Philosophical Studies (2005) 125: 191–217 DOI 10.1007/s11098-004-7812-3
Springer 2005
JASON TURNER
STRONG AND WEAK ...

Author:
Turner Jason

Philosophical Studies (2005) 125: 191–217 DOI 10.1007/s11098-004-7812-3

Springer 2005

JASON TURNER

STRONG AND WEAK POSSIBILITY

ABSTRACT. The thesis of existentialism holds that if a proposition p exists and predicates something of an object a, then in any world where a does not exist, p does not exist either. If ‘‘possibly, p’’ entails ‘‘in some possible world, the proposition that p exists and is true,’’ then existentialism is prima facie incompatible with the truth of claims like ‘‘possibly, the Eiﬀel Tower does not exist.’’ In order to avoid this claim, a distinction between two kinds of world-indexed truth – and two associated kinds of modality – is needed. This paper embodies an attempt to develop a full account of just such a distinction.

1.

PRELIMINARIES

My aim in this paper is very simple: to give an account of how a proposition could be true at (or of) a world without existing in that world. Following Alvin Plantinga (1974, p. 46; cf. van Inwagen, 1986, p. 190), I take it that, if p is a proposition and W a possible world, p is true in W if and only if, were W actual, p would be true, and ‘a exists in W ’ means that, if W were actual, a would exist.1 Furthermore, I accept the thesis that, necessarily,2 a proposition p is true only if it exists. From this it follows quite naturally that a proposition p is true in a possible world W if and only if, were W actual, p would exist and be true; hence, if p is true in a possible world W, p exists in that world. This is of particular concern to those who hold the thesis of existentialism: If a proposition p exists and predicates something of an object a, then in any possible world W, if a does not exist in W, then p also does not exist in W.3

(By ‘a does not exist in W ’ I mean that it is not the case that a exists in W.) Another way of stating this thesis is that there cannot be propositions which refer to (or are intuitively

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‘‘about’’) things that do not exist (Adams, 1981, p. 7). Since some objects exist only contingently, it follows that some propositions – speciﬁcally, the ones ‘‘about’’ these objects – also exist only contingently. While there is good motivation for this position (Adams, 1981, pp. 3–6; Fine, 1985, pp. 155–160), it runs up against problems rather quickly. For surely ð1Þ

Possibly, the Eiffel Tower4 does not exist.

Yet, on the standard interpretation of possibility claims, ð2Þ

Possibly, p iff, for some possible world W; p is true in W:

By our above considerations, (2) entails ð3Þ

Possibly, p iff for some possible world W; were W actual; p would exist and be true:

Finally, (1) and (3) imply ð4Þ

For some possible world W; were W actual; the proposition ‘‘the Eiffel Tower does not exist’’ would exist and be true.5

However, if the proposition is true in W, then presumably the Eiﬀel Tower does not exist in W. Thus, were W actual, a proposition p ‘‘about’’ an object a would exist, but a would not – a ﬂat contradiction of existentialism. Call the above argument the ‘‘possibility argument,’’ since it trades on our commitment to certain intuitively obvious possibility claims. Kit Fine (1985) attempts to circumvent the possibility argument with the following distinction: One should distinguish between two notions of truth for propositions, the inner and the outer. According to the outer notion, a proposition is true in a possible world regardless of whether it exists in that world; according to the inner notion, a proposition is true in a possible world only if it exists in that world. We may put the distinction in terms of perspective. According to the outer notion, we can stand outside a world and compare the proposition with what goes on in the world in order to ascertain whether it is true. But according to the inner notion, we must ﬁrst enter with the proposition into the world before ascertaining its truth. (1985, p. 163)

If Fine is correct, the existentialist need not fear the possibility argument. By distinguishing between two kinds of world-

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indexed truths, we can distinguish between two kinds of possibility. Clearly Fine’s ‘‘inner truth’’ corresponds to ‘‘truth in’’; a proposition p is true ‘‘according to the inner notion’’ with respect to a world6 W if and only if p is true in W. Following Robert Adams (1981, p. 22), I will use the locution ‘‘true at’’ to talk about ‘‘outer truth.’’ Thus, a proposition p is true ‘‘according to the outer notion’’ with respect to W if and only if p is true at W. We can now oﬀer two analyses of possibility: Strong Possibility: p is strongly possible ð}s pÞ iff, for some world W; p is true in W: Weak Possibility: p is weakly possible ð}w pÞ iff, for some world W; p is true at W:

(In a similar manner, p will be strongly necessary ((sp) iﬀ it is true in every world and weakly necessary ((wp) iﬀ it is true at every world.) The existentialist’s resistance to the possibility argument is now clear. ‘Possibly, p’ is systematically ambiguous between strong and weak possibility. Premise (2) is only true for a ‘‘strong’’ notion of possibility; the existentialist will insist that (1) is only true for a notion of ‘‘weak’’ possibility, escaping the argument while preserving our commitment to the apparent truth of (1) (Fine, 1985, p. 163). The in/at distinction has, to at least some minds, strong intuitive appeal. Others, however, ﬁnd it less than perspicuous (cf. Plantinga, 1985, pp. 342–344). After all, unless worlds are large concrete objects, it is not as though we can stand alongside of them as observers, taking notes and comparing them with others in order to discover what we can say about them. It is not enough to simply say that there is a property of being ‘‘true at’’ – an account of this notion needs to be given. It is to this task that the present paper is dedicated. It is not my purpose to defend the thesis of existentialism, except insofar as an account of weak possibility can serve in that defense. Since existentialism is an actualist theory – it tends to be motivated by the belief that there are no non-actual objects or ‘‘mere possibles’’ (Adams, 1981, p. 7) – I will assume the truth of actualism. I will also assume structuralism, the view that

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propositions are structured abstract entities (and that their internal structure captures the syntax of a logical language), and objectualism, which holds that structured propositions have individual objects as their constituents. If we are essentialists about our structuralism – that is, if structured propositions have their constituents essentially – then the theses of actualism, structuralism, and objectualism taken together entail existentialism (cf. Fine, 1980, pp. 160–162), although the converse is not true. It will be convenient to suppose, though, that (strongly) necessarily, propositions exist if and only if all their constituents exist. In other words, in any world W, p exists in W iﬀ each of p’s constituents exist in W. I will take this as deﬁnitive of the brand of existentialism used in this paper. It will also prove useful to assume that properties and relations are necessary entities and that there are no primitive haecceities.7 Since we are assuming actualism, possible worlds will have to be some sort of actualist-friendly entities. We have already ruled out conceptions of possible worlds out of unstructured, non-objectualist propositions, but there are other ways to be an actualist. One way is to simply build the propositions out of structured, objectual propositions, as suggested by Adams (1981, pp. 20–22)8 and an early Fine (1970, pp. 128 and 130– 139).9 Other approaches involve possible worlds that represent ‘‘pictorally’’ (by isomorphism) (cf. Lewis, 1986, pp. 165–174) or that take possible worlds and the truth-in relation as unanalyzed primitives (cf. Lewis, 1986, pp. 174–176; I also take Fine, 1985, pp. 180–183 as suggestive of something along these lines).10 In order to remain as ecumenical as I can about these issues, for the purposes of this paper I will leave possible worlds unanalyzed and take the truth-in relation as primitive; various actualists may ﬁll in the details as best pleases them.

2.

WORLD-DESCRIPTIONS

Let us begin by deﬁning the ‘‘world description’’ of W – intuitively, a proposition that exists in both the actual world and W and is, in some sense, a maximal description of W. A notion of

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entailment will be needed for this deﬁnition. Unfortunately, the traditional analysis: p entails q iff ‘‘(ðp qÞ’’ is true

will not help us here. The necessity of the conditional ‘‘p q’’ must be either weak or strong. We cannot make use of the weak version, however, since we aim to ultimately give a non-circular account of weak modal operators (cf. Crisp, 2002, pp. 41–42). On the other hand, it is not clear that a strongly necessary conditional will be equipped do the work we need. If a constituent of p is also a constituent of every proposition which is a logical construct involving p, then ‘‘(s(p q)’’ will only be true if p and q necessarily exist. For suppose that p was a contingently existing proposition; since it can only fail to exist by the failure of one of its constituents – say, a – to exist, there must be a world where both a and p fail to exist. But, by hypothesis, a is also a constituent of ‘‘p q,’’ so ‘‘p q’’ will fail to exist in that world also. (Similar remarks apply for q.) We might try instead deﬁning entailment as p entails q iff ‘‘ }s ðp & qÞ’’ is true.

