Truth At a World for Modal Propositions 1 Introduction Existentialism is a thesis that concerns the ontological status of individual essences and singular propositions. Let us define an individual essence E as follows: a property of some actual individual a is an individual essence E iff (i) E is essential to a so that it is not possible that a exist and lack E and (ii) E is essentially unique to a so that for any world W and individual b, E is a property of b in W only if a = b. 1 With respect to singular propositions, we will say that a proposition is singular with respect to object o just in case the proposition is directly about o by having o as a constituent. 2 For example, the proposition Socrates does not exist is singular with respect to Socrates insofar as he is a constituent of that proposition. Existentialism maintains that individual essences are ontologically dependent upon their exemplifications and that singular propositions are ontologically dependent upon their constituents. Consequently, in a world in which Socrates never exists, there are no individual essences or singular propositions about him. Some have thought that existentialism is untenable since it cannot account for certain obviously true modal claims. To see this consider Jason Turner s possibility argument: 1. Possibly, Socrates does not exist 2. Possibly, p if and only if p is true with respect to some world W 3. p is true with respect to W if and only if were W actual, p would exist and be true. 1 Here I follow Plantinga (1983, p. 1). 2 See Fitch (2002).
4. Therefore, were W actual, Socrates does not exist would exist and be true. 3 (4) is a straightforward contradiction of the second existentialist thesis that singular propositions are ontologically dependent upon the individuals they involve. Hence, the existentialist would seem forced to deny the obvious: that Socrates might not have existed. One way for the existentialist to resist this argument is to distinguish between two senses in which a proposition can be true with respect to a world: true at a world and true in a world. The existentialist can maintain that the modal operators and are to be analyzed in terms of truth at a world rather than truth in a world: p iff p is true at some world p iff p is true at every world. In light of this distinction, we can see that the above argument turns on an equivocation: (2) is only true when understood as truth at a world while (3) is only true when understood as truth in a world. Therefore, there is no single reading of truth with respect to a world under which all the premises are true. 4 The notion of truth in a world is quite familiar and I will assume unobjectionable. However, the notion of truth at a world is relatively unfamiliar and might seem obscure. Since the existentialist s response is only as good as the notion of truth at a world, we cannot judge its success until we can judge the intelligibility of this notion. In what follows I will be particularly concerned with explaining the truth conditions of truth at a world for modal propositions. Robert Adams (1981) and, 3 Turner (2005, p. 192). 4 See Adams (1981, pp. 18-21) and Fine (1985, p. 163). 2
recently, Turner (2005) have rendered the notion of truth at a world for non-modal propositions relatively clear. However, they disagree as to the proper account for modal propositions. I will be concerned with resolving this dispute. I shall argue that Adams s account is more at home within the existentialist s ontology. I defend this thesis by first showing that existentialism is committed to certain metaphysical claims that Turner s account cannot accommodate, and second by showing that Adams s account can readily accommodate these claims. In the course of this paper we will see that existentialism is committed to rejecting modal logics S5, S4, and B. Some find this is a hard price to pay, but I believe it is worth the cost. Throughout this paper I will assume actualism and property actualism. Let us define actualism as the thesis that everything there is, in the most unrestricted sense consistent, is actual. I will follow Plantinga in defining property actualism as follows: necessarily, for any object x and property P, it s not possible that x should have had P but not existed. 5 2 Truth at a world for modal propositions Recall that existentialism is committed to claiming that for any world W in which Socrates never exists, were W actual, there would be no possibilities whatsoever concerning Socrates. This is because neither Socrates, nor any of his individual essences, nor any singular propositions about him, would exist. It is quite plausible to assume that it is possible that Socrates does not exist. However, the existentialist cannot appeal to the familiar analysis of truth in a world in order to account for this possibility since there is 5 Plantinga refers to this view as serious actualism (1983, p. 11). However, I will follow Fine (1985, p. 163) in referring to it as property actualism. 3
no world W in which the proposition Socrates does not exist is true. In order to accommodate this modal claim the existentialist must appeal to truth at a world. It is possible that Socrates does not exist since the proposition Socrates does not exist is true at some world. But what is it for a modal proposition to be true at a world? Intuitively a proposition is true at a world W if and only if the proposition properly characterizes what the universe would be like were W actual. I will abbreviate the right hand side with simply, the proposition properly characterizes W. With this intuitive characterization of truth at a world, let us turn to Turner s more elaborate account of truth at a world for modal propositions. propositions: Turner proposes a single criterion for truth at a world for modal (C4) If p is of the form q, q, ~ q, or ~ q, and if p is true simpliciter, then for all worlds W, p is true at W. 6 Turner attempts to motivate this criterion as follows: Return to the picture thinking that truth at a world is supposed to capture. We are standing outside of a world, looking into it, and using the propositions, objects, properties, and relations of our own world to describe what we see. It makes sense to think that which predications of a are true at a world is determined solely by things going on in that world how could facts from other worlds ever get into the picture? But we tend to think that modal truths are not made true solely by what is going on in any one world but by what goes on in the entire space of possible worlds. Furthermore, on the model of standing outside of a 6 Turner (2005, p. 205). 4
