Oxford Studies in Metaphysics [Vol. 10]

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OXFORD STUDIES IN METAPHYSICS l!ditoria/ Aduis01y Board

OXFORD STUDIES IN METAPHYSICS

David Chalmers (New York University md Australasian National University) Andrew Cortcns (Boise State University) Tamar Szabo Gendler (Yale University)

Volume 10

Sally Haslangcr (MIT) John Hawthorne (University of Southern California) Mark Heller (Syracuse University) Hud Hudson (Western Washington University) Kathrin Koslicki (University of Alberta)

Edited by

Kris McDaniel (Syracuse University) Brian McLaughlin (Rutgers University) Trenton Merrick requires of the world that R, then if; requires of the world that R. Thus 'Actually grass is green' requires nothing more or less of the world than 'Grass is green'. I won't rely on the correctness of the above definitions exactly as stated; just as dcRosset, so he tells us (2015: 135), docs not rely on his 'rough and ready' definition of analyticity. lfowcver, I shall make two assumptions. First: (schematically) if a sentence requires of the world that R, then, whichever world is actual, that sentence is true only if R. Second: if a sentence is true

(M 1) (M2)

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Someone is married. Someone is married to someone else.

(M 1) explicitly affirms the existence of one person only, whereas (M2) explicitly affirms the existence of two. Nonetheless it seems clear chat the conjunction of (MI) and (M2) requires nothing more of the world (and in particular, is no less ontologically parsimonious) than (M 1) alone, because (Ml) analytically entails (M2). And likewise, we might think, for more metaphysically interesting analytic entailments, if such there be. However, dcRosset denies that DAO and GAO hold in full generality. He suggests that in the metaphysically interesting cases just alluded to, these principles are liable to break down. So even if, to take his leading example, 'There are particles arranged tablewise in location L' analytically entails 'There is a table in L', it remains plausible that the conjunccion of these sentences is less ontologically parsimonious than the former sentence alone. (Here he targets Amie Thomasson in particular. For the relevant sense of 'arranged tablcwise' and the like, sec her 2007: 16-17.) Thus while unQuineanly granting the tenability of the analytic/synthetic distinction, deRossct disputes its metaphysical significance. To this end, he purports to produce counterexamples to DAO and GAO by means of linguistic stipulations. He offers two main candidate counterexamples just to GAO (though he admits that the first of these is not decisive) and one specifically to DAO. I aim to show where these arguments go wrong.

2. 'VERDANTLY*' Here is dcRossct's first candidate counterexample to GAO. (I bypass his initial, illustratively unsuccessful 'verdantly' stipulation.) Suppose we introduce a new sentential operator 'verdantly*' by stipulating the following rules of inference:

P ~Verdantly* P Verdantly* P ~Actually grass is green Verdantly* P ~Possibly P

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john Horden

Deuious Stipu!tztions

'Actually' here has its familiar interpretation from two-dimensional semantics: r Actually (/>' is true at a world iff is true at the actual world. Hence what deRosset calls Steuenson ~- constraint on linguistic stipulation (a generalized version of the constraint proposed by Stevenson, l 961) initially appears to be satisfied:

deRosset are in trouble; but any such objection to those views-which, to be fair, deRosset does not himself raise-seems question-begging. In any case, there is another reason to think that the 'verdantly*' stipulation fails. For 'verdantly*', as defined, is by reasonable standards dearly and fatally nonconservative. N uel Belnap (l 962) recommended a conservativeness constraint on stipulation in response to Arthur Prior's (1960) parody of implicit definition, wherein 'tonk' is defined by the rules:

[AJ linguistic stipulation succeeds only if there is a consistent way to assign truth conditions to sentences containing the introduced term that makes the content of the stipulation true. (20 l 5: l 41)

To see this, suppose grass were not green. Then by the above introduction rule, 'Verdantly* grass is not green' would be true, and so by the first elimination rule, 'Actually grass is green' would be true. But even if grass were not green, grass would still be green at the actual world, so no inconsistency results. If his 'verdantly*' stipulation succeeds, deRosset tells us, we have a counterexample to GAO. For then 'Snow is white' analytically entails 'Actually grass is green'. But 'Snow is white and actually grass is green' requires more of the world than 'Snow is white'. For the latter only requires that snow be white, whereas the former also requires that grass be green. However, deRosset admits some uncertainty as to whether this stipulation succeeds (2015: l43-4). For perhaps when applying Stevenson's constraint we are not entitled to take for granted tl posteriori necessities such as the fact that grass is actually green. Perhaps we should also consider how to evaluate sentences containing 'verdantly*' on the assumption that grass is not actually green. But then consider what truth value should be assigned to Verdantly* grass is either green or not green.

If it is true, the first elimination rule for 'verdantly*' is invalid, because 'Actually grass is green' is by hypothesis false. If, on the other hand, the evaluated sentence is false, the introduction rule is invalid, because 'Grass is either green or not green' is (we may assume) logically true. So the 'verdantly*' stipulation fails to meet Stevenson's constraint on this more stringent construal. DeRosset ultimately leaves open how stringently Stevenson's constraint should be construed, and so admits that his 'verdantly*' stipulation does not provide a decisive counterexample to GAO. Fair enough; though whichever way we construe Stevenson's constraint, I wonder how useful it will be for adjudicating on attempted stipulations in cases where the bounds of conceptual possibility are in dispute. Should we consider how to evaluate sentences containing 'number' or 'proposition' on the assumption that there are only finitely many things, for example? If so, then the broadly Fregean and Carnapian views of abstracta targeted by

P~ P tonk Q P tonk Q ~ Q Thus any sentence analytically entails any other; hence every sentence is analytic. Obviously the stipulation fails; the only question is how. J. T. Stevenson (1961) proposed one constraint in response: every truthfunctional connective must have a consistent truth table (notice that this is significantly weaker than the constrnint deRosset names after Stevenson, on either construal of the latter). Belnap proposed another constraint: the introduction of any new vocabulary must yield a conservative extension of the language. That is to say, the rules stipulated to govern new vocabulary cannot allow the derivation of any sentence of the old language (i.e. any sentence without the new vocabulary) that was not already derivable. 1 In footnotes, deRosset describes this conservativeness constraint, and observes that it is met by his subsequent stipulations-presumably regarding this as a point in their favour-but conspicuously does not claim the same for his 'verdantly*' stipulation. And it is easy to see why: 'Actually grass is green' is a synthetic sentence of English before this stipulation, but would become derivable from any sentence whatsoever (and hence analytic) were the stipulation to succeed. Thus Belnap's constraint is manifestly violated. Admittedly, many regard Belnap's constraint as too strong (see Read, 1988: §9.3; 2000: 125-7; Peacocke, 1993: §§3-4; 2004: 18-21; Prawitz, 1994: 374; Shapiro, 1998: §3). For example, it seems legitimate to add second-order quantifiers or a truth predicate to a first-order system of arithmetic, though doing so enables the derivation of the original system's Godel sentence. Also, abstraction principles such as Hume's Principle The number of Fs

=

the number of Gs iff there are just as many Fs as Gs

if successfully stipulated, allow us to derive results about the (infinite) size of the domain of first-order quantification that were previously statable but perhaps unprovable. Hence Bob Hale and Crispin Wright endorse a weaker 1 This principle of conservativeness, or 'noncreativity', was previously endorsed by others; sec e.g. Frcge, 1914: 208. For a historical discussion, see Urbaniak and l·famari, 2012.