Now the problem is that any proposition entails the existence of any object. The proposition that it is not the case that the Eiﬀel Tower exists is not strongly possible, so for any p, ‘‘p & ~(the Eiﬀel Tower exists’’ is not strongly possible. (I take it that strong possibility is closed under conjunctive elimination.) Since this is clearly unacceptable, we cannot deﬁne entailment in this manner. To avoid these problems, I will employ a notion of ‘‘provisional entailment.’’ To say ‘p provisionally entails q’ is to say that, for all worlds W, if p is true in W, then q is true in W. Thus p’s provisional entailment of q is determined by all p-worlds being q-worlds, even though p and q might not exist in all worlds. We are now prepared to deﬁne world-descriptions: (Df. w) A proposition w is the world description for a world W iﬀdf. w is a proposition and W a world such that (i) w is true in W, and (ii) for all propositions p, if w does not provisionally entail p, w & p is not true in W.

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The intuitive notion is that these world-descriptions oﬀer a ‘‘complete’’ description of a possible world W. It is complete in the sense that (a) it is true in W, and (b) there is nothing that can be added to it (which is not provisionally entailed by it) which would also be true in W. Now, if the in/at distinction is correct, there is still more to be said about W – to wit, those things that are true at W but not in W – so it is not a ‘‘complete’’ description in this sense. But it is as complete a description as can exist in both the actual world and W. There is nothing particularly exceptional about my deﬁnition of a world description; various authors have analyzed possible worlds in similar ways. The structures of these world-descriptions are more interesting. We can see what they would be like by considering how to ‘‘make’’ a world description w. Suppose that the actually existing objects are a1, …, an, and further that, were W actual, only a1, …, ai (where 1 £ i £ n) of these would exist. Suppose further that there would be some other objects, alien to the actual world, were W actual; call them b1, …, bm. Now let w* be a world description of W that exists in W. (The conjunction of all propositions that would exist and be true were W actual would do the job.) The proposition w* will be of the form ‘‘F(a1,…,ai, b1,…,bm)’’ for some formula ‘F.’ Now simply quantify into the b1,…,bm positions to get ‘‘$x1 … $xm(F(a1,…, ai, x1,…xm)).’’ This is our world description of W. Of course, given our existentialism, we could not do this – there is no proposition w* (since there are no b1,…,bm), so we cannot use it to construct any other proposition. But this is no matter; the foregoing is only meant to provide an intuitive picture of a systematic way of ‘‘getting at’’ these worlddescriptions. The propositions exist on their own, without ever having been constructed – provided w exists, it does not need the further existence of the w* that would end up making it true were W actual. It may be objected that world-descriptions somehow underdetermine a world. Suppose that w describes a world – say, W1 – exactly like the actual one but for two extra electrons. If W1 were actual, then there would be two more electrons, e1 and e2. Any diﬀerences in e1 and e2 (besides electron-hood) will be

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purely contingent.11 Call the conjunctions of the properties had by e1 and e2 ‘P’ and ‘Q,’ respectively. Suppose W1 were actual. Then ‘‘P(e1) & Q(e2)’’ would be true; but ‘‘Q(e1) & P(e2)’’ could have been true. Call this world – the one identical to W1 but for the inversion of e1’s properties with those of e2 – W2. The world-description w is apparently true in W2 also, for it will look something like: ‘‘…$x$y… (… & x is an electron & y is an electron & P(x) & Q(y) & x „ y & …),’’ and contain no other e1- and e2-relevant clauses. In this case, though, how could we say w describes just W1 – doesn’t it describe both W1 and W2? Although I readily agree that, were W1 actual, there would be another possible world W2 in which w would be true, I am less certain that there actually is such a W2 for our world description to describe. The only ‘‘diﬀerence’’ between W1 and W2 is a diﬀerence in the distribution of properties over two non-existing objects, e1 and e2. If asked how W1 and W2 diﬀer, someone might say, ‘‘Well, in W1, e1 has property P, but in W2 it is e2 that has P.’’ But given our actualism, there are no e1’s or e2’s to diﬀer in properties in W1 and W2. It is not as though W1 and W2 diﬀer by entering into diﬀerent relations with non-actual objects; there are no non-actual objects for them to enter into relations with. Someone might object that, since I have remained silent about the nature of possible worlds, I am in no position to say there are not two (or more) worlds like W1 and W2 that satisfy a single world description.12 This seems fair. But if there are two divergent possible worlds like W1 and W2, and they are in fact distinct, it seems appropriate that we should not be able to distinguish between them using only existing propositions. The best we can hope to do is give a description of a ‘‘type’’ of world rather than a description of a single world (Adams, 1981, p. 21). Anyone swayed by both actualism and existentialism should think that this is just as it should be. 3.

TRUTH AT A WORLD: NEGATIVE EXISTENCE

We are now in a position to analyze truth at a world. In the next two sections, I provide three conditions suﬃcient for a non-modal13 proposition’s being true at a world. In the sections

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following, I argue ﬁrst that these three taken disjunctively are necessary and then extend the account to modal propositions. We should ﬁrst note that, intuitively speaking, whatever truth at a world is, propositions true in W should also be true at W. I therefore take the following suﬃciency condition to be unimpeachable: ðC1Þ

If p is true in W; then p is true atW:

Since w is true in W, it follows quite simply from (C1) that w is true at W. Now, w gives us as complete a description as possible of all the objects that would exist were W actual. Furthermore, every object that is not a constituent of w is an object that would not exist were W actual.14 If we deﬁne an existence predicate ‘E!’ as follows: ðDf:E!Þ E!ðaÞ ¼df: 9xðx ¼ aÞ

(Lambert, 2001, p. 265), we can oﬀer the following suﬃciency condition for truth at a world: ðC2Þ

If p ¼ ‘‘ E!ðaÞ; ’’ w is W’s world description, and a is not a constituent of w; then p is true at W:

I suspect that (C2) is the most contentious suﬃciency condition I will propose. At ﬁrst glance, it will appear ad hoc and unmotivated. (C2) provides the resister of the possibility argument with exactly what he wants: the result that ‘‘The Eiﬀel Tower does not exist’’ is true at worlds with no Eiﬀel Tower. For clearly, if there is no Eiﬀel Tower in a world W, and w is the world description for W, the Eiﬀel Tower will not be a constituent of w. Yet there is strong intuitive support for (C2). We must not forget the basic picture we are working with: we want to ‘‘stand outside a world and compare the proposition with what goes on in the world in order to ascertain whether it is true’’ (Fine, 1985, p. 163). The objection is that this is just ‘‘picture thinking’’ (Plantinga, 1985, p. 343) – we need a positive account of how to ‘‘stand outside’’ of worlds and look into them. It seems highly plausible that this is exactly what we are doing when we examine the structure of a world description. We are using the tools and resources of our existing propositions to

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‘‘look into’’ a world without having to ‘‘stand inside’’ of it. And since our world-descriptions are maximal in an important sense, if we examine them and do not see a given object a – an object with which we are familiar, since it is actual – I cannot see how one could object to the conclusion that ‘‘a does not exist’’ is a proposition which accurately describes the world in question. One may further object that, even if propositions are structured and contain objects as constituents, there is no clear notion of ‘‘examining’’ a proposition to see if it contains a given object. How does one ‘‘look’’ at a proposition? Note, however, that the language of ‘‘examining’’ is metaphorical, and this objection is mainly epistemic. The notions involved in the ‘‘examination’’ of propositions are all purely logical: for any proposition p and object a, either a is a constituent of p or it is not. The epistemic problem is determining which a’s are constituents of which p’s. But I do not see how this is any more diﬃcult than, say, determining for some possible world W which propositions are true in it. The epistemic diﬃculty in the latter case is not taken as suﬃcient for showing that the notion of truth in a world is unintelligible. I do not see why similar diﬃculties in matching an object a with a proposition p should imply that there is anything untoward about (C2).

4.