world looking into it, it is not implausible to think that we should be able to see the entire space of possible worlds. We can say, of a non-a world W, that at that world it is possible that a is P, precisely because, standing outside of W, we can see other worlds worlds where a exists and is P. 7 From (C4) it follows that propositions such as possibly Socrates exists, and there is an object x that possibly is the brother of Socrates are true at worlds in which Socrates does not exist since they are true simpliciter. Notice that according to (C4) and the definition of, if q, then q. (C4) entails that if some proposition q is true, then it is true at every world and we have defined so that p is true if and only if p is true at every world. This is a formally convenient result for Turner since it validates the characteristic axiom of the strong modal logic S5: p p. S5 has considerable intuitive support and allows for a formally elegant modal logic; it should not be abandoned lightly. 8 Nevertheless, existentialism is committed to rejecting (C4). This principle fails to capture the fact that were some other world in which Socrates does not exist actual, what is possible would have changed: things that are possible would no longer be possible. That is, it fails to capture that the set of the possibilities there are is not identical with the set of the possibilities there could have been. The issues here are very delicate and difficult to sort out. To see the unacceptability of (C4), I will first draw out two shortcomings of this criterion. I will then trace the origin of both these shortcomings to (C4) s failure to take serious the existentialists ontology. I will conclude by suggesting 7 Turner (2005, p. 205). 8 The modal logic developed by Menzel (1991) secures similar (in my opinion problematic) results as (C4). 5
that we stick with Adams s offered criterions for truth-at for modal propositions since they are more at home within the existentialist s ontology. Let us begin with the first shortcoming. Consider the proposition that possibly Socrates exists and some world W in which Socrates does not exist. According to existentialism, were W actual, there would be no possibilities concerning Socrates. The most natural way of expressing this is by maintaining that possibly Socrates exists is false at W and its negation true. It is difficult to grasp in what sense the former proposition properly characterizes W. Instead, the latter would seem to properly characterize W since were W actual there would be no possibilities concerning Socrates. So what are we representing with the claim that possibly Socrates exists is true at W? It obviously cannot represent the following: that were W actual, there would be no possibilities concerning Socrates. It also cannot represent that were W actual, it would be possible that Socrates exists since this would contradict existentialism. The only option left, as far as I can see, is that saying that this proposition is true at W represents the fact that it is actually true. This would seem to be what Turner intends since what determines the truth of a modal proposition at a world is the proposition s truth-value status in the actual world. But if this is what it represents then it is hard to see why such a proposition should be true at W. Truth at W is equivalent to properly characterizes W; it is not equivalent to properly characterizes the actual world. Therefore, it would seem that Turner must chose between giving up existentialism and giving up our categorization of truth at W as equivalent to properly characterizes W. Since Turner developed truth at a world in order to defend existentialism, he must reject the latter requirement. But what else might truth at a world 6
consist in? All those who have undertaken to develop truth at a world have accepted this requirement and with good reason. Truth-at was developed in order to acknowledge and incorporate into our modal logic the fact that certain propositions properly characterize a world even though they are not true in that world. To reject the equivalence is to reject the original intention behind distinguishing two senses of truth with respect to a world. I believe the best way to avoid this problem is to reject (C4). A closely related problem is that by endorsing (C4) we preclude any way of representing the fact that if a Socrates-less world were actual, there would be no possibilities concerning Socrates. This is (C4) s second shortcoming. As we noted above, one of the main implications of existentialism is that de re possibilities concerning contingent objects will vary as worlds contain less objects than the actual world. We intuit that Socrates might not have existed; there is a world in which Socrates does not exist. Existentialism entails that in such a world there are no possibilities concerning Socrates. The existentialist desires a way to make this point perspicuous in her modal logic. The most obvious way to make this point clear is to have a logic in which propositions such as possibly Socrates exist are false at any world in which Socrates does not exist. The negation, it is not possible that Socrates exist properly characterizes such a world since were that world actual, there would be no possibilities concerning Socrates. But Turner s (C4) prevents us from saying this, at least if bivalence is to hold for truth-at. (C4) entails that possibly Socrates exists is true at W since it is actually true. It is not the case that possibly Socrates exists is true at W only on pains of denying bivalence. Hence, in order to allow our modal logic to be capable of representing the fact that were W 7