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John Horden

Devious Stipulations

version of conservativeness (cf. Field, J 980: ch. 1; Wright, 1997: §9; 1999: §2. 5; Schiffer, 200.'3: §2.2), also mentioned by dcRossct (2015: 14 l, n. 19), with the following proviso attached:

Now, he continues, we have a more decisive counterexample co GAO. Por if this stipulation succeeds, then 'Grass is grassgrcen' is thereby guaranteed co be actually true, no matter what colour grass actually has. 3 So 'Snow is white' analytically entails 'Grass is grassgreen'. But 'Snow is white and grass is grassgreen' requires more of the world than 'Snow is white'. Por since grass is actually green, 'grassgrccn', according to the stipulation, expresses the property of being green. So whereas 'Snow is white' only requires that snow be white, 'Snow is white and grass is grassgrcen' also requires that grass be green. Clearly something fishy is going on here. If 'Grass is grassgrccn' is guaranteed to be actually true, no matter what colour grass actually has, then it cannot require of the actual world that grass be green. Even if we grant that the stipulation succeeds, it seems sensible to examine the semantics of the novel term 'grassgreen' a little more closely before we embrace inconsistency. So what is its stipulated meaning? That, I think, depends on whether the stipulation is conditionti! or unconditional. This yields another dilemma. Pirst horn: 'is grassgrecn' is conditionally stipulated to mcy," in B. Nanay (ed.), Perceiving the World (New York: Oxford University Press), 13-32. Cohen, ]. 20l0b. "Perception and Computation," Philosophical !rsues 20(1): 96-124. Dancy, .J. 1986. "Two Conceptions of Moral Realism," Proceedings ofthe Aristotelian Society, supp. vol. 60: 167-87. Reprinted in J. Rachels (ed.), Moral The111y (Oxford: Oxford University Press, l 997), 227-44. Evans, G. 1980. "Things Without the Mind," in Z. van Straaten (ed.), Philosophical Subjects: Essays Presented to P. F. Strawson (Oxford: Oxford University Press), 76-116. Reprinted in Evans Collected Papen (Oxford: Clarendon, 1985), 249-90. Gendler, T. and Hawthorne, J. (eds.) 2006. Perceptutt! Experience (Oxford: Oxford University Press). Gert, J. 2008. "What Colors Could Not Be: An Argument for Color Primitivism," journal of Philosophy 105(3): 128-57. Johnston, M. l 992. "How to Speak of the Colors," Philosophical Studies 68(.3): 221-63. Reprinted in Byrne and Hilbert 1997, 137-72. Johnston, M. l 998. "Is the External World Invisible?" Philosophical Issues 7: 185-98. Johnston, M. 2004. "The Obscure Object of Hallucination," Philosophical Studies 120(1-3): 113-83. Johnston, M. The Manifest, unpublished manuscript. Kalderon, M. E. 2007. "Color Pluralism," Philosophical Review 116(4): 563·-60!. Kalderon, M. E. 2008. "Metamerism, Constancy, and Knowing Which," Mind 117(468): 935-71. Kalderon, M. E. 'Experiential Pluralism and the Power of Perception'. In Themes fi'om Travis. (Oxford: Oxford University Press, forthcoming). Kalderon, M. E. "Color and the Problem of Perceptual Presence," unpublished manuscript. Kelly, S. D. 2004. "Seeing Things in Mcrlcau-Ponty," in T. Carman and M. B. N. Hansen (eds.), The Cambridge Companion to Merleau-Ponty (Cambridge: Cambridge University Press). Kulvicki, ). 2007. "What is What It's Like? Introducing Perceptual Modes of Presentation," Synthese 156(2): 205-29. Levin, J. 2000. "Dispositional Theories of Color and the Claims of Common Sense," Philosophical Studies 100(2): 151-74.

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Htigit Benbaji Ben Gurion Univer:rity

REFERENCES Allen, K. 2007. "The Mind-Independence of Colour," European journal of Philosophy 15(2): 137-58. Allen, K. 2009. "Being Coloured and Looking Coloured," Cmuidian journal of Philosophy 39(4): 647-70. Allen, K. 2010. "Locating The Unique Hues." Rivistrt di Estetica 43(43): 13-28. Austin, J. L. 1962. Sense and Sensibilirt (Oxford: Oxford University Press). Bcnbaji, H. 2010. "Token Monism, Event Dualism, and Ovcrdetermination," Cmaditm journal of Philosophy 40(1): 20. Bcnbaji, H. 2016. Why Color Primitivism AustmLlsirm joumal ofPhilosophy, Vol. 94.2. Bennett, M. R. 2003. Philosophical Fo1mck1tions ofNeuroscience (Oxford: Blackwell). Berkeley, G. 1948-57. "Three Dialogues between Hylas and Philonous." IY1e Works of Georg,e Berkeley, Bishop of Cloyne, ed. A. A. Luce and T. E. Jessop (London: Thomas Nelson and Sons), 9 vols. Boghossian, P. and Vclleman, ]. D. 1989. "Color as a Secondary Quality," Mind 98(389): 81-103. Reprinted in Byrne and Hilbert 1997: 81-104. Brewer, B. 1999. Perception and Retlson (Oxford: Clarendon Press). Broackes, J. 1997. "The Autonomy of Colour," in Byrne and Hilbert 1997, 191-225. Broackes, J. 2007. "Colour, World and Archimedean Metaphysics: Stroud and the Quest for Reality," Erkenntnis 66(1): 27-71. Byrne, A. 2001. "Do Colours Look Like Dispositions? Reply to Langsam and Others," Philosophical Quarterly 51 (203): 238-45. Byrne, A. and Hilbert, D. R. (eds.) 1997. Re11dings on Co/01; Vol. I: The Philosophy of Color (Cambridge, MA: MIT Press). Byrne, A. and Hilbert, D.R. 2007. "Color Primitivism." Erkemztnis 66 (1/2): 73-105. Byrne, A. and Hilbert, D.R. 2011. "Are Colors Secondary Qualities?" in L. Nolan (ed.), Primmy and Secondmy Qualities (Oxford: Oxford University Press). Campbell,]. 1993. 'The Simple View of Colour," in Byrne and Hilbert 1997, 177-90.