TRUTH AT A WORLD: LOGICAL RELATIONS

Although we now have a notion of truth at a world that will help the existentialist escape the possibility argument, there is more to be done. A third suﬃciency condition is needed to account for other propositions which have as constituents objects that do not exist in a world. Taken together, (C1) and (C2) say nothing about the truth at non-Eiﬀel-Tower worlds of propositions like ð5Þ

‘‘The Eiffel Tower is green; ’’

ð6Þ

‘‘Something exists which is shorter than the Eiffel Tower; ’’

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and ð7Þ ‘‘There exists an x such that the Eiffel Tower does not exist and x is green:’’

It would be nice if we could somehow make the at-W truth value of these propositions parasitic on the truth of w and the various ‘‘~E!(a)’’s true at W. An enticing proposal is to say a proposition is true at a world if it follows logically from other propositions true at that world (Adams, 1981, pp. 23–25). What does it mean, though, for one proposition to ‘‘follow logically’’ from another? Logical implication is supposed to be a sentential relation; for proposition-to-proposition relations, we tend to talk about entailment. We saw in section 2 that traditional accounts of entailment cannot help us here. We will have to use some other propositional relation to move from the truth of w and the ‘‘~E!(a)’’s to propositions like (5)–(7). Recall that our propositions are entities with structure – structure which models their logical form. In predicate logic, a sentence ‘p’ is said to follow from another sentence ‘q’ if and only if every interpretation (i.e., function from terms to objects, properties, and relations) that renders ‘q’ true renders ‘p’ true also. Since our propositions have syntactic structure, we can provide interpretations for them as well; these would be functions from propositional elements to objects, properties and relations. (Depending on the form of our objectualism, propositional elements are objects, properties and relations; presumably a proposition p will be true on an interpretation if its propositional elements are mapped to other propositional elements which are constituents of a true proposition with the same structure as p.) I will say that one proposition is ‘‘semantically implied’’ by another if the former is true on every interpretation that renders the latter true. In like manner, I will say a proposition p ‘‘follows from’’ a proposition q in a language L if and only if p is true on every interpretation, under the interpretative rules of L, that renders q true. For illustration, suppose that propositions are ordered pairs consisting of properties, n-ary relations, objects, other propositions, and ordered n-tuples thereof. Thus, the proposition

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‘‘a is red’’ would be something like Æbeing red, aæ, and ‘‘Joe is between Jack and Mary’’ might be Æbetween, ÆJoe, Jack, Maryææ. Likewise, the conjunction ‘‘Jim is hungry and Jack is sick’’ is perhaps ÆCONJ, ÆÆbeing hungry, Jimæ, Æbeing sick, Jackæææ, and ‘‘Someone is tired’’ may similarly be ÆSOME x, Æbeing tired, xææ. Now let p be the proposition ÆCONJ, ÆÆbeing made of glass, the Eiﬀel Toweræ, Æbeing a philosopher, Plantingaæææ. Let q be Æbeing a philosopher, Plantingaæ. Clearly, p is false and q is true. But p semantically implies q, since, for every interpretation on which p is true, q is also true. (For example, take an interpretation that maps being made of glass onto being prime, being a philosopher onto being composite, the Eiﬀel Tower onto the number 2, and Plantinga onto the number 9. On this interpretation, p is true, as is q. Furthermore, it should be clear that there is no interpretation on which p is true and q false.)15 This notion of semantic implication captures the idea behind logical implication but allows us to work directly with propositions. Nonetheless, it will not do to simply say that propositions semantically implied (under the interpretation rules for predicate logic) by propositions true at a world are themselves true at a world. It seems uncontestable that ‘‘"x(x ¼ x)’’ is true in all possible worlds; it will therefore be true at all possible worlds by (C1). But then, by universal instantiation, ‘‘(the Eiﬀel Tower ¼ the Eiﬀel Tower)’’ will be true even at worlds where the Eiﬀel Tower does not exist, and by existential generalization, ‘‘$x(x ¼ the Eiﬀel Tower)’’ will also be true at these worlds (cf. Burge, 1974, p. 311).16 Both of these diﬃculties can be overcome if we help ourselves to a ‘‘negative free logic’’ (Adams, 1981, pp. 25–26 and Lambert, 2001, p. 260). A free logic is a language where the rules of existential generalization and universal instantiation are restricted to avoid paradoxes like the ones above while allowing for non-referring terms. Speciﬁcally, if ‘t’ is a term of the language, one cannot instantiate a universal claim with ‘t’ or generalize to an existential claim from ‘t’ without having ‘E!(t)’ Since the Eiﬀel Tower does not exist at W, if we make semantic implication comply with free logic interpretative rules,

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both steps in the inference from ‘‘"x(x ¼ x)’’ to ‘‘$x(x ¼ the Eiﬀel Tower)’’ are blocked. A negative free logic is a free logic where all n-ary atomic predications containing a term ‘t’ are false if ‘~E!(t)’ is true. This feature accords with a fairly common intuition that sentences about nonexistents are false (Burge, 1974, pp. 309–311, 317–332; Adams, 1981, pp. 23–25).17 We will use this negative free logic in our next suﬃciency condition: (C3)

If a proposition p follows in a negative free logic from propositions true at W; then p is true at w:

Taken together, (C1), (C2), and (C3) ensure that (5) and (6) are false at worlds where the Eiﬀel Tower does not exist, and (7) is true at those worlds if they contain some green item. It may be found surprising that we have to resort to a free logic – a logic designed to make sense of non-referring terms and even allow for intelligible inferences within a Meinongian metaphysic – in order to make sense of a view that springs from a strict actualism. But there is good reason that this should be so. The perspective aﬀorded by truth-at-a-world semantics is one where we talk about other possible worlds, but make use of the things to hand in our own world in order to do so. Some of the things to hand are objects that don’t exist in the world being talked about. This makes them mere ‘‘possibilia’’ at that world. When we enter into the at-a-world discourse, we enter into a realm where we can talk at least partially like Meinongians or possibilists about a given world; this is because we have access to things in our own world which are the nonexistents of the world being talked about. Thus speaking at-a-world allows us to have our cake and eat it too: we get the expressive power of the possibilist or the Meinongian – the ability to speak about things that do not exist in the world in question – without any of the ontological commitment. 5.

NECESSARY CONDITIONS

In section 3, I announced that I would argue that (C1)–(C3), taken disjunctively, would serve as necessary conditions for a

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non-modal proposition’s being true at a world. Why not for the modal ones? Some modal propositions may be intuitively true at a world and yet not follow from any of (C1)–(C3). Suppose, for instance, that a exists in the actual world and has the property expressed by the predicate ‘P,’ but a does not exist in W. It seems, in this case, that ‘‘}$x(P(x))’’18 will be true in W. (Since ‘‘$x(P(x))’’ is true, it should be true in the actual world; from the perspective of W, it will still be true in the actual world (since it exists in W it can have that property there) and by W ’s application of (C1), true at the actual world as well.) When we are speaking ‘‘at’’ W, though, we can make use of a, and so we should want to say that ‘‘}P(a)’’ is true at W. This does not follow from (C1), (C2), or (C3). Nonetheless, for all non-modal propositions p, if p is true at W, then p satisﬁes (C1), (C2), or (C3). I will argue for this by showing that for every non-modal p, either p or its negation is true at W by one of (C1)–(C3), but I will need to assume that for all worlds W and non-modal propositions p, if p exists in W, either p or ~p is true in W.19 Suppose now that p is a non-modal proposition. If p or its negation is true in W, then by (C1) it or its negation will be true at W as well. So, suppose that neither p nor its negation is true in W; then (since all non-modal propositions are either true or false in W) p must not exist in W. Recall that we are assuming that the only way a proposition can fail to exist is by having a constituent that fails to exist. Thus, if p does not exist in W, then at least one of its constituents, say a, must not exist in W. So, by (C2), ‘‘~E!(a)’’ is true at W. Now, p is either atomic or non-atomic. If p is atomic, then ~p is true by (C3) (and the truth of ‘‘~E!(a)’’); if it is non-atomic, then an appropriate induction (see Appendix A) suﬃces to show that either p or ~p is true at W. 6.