actual, there would be no possibilities concerning Socrates, we must reject (C4) (it is really no option at all to reject bivalence for modal propositions). (C4) is not at home with existentialism nor is it consonant with the original intent behind the in/at distinction. Although certain propositions cannot be true in any world, they nonetheless properly characterize some world: the proposition in some sense correctly represents what the universe would be like were that world actual. Truth at a world was introduced by Adams to account for this. We now have a way of positively describing what the universe would be like were some world in which Socrates does not exist actual. But we must not forget that propositions true at worlds are intended to properly characterize those very worlds. As we have seen (C4) violates this requirement. Further, the existentialist desires a way to express her commitment to maintaining that non-qualitative possibilities concerning contingent objects will vary depending on what objects there actually are. Again, (C4) was faulted for precluding the possibility of representing this fact. But surely Turner is right that what determines the truth of a modal proposition is not merely what goes on in any isolated world. What goes on in the entire space of possible worlds is also relevant, but it is not the only thing that is relevant. He is correct that what is possible is partly determined by the entire space of possible worlds, but it is not primarily or fundamentally so determined. It is fundamentally determined by what there actually is. I believe that the shortcomings of Turner s semantics can be partly traced to his failure to take this point seriously. Turner s claim that non-qualitative possibilities are (partly) determined by what goes on in the entire space of possible 8
worlds is irrelevant to whether possibly Socrates exists should be true at a world in which Socrates does not exist. Turner goes awry by illicitly maintaining that one s vantage point affects the ontology of possible worlds. The fact that we can see all possible worlds from our vantage point does not affect what exists in W. But instead, it merely affords us the expressive power desirable of an adequate modal semantics. The truth of propositions, whether it be truth at or truth in W, must be grounded in the ontological fabric of W. When assessing the truth of non-qualitative modal claims at W, we must, so to speak, look through W to the other possible worlds. We must first consider the ontological furniture of W and only then is the entire space of possible worlds relevant. If Socrates fails to exist in W, there should be no true predicative modal propositions directly about him at W. If Socrates exists in W, then, and only then, do we proceed to examine other possible worlds, here we include the actual world, in order to discern whether certain modal claims about Socrates are true. This picture illustrates that, according to existentialism, what non-qualitative possibilities there are depends fundamentally on what there actually is and only secondarily on what goes on in other possible worlds. (C4) ignores this. We ought to adopt a criterion of truth-at for modal propositions that reflects this fact. 3 (C6) and (C7) So we need a set of criterions that allows us to respect the existentialists ontology, capture their commitment to the non-iteration of possibilities while remaining consonant with the purpose behind the in/at distinction. I suggest that we return to Adams s 9
proposed criterions for truth at a world. Let W be a possible world in which an actual individual a does not exist, then: (C6) If p and p are singular propositions about a, then ~ p and ~ p are true at W. (C7) If ~ x1 xn (F(a, x1,, xn) and ~ x1 xn (F(a, x1,, xn) are singular propositions about a, then they are true at W. 9 (C6) and (C7) allow us to represent the existentialist s commitment to the fact that what there is fundamentally determines what is possible. The proposition possibly Socrates exists will be false at any world in which Socrates does not exist. This represents the existentialist s claim that were that world actual, there would be no possibilities about Socrates. By adopting (C6) and (C7) we opt for a modal logic that treats all predicative modal propositions as false at worlds in which the individuals they ascribe properties to do not exist. These criterions also entail the non-iteration of possibilities. Given space limitations I will be content with showing that (C6) and (C7) invalidate the characteristic axiom of B. Similar counterexamples for the axioms of S4 and S5 are readily available. B s characteristic axiom is p p. According to existentialism this axiom has false instances. Letting p stand for Socrates exists, it turns out that p is true and p is false. p is true only if p is true at every world. However, as we have seen, p is false at every world in which Socrates does not exist. So p and ~ p are both true. These are unfortunate results since they make the construction of a modal logic more difficult than we might have originally thought. However, their rejection is necessary if our modal logic is to respect our metaphysics. B, S4, and S5 axioms violate metaphysical principles laid down by the existentialist. Although their rejection is costly, 9 Adams (1981, p. 29). 10
I believe it is worth the cost. Better to have a complicated modal logic that accurately represents our ontology than to have a simple modal logic that fails to represent our ontology. 11
References Adams, Robert 1981. Actualism and Thisness, Synthese 49: 3-41. Fine, Kit. 1985. Plantinga on the Reduction of Possibilist Discourse, Alvin Plantinga ed. James E. Tomberlin and Peter van Inwagen. Boston: D. Reidel, 145-186. Fitch, Greg. 2002. Singular Propositions, The Stanford Encyclopedia of Philosophy, ed. Edward Zalta, URL = http://plato.stanford.edu/entries/propositions-singular Menzel, Christopher. 1991. The True Modal Logic, Journal of Philosophical Logic 20: 331-374. Plantinga, Alvin. 1983. On Existentialism, Philosophical Studies 44: 1-20. Turner, Jason. 2005. Strong and Weak Possibility, Philosophical Studies 125: 191-217. 12