinas (Ver) 2,12. Thomas Aquinas (STh) I,14,13 ad 3. Thomas Aquinas (Ver) 2,12.

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time (in this world) i.e. time indices must be attributed to events of this world (they are in space-time), but not to any action of God. (4) Therefore: God does not know the occurrence of an event as past (for him); nor does he know the occurrence of an event as future (for him). However, God does know the occurrence of an event of this world as past (or future) with respect to some point of reference (or reference system) of this world; or more specifically God does know the occurrence of an event of this world (say a human action) as a past or as a future event with respect to a point of time (say London time) of a group of men in this world. The deeper reason for that is that there is no absolute time as also the Theory of Special Relativity tells us. Time is the time of this world (universe). But we cannot even say that there is one time for the whole world (universe); since every reference system (as a part of the universe) hat its own time. However since God is not identical (or part of) this universe (or not identical with his creation), he is outside time. On the other hand, the time of this world is relative in the sense of the Theory of Relativity. This means: (1) There is no designated point of time, but only time intervals.146 (2) The time scale and simultaneity are not the same in different reference systems which are moving with different inertial movement or acceleration. (3) Time does not pass equably for different reference systems. Whether time passes more slowly or more quickly depends both on the movement of the reference system and on the gravitaional field in which it is located. From this it follows: (4) For a reference system (observer) moving with velocity of light (in vacuum) no time passes. 11.332 The same event may be present and future for two different observers or reference systems.147 146

It may seem that a designated point of time is the point of time of the Big Bang which is – according to the Standard Theory of Cosmology – about 15 billion years in the past. But although the Standard Theory is well supported by the cosmic background radiation (discovered by Penzias and Wilson in 1965), the theory is still controversial because there are competitors that seem at least mathematically correct though without empirical test or confirmation so far. Furthermore, this point of time is not very precise. It rests on a number of theoretical assumptions. For instance, on the assumption that the cosmic background radiation cooled down uniformly to the magnitude 2,7 K at this time. 147 This is a result of the theory of Special Relativity.

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A star explosion of the Sirius (Alpha-Centauri) is present for an observer there (or close to it), but is future for us on earth, since we could observe it only about 4 years later (since the distance is about 4 light years). The respective star may be further away, say 1000 light years. It may even not exist anymore when we observe it in a certain state. Imagine an observer (A) (or reference system), who is locally present whenever a cosmological event like a star explosion is actually taking place. He then knows these events in their actuality; whereas other observers B1… Bn, which are further away, will observe these events in their future; where the time interval depends on the distance the light has to run through. Under this supposition observer A, who sees all these events in its present actuality, will know also (and can predict) the future observations (events) of all the other observers B1… Bn. This shows a consistent possibility that an omnipresent being can know events in their present actuality, which are future events for human observers. Such events are not restricted to cosmology, but include also events like human actions, where the slightest motive might become aware only after some time. However, the human observer cannot see the distant object (or event) until the light rays of it or the respective causal propagation reach his telescope and his eyes with finite velocity (light velocity), which needs time. On the other hand, God does not need for his knowledge light rays or causal propagation coming from the event; and there is neither spatial nor temporal distance between his knowledge and any event, because he is outside space and time. And being omnipresent by his knowledge148, God can know events (states of affairs) of this universe (and of everything he has created) in its present actuality. Under these events there are some which are future events for humans. Thus in this sense God can know events in their actuality which are future events for humans. It is another consequence of the Theory of Special Relativity that time (of the universe) does not "flow equably" as Newton thought149, but that it can be "stretched" and "compressed": Clocks, when transported with high speed, go

148

According to Thomas Aquinas God "is" in all creatures by his power, by his presence and by his essence; by his power in so far everything is subject to his power, by his presence in so far all things are bare and open to his "eyes" (i.e. his knowledge), by his essence, inasmuch as he is the cause of their being. 149 Cf. Newton (Princ), Scholium: "Absolute, true, and mathematical time, of itself, and from its own nature, flows equably without relation to anything external…"

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more slowly relative to those transported with low speed.150 Living organisms (including men), when they are transported with high speed, grow or become older more slowly than others which are "at rest" or are transported with low speed. A consequence of this is the so-called "clock paradox" or "twin paradox": an astronaut starting at the age of 20 for travelling in space with v=12/13c for 20 years (according to his clock and aging process) when coming back meets his twin brother, who became 52 years older meanwhile (according to the time passed on clocks at the earth and according to his aging process). Thus the meeting of the twin brothers is further in the future for the one who stays on earth and nearer in the future for the one who travels; i.e. the space traveller knowing and predicting the meeting 20 years ahead (his time) can know and predict the future event of the meeting 52 years ahead relative to the reference system earth, i.e. relative to the clocks of his twin brother on earth. Imagine now an observer for whom no time passes because he is travelling with velocity of light (in vacuum). He knows events as present, although they are future for different reference systems (observers) moving more slowly. Now for God no time can pass not because he would move with the speed of light but because he is outside time. He can know therefore every event as actual (present), since he is omnipresent (in this world, or in his creation) by his knowledge. On the other hand, these events, which are present and actual to his knowledge, can be future for us and moreover future in a different sense for different observers or reference systems in this world, depending on the distance from the event; and they can also be nearer or further in the future depending on the movement of the observer (reference system) or on the gravitational field in which he is located. 11.333 The observer who belongs to the world (universe) cannot have a complete knowledge about the world (universe). The observer of a domain of reality may have an incomplete knowledge for different reasons: First, because concerning the domain observed only statistical laws are known; and statistical laws do not allow predicting with certainty an individual case (a singular event, a particular microstate… etc.), but only the behaviour of the whole ensemble (the average over the individual cases, the macrostate). Second, because the system observed behaves chaotically (in the sense of dynamical chaos) and therefore no predictions are 150

This effect was first confirmed with atomic clocks in airplanes by Hafele and Keating (1971) and by the Maryland Experiment in 1975/76.

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possible, except for a very short time, although the underlying laws are dynamical laws.151 Third, because neither the initial conditions, nor the development of the system in its long history is known enough in order to make exact predictions about the future as it is the case with the beginning and further evolution of the whole universe. Moreover, there is a further sense, according to which an observer cannot have a complete knowledge about the world (universe). It is because he belongs as a part to this very world which he tries to observe, to describe and to explain. More specifically it can be proved that under certain normal conditions (like a consistency condition and a deterministic viz. dynamical time evolution between the measured state of the system S at t0 and the apparatus state(s) at t1) the apparatus (observer) contained in S cannot measure at time t1 all states of S at t0. In consequence of that, the respective apparatus (observer) which is contained in S cannot distinguish at time t1 certain states of S at t0.152 But he may (in principle) have a complete knowledge about S in an extended system S' of S. Thus if the system S is the whole world, the observer cannot have a complete knowledge of the whole world to which he belongs as a part. Now God is not a part of the world. Therefore he is not subject to this restriction. Moreover, we cannot assume that he has some kind of incomplete knowledge of the sort described. Not belonging to the world means that God can have a "point of view" of the world which is impossible for a human observer. He can know events of this world which happen at a certain point of time relative to a reference system of this world and which are nearer in the future for some observers and more remote in the future for some other observers belonging to this world.

11.4 Answer to the Objections 11.41 “Present” or “actual event” is ambiguous (to 11.11) The expression "an event E (of this world) is actual or present" is ambiguous since it has to be relativised to a reference system (to an observer). In particular we have to distinguish the reference system RI, which is the space151

Recall the respective discussion about dynamical and statistical laws in ch. 7

above. 152

For proof and further discussion see Breuer (1995, IAS) and (1997, IOP).

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time point where E occurs from any other reference system R where E can be observed. If R is distant from RI, an event which is actually occurring in RI may be future (not yet actual) in R. Thus God could easily know an event as actual (present) w.r.t. RI and know that this event is not (yet) actual, but future to us at R. On the other hand concerning events which are not actual (present) in any reference system of this world, God can know them in their cause. This is also possible w.r.t. such events which are not ruled by dynamical laws, as has been elaborated in 11.312 and 11.32 above. Moreover, such events can also have the status of future probabilistic contingency (recall 10.31(15) above). In this case God does not know these events as actually occurring somewhere, but only as possible and possible not or as probable and probable not. But from this it does not follow that his knowledge about these contingent fact is contingent too (cf. 11.46 below). 11.42 Future events ≠ determinate events (to 11.12) It is not correct, as said in the objection, that to know contingent future events (like free human actions and decisions) in their causes would mean that they are determined. As shown in ch. 10 above, neither truth nor knowledge can determine (or change) the ontological status of an event or state of affairs, be it necessary or contingent or future contingent. Therefore, since a wrong assumption is used as premise in objection 11.12, its conclusion is not proved. 11.43 “Foreknows” is inadequate for God (to 11.13) The objection 11.13 contains two flaws. The first is basically the same as in the objection 11.12: knowledge does not determine the ontological status of a state of affairs (ch. 10). Thus since Judas' action of betraying is a future contingent and free action of Judas, this contingent ontological status is not taken away or changed by the fact that someone or God knows that this will happen (at a certain time under certain circumstances in this world). And consequently the responsibility for this action is also not taken away. The second flaw is the expression "foreknows" w.r.t. God. Such an expression fits to human knowledge since man can foreknow in the sense that he knows at time t1 that an event will happen at time t2. But God's activity of knowing cannot receive a time index, since it does not happen at a certain time (cf. ch. 3). Therefore, the expression "foreknows" is inadequate for God; the most adequate expression for God is just knows in the present tense.

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Because of these two errors in the assumptions the conclusion of objection 11.13 is not proved. 11.44 “Is a fact” is ambiguous (to 11.14) To this objection a similar answer can be given as that given to objection 11.11. The expression "is a fact" or "is factual" is ambiguous in a similar way as "is actual". It has to be relativised to a reference system in the same way. An event which is a fact (factually) occurring in one reference system, may be not (yet) factual in a different reference system. But the respective contingent future proposition may be known and can be predicted in many cases. If the respective event is a fact w.r.t. to at least one reference system, then it is possible in principle even for man to know it and to predict it for other reference systems. So much the more it can be known by God for every reference system. If, on the other hand, it is not yet factual in any of the reference systems it may be known in its causes by God. 11.45 Does foreknowledge destroy free will decisions? (to 11.15) In order to show the mistake in objection 11.15 we shall put the argument into symbolic form: (1) gK(aApt3>t0) aAp … a acts (such) that p (2) gKp ⇒ lgKp defended in ch. 2 (3) gKp ⇒ p defended in ch. 1 (4) gK(aApt3>t0) → laApt3>t0 (5) laApt3>t0 → ¬aAFpt3>t0 aAFp … a acts freely that p (6) gK(aApt3>t0) → ¬aAFpt3>t0 (7) aAFpt3>t0 → ¬gK(aApt3>t0) Concerning this argument we shall ask two questions: (a) Is the argument valid, i.e. does the conclusion (7) and also (6) logically follow from the premises (1), (2), (3) and (5). It is easily seen that the answer is: Yes, provided that we use a wellknown principle of Modal Logic which says: If lp (necessarily p) and if p ⇒ q (p necessarily implies q) then lq (necessarily q). The second question (b) is the question whether the premises are true. Since only then the conclusion is proved to be true by this argument. But in this respect we find an important neglect: What kind of action is A in “aApt3>t0”. The time index tells us that the action will take place in the future (point of time t3) relative to the present point of time t0. But it does not tell us whether the action itself is contingent and hardly predictable or determined by

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dynamical laws and therefore strictly predictable. If the action is an action of free will then it is contingent and not determined by dynamical laws. And according to ch. 10 knowledge cannot change the ontological status (contingency or free will action) of state of affairs (here of the free will action). Thus the epistemic operator ‘K’ cannot change the action operator ‘A’. In order to correct the argument by correcting the premises we first have to complete premise (1) by changing ‘A’ (standing for any action, determined or free) to ‘AF’ (standing for a contingent action of free will). Then the corrected argument reads as follows: (1’) gK(aAFpt3>t0) (2’) = (2) and (3’) = (3) (4’) gK(aAFpt3>t0) → laAFpt3>t0 We can easily see now that premise (5) becomes false: “If necessarily person a acts freely that pt3>t0 then person a does not act freely that pt3>t0.” And so we cannot use premise (5) anymore. And hence conclusions (6) and (7) cannot be derived anymore. The whole argument stops then with conclusion (4’): If God knows that person a acts freely that pt3>t0 then necessarily person a acts freely that pt3>t0. This corrects the argument and solves the respective difficulty concerning human freedom and God’s knowledge of contingent future events.153 Observe further that mixed modalities like lmp (necessarily possibly p) or mlp (possibly necessarily p) are wellknown in Modal Logics. For example the principle mp → lmp is used as an axiom, which, when added to system T, leads to system S5. We may therefore correct the argument in 11.15 by rewriting premise (1) with the help of explicitely stating the contingency of that state of affairs p brought about by the action of free will of person a. In abbreviated form premise (1) can then be formulated thus: (1’’) gKopt3>t0 where op ↔def mp ∧ m¬p In this case the argument ends with (4’’): (4’’) gKopt3>t0 → l(opt3>t0) that is: If God knows that the contingent state of affairs p will obtain at t3>t0 then necessarily the contingent state of affairs p will obtain at t3>t0. Also with this interpretation of the first premise the question of compatibility of contingent future states of affairs and knowledge of future states of affairs is resolved.