MODAL PROPOSITIONS

That is the story for non-modal propositions. What about the modal ones, though? Suppose a does not exist in W1 but does exist in W2, where it has the property expressed by ‘‘P.’’ What is the status of ‘‘}P(a)’’ at W1?

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I suggested above that ‘‘}P(a)’’ should be true at W1. Adams disagrees. He writes Suppose a is the actual world premiere of Beethoven’s ninth symphony, and [W1] is a possible world in which neither a nor I would exist. Then … at [W1] it would still be true that a could have been a musical performance and I could not. This diﬀerence between a and me at [W1] could hardly have been determined by our (common) non-existence there plus some propositions that are not about us. To suppose such a diﬀerence at [W1] between two individuals that would not exist in [W1] seems at least uncomfortably close to ascribing properties at [W1] to individuals that would not exist in [W1] (1981, p. 32).

This concern seems to have two parts. The ﬁrst part worries that there are insuﬃcient non-modal propositions true at W1 to ground the truth of ‘‘}P(a).’’ The second part suggests that ‘‘}P(a)’’ illegitimately ascribes a property (being possibly P) to an object (a) in a world (W1) where the object does not exist. I will deal with the latter concern ﬁrst. While it would be true at W1 that a could have been a musical performance and Adams could not have been, this is not the same as claiming that, at W1, a has the property of possibly being a musical performance and Adams has the property of not possibly being a musical performance. To say otherwise is to confuse the de dicto ‘‘}(a has property: P)’’ with the de re ‘‘a has property: possibly P.’’ The former says something about the proposition, whereas the latter makes a claim about the object. The ﬁrst part of the worry stems from the claim that ‘‘what is true about an individual a at a world in which a does not exist must be determined by a’s non-existence there together with propositions, true at that world, that are not about a’’ (Adams, 1981, p. 32, emphasis altered). My response to this claim will depend on what it means for a proposition to be ‘‘about’’ a. If p’s being ‘‘about’’ a is equivalent to its predicating some particular property of a, then this worry collapses into the one above. Since in some important sense, ‘‘}P(a)’’ isn’t about a at all, but about ‘‘P(a),’’ this concern is misguided. A proposition p’s being ‘‘about’’ a might be less direct. After all, there is an intuitive sense in which ‘‘}P(a)’’ is ‘‘about’’ a, at

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least in a way that it is not about b or the Eiﬀel Tower or any other object we may be concerned with. On a slightly weaker ‘‘about’’ relation, is it right to say that propositions about a true at a world W in which a does not exist must be wholly determined by ‘‘~E!(a)’’ and other propositions, not about a, true at W? I cannot see why. 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 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. There is a simple way of making this picture precise. We merely add a fourth suﬃciency condition to the list: ðC4Þ

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:20

This condition allows us to show (with the help of a few assumptions) bivalence for non-quantiﬁed modal propositions (see Appendix B). It has the added bonus of making S5 the propositional logic of weak modality, for it directly entails the validity of the characteristic S5 axiom for weak modal operators.21 A problem remains. Although I have argued ‘‘}P(a)’’ does not predicate anything of a, it does seem to predicate something of the proposition ‘‘P(a).’’ On at least one view of modal operators, they are ‘‘predicate-like’’: ‘}’ and ‘(’, in essence, predicate of propositions that they are possible or necessary, respectively (cf. Bealer, 1993, pp. 7–16). This is problematic. If

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a is a constituent of p and a does not exist in W, then existentialists hold that p also does not exist in W. Suppose that p is possible, though. Then, by (C4), ‘‘}p’’ will be true at W. But if ‘}’ is predicate-like, (C2) should entail that ‘‘~}p’’ is true at W.22 Contradictions are therefore true at W. The foregoing overstates the problem, but only slightly. It is not clear that the non-existence of p at W will entail the truth of ‘‘~E!(p)’’ at W, since it is not clear how to formalize the logical system intended to capture the structure of propositions. (Speciﬁcally, it is not clear whether a single quantiﬁer will range over both propositions and ﬁrst-order ‘‘objects,’’ so it is not clear whether (df. E!) will deﬁne a predicate that can be applied to propositions.) Nonetheless, the problem remains. The intuition behind using a negative free logic is the intuition that objects that do not exist at a world should not have any properties at that world either. If its being true, at a world, that, p is possible is tantamount to p’s having the property of being possible at that world, then propositions which do not exist at a world should not be possible there either. I can see two ways to escape this diﬃculty. The ﬁrst is the rejection of the claim that intensional operators (e.g., modal ones) are predicate-like in the required way. One may, for instance, adopt a ‘‘prosentential’’ position in which propositions do not count as singular terms (cf. Grover et al., 1975). I would like to be able to agree that modal operators are predicate-like, though, and there are diﬃculties with adopting a prosentential approach that I would rather avoid (see Bealer, 1993, fn. 8). This brings us to the second way of avoiding the problem. The intuition that drove our acceptance of a negative free logic when formulating (C3) was similar to Adams’ intuition that a proposition p’s at-W truth value is dependent only on the existence or non-existence of the objects p is ‘‘about’’ and other things true at W. Since it seems that an object’s properties in a world require that object’s existence, if an object does not exist at W it should not have properties there either. We have already faulted this reasoning when the realm of ‘‘objects’’ is extended to include propositions. When we are standing outside of a world, we have all the propositions that

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exist there with us, so to speak, and we can see what makes them true by looking at the entire space of possible worlds. We might say that all propositions that exist, exist at every world – although we still aﬃrm, in keeping with our existentialism, that many of these propositions do not exist in many of these worlds. The solution is therefore to revise our formulation of (C3). Rather than requiring a negative free logic, we should use the interpretative rules of a positive free logic. In a positive free logic, some propositions with non-referring terms turn out true (cf. Antonelli, 2000, p. 278; Lambert, 2001, p. 260). We must be careful, though. If we accept the intuition that non-propositional objects cannot have properties at worlds where they do not exist (see note 17), we will want to ensure that the only predications with non-referring terms that ever turn out true are of the form ‘‘p is necessary’’ or ‘‘p is possible.’’ We can do this by ensuring the interpretative rules conform with the following second-order axiom: ðAÞ

8x8Fðð E!ðxÞ f& FðxÞÞ ððF ¼ is necessaryÞ _ ðF ¼ is possibleÞÞÞ:

Thus no non-propositional objects have properties at worlds in which they do not exist (i.e., there are no true propositions that predicate properties of them at those worlds), but propositions can still be possible or impossible at worlds where they do not exist.23 Ontologically innocent semantics for positive free logics are possible (see Antonelli, 2000). Development of a modal positive free logic with the required expressive ability (including the required axiom) is beyond the scope of this paper. Nonetheless, barring the impossibility of such a logic, we have reason to believe (C4) can successfully account for the truth of modal propositions at worlds where they do not exist. 7.

PARTING REMARKS

One ﬁnal point is in order. After all the foregoing, someone may still insist that they do not understand what this notion of weak possibility is, or what it could mean for a proposition to be true at a world without being true in that world. While I think my account is intuitively plausible, I do not think that it is

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intuitively obvious; someone may, without ﬁnding fault with the speciﬁcs of my analysis, claim that they still ﬁnd truth at a world unintelligible. But what does such an objection amount to? We have given a clear deﬁnition of a relation, R, from propositions to worlds: a proposition p bears R to a world W if and only if it satisﬁes one of (C1), (C2), (C3), or (C4).24 Furthermore, the deﬁnition makes neither explicit nor covert appeal to a prior notion of weak possibility. I do not know what more could be asked of such an account. If anyone remains confused about truth at a world, I would simply say that a proposition is true at a world if and only if it bears R to that world. The objection, however, may be somewhat diﬀerent; perhaps one understands R, but does not see what a proposition’s bearing R to a word has to do with its being true at that world. If this is the complaint, I wonder what conditions a relation from propositions to worlds must satisfy in order to stand a chance of successfully playing the truth-at-a-world role. If the answer is ‘‘none – truth at a world simply makes no sense, so no relation can ever serve,’’ the objection seems little more than a dogmatic assertion that truth at a world cannot make sense, and no proposed analysis will be countenanced. On the other hand, if some list of desiderata is given, then I suggest that my interlocutor has at least a minimal understanding of truth at a world. What we must then do is ﬁnd a relation which satisﬁes all these conditions. Of course, it is possible that, given some (hitherto unspeciﬁed) set of conditions, nothing will be able to satisfy them all. If the conditions truly do capture the notion that we’re trying to get at, and they cannot be jointly satisﬁed, then truth at a world is indeed doomed. But if this is so, I should like to see why. Until then, I can see no reason for rejecting the general notion of truth at a world or the account of it given here.25 APPENDIX A

Claim: If p is a non-atomic, non-modal proposition which does not exist in W, then either p or ~p is true at W by (C1), (C2), or (C3).