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Colon as Primitiue Diipositions

McDowell, J. 1985. "Values and Secondary Qualities," in T. Honderich (ed.), Morality and Objectiuity (Boston: Routledge & Kegan Paul), l l 0-29. McDowell, J. 2004. "Reality and Colours: Comment on Stroud," Philosophy and Phenomenological Research 68(2), 395-400. McDowell, J. 2011. "Colors as Secondary Qualities," in J. Bridges, N. Kolodny and W. Wong (eds.), The Possibility of Philosophical Understanding: Reflections on the Thought o/Bttny Stroud (Oxford: Oxford University Press). McGinn, C. 1983. The Subjectiue View (Oxford: Oxford University Press). McGinn, C. 1996. "Another Look at Colors,".foimutl ofPhilosophy 93(11): 537-53. Macpherson, F. 2006. "Ambiguous Figures and the Content of Experience," Nazis 40(1): 82-117. Martin, M. G. F. 2010. "What's in a Look?" in B. Nanay (ed.), Perceiving the World (New York: Oxford University Press), 160-225. Misccvic, N. 2007. "Is Color-Dispositionalisrn Nasty and Unccologirnl?" Erkenntnis 66(1-2): 203-31. Noc, A. 2004. Action in Perception (Cunbridge, MA: MIT Press). ()'Shaughnessy, B. (2000). Consciousness and the World (Oxford: Oxford University Prc->s). Peacocke, C. 1984. "Color Concepts and Color Experience," Synthese 58(3): 365-82. Peacocke, C. 2008. "Sensational Properties: Theses to Accept and Theses to Reject." Re1me lnterm1tiont1!e de l'hilosophie 62: 7-24. Pettit, P. 2003. "Looks as Powers," Philosophical Issues 13: 221-52. Phillips, I. 2006. "Perception and Context,'' paper delivered at the NPAPC, Warwick, July 2006. Russell, B. 1912. The Problems of Philosopf~y (London: Oxford University Press). Ryle, G. 1949. The Concept of Mind (London: Penguin). Schellenberg, S. 2008. "The Simation-Dependency of Perception," journal of Philosophy I 05(2): 55-84. Sellars, W. 1956. "Empiricism and the Philosophy of Mind," in H. Feig! and M. Scriven, eds., Minnesota Studies in the Philosophy of Science, volume 1, The Foundations of Science and the Concepts of Psychology rmd Psychoanalysis (Minneapolis: University of Minnesota Press), 253-329. Shoemaker, S. 2006. "On the Ways Things Appear," in Gendler and Hawthorne 2006, 461-80. Siewert, C. 2006. "Is the Appearance of Shape Protean?" Psyche 12(3): 1-16. Smith, A. D. 2002. lfJe Problem ofPerception (Cambridge, MA: Harvard University Press). Watkins, M. 2002. Rediscovering Colors: A Study in Pollyrmnrt Rerdism (Dordrecht: Kluwer). Watkins, M. 2010. "A Posteriori Primitivism." Philosophic,tl Studies 150(1 ): 123-37. Westphal, J. 1987. Colour: Some Philosophictil Problems from Wittgenstein (Oxford: Blackwell). Westphal, J. 2005. "Conflicting Appearances, Necessity and the Irreducibility of Propositions About Colours," Proceedings of the Aristotelian Society 105(2): 219-35.

Wiggins, D. 1976. "'fruth, Invention, and the Meaning of Lifr," reprinted in his Needs, Values, Truth: Essays in the Phi/{)sophy ofVrilue (3rd edn) (Oxford: Clarendon, 1998), 87-138. Yablo, S. 1992. 'Mental Causation'. Philosophical Reuiew 101(2): 245-80. Wiggins, D. l 987. "A Sensible Subjectivism?" reprinted in his NeerLr, Vrtlues, huth: Essays in the Philosopl~y of Vrtlue (3rd cdn.) (Oxford: Clarendon Press, 1998), 185-214.

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Are There lne.ffrtb!e Aspects of Reri!ity?

7 Are I'here Ineffable Aspects of Reality? Thom11s Hofweber

l. INTRODUCTION

Should we think that some aspects of reality arc simply beyond creatures like us, in the sense that we are in principle incapable of representing them in thought or language? Or should we think that beings with a mind and language like ours arc able to represent every truth and every fact? In other words, should we think that some truths are ineffable for us: beyond what we can think or say? Whatever the answer is, it likely has substantial consequences. If it is no, i.e. no truth is ineffable for us, then this might shed light on what reality is like, or what our minds are like, or why the two match up so well. If the answer is yes, some truths are ineffable for us, then this might affect our attempts to understand all of reality. In particular, it might affect the project of metaphysics and its ambition to understand all of reality in its grander features. If we had reason to think that only a limited range of facts can be represented by creatures like us then this might give us reason to think that metaphysics in its ambitious form is beyond what we can hope to cany out successfully. In this paper I will argue that the question whether there are any ineffable truths or facts is an important, although somewhat neglected, question whose answer has significant consequences, and I will make a proposal about what the answer is, on what this answer depends, and what follows from it. The paper has four parts: first, I will clarify what is at issue and make the notion of the ineffable more precise in several ways. Second, I will argue that there are ineffable truths using several different arguments. These arguments will rely on a certain hidden assumption which is almost universally made implicitly and accepted by most when made explicit, but which I will critically investigate later in the paper, in part four. A third part will attempt to answer a puzzle about the ineffable connected to the relationship between effable and ineffable truths and why the ineffable seems to be more hidden from us than would be suggested merely by the fact that it is ineffable.

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The solution to this puzzle will make clear how the ineffable is significant. I will argue that it has important consequences for metaphysics in particular, which suggest modesty instead of ambition. In the fourth part I will have a closer look at a hidden assumption that was relied upon until then, but which might well be mistaken. [ will ;trgue that we have good, but not conclusive, reason to think that this assumption is indeed false, and if so then everything changes. On the natural way in which this assumption is false we get no ineffable truths, no modesty in metaphysics, but a form of idealism instead. I will try to make clear that the resulting form of ide;tlism is coherenr, significant, and quite possibly true. Whether the crucial assumption is, in the end, true I won't be able to settle here, but we can see that there will be important consequences either way. I will pick sides at the end. But before we can get to all this we need to get clearer about what is at issue.

2. WHAT IS THE QUESTION? The ineffable naturally appears as a possibility when we think about the relationship between what reality is like, on the one hand, and what we can truly say, on the other. The relationship between these two leads to one unproblematic (for present purposes) area of overlap, and to two more mysterious outlying areas (see Figure 7.1). The area of overlap is a true description of reality: we can truly say something and reality is like that. Although much can and has been said about how this is to be understood in more detail, I will leave it untouched here, since my concern is with the two more problematic cases: first whether

Correct description The ineffable

The non-descriptive

What reality is like

What we can say truly

Figure 7.1 The location of the ineffa.blc

Thomas Hofweber

Are There lnejfi1ble Aspects of Reality?

what we can truly say goes beyond what reality is like, and second whether what reality is like goes beyond what we can truly say, which is our main topic. The first possibility might seem incoherent. How could what we can truly say go beyond what reality is like? If we said it truly then how could reality not be like what we said? Those who think that this option is coherent generally maintain that it only seems incoherent to us because we mistake it with something else: either that reality is different from how we say it is (and thus what we say should be false) or else that we say something truly about something other than reality (which isn't an option, since reality is all-inclusive). Instead, they hold, this option is coherent, since we can say something truly that isn't descriptive at all, neither of reality nor anything else. Some parts of speech aim to describe, while other parts aim to do something else, for example express an attitude of the speaker. Truth applies to both, and thus we can say something truly that goes beyond what reality is like. It is true, but doesn't aim to describe reality, and thus reality isn't required to be as described for it to be true. Exprcssivism about normative discourse combined with minimalism about truth is a paradignntic instance of this approach. The question, of course, still remains whether it indeed is coherent, but since we will not focus on this outside area in our Venn diagram, we don't have to settle this here. This part of our diagram will only have a minor role in what is to come. My main concern is the other outside area: parts of what reality is like that outrun what we can truly say. This is the ineffable, that which we can't say. Herc there should at first be no question about its coherence, but there is a real question about whether there is anything which is ineffable. Is there anything that reality is like that goes beyond what we can say, and thus say truly? If so, how much of it is there? Is it merely a little sliver at the edge of the overlap, maybe something related to the paradoxes, or to consciousness, or is it a vast area, maybe most of what reality is like? What would follow for inquiry in general and philosophy in particular if there were a large area of the ineffable? To make progress on these questions we will first need to clarify the relevant notion of the ineffable, and how this problem is different from a number of other problems in the neighborhood. These problems arc also real and interesting problems, but not the ones I am trying to make progress on here. My discussion here will focus on a notion of the ineffable that is most promising for it being metaphysically significant, in that it captures the sense in which it just might be that minds and languages like ours arc not good enough, in principle, to represent some aspect of reality, and therefore are not good enough for carrying out an ambitious project of metaphysics. Whether or not we arc limited in this way is what I hope to find out. And to do this we should put aside some issues that I will not tty to resolve and focus in on the relevant ones instead.