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The argument discussed by Linda Zagzebski can also be corrected along these lines. Then her premise (7) becomes false and conclusion (8) is no more derivable.

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11.46 Is knowledge of contingent future propositions inconsistent? (to 11.16) For reasons of clarification we shall first put the argument in 11.16 into symbolic form: (1) pt2 is a contingent future state of affairs (2) mpt2 ∧ m¬pt2 (where t2 is in the future w.r.t. t0, presence) (3) mgKpt2 ∧ mgK¬pt2 (4) gKpt2 ∨ ¬gKpt2 (5) (a) gKp → lgKp (b) ¬gKp → l¬gKp (6) lgKpt2 ∨ l¬gKpt2 Zagzebski writes (6) as ¬m¬gKp t2 ∨ ¬mgKpt2 which is logically equivalent to (6). She claims that (6) and (3) contradict each other which is however not the case; since the negation of (6) is: mgKpt2 ∧ m¬gKpt2 As can be seen the second part m¬gKpt2 is not equivalent to mgK¬pt2. In the first, the negation is attached to the action of knowing, in the second, to that what is known. However one can derive a contradiction independently from (3) and (5). From (5b) it follows that mgKp → gKp, by contraposition. And by substituting ¬p for p it follows that mgK¬p → gK¬p. With the help of these we can derive from the two parts of (3): gKpt2 ∧ gK¬pt2 This leads to the contradiction pt2 ∧ ¬pt2 by the principle KT: gKp → p which was defended in ch. 1. Although the claim that (6) and (3) are contradicting each other is too strong, the weaker claim (maybe Zagzebski didn’t want to claim more) that (6) implies the negation of (3) is correct provided we assume the principle of modal logic: “If it holds that p strictly implies q then it holds that mp implies mq”: Since gK¬p strictly implies ¬gKp, m gK¬p implies m ¬gKp and therefore (3) implies (3*): (3*) mgKpt2 ∧ m¬gKpt2 Since (6) implies the negation of (3*) and the negation of (3*) implies the negation of (3), it follows that (6) implies the negation of (3). In this way we may derive a contradiction of the form “(3) and non-(3)”. We are now turning to inspect the premises.

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The premises are (1), (2), (4) and (5). Now (4) is a law of logic, it cannot be false.154 (1) and (2) are assumptions which are accepted since the existence of contingent states of affairs is empirically well confirmed. Denying (1) and (2) would mean to claim some kind of necessity (be it necessity of the past or necessity by deterministic laws) for every state of affairs of this world – which seems absurd. Observe in this connection that, for more accuracy, we could quantify premises (1) and (2) and consequently (3) with the quantifier ‘(∃p)’, i.e. ‘for some p’ and also (4), (5) and (6) with the quantifier ‘(∀p)’, i.e. ‘for all p’. This would not change the situation: (6) still then implies the negation of (3). Premise (5a) was defended at full length in ch. 2. In the argument by Zagzebski premise (5) is supported by the principle of necessity of the past. But what about premise (5b). Is it defensible in a similar way as (5a)? First of all (5b) is independent from (5a). (5a) says that for God’s knowledge knowing and necessarily knowing is equivalent, since lgKp → gKp is valid by the law lp → p of Modal Logic. Now (5b) says that for God’s knowledge also knowing and possibly knowing is equivalent; since p → mp is also a law of Modal Logic. But this does not seem so easily defensible since God’s activity is purely actual without potency it is very questionable what ‘mgKp’ should mean. In any case it is easy to see that without (5b) neither of both contradictions mentioned can be derived. A further consideration concerns (3) which for Zagzebski follows from (2) and the premise: there is (and was in the past) an essential omniscient knower. (3) cannot be derived from (2) mpt2 ∧ m¬pt2 alone. To derive (3) from (2) one needs in addition the two principles of omniscience: p → gKp and ¬p → gK¬p. These principles are discussed at the beginning of the next chapter 12. But there (and in the axiomatic theory, ch. 13) they are not used. Instead of them the following weaker principle is used: God knows everything (every truth) about himself, about Logic and Mathematics and about his Creation. Now, assuming that these stronger principles are strictly valid, (3) follows from (2) with the help of the law of Modal Logic p ⇒ q |- mp → mq. But also here the problem is whether these principles are too strong. Since it is not plausible that from the contingency of p (if p is a future-proposition) which is expressed by mp ∧ m¬p we should derive the contingency of the knowledge of p, expressed by mxKp ∧ m¬xKp. Observe moreover that this would not 154

We assume here Classical Logic in which the tertium non datur is universally valid. An exception would be Intuitionistic Logic where the principle is not universally valid.

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even be generally true for man’s knowledge. As it was shown in ch. 2 (answer to the second objection 2.42) that although what is proved can be a contingent fact, the proof process (with the help of laws and logical derivation rules) is necessary and can be necessarily known; in this case also the so derived conclusion, although itself contingent can be necessarily known. Therefore: Since the move from the contingency of p to the contingency of the knowledge of p is not even generally valid for man’s knowledge it need not be valid for God’s knowledge; i.e. there is no hinderauce that God necessarily knows something which is contingent and necessarily knows that it is contingent. This is also supported by ch. 10, where it has been defended that neither knowledge nor truth can change the ontological status of a state of affairs (here of a contingent state of affairs). Summarizing we may conclude that the argument in 11.46 (ad 11.16) rests on two problematic premises: On (5b) and on the move from “contingent p to contingently known that p”. Therefore the conclusion in 11.16 that God’s knowledge of contingent future states of affairs is inconsistent is not proved by this argument. 11.47 Free actions ≠ non-causal actions (to 11.2) To this argument we should add that also free actions and decisions of man (FADW) are not non-causal. The freely acting and deciding person knows that he is the cause and that he is responsible for these actions and decisions. Moreover, there are different kinds of reasons which are causes without being the only ones and without being determining causes like those listed in 11.312 (2b-d) above. And thus God can know also a certain person as the cause for a contingent action in the future.

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12. Whether God Knows Everything That is True This question is the opposite to the one in chapter 1: Whether everything is true what God knows. Symbolically, the question of chapter 1 is formulated as: ∀p(gKp → Tr(p))? Thus question 12 can be formulated as: ∀p(Tr(p) → gKp)?

12.1 Arguments Contra 12.11 If God is omniscient, then he knows everything what is true, or he knows all the truths. Now all that is true or all the truths can be comprehended in the set of all truths. But there is no set of all truths. Since if we assume one, say Tr = {tr1, tr2, …}, then to each set of the power set of Tr there will correspond some truth (for example to θ the true proposition tr1 ∉ θ) and by Cantor's argument the power set of Tr is larger than Tr. Therefore God cannot know everything what is true and consequently he cannot be omniscient. 12.12 To know everything that is true, is to know an infinite number of propositions. Thus if God knows everything that is true, he must know an infinite number of propositions. Now to know an infinite number of propositions seems to be possible only by knowing the axioms (the axiomatic system) from which they follow by the laws of logic. But everything that is true, i.e. all true propositions, are not axiomatisable.155 Therefore God cannot know everything that is true. 12.13 To know everything what is true, means to be able to effectively list all true propositions. Thus if God knows everything that is true, he is able to effectively list all true propositions. Now all those propositions which can be effectively listed are recursively enumerable. But (the set of) all true propositions are not recursively enumerable.156 Therefore God cannot know everything that is true.

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This is a result (theorem) of metamathematics by Tarski and others; cf. 12.342(2). This is a result (theorem) of metamathematics by Tarski and others; cf. 12.342(2).

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12.14 To know everything that is true means to know also all the logical consequences of true propositions according to the principle DO and DI (recall ch. 1). But since there are a lot of superfluous irrelevant and redundant propositions among the consequences of true propositions, it means to know also a lot of superfluous, irrelevant and redundant truths. But this does not seem to be compatible with the perfect knowledge of God. Therefore God cannot know everything that is true. 12.15 To have a perfect and comprehensive knowledge of everything that is true, means to know of the true propositions that they are true and of the false propositions that they are false (not true). But this means to be able to effectively list all true propositions (theorems) on the one hand, and to effectively list all false propositions (non-theorems) on the other hand in which case both, the system of the true propositions (theorems) and the system of the false propositions (non-theorems), are recursively enumerable. This again implies that the system of all true and all false propositions is decidable and recursive. But it is known by proof that the system of all true and all false propositions is not decidable and not recursive.157 Therefore one cannot know of all true propositions that they are true and of all false propositions that they are false. And thus God cannot know them all and so he cannot be omniscient.

12.2 Arguments Pro Even if in the infinite series of numbers there is no highest number, for whom whose knowledge is infinite, the infinite need not be incomprehensible.

12.3 Proposed Answer 12.31 Introduction The set theoretical paradoxes show that one has to be careful with expressions like 'everything', 'all' and more specifically with expressions like 'all sets', 'all truths', 'all predicates', 'all functions' … etc. As is well-known, such expressions – when used uncritically and unrestrictedly – lead to contradictions. To avoid such contradictions, mainly three methods have been used in set theory: to incorporate a type theory, to restrict the axiom of comprehension by the axiom of separation or finally to distinguish sets (which can be members) from classes (which cannot be members). The first 157

This is a result (theorem) of metamathematics by Gödel and Church; cf. 12.342(2).

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method was applied by Russell and Quine, the second by Zermelo and Fraenkel, the third by von Neumann, Bernays and Gödel. The problems just mentioned have nothing specifically to do with omniscience or God's knowledge. They always come up when expressions like 'all sets', 'all truths' are used uncritically or unrestrictedly. Therefore they are related to omniscience or to God's knowledge only to that extent to which man uses such expressions uncritically in the application to God or to His knowledge. There are, it seems, two ways to avoid the above mentioned contradictions when speaking of God's knowledge or of omniscience: (1) To use such expressions in one of the restricted ways as is done in set theory or in a consistent theory of truth. (2) Not to use such expressions w.r.t. question 12, but to enumerate the different domains which are comprehended by God's knowledge. In this chapter we shall apply the second possibility. This does not mean however that we find way (1) inapplicable. Recall however the problems discussed in section 11.46 above. In the appendix of my book on evil158 I have used ∀p(p ↔ gKp) as a definiens for being omniscient in the sense of having both, sound and complete knowledge, which is represented by ∀p(gKp → p) and ∀p(p → gKp), where the latter is an answer to question 12. And then it is shown there that there are consistent (axiomatic) theories which have the following thesis among their theorems: God exists, God is omniscient, God is omnipotent, God is normative and volitive consistent, whatever God wills or causes is good, whatever God can will or can cause is good, there are evils of different kinds, including moral evils. One could also start with the conjunction ∀p[(gKp → p) ∧ (gKp ∨ gK¬p)] using it as a definiens for omniscience. From this one can derive ∀p(p → gKp). On the other hand way (2) by describing the most important domains of God's knowledge, needs to observe the critical restrictions also w.r.t. the truths of each domain. But way (2) is more informative than way (1) in the sense that it incorporates the different domains and gives some details about them. In general, paradoxes of set theory or of the theory of truth (for example Liar paradoxes159) only show that our way of expressing some more complicated propositions (like those involving self reference) is insufficient or makes false hidden assumptions or does not make explicit some correct hidden presuppositions. That is, the difficulties are on the side of our imperfect 158