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Assume (for the sake of simplicity) that the sentences of a ﬁrst-order negative free logic capture the syntactic structure of propositions. Then we need only concern ourselves to a set of primitive operators in the language, say, ‘‘~,’’ ‘‘&,’’ ‘‘",’’ and ‘‘=.’’ If p contains deﬁned operators, then there will be a proposition q logically equivalent to p that contains only primitive operators; by (C3), p will have the same at-W truth value as q. So without loss of generality suppose p has only primitive operators. Assume further that propositional constituents that play the logical operator role (if such there be) exist necessarily and that constituents of p are constituents of every logical construct involving p. I will prove the claim by induction on the number n of logical operators in p: Case n ¼ 1: Case 1: p ¼ ~q for some q. Since p has only one operator (n ¼ 1), q is atomic. Since p does not exist in W, some constituent a of p does not exist in W; hence, some constituent a of the atomic q does not exist in W, and so by (C2) and (C3), ~q is true in W. Case 2: p ¼ q & r for some q and r. Since n ¼ 1, q and r are atomic. Since p does not exist in W, some constituent a of either q or r does not exist in W. Suppose (without loss of generality) that a is a constituent of q; then ~q is true at W by (C2) and (C3). But, if ~q is true at W, then ~(q & r) is true at W by (C3); hence, ~p is true at W. Case 3: p ¼ ‘‘"xF(…x…)’’ for some formula ‘F.’ Since n ¼ 1, if b is any object that exists in W, ‘‘F(…b…)’’ is an atomic proposition. Since p does not exist in W, some constituent a of ‘‘F(…b…)’’ does not exist in W. So, ‘‘F(…b…)’’ has a constituent, a, which does not exist in W. Hence, ‘‘~F(…b…)’’ is true at W by (C2) and (C3). Therefore, ‘‘~"xF(…x…)’’ ¼ ~p is true at W by (C3). Case 4: p ¼ ‘‘a ¼ b’’ for some objects a and b. Since p does not exist in W, either a or b does not exist in W. Hence, by (C2) and (C3), ~p is true at W. Case n > 1. Inductive hypothesis: Suppose that, if q is a proposition with m < n operators, either q or ~q is true at W.

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Case 1: p ¼ ~q for some q. Since p has n operators, q has n ) 1 (

Springer 2005

JASON TURNER

STRONG AND WEAK POSSIBILITY

ABSTRACT. The thesis of existentialism holds that if a proposition p exists and predicates something of an object a, then in any world where a does not exist, p does not exist either. If ‘‘possibly, p’’ entails ‘‘in some possible world, the proposition that p exists and is true,’’ then existentialism is prima facie incompatible with the truth of claims like ‘‘possibly, the Eiﬀel Tower does not exist.’’ In order to avoid this claim, a distinction between two kinds of world-indexed truth – and two associated kinds of modality – is needed. This paper embodies an attempt to develop a full account of just such a distinction.

1.

PRELIMINARIES

My aim in this paper is very simple: to give an account of how a proposition could be true at (or of) a world without existing in that world. Following Alvin Plantinga (1974, p. 46; cf. van Inwagen, 1986, p. 190), I take it that, if p is a proposition and W a possible world, p is true in W if and only if, were W actual, p would be true, and ‘a exists in W ’ means that, if W were actual, a would exist.1 Furthermore, I accept the thesis that, necessarily,2 a proposition p is true only if it exists. From this it follows quite naturally that a proposition p is true in a possible world W if and only if, were W actual, p would exist and be true; hence, if p is true in a possible world W, p exists in that world. This is of particular concern to those who hold the thesis of existentialism: If a proposition p exists and predicates something of an object a, then in any possible world W, if a does not exist in W, then p also does not exist in W.3

(By ‘a does not exist in W ’ I mean that it is not the case that a exists in W.) Another way of stating this thesis is that there cannot be propositions which refer to (or are intuitively

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‘‘about’’) things that do not exist (Adams, 1981, p. 7). Since some objects exist only contingently, it follows that some propositions – speciﬁcally, the ones ‘‘about’’ these objects – also exist only contingently. While there is good motivation for this position (Adams, 1981, pp. 3–6; Fine, 1985, pp. 155–160), it runs up against problems rather quickly. For surely ð1Þ

Possibly, the Eiffel Tower4 does not exist.

Yet, on the standard interpretation of possibility claims, ð2Þ

Possibly, p iff, for some possible world W; p is true in W:

By our above considerations, (2) entails ð3Þ

Possibly, p iff for some possible world W; were W actual; p would exist and be true:

Finally, (1) and (3) imply ð4Þ

For some possible world W; were W actual; the proposition ‘‘the Eiffel Tower does not exist’’ would exist and be true.5

However, if the proposition is true in W, then presumably the Eiﬀel Tower does not exist in W. Thus, were W actual, a proposition p ‘‘about’’ an object a would exist, but a would not – a ﬂat contradiction of existentialism. Call the above argument the ‘‘possibility argument,’’ since it trades on our commitment to certain intuitively obvious possibility claims. Kit Fine (1985) attempts to circumvent the possibility argument with the following distinction: One should distinguish between two notions of truth for propositions, the inner and the outer. According to the outer notion, a proposition is true in a possible world regardless of whether it exists in that world; according to the inner notion, a proposition is true in a possible world only if it exists in that world. We may put the distinction in terms of perspective. According to the outer notion, we can stand outside a world and compare the proposition with what goes on in the world in order to ascertain whether it is true. But according to the inner notion, we must ﬁrst enter with the proposition into the world before ascertaining its truth. (1985, p. 163)

If Fine is correct, the existentialist need not fear the possibility argument. By distinguishing between two kinds of world-

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indexed truths, we can distinguish between two kinds of possibility. Clearly Fine’s ‘‘inner truth’’ corresponds to ‘‘truth in’’; a proposition p is true ‘‘according to the inner notion’’ with respect to a world6 W if and only if p is true in W. Following Robert Adams (1981, p. 22), I will use the locution ‘‘true at’’ to talk about ‘‘outer truth.’’ Thus, a proposition p is true ‘‘according to the outer notion’’ with respect to W if and only if p is true at W. We can now oﬀer two analyses of possibility: Strong Possibility: p is strongly possible ð}s pÞ iff, for some world W; p is true in W: Weak Possibility: p is weakly possible ð}w pÞ iff, for some world W; p is true at W:

(In a similar manner, p will be strongly necessary ((sp) iﬀ it is true in every world and weakly necessary ((wp) iﬀ it is true at every world.) The existentialist’s resistance to the possibility argument is now clear. ‘Possibly, p’ is systematically ambiguous between strong and weak possibility. Premise (2) is only true for a ‘‘strong’’ notion of possibility; the existentialist will insist that (1) is only true for a notion of ‘‘weak’’ possibility, escaping the argument while preserving our commitment to the apparent truth of (1) (Fine, 1985, p. 163). The in/at distinction has, to at least some minds, strong intuitive appeal. Others, however, ﬁnd it less than perspicuous (cf. Plantinga, 1985, pp. 342–344). After all, unless worlds are large concrete objects, it is not as though we can stand alongside of them as observers, taking notes and comparing them with others in order to discover what we can say about them. It is not enough to simply say that there is a property of being ‘‘true at’’ – an account of this notion needs to be given. It is to this task that the present paper is dedicated. It is not my purpose to defend the thesis of existentialism, except insofar as an account of weak possibility can serve in that defense. Since existentialism is an actualist theory – it tends to be motivated by the belief that there are no non-actual objects or ‘‘mere possibles’’ (Adams, 1981, p. 7) – I will assume the truth of actualism. I will also assume structuralism, the view that