Ineffable feelings. First, there is a common use of'\vords are not enough" to articulate the limits of language, but these are not the limits I am concerned with here. Take, for example "I can't tell you how happy I am to see you!" "Words arc not enough to say how glad [am to see you!" It would be beside the point to answer "Are you very, vety happy, or even happier than that?" It is not that the first speaker has a degree of happiness such that no words can pick out that degree of happiness. After all,"[ am maximally happy" would certainly be good enough to do that. Where words give out is not in describing the degree of happiness, but rather in giving the hearer a sense of what it feels like to be that happy. Words are not enough in getting the hearer to feel the way the speaker feels, or at least give them a sense of what such a feeling is like. But they are enough for describing how happy someone is: very, very happy. That words are not enough to transmit feelings in this sense is notable, but not a limitation oflanguagc in capturing what reality is like. Words might also not be enough to get you on the last Hight to Raleigh, in the sense that no matter what words you utter, you won't get on that flight. This limitation oflanguage is not one in its descriptive power, but in its limited effects to produce feelings or get an airline seat, a limitation I can happily accept and which isn't my concern here. What I am concerned with here is whether there are any facts, any truths, or any true propositions, such that we cannot, in principle, state or represent these facts, truths, or true propositions in our language.

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Ineffable objects. The notion of the ineffable is often tied to objects, and as such it is seen as problematic and paradoxical. An incfE.1.ble object is usually understood in one of two ways. It either is one that we can't talk about at all, or it is one about which nothing can be truly said. An example of the former is sometimes taken to be God when God replies to Moses' question about what his name is with "I am who I ~tm," and leaves it more or less at that. 1 One possible lesson of that is that God cm't be named, although this seems somewhat incoherent, since I just named God with "Goel." It is God, after all, who is supposed to be unnameable. Another lesson might be that God shouldn't be named, which wouldn't make God ineffable, of course, just normatively out of the naming game. It wouldn't be a limitation on our representational capacities, just on how we should employ them. On the other conception of ineffable objects, as ones about which nothing can be truly said, it also is generally taken to lead to a paradoxical 2 conclusion. After all, can't we at least truly say about ineffable objects that nothing can be truly said of them? In this sense ineffable objects can be tied

1

Exodus 3: 13-15. For a discussion of the ineffable in that sense, and a form of an embrace of the apparent paradoxes, sec Priest (2002). 2

Ihomas Hojiveber

Are There Ineffable AJpects of Reality?

to our main concern: ineffable truths or facts. If nothing can be truly said about an object o then any fact involving o should be ineffable. But the latter, ineffable facts, are not paradoxical. It is not required that notl:ing can be truly said about ineffable facts, only that ineffable facts can't be effed, that is, one can't utter a sentence such that this sentence expresses, states, or represents, that fact. I can't state the fact in question, but I might well be able to say true things about the fact, including that I can't state it. Ineffable facts or truths are not paradoxical, although ineffable objects, that is, objects about which nothing can be truly said, do seem to be paradoxical. An ineffable fact is simply a case where what is true outruns what we can truly say. It is not a paradox, but whether this is ever so is our concern here.

rather whether there is a conceptual representation of every fact or trmh. Conceptual representations arc paradigmatically the kind of representation we have in thought or language. The ineffable concerns the limits of conceptual representation, not the limit of representation more generally.

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A gap between language and thought? A third topic connected to the ineffable is the question whether there are certain facts or propositions that one can think, but one can't say. That is, are there certain contents that our thoughts can have, but there is no utterance of a sentence that has that same content? Some think that there are. One candidate for this are thoughts that involve phenomenal concepts. Maybe such concepts allow us to think thoughts that we can't put in language. Another, more traditional, example is a version of mysticism. According to it we can attain insights by various means like fasting or mediation, but we can't communicate them to others after we achieve them. These insights are not supposed to be feelings, but instead have propositional content. They are thoughts with contents that can be true or false. However, due to the nature of these contents they cannot be put into language. Although the mystic can think a thought wit~ that content, they can't communicate it with language. You have to meditate/fast/etc. to gain that insight. Whether either one of these cases obtains is controversial, of course, but this controversy does not matter now. I am not concerned with whether there are some limitations of language that are not limitations of thought. Instead I am concerned with whether there are certain facts or truths that are simply beyond us in either way, be it thought or language. I want to find out whether there are truths that we cannot represent at all, be it in language or be it in thought. Thus from now on I will take the ineffable to be that which we can neither think nor say. Whether there is a gap between language and thought thus won't matter for what is to come, interesting as the question is otherwise. Conceptual representation vs. other representation. Our issue here is not whether we can represent everything in some way or other, but rather whether for every fact or truth we can have a conceptual representation of that fact or truth. It might well be that something ineffable is going on right over there, and I could pull out my camera and take a picture of it, and thus represent it in some way. The issue is not whether I can always do that, but

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Fine vs. coarse contents. If the proposition that I am hungry now is different from the proposition that TH is hungry at t then this truth likely is ineffi1ble for everyone but me right now. You would only be able to express it if you were me and even then only at that particular time. This would mueness of (E') amounts indeed to a reductio of quantifier vagueness. b A sep~rate, and more crucial issue concerns the reductio step at the very end of Siders argument. Indeed, reductio ad absurdum is a valid form of

10 9

3z(x

B~; then, as I~o~'.~1an [ 17, p.

Since a statement is vague just in case it is neither true nor false, (l) and (9) are jointly inconsistent. By reductio ad absurdum, we can discharge the main premise (1) and infer its negation, namely 10.

207

Vague Edstence

Alessandro Torztt

206

11

Sider [29, p. 390]. Fora discussion, see Williams [33], Barnes [3], Heck (15], Akiba [lJ [2]. Cf. Lewis (19].