Weingartner (2003, Evil), p. 137. Cf. ch. 1 for a specific case. For a simple solution of different kinds of Liar paradoxes cf. Weingartner (2000, BQT), ch. 7. cf. Plantinga/Grim (1993, TOC) 159

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knowledge and there is not enough reason to blame reality for it, much less to blame God. If God is infinite, then it is not astonishing that even many descriptions from many different points of view will give – if correct – only a tiny and very partial picture of Him via analogy. 12.32 God's knowledge about himself There are three comprehensive domains of God's knowledge: God's knowledge about Himself, God's knowledge about his creation and God's knowledge about Logic and Mathematics. The latter domain does not necessarily seem to be included in the two former domains. We can say, therefore, God is omniscient iff he knows all the truths about Himself, about creation and about logic and mathematics. God's knowledge about Himself can be considered under two aspects: first (1), insofar as it is about himself independently of his relation to creation, second (2), insofar it is knowledge about his relation to creation. And the latter may be subdivided again into two subdomains: (2a) Into his knowledge about his relation to creation independently of this particular creation he has been doing; and to this pertains his knowledge about his omnipotence. (2b) Into his knowledge about his relation to this particular creation; and to this subdomain belongs his knowledge about his being the first cause, and further about his love, justice, mercy, providence, conservation and government w.r.t. his creation. ad (1) Concerning God's knowledge about himself, independently of his relation to creation, we have to say that it is necessary in two ways: First, in the sense that whatever he knows about himself, he necessarily knows, and second in the sense that whatever he knows about himself is something which is necessarily the case. Under the assumption that to speak about God (independently of his relation to creation) means the same as to speak about his essence, we may formulate what has been said above in the following way: If p belongs to the theorems (true propositions) about God's essence, then both: God necessarily knows that p and God knows that necessarily-p. Symbolically: ∀p[pεTg's Essence → (lgKp ∧ gKlp)]160 160

As was pointed out already in ch. 7.43, to use propositions in such formulations does not mean that God's thinking or knowing uses propositions. We cannot assume of him that he has to split up subject and predicate and to affirm a property of an individual or such similar things which are basic for human knowledge. But we humans can only say something true about God by forming propositions.

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We do not assume the opposite implication to be valid, too. For example a simple logical tautology like p → p or a proposition of the multiplication table are such that both lgKp and gKlp will hold. But to say that these propositions are theorems about God's essence would amount to a very special view about Logic and Mathematics which we do not presuppose here. All theorems about God's essence God not only knows with necessity, but also with perfection, i.e. in a most perfect way. What it means to know something in a perfect way we may understand in an analogous way with respect to our (human) knowledge: Among the propositions which are in principle provable (demonstrable) those are most perfectly known to us (for example to a mathematician) which have been in fact proved – compared to those which could not be proved so far such that only probable reasons were given. Now although God does not need to make a proof in order to know perfectly, – his knowledge is not knowledge with the help of proofs anyway – he nevertheless knows himself in the most perfect way. And similarly to what has been said above about necessity, it holds for perfection: If p belongs to the theorems (true propositions) about God's essence, then both: God most perfectly knows that p and God knows that what he knows is most perfect. Moreover, we may add that everything what is true about God's essence, God knows also in a most complete way. Again, to understand what it means to know something in a complete way, we may use an analogy w.r.t. human knowledge. An axiom system or a theory (consisting of laws + initial conditions) about a domain D is complete iff all true propositions about D are derivable from the axiom system or theory. Thus to have complete knowledge about D means to know the complete axiom system or theory plus all the true propositions about D which are derivable from it; but since they include also the axioms or the laws (+ initial conditions) of the theory we may just say it means to know all true propositions about the domain D. Thus God has complete knowledge about his essence means that he knows all the true propositions about his essence.161 ad (2a) If God necessarily perfectly and completely knows himself (viz his essence), it follows that he must also necessarily and perfectly and completely know his power and that means also to know to what his power extends.162 Now his power extends first of all to the whole creation, i.e. to this particular universe and to all other spiritual creatures. But it extends further to the 161

Concerning irrelevant and redundant consequences of true propositions see ch. 7.42 and the answer 12.44 to objection 12.14 below. 162 Cf. the argument in Thomas Aquinas (STh) I, 14,5.

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possible creation, to creatures he does not create (cause), but can create (cause). According to Thomas Aquinas, the limit for this domain of possibilities is only logical inconsistency.163 But Thomas Aquinas has certainly in mind that to create (cause) something inconsistent with his essence or with his goodness or with his commandments, would be a logical inconsistency for God. ad (2b) Concerning his particular creation, he necessarily knows his being the first cause, his love, justice, mercy, providence, conservation and government. However, it has to be observed that the creation is not a necessary outcome of his essence, but a free decision and therefore the necessity does not refer to his action of creation. Although he necessarily knows that he is acting as a creator, it does not follow from that that he necessarily is acting as a creator. 12.33 God's knowledge about his creation God's knowledge about his creation can be divided into two domains: into his knowledge about the universe and into his knowledge about other creatures, especially spiritual creatures, like the angels. According to chapter 2. we have to say that whatever God knows, he necessarily knows. Therefore it must hold: Whatever God knows about the universe, he necessarily knows. Again from this it does not follow that what happens in the universe, happens with necessity. Since some events are governed by dynamical laws and these happen by a certain kind of conditional necessity and the laws of nature themselves hold also with a certain kind of (natural) necessity. But other events happen in most cases according to statistical laws and these happen with a weaker kind of necessity. Still other events happen rather accidentally and still others according to free will decision; both of these events do not occur necessarily (recall ch. 10 for details). To put it in other words, we may say that God has chosen necessary laws and necessary causes for some events in the universe and statistical laws and causes with weaker necessity for other events and still other causes like consideration and balancing motives for free will decisions… etc. Analogously we can say also concerning the other creatures that everything God knows about them (who are also his creatures) he necessarily knows about them. Although not everything what happens among them happens necessarily; only some events concerning these 163

Cf. Thomas Aquinas (STh) I, 25,3; "Whatever implies a contradiction does not come within the scope of divine omnipotence."

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creatures will happen necessarily, but others will not happen necessarily. Accordingly, we may first say that God's knowledge about his creation encompasses the knowledge of all the laws concerning our universe and the other creation, it encompasses further the knowledge of all the law governed states and events including initial and random conditions and the knowledge of the constants of nature. Some have said that God's knowledge about his creatures is law-like knowledge such that God cannot know singular events which are not happening according to laws or he could not know individuals as individuals. But this would mean that God has only imperfect knowledge about his own creatures which is rather absurd. And in chapter 7 it was already defended in detail that God knows also singular truths. Therefore we have to say that God necessarily knows all events of this universe and all events of his other creatures be they ruled by laws or not. God's knowledge about the universe includes also his knowledge about mankind and about each individual human person. And this means that God necessarily knows all past and present events including the respective human actions (where 'past' and 'present' refer to the time of the earth); this was shown in general already in chapter 4. But God may know also future states of affairs including free decisions of man. With respect to both past (present) and future human actions it is important to realise that God knows all the capabilities and abilities of every particular human person. The knowledge of contingent future states of affairs is a kind of knowledge which is specifically possessed by God. Since to predict the future with the help of dynamical or statistical laws is also an ability of man. And we could imagine that this ability may be possessed in a still much more perfect way by spiritual creatures like angels. But to know future free decisions of individual persons, is something which belongs to God alone. Otherwise his knowledge would not completely surpass the knowledge of man and spiritual creatures. How this is possible was shown in more detail in chapter 11 above. Thus we have to say that God necessarily knows all truths about his creatures. 12.34 God's knowledge about Logic and Mathematics It is well-known that human knowledge about logics and mathematics is limited. Special limitations are expressed by the so-called limitative theorems in the foundations of logics and mathematics. On the assumption that God is the creator of the universe including man and consequently also the creator of

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the human mind and brain, we have to say that God's knowledge supersedes essentially – not only gradually – human knowledge. And this will hold not only for knowledge in general, but specifically also for knowledge about logics and mathematics. Therefore God's knowledge about logics and mathematics – if it is limited at all – cannot be limited in the same way as human knowledge about logics and mathematics. 12.341 Leibniz's idea of human knowledge concerning Logic and Mathematics Leibniz thought that human knowledge is not limited in the way which was discovered only in the last century. His view was that concerning the socalled (by him) veritees de raison (truths of reason) man can (in principle) build up scientific systems which have the following three properties below.164 The two huge domains of science which consist only of truths of reason are logics and mathematics on the one hand and metaphysics on the other. The three properties are: (1) A general conceptual framework which contains all the important basic terms (basic concepts) and all the derived terms (derived concepts) which are built up from the basic terms by definitions. The general conceptual framework is called characteristica universalis. In fact Leibniz thought the characteristica universalis can be made still more precise by mathematising it: A scientific term (concept) can be first analysed as being either basic (primitive) or else being reducible to a basic (primitive) term with the help of a chain of definitions. The mathematisation then proceeds in two steps: First every basic (primitive) term can be represented by a characteristic basic number (or by a characteristic pair of basic numbers). Secondly every compound term can be represented by a characteristic number which is equal to the result of applying a certain mathematical function to numbers (pairs of numbers) which represent primitive terms. (2) The scientific systems of truths of reason (i.e. Logics, Mathematics and Metaphysics) can be built up more geometrico, that is as axiom systems. Every truth of these systems is finitely analytic in the following sense: every truth can be traced back in a finite number of steps (in which also definitions may be involved) to the axioms. And the axioms themselves are finite in number. In modern terms Leibniz would have to say that all these systems are finitely axiomatisable 164

For details cf. Rescher (1979, LIP) and Weingartner (1983, IMS).

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(which is not correct, see below): This, however, does not hold for the scientific systems of physics or of that of jurisprudence and ethics. Since here contingent premises enter, the systems are not finitely analytic (finitely axiomatisable) for man. Only God knows the true axioms. (3) The scientific system of truths of reason (i.e.) Logics, Mathematics and Metaphysics, when built up more geometrico, i.e. as axiom systems, have the further properties of being consistent and complete. The first is implied by Leibniz's principle: All finitely analytic propositions are necessarily true. As has been said above, a finitely analytic proposition is one that can be traced back to or derived from the axioms in a finite number of steps. This principle is therefore a soundness principle for an axiomatic system. The second property, the completeness of the axiom system is implied by Leibniz's principle of sufficient reason which reads in its logical form: Every truth has its proof (from the axioms + definitions).165 That means that Leibniz claimed for the domains of logic, mathematics and metaphysics completeness in the following sense: Every truth in the domain of logic (in the domain of mathematics, in the domain of metaphysics) is derivable from the axioms of the system of logic (mathematics, metaphysics) in a finite number of steps. Further the respective axioms can be found in principle by man for these domains of truths of reason. Leibniz did not claim completeness for the domain of physics, nor for that of jurisprudence and ethics. In these domains only God knows the right axioms, whereas for humans many truths in these domains are infinitely analytic for which an infinite number of steps would be necessary such that man is not able in principle to trace back (or derive) the respective truth to (from) the axioms. Whether Leibniz thought that the scientific (axiomatic) systems of logic, mathematics and metaphysics are also decidable, is not an easy question. On a closer look however it seems that he claimed decidability only for parts. Thus he found himself a mathematical decision procedure for syllogistics.166 He seems to have hoped to extend such a decision method to other domains of logic and to parts of mathematics, but he did not really claim decidability of the full domains.

165 166

LLA).

Leibniz (GP) 2, p. 62. Leibniz (OF), p. 77-84. Cf. Weingartner (1983, IMS), p. 175 and Marshall (1977,

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If we now ask the question which of the features of Logic and Mathematics pointed out by Leibniz could illuminate our view concerning God's knowledge of Logic and Mathematics, we may give the following answer: Concerning the general conceptual framework, God will of course know what the correct (explicit) definitions of all the different logical and mathematical concepts and structures are, but we cannot therefore assume (i.e. it does not follow from that) that God thinks in definitions (as we know them) or that he needs definitions and splitting up definiendum and definiens in order to think.167 Concerning axiomatisability the comment is similar: Although God will know the axioms of those systems of logic and mathematics which are axiomatisable even if these were infinitely many axiom schemata; but from this it does not follow that God's knowledge of the truths of logic and mathematics consists of knowing axiom systems, derivation rules and derived theorems. He will know these truths in one action of knowing and not after one another, if he is outside time as was defended in ch. 3. Also concerning consistency and completeness we have to say that God will know the consistency of those systems which are in fact consistent, even if man cannot prove this with the help of the respective system – according to the limitations proved by Gödel's second incompleteness theorem – and may not be able to prove it at all. Similarly, God will know which systems are complete and which are not, even if man will not find out this for all possible systems. But we must not assume that God knows this by making the usual consistency – or completeness proofs. He does not need to make proofs (á la mathematicians or logicians) in order to know; which is supported by being outside time and by thinking purely actually and not in a discursive way. 12.342 The limitations discovered in the 20th century. We shall discuss these limitations in three steps similar to those of Leibniz.