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propositions are structured abstract entities (and that their internal structure captures the syntax of a logical language), and objectualism, which holds that structured propositions have individual objects as their constituents. If we are essentialists about our structuralism – that is, if structured propositions have their constituents essentially – then the theses of actualism, structuralism, and objectualism taken together entail existentialism (cf. Fine, 1980, pp. 160–162), although the converse is not true. It will be convenient to suppose, though, that (strongly) necessarily, propositions exist if and only if all their constituents exist. In other words, in any world W, p exists in W iﬀ each of p’s constituents exist in W. I will take this as deﬁnitive of the brand of existentialism used in this paper. It will also prove useful to assume that properties and relations are necessary entities and that there are no primitive haecceities.7 Since we are assuming actualism, possible worlds will have to be some sort of actualist-friendly entities. We have already ruled out conceptions of possible worlds out of unstructured, non-objectualist propositions, but there are other ways to be an actualist. One way is to simply build the propositions out of structured, objectual propositions, as suggested by Adams (1981, pp. 20–22)8 and an early Fine (1970, pp. 128 and 130– 139).9 Other approaches involve possible worlds that represent ‘‘pictorally’’ (by isomorphism) (cf. Lewis, 1986, pp. 165–174) or that take possible worlds and the truth-in relation as unanalyzed primitives (cf. Lewis, 1986, pp. 174–176; I also take Fine, 1985, pp. 180–183 as suggestive of something along these lines).10 In order to remain as ecumenical as I can about these issues, for the purposes of this paper I will leave possible worlds unanalyzed and take the truth-in relation as primitive; various actualists may ﬁll in the details as best pleases them.

2.

WORLD-DESCRIPTIONS

Let us begin by deﬁning the ‘‘world description’’ of W – intuitively, a proposition that exists in both the actual world and W and is, in some sense, a maximal description of W. A notion of

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entailment will be needed for this deﬁnition. Unfortunately, the traditional analysis: p entails q iff ‘‘(ðp qÞ’’ is true

will not help us here. The necessity of the conditional ‘‘p q’’ must be either weak or strong. We cannot make use of the weak version, however, since we aim to ultimately give a non-circular account of weak modal operators (cf. Crisp, 2002, pp. 41–42). On the other hand, it is not clear that a strongly necessary conditional will be equipped do the work we need. If a constituent of p is also a constituent of every proposition which is a logical construct involving p, then ‘‘(s(p q)’’ will only be true if p and q necessarily exist. For suppose that p was a contingently existing proposition; since it can only fail to exist by the failure of one of its constituents – say, a – to exist, there must be a world where both a and p fail to exist. But, by hypothesis, a is also a constituent of ‘‘p q,’’ so ‘‘p q’’ will fail to exist in that world also. (Similar remarks apply for q.) We might try instead deﬁning entailment as p entails q iff ‘‘ }s ðp & qÞ’’ is true.

Now the problem is that any proposition entails the existence of any object. The proposition that it is not the case that the Eiﬀel Tower exists is not strongly possible, so for any p, ‘‘p & ~(the Eiﬀel Tower exists’’ is not strongly possible. (I take it that strong possibility is closed under conjunctive elimination.) Since this is clearly unacceptable, we cannot deﬁne entailment in this manner. To avoid these problems, I will employ a notion of ‘‘provisional entailment.’’ To say ‘p provisionally entails q’ is to say that, for all worlds W, if p is true in W, then q is true in W. Thus p’s provisional entailment of q is determined by all p-worlds being q-worlds, even though p and q might not exist in all worlds. We are now prepared to deﬁne world-descriptions: (Df. w) A proposition w is the world description for a world W iﬀdf. w is a proposition and W a world such that (i) w is true in W, and (ii) for all propositions p, if w does not provisionally entail p, w & p is not true in W.

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The intuitive notion is that these world-descriptions oﬀer a ‘‘complete’’ description of a possible world W. It is complete in the sense that (a) it is true in W, and (b) there is nothing that can be added to it (which is not provisionally entailed by it) which would also be true in W. Now, if the in/at distinction is correct, there is still more to be said about W – to wit, those things that are true at W but not in W – so it is not a ‘‘complete’’ description in this sense. But it is as complete a description as can exist in both the actual world and W. There is nothing particularly exceptional about my deﬁnition of a world description; various authors have analyzed possible worlds in similar ways. The structures of these world-descriptions are more interesting. We can see what they would be like by considering how to ‘‘make’’ a world description w. Suppose that the actually existing objects are a1, …, an, and further that, were W actual, only a1, …, ai (where 1 £ i £ n) of these would exist. Suppose further that there would be some other objects, alien to the actual world, were W actual; call them b1, …, bm. Now let w* be a world description of W that exists in W. (The conjunction of all propositions that would exist and be true were W actual would do the job.) The proposition w* will be of the form ‘‘F(a1,…,ai, b1,…,bm)’’ for some formula ‘F.’ Now simply quantify into the b1,…,bm positions to get ‘‘$x1 … $xm(F(a1,…, ai, x1,…xm)).’’ This is our world description of W. Of course, given our existentialism, we could not do this – there is no proposition w* (since there are no b1,…,bm), so we cannot use it to construct any other proposition. But this is no matter; the foregoing is only meant to provide an intuitive picture of a systematic way of ‘‘getting at’’ these worlddescriptions. The propositions exist on their own, without ever having been constructed – provided w exists, it does not need the further existence of the w* that would end up making it true were W actual. It may be objected that world-descriptions somehow underdetermine a world. Suppose that w describes a world – say, W1 – exactly like the actual one but for two extra electrons. If W1 were actual, then there would be two more electrons, e1 and e2. Any diﬀerences in e1 and e2 (besides electron-hood) will be

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purely contingent.11 Call the conjunctions of the properties had by e1 and e2 ‘P’ and ‘Q,’ respectively. Suppose W1 were actual. Then ‘‘P(e1) & Q(e2)’’ would be true; but ‘‘Q(e1) & P(e2)’’ could have been true. Call this world – the one identical to W1 but for the inversion of e1’s properties with those of e2 – W2. The world-description w is apparently true in W2 also, for it will look something like: ‘‘…$x$y… (… & x is an electron & y is an electron & P(x) & Q(y) & x „ y & …),’’ and contain no other e1- and e2-relevant clauses. In this case, though, how could we say w describes just W1 – doesn’t it describe both W1 and W2? Although I readily agree that, were W1 actual, there would be another possible world W2 in which w would be true, I am less certain that there actually is such a W2 for our world description to describe. The only ‘‘diﬀerence’’ between W1 and W2 is a diﬀerence in the distribution of properties over two non-existing objects, e1 and e2. If asked how W1 and W2 diﬀer, someone might say, ‘‘Well, in W1, e1 has property P, but in W2 it is e2 that has P.’’ But given our actualism, there are no e1’s or e2’s to diﬀer in properties in W1 and W2. It is not as though W1 and W2 diﬀer by entering into diﬀerent relations with non-actual objects; there are no non-actual objects for them to enter into relations with. Someone might object that, since I have remained silent about the nature of possible worlds, I am in no position to say there are not two (or more) worlds like W1 and W2 that satisfy a single world description.12 This seems fair. But if there are two divergent possible worlds like W1 and W2, and they are in fact distinct, it seems appropriate that we should not be able to distinguish between them using only existing propositions. The best we can hope to do is give a description of a ‘‘type’’ of world rather than a description of a single world (Adams, 1981, p. 21). Anyone swayed by both actualism and existentialism should think that this is just as it should be. 3.