Vcigue Evistence

Alessandro Torw

208

209

inference for bivalent languages: an inconsistent condition cannot be true, hence it must be false. When the language is not bivalent, on the other hand, reductio ad ahsurdum fails, since being non-true is a weaker condition than being false. In particular, reductio is invalid in the case of vague languages: if a condition is inconsistent, it must be either false or indeterminate. It follows that Sider's final reductio step is valid insofar as the language in which the argument is formulated, viz. the metalanguage of ~xef;.' is perfectly precise. Since the main premise ( l) is equivalent to its precisihcanonal truthcondition (2), one is vague just in case the other is. Now, (2) docs two things: it quantifies over precisifications, and it says of the obje~t-.L~ngua?e sentence 3xq> that it is true in some but not all of them. By definmon, for every precisification s, truth-in-s is determinate; therefore, if there is any vagueness at all in (2), it must come down to what precisifications there are. The moral is that, if quantification over precisifications is precise, so that there is exactly one candidate set of precisifications for the object-language, . goes throug h- L'f not, not. 12 Si. der 's reductw Let me flesh out this point a bit further. It is not at all uncommon for a vague term (and, a fortiori, a vague language) to be asso~iat~d ";'}th m~ltip~e sets of precisifications. Take for instance the color ad3ecuve blue. 1 hts could refer to the spectrum of visible light with a wavelength between 450 and 490 nanometers-each single value specifying a precisification. But there is nothing about that very interval that makes it the right one. If the set of precisifications of "blue" can be 450-490nm, then the intervals 455-495nm or 445-485nm will work just as fine, as well as many other intervals in that neighborhood. Which is to say, none of them is a better candidate set of precisifications for the meaning of "blue." We must conclude that the term "blue" is second-order vague, insofar as the extent of its vagueness is itself vague. Could it be the case that second-order vaguen.ess affects the language of 3xef;? From what we have seen so far, nothmg prevents such a possibility. In a scenario of this sort, quantifyin.g over precisifications in Sider's argument would be vague and the reductw step,

that ( 1) is false and, therefrire, that vague existence has been disproved. In particular, even though the quantifier :3 cannot be definitely vague, there is a prima fade possibility that it might be second-order vague. In this case, there would be a set of precisifications that makes 3xef; vague, and another set that makes 3x' is vague-, is sharp. Sider's argumentative strategy could be iterated at any order-and so the relevant rejoinder. Consequently, as long as quantifying over precisifications,, is

us that 1.

r 3xef;' is vague

cannot be true, provided that ef; is precise and 3 is unrestricted. If the metalanguage of 3 is not perfectly precise, however, there is no guarantee

12 The i' is vague-, is false

Now, by virtue of Sider's first argument, the first conjunct of (2*) entails 3*

On 4*

In some precisification 2 r l_-, is true the other hand, precisifications2 being classical, it is the case that

Tn all precisifications2 r l_-, is false

hence a contradiction. If we could now apply reductio ad ahsurdum, we would be able to infer 5*

rr 3xef;' is vague-, is not vague

21 l

Alessandro Torzrl

Vtigue E·dstence

vague at every order n, neither side will get the upper hand, and the possibility of vague existence will remain neither proved nor disproved. The dialectics prompts the following moral. A precisificational semantics is a framework in which truth, falsity, and vagueness for a language L arc formulated, within some relevant metalanguage L , via the notion of truth-in-a-precisification-of L. Since precisifications arc ways of making a language precise, it cannot be vague whether an L-statcment is true or false in a given precisification. This account is perfectly compatible, however, with there being a vague set of precisifications of L. When that happens, the prccisificational truth-conditions for ,L-statemcnts can be vague, insofar as they arc formulated by means of L -statements of the form "there is a prccisification .... " Crucially, Sider's reductio of vague existence requires that the truth conditions for 3xcp be formulated in a perfectly precise metalanguage. (Or at least that a precise metalanguage could be found somewhere up in the hierarchy.) The existence of such a language is neither guaranteed nor required by a precisificational account of vagueness. Therefore, pace Sider, if what precisifications there are is vague at all orders, vague existence will remain an open possibility. The above discussion is complicated by the use of a hierarchy of metalanguages, each expressing whether sentences of the relevant object language are true, false, or indeterminate. Matters can be simplified as follows. Given sentence q> in an object language L, in lieu of the metalanguage expression

Which is to say, existence cannot be said to be definitely vague, no matter what the vagueness order is (provided that cp is precise). I will henceforth refer to this condition as Sider-determinacy. Let us stipulate that an existence statement r 3xcp' is anti-Sider-detennintfte just in case, for all n, r !'1 3xq>' is not false. I have argued in the foregoing discussion that precisificational truth conditions need not be given in a precise metalanguage and, in particular, that the set of precisifications can in principle be vague at each order in an infinite hierarchy of metalanguages. Insofar as, for all n, r !" 3xcp' can be vague, it follows that Sider-determinacy is compatible with anti-Sider-determinacy. When an existence statement is both Sider-determinate and anti-Sider-determinate, I will say that it expresses an instance of super-vrigue existence. For example, it might be that it is vague whether there exists the sum of F and G, and it is vague whether it is vague whether there exists the sum of F and G, and it is vague whether it is vague whether there exists the sum of F and G, etc. In the next section, I will show how super-vague existence can be accommodated within a precisificational model theory. But before that, I wish to address a potential objection.

210

r cp' is true we will use

where !:-. is an object-language sentential operator with the intended meaning "it is definitely the case that." Also, let's define the expression Icp ("it is vague whether cp") as •1:-.cp /\ •1:-.•cp. By iterating these newly introduced sentential operators, it is now possible to reduce the hierarchy of metalanguage truth/falsity/vagueness predicates to the object language L. For instance, the above condition l*

rr

3xcp'

is vague' is vague

translates into L as

II 3xcp. [n general, we can express that cp is n-th order vague simply by iterating the I operator n times. Now, let!" be short for the concatenation What n the generalized reductio has shown is that, for all n, r [" 3xcp' is not true.

U·

1.3 Metalanguage objection In the above reconstruction of Skier's argument against vague existence, I pointed out that the final reductio step requires the metalanguage of 3xcp (viz. the language in which the argument is formulated) to be perfectly precise. The same problem cropped up, mutatis mutandis, in the higherorder generalization of the argument. Sider touches on the issue of a precise metalanguage as he describes what it means to give an account of vague statements such as "5 is bald": When confronted with vagueness, I retreated to a relatively precise background language to describe the relevant facts. In this background !tzng1Mge I quantified over the various sets containing persons with different numbers of hairs, and said that the referent of S was in some but not all of these sets. [... ]Moreover, in principle one could describe the sets with perfect precision by retreating to a background language employing only the vocabulary of fondamental physics. (Sider [26, p. 139], emphasis added)

It is tempting to interpret the passage as entailing that the precisificational truth-conditions of 3x can be assumed to be perfectly sharp and, therefore, that Sider's reductio of vague existence is valid. Consequently, existence cannot be super-vague, since it is definitely not vague. The above passage, however, does not license this conclusion. Recall from the discussion in Section 1.2 that one thing is to say that (i) "true-in-s" is

Alessandro Torvz

212

sharp, for s a given precisification; and another thing is to say th~t. (ii) "true in some precisification" is sharp. Indeed, (i) is true by defin1t1on, insofar as each precisification is a classical interpretation described in the metalanguage with perfect precision. But (i) can obtain in the absence of (ii), namdv, if it is indeterminate what precisificuions there f'i•!::,. fails. On the model-theoretic side, the failure of either condition corresponds to a non-transitive or non-euclidean admissibility relation: see Williamson [37, p. 133]. In fact, higher-order vagueness has its modal counterpart in the contingency of contingency, which also takes place only in systems weaker than 55.

216

Vtigue E' is NS-valid iff it is a NS-consequence of the empty set.