167

The definitions which are at stake here are not those which are arbitrarily introduced into a context like definitions as convenient abbreviations (for example DNA), but those which are true statements describing the essential features of the concept or structure in question, like the definition of circle. That the important and interesting definitions in the sciences are true or false can be substantiated with a number of reasons. Cf. for definitions in mathematics: Kreisel (1981, BMD). For definitions in general: Weingartner (1989, DRV) and (2000, BQT) ch. 5 (Are definitions true or false?). See also 12.342(1) below.

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(1) There are limitations to a most universal conceptual framework as Leibniz's characteristica universalis. Important examples are the following ones: (a) Tarski discovered that the concept of truth (or the truth predicate) when applied to a (sufficiently rich) formalised system of sentences of language level n is not definable in n, but in language level n+1 (metalanguage including language n). We have to add: provided that the underlying logic is two-valued classical predicate logic with identity (PL1=).168 Or in other words: The notion of truth (the set of all true sentences) of a consistent formalised system containing recursive number theory is not definable in this system. (b) Not every concept or sentence can be arbitrarily expressed or represented. For example if the axiom system of ZF set theory is consistent, it can be proved that there is a sentence of set theory which is not arithmetically expressible.169 A further related theorem is Tarski's Undefinability Theorem.170 (c) Even very precise equivalence-transformations or coding systems are not completely unique. For example, one can show that if two languages S1 and S2 are logically equivalent, the usual definition of verisimilitude (and other definitions) are not invariant w.r.t. to a transformation from S1 to S2; such that in S1 A > B (A is nearer to the truth than B) holds whereas in S2 A < B holds.171 Another example is the fact that even so precise coding systems like Gödel numbers are not completely unique: For instance one can show that the existence of ungrounded and paradoxical sentences (in the sense of Kripke) is not invariant against the kind of Gödel numbering which is chosen if instead of the strong 3-valued logic of Kleene the weak 3-valued logic of Kleene is taken.172 168

Cf. Tarski (1935, WBF). It is known that by deviating from PL1=, for instance by dropping negation (Myhill, 1950, SDT) or by introducing truth value gaps (Kripke, 1975, OTT) or by constructing a logic with independent quantifiers, which is weaker than PL1= (Hintikka, 1996, PMR) the truth predicate is definable in the object language. 169 A sentence of set theory is arithmetically expressible iff it is demonstrably equivalent (within set theory under ZF) to some sentence of elementary arithmetic. 170 Cf. Fraenkel et al. (1973, FST), p. 312. 171 Cf. Schurz (1990, SAE). For another example cf. Weingartner (2000, BQT), p. 178 (where it should read in line 7 from below: (p ∧ ¬ q)* < (¬p ∧ ¬q)*). 172 Cf. Cain, Damnjanovic (1991, WKS). Cf. also Weingartner (1997, LCD). These and related problems were observed by Kreisel much earlier. Cf. Kreisel (1953, PHk), Kreisel, Takeuti (1974, FSR).

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After giving these examples for limits to a most universal conceptual framework we may ask whether God's knowledge could be subject to such limitations. First of all, if God is the creator of the world including man, then he will know the limits of the human mind concerning universal conceptual frameworks. Secondly, he will know that and how man needs conceptual frameworks to think and especially to solve more complicated problems. But from all this it does not follow that God himself is bound to conceptual frameworks or coding systems in order to think. Using conceptual frameworks or coding systems implies that such a thinking is discursive, i.e. involves succession; since the conceptual framework or the coding system is first understood and constructed and then used for the proof. But God's knowledge is not discursive and does not involve succession.173 Therefore God's thinking does not need or use conceptual frameworks or coding systems. A further support for this is that discursive and successive thinking needs time. But if God is outside time, then there cannot be discursion or succession in his actions. Concerning the specific limitations above it can be said: Although God will know Tarski's result (like he will know any other correct result proved by man), his knowledge is not bound to the specific limitation since he can comprehend arbitrarily many language levels even an infinite number of them. That every system of language (be it natural or scientific) has its limits of expressibility means a restriction for humans in so far as they have to use systems of signs (of languages) in order to think in a precise way. But we cannot assume of a most perfect being that he would need such scientific language systems, like that of arithmetic or set theory, in order to think more precisely. The first example in (c) just shows that logical equivalence is not a very strong notion if the usual First Order Predicate Logic with identity is used. The second shows that even Gödel numbers are not completely unique when applied to certain domains. But why should this be a problem for the knowledge of a most perfect being who does not need to use either logical equivalences or some coding system in order to know and to comprehend. (2) There are limitations concerning axiom systems. (a) First of all the system of Classical two-valued Propositional Logic is axiomatisable, decidable and complete. First Order Predicate Logic is axiomatisable and complete, but not decidable. Also Neumann-Bernay's 173

Cf. Thomas Aquinas (STh) I, 14,7: "In the divine knowledge there is no discursion."

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and Gödel's set theory are finitely axiomatisable, but Zermelo Fraenkel set theory ZF is not. (b) The true sentences (wffs) of elementary arithmetic (Peano Arithmetic) are not axiomatisable (and not decidable). Zermelo-Faenkel (ZF) set theory and Neumann-Bernays (NB) and Gödel (G) set theories are all essentially undecidable; i.e. they are undecidable and every consistent extension of them is also undecidable.174 (c) No axiomatisation of mathematics can exactly capture all true statements of mathematics, since it cannot even capture all true statements of elementary arithmetic. More generally: No axiomatisation can capture all statements which are true; i.e. the set of true sentences is not axiomatisable and therefore also not recursively enumerable. We may ask now the question in what sense God must be free from these and similar limitations concerning axiomatisablitiy and decidablity: (i) Even if the system is axiomatisable and may be decidable man knows and comprehends (to some extent) the axioms and several theorems. But he is not able to comprehend all the theorems since they are infinite in number; although any specific theorem (if the system is decidable, then also any specific non-theorem) can be mechanically calculated. However, of God we must assume that he can comprehend infinitely many theorems and axioms at once. (ii) Man cannot comprehend at once many things: he understands different subjects (theorems) after one another and proceeds from the axioms (premises) to the theorems (conclusion). However, this cannot hold for God for two reasons: if God is outside time, he does not understand one thing (theorem) after the other. Further to proceed from axioms or from one theorem to the other would mean to proceed from not-knowing (the new theorem to be proved) to knowing it (when the proof is established). But this is impossible for the knowledge of God. 174

A formalised system of sentences is axiomatisable if its sentences can be effectively listed, i.e. if they are recursively enumerable; in this case the sentences of the system are provable (under some specified rules of proof) from a finite subset called the axioms. The system is decidable if both its theorems and its non-theorems can be effectively listed or if both its theorems and its non-theorems are recursively enumerable. In this latter case the system is recursive. In other words: a system S is decidable if there exists an effective, uniform method (decision method) of determining whether a given sentence of S is valid in S (otherwise undecidable).

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(iii) For man some proposition may be provable or knowable in principle, but not (yet) proved and not (yet) known. For God there is not such a distinction: what is provable or knowable, is known by him. (iv) Some systems are axiomatisable, some others even decidable; that is their theorems are recursively enumerable or even recursive. That means there is a mechanical procedure or computer program to calculate the theorems or both the theorems and non-theorems. Other richer systems, especially those of higher order logic and many systems in mathematics175 are undecidable and also not axiomatisable. In these latter cases it is more complicated to get a suitable or sufficient knowledge of the theorems of the system. However, we cannot assume that an immaterial perfect being would need a computer program or a mechanical procedure in order to know the theorems; and this holds independently of whether the system is axiomatisable or decidable or neither. Thus God also does not need an axiomatisation or partial axiomatisation in order to know the mathematical theorems. He can know all theorems of mathematics without proof or axiomatisation. (3) There are limitations concerning consistency and completeness. (a) There are finitary proofs of consistency of (classical two-valued) Propositional Logic and of some very weak systems of set theory (for example: the simple theory of types without an axiom of infinity). But the consistency of a system of set theory which is rich enough to be sufficient for a reasonable part of mathematics (even for the very restricted part of Peano arithmetic) cannot be proved by finitary means, since it cannot be proved even in the theory itself. 176 There are, however, relative consistency proofs in a twofold sense: First, in the sense that in a stronger theory one may prove the consistency of a weaker one; for example in the set theory of Quine-Morse the consistency of the set theory of Zermelo Fraenkel can be proved. Secondly, in the sense that if one theory is consistent, the stronger one which is received by adding an additional axiom is also consistent: Thus for example if Gödel's

175

Exceptions are for example Elementary Geometry and the Theory of Real Closed Fields which are both decidable. 176 This is a result of Gödel's second incompleteness theorem: the consistency of a sufficiently rich theory cannot be proved with the means of that theory.

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set theory is consistent, then so is the theory extended by the Axiom of Choice and the Continuum Hypothesis.177 In a more general sense we have to say that for such rich theories there are no direct or absolute consistency proofs. From this it also follows that for the system of (all) established mathematical theorems there cannot be a consistency proof, since we cannot imagine a stronger mathematical system. And by weaker means the poof cannot be carried out as it follows from Gödel's second incompleteness theorem. Moreover it follows that a consistency proof of all scientific well-confirmed scientific results is impossible for man; it transcends man's ability. (b) A scientific theory T which describes a domain D of objects (maybe also conceptual) or of reality is called complete if every true statement about D is derivable from T.178 Classical two-valued Propositional Logic and First Order Predicate Logic are complete in all three senses. But stronger theories are usually incomplete. Thus Leibniz was right w.r.t. the completeness of Logic if he meant Syllogistics and even if he would have meant First Order Predicate Logic (which he did not know). But he was not correct w.r.t. mathematics since already elementary arithmetic (Peano Arithmetic) is incomplete. We may now ask again the question in what sense God is not affected by these and similar limitations concerning consistency and completeness. (i) For man finitary consistency proofs might be more transparent and better comprehendible. For God, however, there is no difficulty with infinitary means, first of all because he does not need any "means" or "proof procedures" in order to know; secondly, because he can comprehend infinitely many levels of stronger systems which include the consistency of the weaker systems. But again he does not need such levels of systems in order to know. (ii) For human knowledge most domains D of knowledge are incomplete in the semantic sense (see above); thus the domain of mathematics, of physics, of cosmology, of biology etc. are all incomplete. However, of God we have to assume that he knows all the truths belonging to such a domain D.

177

Cf. Fraenkel et al. (1973, FST) ch. V. and Gödel (1940, CAC). This is called semantic completeness. A theory T is called formally complete, if there is no proper consistent extension of T (with the same vocabulary). T is complete concerning derivability if for every (closed) sentence s well-formed in T, either s or non-s is derivable from (the axioms of) T. 178

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(iii) Man cannot comprehend the infinite completely, but only "potentially". That is by some recursive procedure like "1, 2, 3, … and so on" or by an instruction which leads to an infinite series like (α) 0 is a figure (β) if x is a figure, then so is +x+ However, we have to assume that God can comprehend the infinite completely and without proceeding from one step to the other. Moreover, it is known that beyond the first level of the infinite in the sense of the denumerable infinite there are higher levels of the transfinite realm which are ordered by the continuum hypothesis. Although in this transfinite domain there seems to be considerable freedom concerning the further details of the ordering relations.179 This shows that man is rather unsure about the ordering laws in this domain. Also here we have to assume that God does not need to proceed level by level and does not need the continuum hypothesis or another ordering principle for comprehending the transfinite domain (even if he does know what principles are used by human mathematicians).