TRUTH AT A WORLD: NEGATIVE EXISTENCE

We are now in a position to analyze truth at a world. In the next two sections, I provide three conditions suﬃcient for a non-modal13 proposition’s being true at a world. In the sections

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following, I argue ﬁrst that these three taken disjunctively are necessary and then extend the account to modal propositions. We should ﬁrst note that, intuitively speaking, whatever truth at a world is, propositions true in W should also be true at W. I therefore take the following suﬃciency condition to be unimpeachable: ðC1Þ

If p is true in W; then p is true atW:

Since w is true in W, it follows quite simply from (C1) that w is true at W. Now, w gives us as complete a description as possible of all the objects that would exist were W actual. Furthermore, every object that is not a constituent of w is an object that would not exist were W actual.14 If we deﬁne an existence predicate ‘E!’ as follows: ðDf:E!Þ E!ðaÞ ¼df: 9xðx ¼ aÞ

(Lambert, 2001, p. 265), we can oﬀer the following suﬃciency condition for truth at a world: ðC2Þ

If p ¼ ‘‘ E!ðaÞ; ’’ w is W’s world description, and a is not a constituent of w; then p is true at W:

I suspect that (C2) is the most contentious suﬃciency condition I will propose. At ﬁrst glance, it will appear ad hoc and unmotivated. (C2) provides the resister of the possibility argument with exactly what he wants: the result that ‘‘The Eiﬀel Tower does not exist’’ is true at worlds with no Eiﬀel Tower. For clearly, if there is no Eiﬀel Tower in a world W, and w is the world description for W, the Eiﬀel Tower will not be a constituent of w. Yet there is strong intuitive support for (C2). We must not forget the basic picture we are working with: we want to ‘‘stand outside a world and compare the proposition with what goes on in the world in order to ascertain whether it is true’’ (Fine, 1985, p. 163). The objection is that this is just ‘‘picture thinking’’ (Plantinga, 1985, p. 343) – we need a positive account of how to ‘‘stand outside’’ of worlds and look into them. It seems highly plausible that this is exactly what we are doing when we examine the structure of a world description. We are using the tools and resources of our existing propositions to

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‘‘look into’’ a world without having to ‘‘stand inside’’ of it. And since our world-descriptions are maximal in an important sense, if we examine them and do not see a given object a – an object with which we are familiar, since it is actual – I cannot see how one could object to the conclusion that ‘‘a does not exist’’ is a proposition which accurately describes the world in question. One may further object that, even if propositions are structured and contain objects as constituents, there is no clear notion of ‘‘examining’’ a proposition to see if it contains a given object. How does one ‘‘look’’ at a proposition? Note, however, that the language of ‘‘examining’’ is metaphorical, and this objection is mainly epistemic. The notions involved in the ‘‘examination’’ of propositions are all purely logical: for any proposition p and object a, either a is a constituent of p or it is not. The epistemic problem is determining which a’s are constituents of which p’s. But I do not see how this is any more diﬃcult than, say, determining for some possible world W which propositions are true in it. The epistemic diﬃculty in the latter case is not taken as suﬃcient for showing that the notion of truth in a world is unintelligible. I do not see why similar diﬃculties in matching an object a with a proposition p should imply that there is anything untoward about (C2).

4.

TRUTH AT A WORLD: LOGICAL RELATIONS

Although we now have a notion of truth at a world that will help the existentialist escape the possibility argument, there is more to be done. A third suﬃciency condition is needed to account for other propositions which have as constituents objects that do not exist in a world. Taken together, (C1) and (C2) say nothing about the truth at non-Eiﬀel-Tower worlds of propositions like ð5Þ

‘‘The Eiffel Tower is green; ’’

ð6Þ

‘‘Something exists which is shorter than the Eiffel Tower; ’’

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and ð7Þ ‘‘There exists an x such that the Eiffel Tower does not exist and x is green:’’

It would be nice if we could somehow make the at-W truth value of these propositions parasitic on the truth of w and the various ‘‘~E!(a)’’s true at W. An enticing proposal is to say a proposition is true at a world if it follows logically from other propositions true at that world (Adams, 1981, pp. 23–25). What does it mean, though, for one proposition to ‘‘follow logically’’ from another? Logical implication is supposed to be a sentential relation; for proposition-to-proposition relations, we tend to talk about entailment. We saw in section 2 that traditional accounts of entailment cannot help us here. We will have to use some other propositional relation to move from the truth of w and the ‘‘~E!(a)’’s to propositions like (5)–(7). Recall that our propositions are entities with structure – structure which models their logical form. In predicate logic, a sentence ‘p’ is said to follow from another sentence ‘q’ if and only if every interpretation (i.e., function from terms to objects, properties, and relations) that renders ‘q’ true renders ‘p’ true also. Since our propositions have syntactic structure, we can provide interpretations for them as well; these would be functions from propositional elements to objects, properties and relations. (Depending on the form of our objectualism, propositional elements are objects, properties and relations; presumably a proposition p will be true on an interpretation if its propositional elements are mapped to other propositional elements which are constituents of a true proposition with the same structure as p.) I will say that one proposition is ‘‘semantically implied’’ by another if the former is true on every interpretation that renders the latter true. In like manner, I will say a proposition p ‘‘follows from’’ a proposition q in a language L if and only if p is true on every interpretation, under the interpretative rules of L, that renders q true. For illustration, suppose that propositions are ordered pairs consisting of properties, n-ary relations, objects, other propositions, and ordered n-tuples thereof. Thus, the proposition

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‘‘a is red’’ would be something like Æbeing red, aæ, and ‘‘Joe is between Jack and Mary’’ might be Æbetween, ÆJoe, Jack, Maryææ. Likewise, the conjunction ‘‘Jim is hungry and Jack is sick’’ is perhaps ÆCONJ, ÆÆbeing hungry, Jimæ, Æbeing sick, Jackæææ, and ‘‘Someone is tired’’ may similarly be ÆSOME x, Æbeing tired, xææ. Now let p be the proposition ÆCONJ, ÆÆbeing made of glass, the Eiﬀel Toweræ, Æbeing a philosopher, Plantingaæææ. Let q be Æbeing a philosopher, Plantingaæ. Clearly, p is false and q is true. But p semantically implies q, since, for every interpretation on which p is true, q is also true. (For example, take an interpretation that maps being made of glass onto being prime, being a philosopher onto being composite, the Eiﬀel Tower onto the number 2, and Plantinga onto the number 9. On this interpretation, p is true, as is q. Furthermore, it should be clear that there is no interpretation on which p is true and q false.)15 This notion of semantic implication captures the idea behind logical implication but allows us to work directly with propositions. Nonetheless, it will not do to simply say that propositions semantically implied (under the interpretation rules for predicate logic) by propositions true at a world are themselves true at a world. It seems uncontestable that ‘‘"x(x ¼ x)’’ is true in all possible worlds; it will therefore be true at all possible worlds by (C1). But then, by universal instantiation, ‘‘(the Eiﬀel Tower ¼ the Eiﬀel Tower)’’ will be true even at worlds where the Eiﬀel Tower does not exist, and by existential generalization, ‘‘$x(x ¼ the Eiﬀel Tower)’’ will also be true at these worlds (cf. Burge, 1974, p. 311).16 Both of these diﬃculties can be overcome if we help ourselves to a ‘‘negative free logic’’ (Adams, 1981, pp. 25–26 and Lambert, 2001, p. 260). A free logic is a language where the rules of existential generalization and universal instantiation are restricted to avoid paradoxes like the ones above while allowing for non-referring terms. Speciﬁcally, if ‘t’ is a term of the language, one cannot instantiate a universal claim with ‘t’ or generalize to an existential claim from ‘t’ without having ‘E!(t)’ Since the Eiﬀel Tower does not exist at W, if we make semantic implication comply with free logic interpretative rules,

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both steps in the inference from ‘‘"x(x ¼ x)’’ to ‘‘$x(x ¼ the Eiﬀel Tower)’’ are blocked. A negative free logic is a free logic where all n-ary atomic predications containing a term ‘t’ are false if ‘~E!(t)’ is true. This feature accords with a fairly common intuition that sentences about nonexistents are false (Burge, 1974, pp. 309–311, 317–332; Adams, 1981, pp. 23–25).17 We will use this negative free logic in our next suﬃciency condition: (C3)

If a proposition p follows in a negative free logic from propositions true at W; then p is true at w:

Taken together, (C1), (C2), and (C3) ensure that (5) and (6) are false at worlds where the Eiﬀel Tower does not exist, and (7) is true at those worlds if they contain some green item. It may be found surprising that we have to resort to a free logic – a logic designed to make sense of non-referring terms and even allow for intelligible inferences within a Meinongian metaphysic – in order to make sense of a view that springs from a strict actualism. But there is good reason that this should be so. The perspective aﬀorded by truth-at-a-world semantics is one where we talk about other possible worlds, but make use of the things to hand in our own world in order to do so. Some of the things to hand are objects that don’t exist in the world being talked about. This makes them mere ‘‘possibilia’’ at that world. When we enter into the at-a-world discourse, we enter into a realm where we can talk at least partially like Meinongians or possibilists about a given world; this is because we have access to things in our own world which are the nonexistents of the world being talked about. Thus speaking at-a-world allows us to have our cake and eat it too: we get the expressive power of the possibilist or the Meinongian – the ability to speak about things that do not exist in the world in question – without any of the ontological commitment. 5.