2.3 Negative supervalnationary semantics A negative superualuationmy (NS) model M* for LL\ .is a pair (.P ,a*) where P is a variable domain frame and a* an interpretation function such that, for every point t ES, (i) a*(=', t) is the identity relation over Dom( t) and (ii) for Pan n-ary predicate, a*(P,t) ~ Dom(t)". . . . Let VAR be the set of variables in £ 6 and S the set of specificauon pomts. A value assignment for VAR over M* is a set of partial functions { g,}, Es

(Eq l)

such that:

l. 2.

3.

g, : VAR_, Dom(t) U{t } sis a total function/: VAR_, U if dx)~nd g,(x) are both defined, then t,(x)

= t,(x)

The first condition allows a variable to be undefined at some precisifications, whereas the second condition forces a variable to be defined at some precisification. Consequently, negative supervaluat.ionary sem.an~ics is r~ot a framework for definitely non-referring terms, unlike free logic. fhe thm{ condition guarantees that variable assignments are rigid across specification points. . . . Truth-at-a-points (local truth) for cf> under a variable assignment { t,}, Es m an NS-model JVI*, written ( M* ,s, {g,}, Es) I= N5cf>, is defined recursively thus: (at)

If cf>= P(x 1 , .. .,x,,), then (M*, s, {0,Es) I= NS iff t, is defined for all i E { 1,..., n} and (t,(x1), .. ., t,(x,,)) E a*(P, s)

(...,) If cf> = •if1, then ( JVI*, s, {0 s Es) I= N5cf> iff ( .M*, s, {0 , Es) I= Nsifi (A) If cf>=(i/;Ax), then (M*, s,{0,Es)l=N,cf> iffboth (M*, s,{0,Es) (V)

Let us now turn to the key properties of negative supervaluationa1y semantics. Standard prccisificational theories of vagueness, such as Fine's specification space semantics or Williamson-style constant domain semantics, have the virtue of being classical in a precise sense: given a purely extensional language, classical consequence (I= c) and supcrvaluationary consequence (I= sv) coincide. Namely, for cf> and r formulated in a first-order language£ without definiteness operator, 15

l=Nsi/1 and (M*, s,{g,LEs)l=NsX . , If cf>= Vxtf;, then (M*,s, {0,Es) I= Nscf> iff, for every {t JsES such that {, is defined on x and differs from

t,

at most on x,

(M*, s, {{J,Es) I= NSV1 (!'::..) If cf>= l'::.ifi, then (M*,s, {0,Es) I= Nscf> iff, for every t such that sRt, (M*, t, {0,Es) I= NSV1 A few more definitions are needed. A formula cf> is true in a NS-model M* relative to a variable assignment { g,}, Es (i.e. ( M*, {g,}, Es) I= Nscf>) iff, for every s ES,

(M*, s, {t,LEs) I= NS· A formula cf> is true in a NS-model M* (i.e. M* I= Nscf>) iff, for every {g,},E5" (M*, {g,},ES) I= NS·

r I= c

(K) I= N.'>6( ·---+ 1/;) __, (6 ---> A

-+

l':i 3xcf>

fails, too. (Proof in Appendix B.6.) We can finally return to the main point, quantifier vagueness. Our goal is to find some NS-model of super-vague existence. First of all, we need to check that the following condition holds: Sider-determinacy:

for all A1*, { t,}, Es and n ) 1: ( M*, { t,}, Es) Jz'.' NS !'1 :Jx(x = y)

for otherwise negative supervaluationary semantics would be inconsistent with Sider's result, or with the higher-order generalization of it. Second, we would have to show anti-Sicler-determinae,y:

for some M* and { t,} s Es and for all n) 1: (M*, {0,Es) Jz'.' •!" 3x(x = y)

A proof that both conditions hold in negative supervaluationary semantics can be found in Appendix B.7. We can conclude that super-vague existence is NS-satisfiable. Let's recap. Sider argued that

(V) 20 One such framework, involving the use of possibilist quantification, is defended in Linsky and Zalta [20 J [21 J, Williamson [36 J. An alternative sem, which paraphrases away 6. (as well as any other expression defined via tJ.) by quantifying over precisifications which are extensional, set-theoretic objects. Thus, the language of Sicler's proof being 6.-free, reductio ad absurdum appears to be valid after all. What undermines the above objection is the tacit assumption that, if the semantics for the language of Sider's proof is supervaluationary, then it has to be some kind of standmd supervaluationism, such as Fine's specification space semantics or Williamson-style constant domain semantics. I claim instead that, if Sider' s proof is coached in a vague language, we should model it via negative supervaluationism. The reason for this choice is quite straightforward. I argued in Section 1.2 that if Sider's language lacks

225

Ted doesn't exist F= Ns.l since names and free variables cannot be definitely non-referring in a NSmoclel. However, it doesn't follow that

F= NITeel exists for otherwise existence would always be determinate in negative supervaluationism, which we know not to be the case clue to the NS-satisfiability of super-vague existence (Appendix B.7). It is worth noting that, even though we could give a Sider-style argument against vague existence which doesn't employ reductio ad absurdum, the new argument would still have to be formulated in a precise language, in order to be valid. For example, we could give a proof by cases: l.

r 3xcf>' is either vague or not vague.

ii. Suppose r 3xcf>' is vague. Therefore, it is true at some precisification, and so true at all precisifications, which means that it is true (cf. steps 3-9 in my reconstruction of Sider's argument, Section 1.1). But if r 3xcf>' is true, then it is not vague. iii. Supposer 3xcf>' is not vague. Therefore, it is not vague. iv. Thus, r 3xcf>' is not vague (proof by cases) The argument has the form: i. ii. iii. iv.

pV•p pF=q -pF=q F=q

Now, in standard supervaluationism, proof by cases behaves just like reductio ad absmdum: although invalid in general, it is valid for arguments stated in a Li-free language. But again, if the language in which the above proof is stated is vague due to quantification over precisification, the correct framework is negative supervaluationism, with the relevant consequence relation F= NS which does not validate proof by cases even for Li-free languages, as the following simple counterexample will show:

226

Vrtgue E;;istence

Alessandro Torvi

i. a = ilV t1 'F a ii. a= a F N'i 3x(x =a) iii. a f a F= NS 3x(x = a) iv. F NS 3x(x =a)

l ndecd, in some model the three premises arc true, 23 whereas the conclusion is untrue.

2.5 Inferentialist objection have modeled super-vague existence by means of precisified quantifiers.

I will now consider an objection, which is an adaptation of an argument originally formulated by Williamson [34], purporting to show that there cannot be multiple prccisifications of the existential quantifier. Given a language £ with vague 3, define a new language £' in which 3 is replaced with two precisifications 3 1 and 3 2 . For instance, 31 could be the ontologically sparse quantifier of the mereological nihilist, w~ereas 32 is the promiscuous quantifier of the universalist. (Likewise, in {-- there will be "cornposition 1" and "composition 2 "). Now, let cf>(x) be a£ formula. From 31xcf>(x) we can deduce cf>(z) by cxistential 1 instantiation, where z is chosen so that it does not occur free in cf>(x). By existential2 generalization, 32xcf>(x) follows from cf>(z). Hence, there exists a deduction of 32xcf>(x) from 31xcf>(x). Since we can produce the same kind of argument running in the opposite direction, the two quantifiers are equivalent, which contradicts the initial assumption that 3 1 and :3 2 are distinct precisifications. To this argument I offer a two-tiered reply. For reasons that will soon become clear, I take the second part of my reply to be the more enlightening one. Firstly, the objection assumes that the precisified quantifiers are classical, in the sense that for each :J,,, the rules of generalization,, and instantiation,, are the classical ones. But this assumption is unwarranted. Recall that a precisification of the language is identified with a particular specification point in a model of negative supervaluationary semantics. Local truth, i.e. truth at a specification point, is defined in terms of negative semantics. Moreover, negative free logic is sound and complete with respect to negative semantics. Therefore, the generalization and instantiation rules for a precisified quantifier are the ones of negative free logic: • from cf>(z) /\ 3,,x(x

z) infer 3,,xcf>(x)

• from rand 3,,xcf>(x) infer cf>(z) /\ 3,,x(x = z), where z does not occur free in r or cf>(x). 23

a

=J

a is an NS-inconsistency, hence 3x(x = a) follows from it trivially.