12.4 Answer to the Objections 12.41 (to 12.11) Since the set of all truths is infinite, there is a similar situation as with the set of all (natural) numbers; you can always add one, i.e. there is no highest number and similarly there is no last (ultimate) truth. Or if transcending the denumerable domain, there are always higher powers or levels. But from this one cannot conclude that God could not know this. On the contrary we have to assume that he, as an infinite being, possessing infinite knowledge, can comprehend infinitely many numbers and truths and also infinitely many powers of the continuum (cf. 12.342(2)). 12.42 (to 12.12) Since man cannot know an infinite number of propositions completely, he can help himself by knowing the axioms (premises) of the system and the deduction rules for deriving the theorems. If the system is complete, any specific theorem can be derived although nobody can comprehend or infer the infinite number of all theorems of the respective axioms. If the system is not axiomatisable, only a part of the theorems can be derived from the axioms or axiom schemata. But all these restrictions do not hold for God (cf. 12.342(2)), who can comprehend infinitely many theorems 179

Cf. the remarks of Gödel in the Addenda of his (1940, CAC), p. 70.

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and who does not need axioms or derivation rules in order to know these truths. 12.43 (to 12.13) The answer to this objection is similar to that of the former: God does not need the set of true propositions to be recursively enumerable (to be effectively listed) in order to know them. This (axiomatisability) is certainly a help for man, but God does not need it. 12.44 (to 12.14) It is true that among the consequences of true propositions there are a lot of superfluous, irrelevant and redundant true propositions. Now although such irrelevant truths distract man and sometimes lead also to paradoxes which need to be solved by man, such irrelevant truths do not distract or disturb God in his thinking. Likewise, God cannot be seduced or misled by his knowledge of the moral evils committed by man. 12.45 (to 12.15) The answer to this objection is similar to that of objection 12.12. and 12.13: A system or merely set of truths need not to be decidable in order to be known by God. God does not need any mechanical procedure of calculation or a computer program in order to know (cf. also 12.342(2) above).

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13. A Theory of Omniscience 13.1 Introduction The purpose of the following formal system is to show that – contrary to opposite claims – there are theories of omniscience which are built up axiomatically and are appearently consistent containing the following important theorems: (1) God (as a person) exists (2) God is omniscient (3) God is omnipotent (4) What ever God knows ist true (5) Whatever God knows he necessarily knows (6) God’s knowledge is tenseless and deductively infallible (7) God knows all theorems about himself (8) God knows all theorems about logic and mathematics (9) God knows all theorems about the universe (i.e. about its laws, states, initial conditions, constants and events) (10) God knows all past, present and future events relative to a reference frame of the universe (11) God knows all universal and all singular truths about the universe (12) Although God is omniscient and omnipotent he is neither allwilling nor allcausing (13) God knows all moral evil of this world but neither he can will it nor he can cause it; he permits it (under the condition of creating man with free will) The unerlying logic is (two valued) Propositional Logic extended by the Modal System T (of Feys) + epistemic and other operators and a small part of Predicate Logic (of First Order). The terminology is as follows: (1) The copula ‘is’ is expressed by two primitives ∈ and e where ∈ is used for individual variables (representing individuals) and e for propositional variables representing states of affairs. For example: ‘x ∈ OS’ for ‘x is omniscient’ or ‘x ∈ H’ for ‘x is human’; ‘p e ME’ for ‘the state of affairs p is a moral evil’. (2) With respect to states of affairs also the set theoretical elementhood relation ε is used as for example: ‘p ε T(LM)’, ‘p ε T(CR)’, ‘p ε Tg’s essence’,

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‘p ε Tg’s Commands’ standing respectively for ‘the state of affairs p is an element of the theorems of Logics and Mathematics’, ‘p is an element of the theorems of creation’, ‘p is an element of the theorems of God’s essence’, ‘p is an element of the theorems of God’s Commands’. (3) There are different operators attached to propositional variables by which new statements (true or false) are formed. Examples are: K, W, C, CW, CC, A, P, SW, SA representing respectively: knows, wills, causes, can will, can cause, acts, permits, should will, should act. (4) There are the modal operators ‘’ (standing for ‘necessary’) and ‘m’ (standing for ‘possible’) in accordance with the underlying system of Modal Logic T of Feys. (5) Universal and existential quantifiers are used for both types of variables: ∀x, ∃x, ∀p, ∃p. The quantifier ‘E!x’ used in Axiom A1 reads: there is exactly one (i.e. at least one and at most one) x. (6) The formal system proposed here consists of axioms A1-A8, definitions D1-D25 and theorems T1-T135.

13.2 Theory of Omniscience D1

x ∈ Person ↔ [(∃p)xKp ∧ (∃p)xWp] K … knows (that) x ∈ … x is (has)

D2

x=g ↔ x ∈ Person ∧ x ∈ OS ∧ x ∈ OM ∧ x ∈ AG ∧ x ∈ CT g … God OS… omniscient CT… creator

A1

W … wills (that)

OM … omnipotent AG … allgood

E!x(x ∈ Person ∧ x ∈ OS ∧ x ∈ OM ∧ x ∈ AG ∧ x ∈ CT)

The following theory is concerned with God as an omniscient (OS) being. The other attributes, personality, omnipotence (OM), allgoodness (AG) and being a creator (CT) will be treated only insofar as they contribute to or complement the many aspects of omniscience. In this respect the relations of

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God’s knowledge to his will and power and the relations of his knowledge to him as a cause of creation are of special importance. T1

g ∈ Person ∧ g ∈ OS ∧ g ∈ OM ∧ g ∈ AG ∧ g ∈ CT

A1, D2

13.21 Definitions of Omniscience and of Omnipotence D3

g ∈ OS ↔ ∀p(gKp → p) ∧ ∀p[(p ε T(g) ∨ p ε T(LM) ∨ p ε T(CR)) → gKp] ∧ ∀p(gKp → gKp) ∧ ∀p(gKp → ¬(∃t)gKtp)

Justification of the definiens of D3 (omniscience) (1) ∀p(gKp → p). Assume: ¬∀p(gKp → p). Then ∃p(gKp ∧ ¬p), i.e. there is a state of affairs such that God knows that it obtains but it does not obtain. This is inconsistent with the concept of (strong) knowledge cf. the discussion in chapter 1.3 and also inconsistent with a perfect being. Therefore ∀p(gKp → p) (2) ∀p(p ε T(g) → gKp). Assume ¬∀p(p ε T(g) → gKp). Then ∃p(p ε T(g) ∧ ¬ gKp), i.e. there is a state of affairs about God himself although God has no knowledge about it. This is inconsistent with the concept of omniscience and in general with the concept of a perfect being. Therefore (2) holds. (3) ∀p(p ε T(LM) → gKp). Assume: ¬∀p(p ε T(LM) → gKp). Then ∃p(p ε T(LM) ∧ ¬gKp), i.e. some theorems of logic or mathematics God would not know. Since this holds also for man, for logicians and mathematicians, God’s knowledge would not differ essentially from that of humans. But this seems to be absurd, especially if we think of God as the creator of man. Therefore (3) must hold. (4) ∀p(p ε T(CR) → gKp). Assume ¬∀p(p ε T(CR) → gKp). Then ∃p(p ε T(CR) ∧ ¬gKp), i.e. some true propositions about (his own) creation God would not know. Again this holds for man: some (in fact many) true propostitions about the universe humans do not know. But if God is the creator of the universe we cannot assume that he has a human and fallible knowledge concerning the universe. (5) ∀p(gKp → lgKp). Assume ¬∀p(gKp → lgKp). Then it follows that (∃p)(gKp ∧ m¬gKp), i.e. for some states of affairs p, God knows that p but possibly does not know that p. Now this combination – to know in fact that p but possibly not to know it – is impossible for God; although it is very often

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the case with man: in all cases in which we learn something new in the sciences we can say that we in fact know it now but (since we did’nt know it before) we possibly do not know it. (6) ∀p(gKp → ¬(∃t)gKtp). Assume ¬∀p(gKp → ¬(∃t)gKtp). Then it would hold that (∃p)(gKp ∧ (∃t)gKtp), i.e. God’s knowing would be at a certain time; but which time, we can ask. Especially if we assume with the Special and General Theory of Relativity that spacetime belongs to this world (universe) and is bound to this finite universe. Thus if God does not belong to this world (universe) – although we assume that he has created it – then he and his knowledge (and his will) must be outside time; i.e. we cannot attribute a timeindex to his activity (knowing and willing). To make this argument more transparent remember that already on our earth (as a reference system) we distinguish London-time from Tokyo-time. Moreover time does not “flow equably” (Newton, Principia, Scholium) everywhere but runs faster or more slowly depending on the velocity of the reference system (or of the respective atomic or biological clock, as in animals or human individuals). Thus every reference system (system of stars, similar or larger than our planetary system with the sun) has its own time; therefore the concept of simultaneity is relative to the distance and to the velocity. Moreover: For an observer travelling with light velocity (in vacuum) time does not pass away. All this shows clearly the absurdity of attributing a time of any such reference systems of this world to (the actions of) God. (Cf. ch. 3 of this book). D3.1 D3.2 D3.3 D3.4 D3.5

g ∈ SK ↔ g ∈ CK ↔ g ∈ NK ↔ g ∈ TK ↔ g ∈ OS ↔

T(g) … SK … CK … LM … NK … CR … TK …

∀p(gKp → p) ∀p[(p ε T(g) ∨ p ε T(LM) ∨ p ε T(CR)) → gKp] ∀p(gKp → lgKp) ∀p(gKp → ¬(∃t)gKtp) g ∈ SK ∧ g ∈ CK ∧ g ∈ NK ∧ g ∈ TK

the states of affairs (theorems) about God sound knowledge complete knowledge logics and mathematics necessary knowledge creation (and creatures) tenseless knowledge

D4 g ∈ OM ↔ ∀p(gWp → gKp) ∧ ∀p[gCWp ↔ Cons(p) ∧ Cons({p} ∪ Tg’s Essence) ∧

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Cons({p} ∪ Tg’s Commands)] Cons(p) … p is consistent Cons({p} ∪ Tg’s Essence) … p is consistent with the theorems of God’s essence Cons({p} ∪ Tg’s Commands) … p is consistent with the theorems of God’s commands CW … can will CC … can cause, can make, can bring about P … permits (does not prevent) Lg … logic Math … mathematics U … universe OC … other creatures NC … normative and volitive consistent G … good The will of God is understood in such a way that his will is always fulfilled, i.e. never fails. This is expressed in the first part of the definition D4 of omnipotence: ∀p(gWp → gKp) from which it follows with the help of gKp → p (D3.1) that ∀p(gWp → p). Observe however that expressions like “God wills that man obeys his ten commandments” are not formulated in a correct way since by the above principle: if God wills that, then man will always obey his ten commandments; but this is not the case, as we know. Therefore, if God’s will is applied to human actions of free will the correct formulation is that God wills that man should (ought to) obey his ten commandments, since God does not destroy the freedom of man. This is formulated in D13 and also in D15. On the other hand this does not hinder that in some cases God wills that the human person wills something and in these cases this is not a free will decision but may be some inclination (natural right) which is genetically inborn or a result of environement conditions or of education. Justification of the definiens of D4 (omnipotence) (1) ∀p(gWp → gKp), i.e. whatever God wills that it happens he knows that it happens. We shall first examine the following important consequence of it: ∀p(gWp → p), i.e. whatever God wills that it happens, is the case. Assume the contrary: For some states of affairs p, God wills that p occurs but p does not occur: ∃p(gWp ∧ ¬p). In this case God could’nt be almighty (omnipotent). We know that this happens very often with humans: they will that something happens but it does not happen, i.e. their will is not (always) fulfilled. But this is impossible for an omnipotent being.

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In order to justify (1) now assume the contrary: For some states of affairs which obtain by God’s will it would hold that God would not know that they obtain. This is completely impossible for a perfect being; such a case would even be almost impossible for man: that he brings something about with his will but would not know that it is the case. Therefore (1) must hold. (2) ∀p[gCWp → Cons(p)], i.e. if God can will (or can cause) that p, then p must be consistent; or: Whatever God can will (can cause) is consistent. Assume the contrary: then God’s will and God’s power would be inconsistent which is impossible for a necessary being. Observe that “God can will that p” and “God can cause that p” are equivalent provided that p belongs to the states of affairs of creation or to possible alternatives (of creation) compatible with God’s essence and Commands. The same holds for “God wills that p” and “God causes that p”. But in general only gCCp → gCWp and gCp → gWp hold universally; since God can will something of himself (his goodness or his existence) but he cannot cause it (see D5 below). (3) ∀p[gCWp → Cons({p} ∪ Tg’s Essence)], i.e. whatever God can will (can cause) is consistent with his essence or with his nature. Assuming the contrary would mean that his will or his power (which belong to his essence) is inconsistent (incompatible) with his essence which is impossible. (4) ∀p[gCWp → Cons({p} ∪ Tg’s Commands)], .i.e. whatever God can will (can cause) is consistent with his Commands (towards man). Assuming the contrary would mean that his will or power would be inconsistent in the sense that it would be contrary to his commands which express already his will and power towards man. A2 ∀p(gWp → gCWp) Whatever God wills he can will; the opposite does not hold because his power (what he can will) exceeds his actual (factual) willing. A3 ∀p(gCp → gCCp) Whatever God causes he can cause; the opposite does not hold, because his power (what he can cause) exceeds his actual (factual) causing. D5 gCp ↔ p ε T(CR) ∧ gWp God causes that p iff p belongs to the theorems of his creation and God wills that p. D5.1 gCCp ↔ ¬(p ε T(g)) ∧ ¬(p ε T(LM)) ∧ gCWp

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Observe that ‘God can cause that p’ (gCCp) implies via D5.1 and D4 that p must be consistent itself and consistent with (the theorems of) God’s essence and God’s commands (cf. T106) D6 g ∈ AG ↔ g ∈ NC ∧ ∀p(gWp → p e G) D6.1 g ∈ NC ↔ ∀p[(p ε Tg’s Will w.r.t. man) → gPp] D7 g ∈ CT ↔ ∀p[(p ε T(CR) ∧ gCCp) → gCp] D8 p ε T(g) ↔ (p ε Tg’s Essence ∨ p ε Tg’s Relation to CR) D9 p ε T(LM) ↔ (p ε T(Lg) ∨ p ε T(Math)) D10 p ε T(CR) ↔ (p ε T(U) ∨ p ε T(OC)) D11 p ε Tg’s Essence ↔ (lgKp ∧ lgWp ∧ gKlp) D12 p ε Tg’s Relation to CR ↔ p ε Tg’s Knowledge about CR ∨ p ε Tg’s Will about CR D12.1 p ε Tg’s Knowledge about CR

↔

[gK(p ε T(CR)) ∧ gKp ∧ lgKp ∧ gK¬lp]

D12.2 p ε Tg’s Will about CR ↔ [(gWp ∧ ¬lgWp ∧ gW¬lp) ∨ (p ε Tg’s Will w.r.t. man)] D13 p ε Tg’s Commands ↔ ∀x∈ H(gW(xSWp) ∧ gW(xSAp)) D14 gPp ↔

¬gW¬p

D15 p ε Tg’s Will w.r.t. man ↔ ∀x∈ H[gW(xWp) ∨ gW(xAp) ∨ gW(xSWp) ∨ gW(xSAp)] SW … should (ought to) will that SA … should (ought to) act (in such a way) that T2 g ∈ Person T3 (∃p)gKp ∧ (∃p)gWp

T1 T1, D1

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13.22 God’s Knowledge

T4 g ∈ OS T1 T5 ∀p(gKp → p) T4, D3 Whatever God knows is true (is the case) T6 g ∈ SK T5, D3.1 God has sound knowledge T7 ∀p[(p ε T(g) ∨ p ε T(LM) ∨ p ε T(CR)) → gKp] T4, D3 If p is a theorem about God or a theorem about logic or mathematics or a theorem about creation then God knows that p T8 g ∈ CK T4, D3.2 God has complete knowledge T9 ∀p(p ε T(g) → gKp) T7 God knows everything about himself T10 ∀p(p ε Tg’s Essence → gKp) T9, D8 God knows everything about his essence T11 ∀p(p ε Tg’s Relation to CR → gKp) T9, D8 God knows everything about his relation to his creation T12 ∀p[p ε Tg’s Essence → (lgKp ∧ lgWp ∧ gKlp)] D11 Whatever belongs to God’s essence, God necessarily knows and necessarily wills and of it God knows that it is necessarily the case. T13 ∀p[(lgKp ∧ lgWp ∧ gKlp) → p ε Tg’s Essence] D11 T14 ∀p(p ε T(LM) → gKp) T7 T15 ∀p(p ε T(Lg) → gKp) T14, D9 God knows all theorems of Logic T16 ∀p(p ε T(Math) → gKp) T14, D9 God knows all theorems of Mathematics T17 ∀p(p ε T(CR) → gKp) T7 God knows all theorems (truths) about creation and creatures T18 ∀p(p ε T(U) → gKp) T7, D10 God knows all theorems about the universe T19 ∀p(p ε T(OC) → gKp) T7, D10 God knows all theorems about other creatures T20 ∀p(gKp → lgKp) T1, D3 Whatever God knows he necessarily knows

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T21 g ∈ NK T20, D3.3 God has necessary knowledge T22 ∀p(gKp → ¬(∃t)gKtp) T1, D3 Whatever God knows he does not know at some time T23 g ∈ TK T22, D3.4 God has tenseless knowledge D16 x ∈ LO ↔ ∀p(p ε T(Lg) → xKp) LO … logically omniscient T24 g ∈ LO T15, D16 God is logically omniscient A4 ∀p[p├ q → (gKp ├ gKq)] p ├ q means that p → q is a theorem God knows all the logical consequences of what he knows T25 ∀p[(p ├ q ∧ gKp) → gKq]

A4

D17 x ∈ LI ↔ ∀p[(p├ q ∧ xKp) → xKq] LI … logically (or deductively infallible) T26 g ∈ LI D17, T25 God is logically (or deductively) infallible A5 ∀p[(p ε Tg’s Essence ∨ p ε T(LM)) → p is timeless] Theorems about God’s essence and theorems of logic and mathematics are timeless (are not bound to the time of this world (universe)) T27 ∀p(p ε Tg’s essence → p is timeless) A5 T28 ∀p(p ε T(LM) → p is timeless) A5 T29 ∀p[gK(p ε Tg’s Essence) → gK(p is timeless)] T27, A4 T30 ∀p[gK(p ε T(LM) → gK(p is timeless)] T28, A4 13.23 God’s Knowledge of the Universe D18 p ε T(U) ↔ p ε T-Law(U) ∨ p ε T-State(U) ∨ p ε Init(U) ∨ p ε T-Const(U) ∨ p ε Event(U)

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p ε T-Law(U) … p belongs to the law-theorems or law statements of the universe U p ε T-State(U) … p belongs to the theorems describing a state of the universe U or of a part of it p ε T-Init(U) … p belongs to the theorems desrcibing an initial condition of the universe or a part of it p ε T-Event(U) … p belongs to the theorems describing an event (or process) of the universe U p ε Const(U) … p belongs the theorems about the value of a natural constant of the universe U D19 p ε T-State(U) ↔ (∃t,∃f)pt,f where pt,f describes a (physical) system belonging to U at time t relative to a reference frame f D20 p ε T-Event(U) ↔ ∃S1(t1f) ∃S2(t2f) S1, S2 ε State(U) ∧ p(S1,S2) where p(S1,S2) describes the transition from state S1 to state S2 D21 p ε T-Init(U) ↔ ∃S1(t1f) ∃S2(t2f) S1, S2 ε State(U) ∧ p(C(S1,S2)) where p(C(S1,S2)) describes S1 as a causal condition, together with a law, for S2 T31 ∀p[(p ε T-Law(U) ∨ p ε T-State(U) ∨ p ε T-Init(U) ∨ p ε T-Const(U) ∨ p ε T-Event(U)) → gKp] T18, D18 God knows all the theorems about the laws, states, initial conditions, constants and events of the universe T32 ∀p(p ε T-Law(U) → gKp) T18, D18 T33 ∀p(p ε T-State(U) → gKp) T18, D18 T34 ∀p(p ε T-Init(U) → gKp) T18, D18 T35 ∀p(p ε T-Const(U) → gKp) T18, D18 T36 ∀p(p ε T-Event(U) → gKp) T18, D18 A6 ∀p[(p ε T-Law(U) ∨ p ε T-Const(U)) → (∀t∀s)pt,s] Laws of Nature (of the universe) and Constants of Nature (of the universe) hold always and everywhere viz. are space-time invariant (‘s’ stands for the space conditions). We have to add here critically: According to our knowledge today. The question whether some constants of nature change very slowly has been severely tested by experiments within the last decades. So far no violation of their constancy was discovered (within the respective degree of accuracy). If some fundamental constants like α or G would change then also laws of nature

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would not be completely time-translation-invariant, since such constants enter fundamental laws of nature. (Cf. Mittelstaedt/Weingartner (2005, LNt) ch. 8.2). T37 ∀p(p ε T-Law(U) → (∀t,∀s) pt,s) A6 T38 ∀p(p ε T-Const(U) → (∀t,∀s) pt,s) A6 T39 ∀p[gK(p ε T-Law(U)) → gK((∀t, ∀s)pt,s)] A4, T37 T40 ∀p[gK(p ε T-Const(U)) → gK((∀t, ∀s)pt,s)] A4, T38 T41 ∀p[p ε T-State(U) → (∃t,∃f)pt,f ε T-State(U)] D19 T42 ∀p[p ε T-Event(U) → ∃S1(t1f) ∃S2(t2f) S1, S2 ε State(U) ∧ p(S1,S2) D20 T43 ∀p[p ε T-Init(U) → ∃S1(t1f) ∃S2(t2f) S1, S2 ε State(U) ∧ p(C(S1,S2)) D21 T44 ∀p∀t∀f(pt,f ε T-State(U) → gKpt,f) T33 Instantiation T45 ∀p∀t1∀t2∀f[p(S1,S2) ε T-Event(U) → gKp(S1,S2)] T36 Instantiation T46 ∀p∀t1∀t2∀f[p(C(S1,S2)) ε T-Init(U) → gKp(C(S1,S2))] T34 Instantiation T47 (∀t,∀f)[pt,f → p ε T-State(U)] D19 T48 ∀S1(t1,f)∀S2(t2,f)[S1,S2 ε State(U) ∧ p(S1,S2) → p ε T-Event(U)] D20 T49 ∀S1(t1,f)∀S2(t2,f)[S1,S2 ε State(U) ∧ p(C(S1,S2)) → p ε T-Init(U)] D21 T50 (∀t,∀f)[pt,f → gKpt,f] T47, T44 God knows any (singular) state occurring at time t relative to reference frame f (of the universe). For the proof of T50 observe that T41 is equivalent to ∃t∃f[p ε T-State(U) → pt,f ε T-State(U)] which instantiates to: p ε T-State(U) → pt,f ε T-State(U). Applying universal instantiations of T47 and T44 leads to T50 T51 ∀S1(t1,f)∀S2(t2,f)[S1,S2 ε State(U) ∧ p(S1,S2) → gKp(S1,S2)] T48, T36 T52 ∀S1(t1,f)∀S2(t2,f)[S1,S2 ε State(U) ∧ p(C(S1,S2)) → gKp(C(S1,S2))] T49, T34 T53 ∀t ≤ t0∀f[pt≤t0,f → gKpt≤t0,f] T50 Instantiation where t0 ist the present time relative to a reference frame f; God knows all past and present states (of the universe); viz. God knows all singular truths concerning past and present time (in this world) T54 ∀t > t0∀f[pt>t0,f → gKpt>t0,f] T50 Instantiation God knows all future states (of the universe); viz. God knows all singular truths in the future T55 ∀S1(t1≤t0,f)∀S2(t2≤t0,f)[S1,S2 ε State(U) ∧ p(S1,S2) → gKp(S1,S2)] T51 Instantiation God knows all past and present events (of the universe); God knows all singular truths describing past and present events