227

With that being said, it is easy to see what goes wrong in the inferentialist objection. Recall that :31 is the less promiscuous quantifier, which docs not support compo~ition 1, whereas :32 allows unrestricted cornposition 2 . Let cf>(x) be the £ formula ".x- is a mereologica[ compound 2 ," where "compound2" is the univcrs z = z is a theorem of negative free logic, from the first conjunct of (i) it follows that (ii)

z

f z.

From (ii) and the second conjunct of (i) we can infer (iii)

32x(x =J x ).

But (iv)

V2x(x = x)

is a theorem of negative free logic. Hence, the claim (i) that 3 1 is a restriction of 3 is inconsistent, provided that the precisified quantifiers obey negative free logic. But according to negative supervaluationary semantics, precisified quantifiers do obey negative free logic. We must conclude

21

Cf. Turner [30, pp. 25-6].

228

Alessandro Tor:ztt

Vt1gue Edstence

that it not possible to define :3 and, therefore, that we cannot use both 31 and ch within the same language, as the infercntialist objection presupposes. 25 The moral is that quantifiers behave differently from non-logical predicates in one key respect. If we speak a language where "bald" is vague, we can define a new language in which the original predicate is replaced with a multiplicity of precise prcdiGJ.tes "bald 1," "bald2" etc. This cannot be done with quanti-ficrs on pain of inconsistency. I hope it is now clear that the deeper reason why the infercntialist objection is unsound is that it assumes that quantifiers governed by negative free logic can coexist in a single language.

If existence is super-vague, we ought to accept that composition might be super-vague (i.e. vague at l'i.l'i.. Let n be the cardinality of the domain of the largest complete specification point in a space. (If the cardinalities had no upper bound, it would not be the case that there definitely arc finitely many objects.) Notice that the domain of the base point@ is a subset of the cardinality ofany complete specification point, since if 3y(x "= y) is true at l'i.•l'i. arc supervaluationarily valid in Fine's model. As to (a),

231

Alessandro Torza

Vi1gue hxistence

suppose that f'..

x

cf y)

For suppose a variable assignment maps x and y to the same object a at point t, and let sRt, where a~Dom(s ). Since the assignment is rigid, x and y will fail to refer at s, and so the antecedent of (DD ) must be true at that point. The consequent, on the other hand, is false at s.

Vtigue hY:istence

B.6 NS and the importation schema (IM)

Akiba, Ken (2004). "Vagueness in the World," Notis 38(3): 407-29. Barnes, Elizabeth (2009). "Indeterminacy, Identity and Counterparts: Evans Reconsidered," Synthese 168(1): 81-96. Barnes, Elizabeth (2012). "Metaphysically Indeterminate Existence." Philosophical Studies 166(3): 495-510. Boolos, George (1984). "'I'o Be ls To Be a Value of a Variable (or rn Be Some V11lues of Some Variables)," Journrd ofl'hifosophy 8 l: 430-50. Braun, David and Theodore Sider (2007). "Vague, So Untrue," Notis 41(2): 133-56. Burge, Tyler (1974). "Truth and Singular Terms," Nous 8: 309-25. Carmichael, Chad (2011). "Vague Composition without Vague Existence," Notis 4 5 (2): 315-27. Carnap, Rudolf ( 1950). "Empiricism, Semantics and Ontology." Reprinted (1956) in Meaning rmd Necessity, 2nd cdn. Chicago: University of Chicago Press: 205-21. Chalmers, David, David Manley and Ryan Wasserman, eds. (2009). Metametaphysics: New Essays on the Foundations of Ontology. Oxford: Oxford University Press. Donnelly, Maureen (2009). "Mcrcological Vagueness and Existential Vagueness." Synthese 168(1): 53-79. Evans, Gareth (1978). "Can there be Vague Objects?", Analysis 38(4): 208. Fine, Kit (1975). "Vagueness, 'fruth, and Logic," Synthese 30 (3-4), 265-300. Carson, James (2001). "Quantification in Modal Logic," in Dov Gabbay and Franz Guenthner (eds.) Handbook of Philosophical Logic, second edition, vol. 3. Dordrecht: Reidel: 267-324. Heck, Richard (1998). "That There Might Be Vague Objects (So Far As Concerns Logic)," Monist 81 (2): 27 4-96. Keefe, Rosanna (2000). Theories of Vagueness. Cambridge: Cambridge University Press. Korman, Daniel (2010). "The Argument from Vagueness," Philosophy Compms 5 (10): 891-901. Lewis, David (1986). On the Plum!ity of Worlds. Oxford: Basil Blackwell. Lewis, David (1988). "Vague Identity: Evans Misunderstood," Analysis 48(3): 128-30. Linsky, Bernard and Edward Zalta (1994). "In Defense of the Simplest Quantified Modal Logic," Philosophical Perspectives 8: Philosophy of Logic and Lt1ngu,1ge, Atascadero, CA: Ridgcview: 431-58. Linsky, Bernard and Edward Zalta ( 1996). "In Defense of the Contingently Nonconcretc," Phiwsophicril Studies 84(2-3): 283-94. Nolt, John (2006). "Free Logics" in Dale Jacquette (ed.), Philmophy of Logic. Ht1ndbook of the Philosophy of Science. Amsterdam: North Holland: 1023-60. Quine, W. V. 0. (1951). "Two Dogmas ofEmpiricism." Philosophiet1l Review60(1): 20-43. Sainsbury, Richard (2009). Fiction and Fictionalism. London: Routledge. Sider, Theodore (2001). Four-Dimensionalism. Oxford: Oxford University Press. Sider, Theodore (2003). "Against Vague Existence," Philosophical Studies 114(1): 135-46. Sider, Theodore (2007). "Nco-Fregcanism and Quantifier Variance," Proceedings of the Aristotelian Society 81 (l ): 201-32.

The importation schema: (IM)

3xf..

/.\ 3xrp

is not NS-valid. To construct a counterexample, let /vi* be a NS-model with specification points sand t where sRt, Dom(s) cc= {a} Rescher (ed.), Essays in Honor of Carl C7. Hempel, Dordrccht: Reidel, 104--13.