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T56 ∀S1(t1>t0,f)∀S2(t2>t0,f)[S1,S2 ε State(U) ∧ p(S1,S2) → gKp(S1,S2)] T51 Instantiation God knows all future events (of the universe); viz. God knows all singular truths describing future events T57 ∀S1(t1≤t0,f)∀S2(t2≤t0,f)[S1,S2 ε State(U) ∧ p(C(S1,S2)) → gKp(C(S1,S2))] T52 Instantiation God knows all past and present initial conditions (of the universe); viz. God knows all singular truths describing past and present initial conditions T58 ∀S1(t1>t0,f)∀S2(t2>t0,f)[S1,S2 ε State(U) ∧ p(C(S1,S2)) → gKp(C(S1,S2))] T52 Instantiation God knows all future initial conditions (of the universe); viz. God knows all singular truths describing future initial conditions T59 ∀p[¬p ε T(g) ∨ ¬p ε T(LM) ∨ ¬p ε T(CR)) → gK¬p] T7, ¬p/p God knows what is not (the case); what is not the case about himself, what is not the case about Logics and Mathematics and what is not the case about creation T60 ∀p[¬p ε T-Law(U) ∨ ¬p ε T-State(U) ∨ ¬p ε T-Init(U) ∨ ¬p ε T-const(U) ∨ ¬p ε T-Event(U)) → gK¬p] T31, ¬p/p God knows what is not (the case) concerning the universe; what is not the case about laws, what is not the case about states, what is not the case about initial conditions, what is not the case about constants of nature, what is not the case about events D22 Mut-K(x) ↔ ∃t1∃t2(t1≠t2 ∧ xKt1p ∧ xKt2¬p) Mut-K(x) … the knowledge of x is mutable D22.1 ¬Mut-K(x) ↔ ∀t1∀t2[(t1≠t2 → ¬(xKt1p ∧ xKt2¬p)] ¬Mut-K(x) … the knowledge of x is immutable (i.e. does not change) T61 ∀p[gKp → ¬Mut-K(g)] T22, D22.1 God’s knowledge is immutable T62 ∀p(p → ¬gK¬p) T5, ¬p/p, Contraposition T63 ∀p(gKp → ¬gK¬p) T62, T5 T64 ∀p(gK¬p → ¬gKp) T63 T65 ∀p(¬gKp ∨ ¬gK¬p) T63 Observe that ∀p(gKp ∨ gK¬p) does not follow from T5 or T63. This thesis implies together with with T5 the thesis ∀p(gKp ↔ p). The latter

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is also used in the literature as a definition of omniscience. Although we think that this strong thesis is a possible thesis (or definition) of omniscience if the universal quantifier in “for all states of affairs p: if p then God knows that p” is taken with respective care, we proposed a weaker but at the same time much more detailled definition of omniscience; its respective thesis which replaces ∀p(p → gKp) is T7. Cf. the problems discussed in section 11.46 above. 13.24 God’s Knowledge and Will T66 ∀p(gWp → gKp) T1, D4 Whatever God wills (that it is the case) he knows (that it is the case) T67 ∀p(gWp → ¬gK¬p) T66, T63 T68 ∀p(¬gKp → ¬gWp) T66 T69 ∀p(gK¬p → ¬gWp) T67 T70 ∀p(gWp → p) T5, T66 Whatever God wills is the case; or God’s will is always fulfilled T71 ∀p(p → ¬gW¬p) T70, ¬p/p, Contrapos. T72 ∀p(gWp → ¬gW¬p) T70, T71 T73 ∀p(p → gPp) D14, T71 Everything which occurs (which is a fact) is permitted (not prevented) by God; i.e. God does not will that it does not occur (cf. T71) T74 ∀p(gKp → ¬gW¬p) T5, T71 T75 ∀p(gKp → gPp) T5, T73 Whatever God knows to be the case (i.e. what is a fact) God permits (or does not prevent) to be the case T76 ∀p[(p ε Tg’s Will w.r.t. man) → gPp] T1, D6.1 T77 ∀p[(p ε Tg’s Commands) → (p ε Tg’s Will w.r.t. man)] D13, D15 T78 ∀p(p ε Tg’s Commands → gPp) T76, T77 Everything that belongs to (the theorems of ) God’s commands is permitted (not prevented) by God T79 ∀p(p ε Tg’s Commands → ¬gW¬p) T78, D14 T80 ∀p(¬p ε Tg’s Commands → ¬gWp) T79, ¬p/p If ¬p belongs to God’s commands then it is not the case that God wills that p T81 ∀p(¬p ε Tg’s Will w.r.t. man → gP¬p) T76, ¬p/p T82 ∀p(gCp → gWp) D5

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T83 ∀p(gCCp → gCWp) D5.1 Observe that the opposite implications of T82 and T83 do not hold; since God(necessarily or by his own nature) wills (and can will) his own existence and his goodness but he does not (and cannot) cause it. As a first cause he is only related to his creation and creatures but with his will he is related to both, to himself and to his creation and creatures. In my book on evil (2003, EDK) I used the two definitions gWp ↔ gCp and gCWp ↔ gCCp. For the reasons given above they are too strong; only the implications expressed in T82 and T83 are correct. However for the derivation of theorems in my book on evil (2003, EDK) I used in fact only T82 and T83 and not the opposite implications. Thus no theorem which is too strong was in fact derived in (2003, EDK) T84 ∀p(gCp → gCWp) D5, A2 T85 ∀p(¬gWp → ¬gCp) T82 T86 ∀p(¬gCWp → ¬gCCp) T83 T87 ∀p(¬gCWp → ¬gCp) T84 T88 ∀p[(p ε T(CR) ∧ gWp → (¬(p ε T(g)) ∧ ¬(p ε T(LM)))] A3,D5, D5.1 T89 ∀p[gCp → [Cons(p) ∧ Cons({p} ∪ Tg’s Essence) ∧ Cons({p} ∪ Tg’s Commands)]] T84, D4 Whatever God causes (T89) and whatever God can cause (T90) is consistent and consistent with his essence and his commands. On the other hand if any state of affairs is inconsistent in itself or inconsistent with God’s essence or commands he cannot cause it (and does not cause it). T90 ∀p[gCCp → [Cons(p) ∧ Cons({p} ∪ Tg’s Essence) ∧ Cons({p} ∪ Tg’s Commands)]] T83, D4 13.25 God’s Knowledge and Will in Relation to Moral Evil D23 p e ME ↔ [p e E ∧ ¬Cons({p} ∪ Tg’s Commands)] p is a moral evil iff p is an evil and p is inconsistent with the theorems of God’s commands D23.1 p e E ↔ p is some lack, defect, absence, privation or deficit of some particular good which either ought to be present in a subject or organism or is acceptable to be absent in order to achieve another higher good D24 ¬Cons({p} ∪ Tg’s Commands) ↔ (∀x∈H)(gWxSW¬p ∧ gWxSA¬p)

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T91 ∀p[p e ME → ¬Cons({p} ∪ Tg’s Commands)] D23 T92 ∀p[¬p ε Tg’s Will w.r.t. man ↔ ∀x∈H[gW(xW¬p) ∨ gW(xA¬p) ∨ gW(xSW¬p) ∨ gW(xSA¬p)]] D15, ¬p/p T93 ∀p[¬Cons({p} ∪ Tg’s Commands) → ¬p ε Tg’s Will w.r.t. man] D24, T92 T94 ∀p[p e ME → ¬p ε Tg’s Will w.r.t. man] D23, T93 T95 ∀p[¬p ε Tg’s Will w.r.t. man → ¬(p ε Tg’s Will w.r.t. man)] Indirect Proof: Assume the contrary: ¬p ε Tg’s Will w.r.t. man ∧ p ε Tg’s Will w.r.t. man. Then ¬p and p belong to the theorems of God’s will w.r.t. man, such that God’s will is inconsistent which is impossible. Therefore T95. T96 ∀p[p e ME → ¬(p ε Tg’s Will w.r.t. man)] T94, T95 T97 ∀p(p e ME → gP¬p) T94, T76 T98 ∀p(p e ME → ¬gWp) T97, D14 T99 ∀p(p e ME → ¬gCp) T98, T85 T100 ∀p[(p ∧ p e ME) → (¬gWp ∧ ¬gW¬p)] T98, T71 If p is a moral evil that occurs, then neither God wills that p occurs nor God wills that p does not occur (otherwise it would not occur) T101 ∀p(p → ¬gC¬p) T71, T85, ¬p/p T102 ∀p[(p ∧ p e ME) → (¬gCp ∧ ¬gC¬p)] T99, T101 If p is a moral evil and p occurs then neither God causes that p occurs nor God causes that p does not occur. T103 ∀p[(p ∧ p e ME) → p ε T(CR)] Indirect Proof: Assume (p ∧ p e ME) ∧ ¬(p ε T(CR)). Then p ε T(g) or p ε T(LM). But it is impossible that p ε T(g) (i.e. that God commits a moral evil), i.e. ¬(p ε T(g)). Then p ε T(LM). But this is also impossible because the state of affairs of an occuring moral evil cannot be a theorem of logics and mathematics (see definitions D3.2 and D7). The only possibility left is then that p ε T(CR), i.e. that p belongs to the theorems about creation or creatures (viz. man). Or in other words: Moral evil that occurs belongs to this world (because it is caused by inhabitans of this world). T104 ∀p[(p ∧ p e ME) → gKp] T103, T17 T105 ∀p[(p ∧ p e ME) → (gKp ∧ ¬gWp ∧ ¬gW¬p)] T104, T100 If p is a moral evil and if p occurs then God knows that p occurs but neither God wills that p occurs nor God wills that p does not occur. That means that God does not engage his will in moral evil that occurs (except

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in the sense of permittance). In other words: the case of occuring moral evil shows that with respect to these states of affairs God’s knowledge exceeds God’s will. T106 ∀p[gCWp → Cons({p} ∪ Tg’s Commands)] T1, D4 If God can will p then p is consistent with (the theorems of) his commands. T107 ∀p(p e ME → ¬gCWp) T106, T91 If p is a moral evil then God cannot will that p occurs (cf. T98). T108 ∀p(p e ME → ¬gCCp) If p is a moral evil then God cannot cause that p occurs (cf. T99) T109 ∀p[(p ∧ p e ME) → (gKp ∧ ¬gCWp ∧ ¬gCCp)] T104, T107, T108 If p is a moral email and if p occurs, then God knows that p occurs but neither God can will that p occurs nor God can cause that p occurs. In other words the case of occuring moral evil shows that with respect to these states of affairs God’s knowledge exceeds God’s power. T110 ∀p[(p ∧ p e ME) → (gKp ∧ ¬gCp ∧ ¬gC¬p)] T102, T104 A7 ∃p(p ∧ p e ME) There is moral evil (which occurs) T111 ∃p(p ∧ ¬gWp) A7, T98 T112 ¬(∀p)(p → gWp) T111 God does not will everything that is the case T113 ¬(∀p)(p → gCp) T99, A7 God does not cause everything that is the case; for example God does not cause moral evil of free decisions of man T114 ¬(∀p)(p → gCWp) A7, T107 God cannot will everything that is the case; for example God cannot will (and cannot cause) moral evil T115 ¬(∀p)(p → gCCp) A7, T108 God cannot cause everything that is the case D25 x ∈ AW ↔ ∀p(xWp ∨ xW¬p) x is allwilling iff for all states of affairs p: x wills that p or x wills that ¬p D25.1 x ∈ AC ↔ ∀p(xCp ∨ xC¬p) x is allcausing iff for all states of affairs p: x causes that p or x causes that ¬p

169

T116 ¬(∀p)(gWp ∨ gW¬p) God is not allwilling A7, T100: ∃p(¬gWp ∧ ¬gW¬p) T117 ¬(∀p)(gCp ∨ gC¬p) God is not allcausing A7, T102: ∃p(¬gCp ∧ ¬gC¬p) T118 g ∈ OS ∧ g ∈ OM ∧ ¬g ∈ AW ∧ ¬g ∈ AC

T1, T116, T117, D25, D25.1 God is omniscient and almighty but neither allwilling nor allcausing T119 ∃p(gKp ∧ ¬gWp ∧ ¬gW¬p) T105, A7 For some states of affairs it holds: God knows them, but neither God wills that they occur nor wills that they do not occur. In other words: For some p, God knows that p and permits that p but does not will that p (see D14) T120 ¬(∀p)[gKp → (gWp ∨ gW¬p)] T119 Not everything which is known by God is subject to his will in the sense that it is either willed to occur or willed not to occur. But observe that it holds: Everything is either willed or permitted by God: T121 ∀p[p → (gWp ∨ gPp)] T73 Everything what is the case is either willed or permitted (not prevented) by God T122 ∀p[gKp → (gWp ∨ gPp)] T5, T121 Everything which is known to be the case by God is either willed or permitted (not prevented by him) T123 ∃p(gKp ∧ ¬gWp) T119 T124 ¬(∀p)(gKp → gWp) T123 Not everything which is known by God is also willed by him T125 ∃p(gKp ∧ ¬gCp ∧ ¬gC¬p) T110, A7 For some states of affairs it holds: God knows them, but neither God causes them to occur nor causes them not to occur T126 ¬(∀p)[gKp → (gCp ∨ gC¬p)] T125 Not everything known by God is subject to his causation T127 ¬(∀p)[gKp → gCp] T125 Not everything which is known by God is caused by God T128 ∃p(gKp ∧ ¬gCWp) T109, A7 T129 ∃p(gKp ∧ ¬gCCp) T109, A7 T130 ¬(∀p)(gKp → gCWp) T128 Not everything which is known by God can be willed by him

170

T131 ¬(∀p)(gKp → gCCp) T129 Not everything which is known by God can be caused by him. Observe that this is not a weakness of power but a sign of perfection: he cannot cause moral evil T132 ∀p[(p ∧ p e ME) → gPp] T100, D14 If moral evil (in fact) occurs then God permits (does not prevent) it; since he created man with free will and is normative and volitive consistent (NC, cf. D6.1 and D15). But observe that from T132 it does not follow that God permits (does not prevent) every moral evil; i.e. ∀p(p e ME → gPp) is not a theorem. Thus God may prevent some moral evil in a way which does not take away free will from the acting human person. T133 ∀p[p e ME → (¬gWp ∨ lgWp ∨ ¬gW¬lp)] T98 T134 ∀p[p e ME → ¬(gWp ∧ ¬lgWp ∧ gW¬lp)] T133 T135 ∀p[p e ME → ¬(p ε Tg’s Will about CR)] D12.2, T134, T96 If p is a moral evil, then it is not the case that p belongs to the theorems of God’s will about creation and creatures. Observe however that it does not follow from that, that in this case God wills that non-p; since then moral evil would never occur (because his will is always fulfilled). Therefore if moral evil (in fact) occurs, God permits it (does not prevent it), cf. T132. 13.26 God knows his activities A8 gOp → gK(gOp) Where O is one of the operations W, CW, C, CC, A, P. T136 gK(gOp) → gOp T5 T137 gOp ↔ gK(gOp) A8, T136 God has knowledge of all his activities represented by W, CW, C, CC, A and P. T138 gWp ↔ gK(gWp) A8 T139 gCWp ↔ gK(gCWp) A8 T140 gCp ↔ gK(gCp) A8 T141 gCCp ↔ gK(gCCp) A8 T142 gAp ↔ gK(gAp) A8 T143 gPp ↔ gK(gPp) A8 T144 gCWp ↔ [Cons(p) ∧ Cons({p} ∪ Tg’s Essence) ∧ Cons({p} ∪ Tg’s Commands)] T1, D4

171

T145 gCWp ↔ gK[Cons(p) ∧ Cons({p} ∪ Tg’s Essence) ∧ Cons({p} ∪ Tg’s Commands)] T139, T144 God can will that p (or has power w.r.t. p) iff God knows that p is consistent and that p is consistent with his Essence and with his Commands. T146 gCCp → gK[Cons(p) ∧ Cons({p} ∪ Tg’s Essence) ∧ Cons({p} ∪ Tg’s Commands)] T83, T145 If God can cause that p (or has the power to cause that p) then God knows that p is consistent and p is consistent with his Essence and with his Commands. T147 gK(gWp) → gKp T66, T138 T148 gK(gCp) → (gWp ∧ gKp) T82, T66, T140 T149 p → gK(gPp) T73, T143 T150 gK(gCp) → (gK(gCCp) ∧ gK(gCWp)) A2, A3, T83, T141

173

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183

Subject Index actual event, ambiguity of, 127 all, 136 allcausing, 46, 58, 169 allcausing God, 46, 59, 169 allwilling, 46, 59, 169 allwilling God, 48, 59, 169 anthropic principle, 46 A-propositions, 68, 74 arithmetically expressible, 145 atemporal, 107 axiom systems, limitations of, 146 axiomatisable, 135, 144, 148, 150 causal relation, 44, 53, 58, 60, 118 causing and willing, 44, 47 chance, 38, 45 characteristica universalis, 142, 145 closed propositions, 112 coding system, 145 complete knowledge, 126, 143, 149 completeness, 143, 149 consistency, 44, 47, 148 constant, 162 contingency, 100, 102, 105, 131 contingent, 16, 100, 105 contingent status, 105, 131 contribution of imperfect creatures, 56 copula 'is', 153 counterfactuality, 53, 85, 93, 120 counterfactuals, knowledge of, 93, 95, 120 creation, 140, 159 decidable, 136, 143, 147, 151 degrees of freedom, 71 determinably false, 107 determinably true, 107, 110 determinism, 106, 128

discursive knowledge, 73, 81 Divine Liar, 1, 10, 12 dynamical law, 69, 72, 88 elementhood relation, 153 eternity, 31 event, 162 fact, 50, 129 FADW, 120 finitely analytic, 143 foreknowledge, 116, 128 free action or decision of will, 45, 121, 129, 133 future contingency, 101, 105, 108, 110, 123, 131, 141 future contingency, consistency of, 131 future necessity, 101 future probabilistic contingency, 101, 105 future states of affairs, 116, 119, 125, 131, 141, 164 goal, subordinating under, 44 Gödel numbers, 145 God's commands, 159 God's essence, 138, 159 God's knowledge about the universe, 161 God's knowledge and truth, 113, 137, 160 God's power, 41, 48, 139 God's will, 43, 46, 47, 58 God's will about creation, 159 God's will w.r.t. man, 57, 159 immutable, 81, 83, 164 impossible, 86 impossible, logically, 86 incompatible with laws of nature, 86 indeterminism, 45, 70

184

infinite, 150 infinitely analytic, 143 initial condition, 46, 162 initial state, 46 irreflexivity, 60, 118 irrelevant truths, 67, 73, 136, 151 KCH, 80 knowing and causing, 55, 58, 168 knowing at some time, 26, 35, 76, 82, 110, 124 knowledge, 3, 7, 42, 52, 135, 138 knowledge about creation, 19, 140, 159, 161 knowledge about himself, 19, 138, 170 knowledge about logic and mathematics, 141 knowledge and belief, 7, 15 knowledge and falsity, 91 knowledge and power, 42, 47, 121 knowledge and truth predicate, 108, 145 knowledge and will, 19, 23, 47, 55, 165 knowledge, change of, 79, 81, 83, 164 known as actual, 115, 122, 125, 128 known in their causes, 115, 117, 120, 128 KT, 4 law, 69, 162 law of entropy, 71 law without law, 71 law-necessity, 108 learning process, 56, 87 logic and mathematics, 142 logic and mathematics, limitations of, 145

logical and deductive infallibility, 6, 89 logical and deductive omniscience, 5, 89 logical necessity, 99 logically determined, 98 mathematical necessity, 99 mathematisation, 142 microstates, not realised, 88 middle knowledge, 95, 120 moral evil, 166 more geometrico, 142 mutable, 81, 83, 164 natural necessity, 99 natural necessity, conditional, 99, 101, 118 necessarily contingent, 15, 52 necessary cause, 61 necessary knowledge - knowledge of the necessary, 22, 97, 139 necessary status, 102 necessity and time, 110 non-contradiction, principle of, 86 normative and volitive consistent, 57, 159 normatively inconsistent, 46 not actual, 90 NW, 86 omnipotence, 43, 47, 49, 139 omnipotence, definition of, 43, 156 omnipresent beinig, 125 omniscience and freedom, 53, 62, 116, 129 omniscience and necessity, 2, 15, 19, 115, 118 omniscience, definition of, 155 omnitemporal contingency, 102, 107, 110 operators, 154

185

paradoxes, 136 past and present, 37, 123, 127, 163 past necessity, 100 person, 154 possibility and time, 87, 110 probabilistic contingency, 100, 105 providence, 44, 48, 51 recursive, 136 recursively enumerable, 135, 151 seabattle, 106 singular truths, 67, 69 state, 162 states of affairs, different kinds, 99 states of affairs, not realised, 50, 87 states of affairs, status of, 99, 104, 109, 113 statistical law, 69, 71, 72, 74, 88 sufficient cause, 61 syllogism, 143

Tarski's truth condition, extension of, 11 time as chronological order, 29, 82 time of this world, 26, 32, 82, 125 time, analysis of, 26, 124 time, biological and psychological, 33, 82 transitivity, 53, 59, 118 true justified belief, 2, 13 truth predicate, 103, 107, 109, 145 truth predicate, time index of, 103, 107, 109 truth, reason for, 112 truths, set of, 135, 147, 150 twin paradox, 126 universe, 161 unpreventable necessity, 100 veritees de raison, 142 what is not, 85, 90 what is not, ambiguity of, 90

186

187

Name Index Anscombe, G.E.M., 106 Aristotle, 10, 16, 27, 44, 67, 85ff., 90, 97, 106, 110, 112, 118, 121 Augustine, 31, 48, 54, 61, 67, 68, 74f. Barrow, J., 46 Batens, D., 86 Baumgartner, H.M., 33 Boethius, 31, 106 Boltzmann, L., 71 Breuer, Th., 127 Cain, J., 145 Casati, G., 72 Castaneda, H.N., 83 Chirikov, B., 72 Chisholm, R., 4 Church, A., 136 Craig, W.L., 54, 95 Damnjanovic, Z., 145 Devalla L., 115 Dobzhansky, Th., 7 Dworkin, R., 115 Eccles, J., 61 Ellis, G.F.R, 27, 119 Feys, R., 16, 108, 114, 153f. Fraenkel, A., 8, 137, 145, 147, 149 Gale, R., 68, 75f. Galles, D., 60 Gettier, E., 2, 14 Gödel, K., 8, 136f., 144ff. Grim, P., 1f., 9f., 13 Hafele, J.C., 126 Hausman, D.M., 93 Hawking, S.W., 27, 119 Hintikka, J., 4ff., 10, 16, 87, 106, 145 Hunt, D.P., 21

Inwagen, P. van, 45 Jammer, M., 27 Keating, R., 126 Kreisel, G., 144f. Kretzmann, N., 79f., 82 Kripke, S., 10, 145 Kutschera, F. von, 83 Leftow, B., 76, 85, 91f. Leibniz, G.W., 142ff., 149 Lenzen, W., 6 Lewis, D., 93 Lukasiewicz, J., 106 Marshall, D., 143 Matthew, 94 Maxwell, J.C., 99 Maynard-Smith, J., 7 Mittelstaedt P., 27f., 46, 53, 61, 69, 71f., 87f., 93, 99, 107, 119, 163 Molina, 95, 120 Moore-Ede, M.C., 33 Myhill, J., 145 Newton, I., 27, 69, 93, 99, 125, 156 Pearl, J., 60 Penzias, A., 33, 124 Pike, N., 54, 63ff. Plantinga, A., 65 Plotinus, 23 Popper, K.R., 61 Prigogine, I., 73 Prior, A.N., 29, 61, 106 Rescher, N., 86, 106, 142 Ruse, M., 119 Schrödinger, E., 71, 99 Schurz, G., 74, 145 Schuster, G., 72 Simmons, K., 13 Spinoza, B., 44, 53 Stump, E., 80 Sulzmann, F.M., 33

188

Takeuti, G., 145 Tarski, A., 10f., 90, 135, 145f. Thomas Aquinas, 3, 6f., 13f., 20f., 23, 28, 31f., 37, 40f., 43, 48, 51, 54, 57, 60f., 73, 76, 89f., 95, 98, 106f., 110, 115ff., 122f., 125, 139f., 146 Tipler, F., 46 Treismann, M., 33 Urquhart, A., 177

v. Neumann, J., 8, 14, 137, 146f. Van Benthem, I., 29 Weinberg, St., 33 Wheeler, A., 70f. William of Ockham, 116 Wilson, R., 33, 124 Wright, G.H. von, 16, 108, 114 Zagzebski, L, 116f. 130ff.

PhilosophischeAnalyse PhilosophicalAnalysis 1 Herbert Hochberg Russell, Moore and Wittgenstein The Revival of Realism

8 Rafael Hüntelmann Existenz und Modalität Eine Studie zur Analytischen Modalontologie

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9 Andreas Bächli / Klaus Petrus Monism

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