278

Alexander Pruss

Kleinschmidt, Shieva. (2013). ''Reasoning without the Principle of Sufficient Reason." In Tyron Goldschmidt (ed.), The Puzzle oj'Existence: Why Is There Something R11ther Tlvm Nothing? New York: Routledge, 64-79. Koons, Robert. (2006). "Oppy and Pruss on Cosmologic;1l Arguments." Presented ar rhe Central Division Meeting of the American Philosophical Association, Chicago, April 2006. Leslie, John. (l 979). Vdue 1md E>:istmce. Totowa, NJ: Rowman and Littlefield. Pruss, Alexander R. (200 l). "The Cardinality Objection to David Lewis's Modal Realism." Philosophical Studies 104(2): 167-76. Pruss, Alexander R. (2006). The Principle of Sufficient Reason: A Re11ssessment. New York: C.unbridge University Press. Pruss, Alexander R. (2013). "On the Law of Large Numbers for Nonmeasurable Identically Distributed Random Variables." Bulletin of the Polish Academy of Sciences--Mathematics 61 (2): 161-8. Rescher, Nicholas N. (1995). Sritisfjing Re11son: Studies in the Theo1y of Knowledge. DordrechtlBosron/London: Kluwer. Rcscher, Nicholas N. (2010). Axiogenesis: An Essay in Metaphysical Optima/ism. Lanham, MD: Lexington. Salmon, Wesley C. (1989). Four Decades of Scientific Explanation. Minneapolis: University of Minnesota Press. Sierpit1ski, Wadaw. (1938). "Sur un problcmc conccrnant lcs famille d'ensembles parfaits." Fundamenta Mathematime 31 ( 1): 1-3. Taylor, Richard. (1974). Metaphysics. Englewood Cliffs, NJ: Prentice-Hall. Whitcomb, Kathleen M. (2005). "Quasi-Bayesian Analysis Using Imprecise Probability Asscsscmenrs and the Gcncmlized Bayes' Ruic." Theo1y and Decision 58(2): 209-38.

12 Grounding Ground Jon Erling Lit/and

1. INTRODUCTION Ifr's being the case grounds ef/s being the case, what grounds that r's being the case grounds ¢' s being the case? 1 This is the Problem of Iterated Ground. Dasgupta (2014b), Bennett (2011), and deRosset (2013) have grappled with this problem from the point of view of metaphysics. But iterated ground is a problem not just for metaphysicians: the existing logics of ground 2 have had nothing to say about such iterated grounding claims. In this paper I propose a novel account of iterated ground and develop a logic of iterated ground. The account-what I will call the Zero-Grounding Account (ZGA for short)-is based on three mutually supporting ideas: (i) taking non-factive ground as a primitive notion of ground; (ii) tying nonfactive ground to explanatory arguments; and (iii) holding that true non-factive grounding claims are zero-grounded (in Fine's sense). A notion of ground is factive if the truth of"f grounds .

If E; is an explanatory argument from 6; to rP; f(ir each i, and Dis an explanatory argument to

n

1

...

P1

293

I'

])

"' is an explanatory argument from /1. 0 , /1. 1, .. ., ['tor/>. Figure 12.1 Arguments explanatory aud plain

to

58 For an application of, in effect, hypothetical arguments in a different context sec Schroeder-Heister (1984).

Jon fa-ling Ut!,md

Grounding Ground

Arguments of this form arc explanatory. What this says is that by using a hypothetical explanatory argument from 6 to cp one explanatorily infers cp from 6. Even though hypothetical arguments can be assumed and discharged in the course of an argument, they ;1re not premisses on which the (sub)conclusions depend. (We assume an argument, not the existence of an argument.) One may think of the hypothetical arguments as assumed rules of inference. lf follows from

[i\ If-- e 1

4

- - - - 3 , =?-!

5

/\ =}

cf>

-------

1) f-- .;,, for each •/J. This, as we argued in §6.l

303

p

--1 p

p J\p p=?p!\p =}

(p

=}

p J\f')

-1

p ---V-l pvp

/\-[

I, =?-I

I,=?-!

=}

=}-[ =}

(p

=}

p v p)

=?-!

Figure 12.2 Different ways

42 Dasgupta (2014b, p.573) criticizes the views of deRosset (2013) and Bennett (2011) along these lines.

304

Jon Erling Lit/and

Grounding Ground

--2

--2 _ _ _-'_/\-!

~----~/\-!

__/'_0_'1_ _____ 1 7 =>-I p, '/ => p J\ '/ ,-~-I =>(p, '/ ='> p J\ q)

--~--1,2,=>-l

__.!~s_=::__~!:_!_ ____ =>-I =>

(1; s => r J\ s)

Figure 12.3 Sarne way

ground-the empty one--but they also seem to be grounded in the empty ground in the same way. We can make sense of this in terms of the machine picture. In the two cases depicted in Figure 12.3 the same mechrmisms are applied (and they are applied in the same order) but the applications of the mechanisms differ since p,q are different propositions from r,s. (It is for this reason that we have taken an arc A to represent the application of a mechanism not the mechanism itself.) An advantage of the framework of explanatory arguments is that it promises us the means for defining the notion of a way of grounding. For consider a particular explanatory argument. Uniformly replace items in that argument with schematic letters. This gives us an argument farm. If every argument of that form is explanatory the argument form is an explanritory argument form. We may identify the ways of grounding with the explanatory argument forms. As is easily seen, the two arguments in Figure 12.3 have the same explanatory form. We can now deal with a further objection to the 7GA. Dasgupta (20 I 4b, pp. 531-2) observes that there are patterns in grounding. (For instance, all conjunctions are alike in terms of how they are grounded.) An account of ground should provide us with an account of these patterns. Armed with the notion of a way of grounding we have a ve1y simple explanation: the patterns in grounding are the result of different propositions being grounded in the same way.'u

10. COMPARISON WITH THE STRAIGHTFORWARD ACCOUNT The SFA, recall, holds that when r < q,, then r < (r < q,); unlike the 7CA, however, the SFA does not hold that f < if, is partly grounded in I' ==? if,. Why should we prefer the 7GA to the SFA? The question has a false

305

presupposttton: on my favored way of conceiving of the relationship between the SFA and the 7CA they are not in competition. Let us first observe that the logical techniques developed for the 7GA can equally well be employed in developing a logic of iterated ground that is in accord with the SFA. A defender of the SFA could, for example, give an introduction rule for an operator < + as follows:

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/'!.) is also the case, then •(L'I , the machine churns out the truth •( L'I < t- ef>). Holding that factive ground behaves like < 1 does not obviate the need for zero-grounding. 50

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45 I should stress that negated-more generally, embedded-grounding claims cause problems even if one docs not accept the present framework of explanatory arguments. 46 As admitted by deRosset (2013, p. 16). 47 The simplest way of dealing with negation in the logic of ground is to follow a broadly "bilatcralist" strategy and give separate introduction and elimination rules for negated and unnegated propositions; for this strategy, sec, e.g. Fine (20 l 2a, p. 63) A different, perhaps preferable, treatment of negation is pursued in Fine (2016). 48 How docs the machine know that it has inspected all the arcs? We might imagine that in running the simulation the arcs arc ordered in such a way that each arc occurs infinitely ofren but such that if an arc A with tail /I,. occurs both at position t and at a later position /, then all arcs with tail D. have occurred at least once before/. If, having found no arc B with tail /I,. and head , the machine inspects an arc it h \.'f; is a fonction from the propositional atoms into the vertices of Q. We here demand that V has greater cardinality than the set of sentence letters. We extend the interpretation function from the atomic letters to arbitrary formulae in the obvious way.

Propositon B.6. let Q == ( V,F,A,t,h) be rt hype1gmph. There is rt graph with operrttors Q 1· (V 1, F 1, A+,=..'- ,