NECESSARY CONDITIONS

In section 3, I announced that I would argue that (C1)–(C3), taken disjunctively, would serve as necessary conditions for a

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non-modal proposition’s being true at a world. Why not for the modal ones? Some modal propositions may be intuitively true at a world and yet not follow from any of (C1)–(C3). Suppose, for instance, that a exists in the actual world and has the property expressed by the predicate ‘P,’ but a does not exist in W. It seems, in this case, that ‘‘}$x(P(x))’’18 will be true in W. (Since ‘‘$x(P(x))’’ is true, it should be true in the actual world; from the perspective of W, it will still be true in the actual world (since it exists in W it can have that property there) and by W ’s application of (C1), true at the actual world as well.) When we are speaking ‘‘at’’ W, though, we can make use of a, and so we should want to say that ‘‘}P(a)’’ is true at W. This does not follow from (C1), (C2), or (C3). Nonetheless, for all non-modal propositions p, if p is true at W, then p satisﬁes (C1), (C2), or (C3). I will argue for this by showing that for every non-modal p, either p or its negation is true at W by one of (C1)–(C3), but I will need to assume that for all worlds W and non-modal propositions p, if p exists in W, either p or ~p is true in W.19 Suppose now that p is a non-modal proposition. If p or its negation is true in W, then by (C1) it or its negation will be true at W as well. So, suppose that neither p nor its negation is true in W; then (since all non-modal propositions are either true or false in W) p must not exist in W. Recall that we are assuming that the only way a proposition can fail to exist is by having a constituent that fails to exist. Thus, if p does not exist in W, then at least one of its constituents, say a, must not exist in W. So, by (C2), ‘‘~E!(a)’’ is true at W. Now, p is either atomic or non-atomic. If p is atomic, then ~p is true by (C3) (and the truth of ‘‘~E!(a)’’); if it is non-atomic, then an appropriate induction (see Appendix A) suﬃces to show that either p or ~p is true at W. 6.

MODAL PROPOSITIONS

That is the story for non-modal propositions. What about the modal ones, though? Suppose a does not exist in W1 but does exist in W2, where it has the property expressed by ‘‘P.’’ What is the status of ‘‘}P(a)’’ at W1?

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I suggested above that ‘‘}P(a)’’ should be true at W1. Adams disagrees. He writes Suppose a is the actual world premiere of Beethoven’s ninth symphony, and [W1] is a possible world in which neither a nor I would exist. Then … at [W1] it would still be true that a could have been a musical performance and I could not. This diﬀerence between a and me at [W1] could hardly have been determined by our (common) non-existence there plus some propositions that are not about us. To suppose such a diﬀerence at [W1] between two individuals that would not exist in [W1] seems at least uncomfortably close to ascribing properties at [W1] to individuals that would not exist in [W1] (1981, p. 32).

This concern seems to have two parts. The ﬁrst part worries that there are insuﬃcient non-modal propositions true at W1 to ground the truth of ‘‘}P(a).’’ The second part suggests that ‘‘}P(a)’’ illegitimately ascribes a property (being possibly P) to an object (a) in a world (W1) where the object does not exist. I will deal with the latter concern ﬁrst. While it would be true at W1 that a could have been a musical performance and Adams could not have been, this is not the same as claiming that, at W1, a has the property of possibly being a musical performance and Adams has the property of not possibly being a musical performance. To say otherwise is to confuse the de dicto ‘‘}(a has property: P)’’ with the de re ‘‘a has property: possibly P.’’ The former says something about the proposition, whereas the latter makes a claim about the object. The ﬁrst part of the worry stems from the claim that ‘‘what is true about an individual a at a world in which a does not exist must be determined by a’s non-existence there together with propositions, true at that world, that are not about a’’ (Adams, 1981, p. 32, emphasis altered). My response to this claim will depend on what it means for a proposition to be ‘‘about’’ a. If p’s being ‘‘about’’ a is equivalent to its predicating some particular property of a, then this worry collapses into the one above. Since in some important sense, ‘‘}P(a)’’ isn’t about a at all, but about ‘‘P(a),’’ this concern is misguided. A proposition p’s being ‘‘about’’ a might be less direct. After all, there is an intuitive sense in which ‘‘}P(a)’’ is ‘‘about’’ a, at

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least in a way that it is not about b or the Eiﬀel Tower or any other object we may be concerned with. On a slightly weaker ‘‘about’’ relation, is it right to say that propositions about a true at a world W in which a does not exist must be wholly determined by ‘‘~E!(a)’’ and other propositions, not about a, true at W? I cannot see why. 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 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. There is a simple way of making this picture precise. We merely add a fourth suﬃciency condition to the list: ðC4Þ

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:20

This condition allows us to show (with the help of a few assumptions) bivalence for non-quantiﬁed modal propositions (see Appendix B). It has the added bonus of making S5 the propositional logic of weak modality, for it directly entails the validity of the characteristic S5 axiom for weak modal operators.21 A problem remains. Although I have argued ‘‘}P(a)’’ does not predicate anything of a, it does seem to predicate something of the proposition ‘‘P(a).’’ On at least one view of modal operators, they are ‘‘predicate-like’’: ‘}’ and ‘(’, in essence, predicate of propositions that they are possible or necessary, respectively (cf. Bealer, 1993, pp. 7–16). This is problematic. If

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a is a constituent of p and a does not exist in W, then existentialists hold that p also does not exist in W. Suppose that p is possible, though. Then, by (C4), ‘‘}p’’ will be true at W. But if ‘}’ is predicate-like, (C2) should entail that ‘‘~}p’’ is true at W.22 Contradictions are therefore true at W. The foregoing overstates the problem, but only slightly. It is not clear that the non-existence of p at W will entail the truth of ‘‘~E!(p)’’ at W, since it is not clear how to formalize the logical system intended to capture the structure of propositions. (Speciﬁcally, it is not clear whether a single quantiﬁer will range over both propositions and ﬁrst-order ‘‘objects,’’ so it is not clear whether (df. E!) will deﬁne a predicate that can be applied to propositions.) Nonetheless, the problem remains. The intuition behind using a negative free logic is the intuition that objects that do not exist at a world should not have any properties at that world either. If its being true, at a world, that, p is possible is tantamount to p’s having the property of being possible at that world, then propositions which do not exist at a world should not be possible there either. I can see two ways to escape this diﬃculty. The ﬁrst is the rejection of the claim that intensional operators (e.g., modal ones) are predicate-like in the required way. One may, for instance, adopt a ‘‘prosentential’’ position in which propositions do not count as singular terms (cf. Grover et al., 1975). I would like to be able to agree that modal operators are predicate-like, though, and there are diﬃculties with adopting a prosentential approach that I would rather avoid (see Bealer, 1993, fn. 8). This brings us to the second way of avoiding the problem. The intuition that drove our acceptance of a negative free logic when formulating (C3) was similar to Adams’ intuition that a proposition p’s at-W truth value is dependent only on the existence or non-existence of the objects p is ‘‘about’’ and other things true at W. Since it seems that an object’s properties in a world require that object’s existence, if an object does not exist at W it should not have properties there either. We have already faulted this reasoning when the realm of ‘‘objects’’ is extended to include propositions. When we are standing outside of a world, we have all the propositions that

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exist there with us, so to speak, and we can see what makes them true by looking at the entire space of possible worlds. We might say that all propositions that exist, exist at every world – although we still aﬃrm, in keeping with our existentialism, that many of these propositions do not exist in many of these worlds. The solution is therefore to revise our formulation of (C3). Rather than requiring a negative free logic, we should use the interpretative rules of a positive free logic. In a positive free logic, some propositions with non-referring terms turn out true (cf. Antonelli, 2000, p. 278; Lambert, 2001, p. 260). We must be careful, though. If we accept the intuition that non-propositional objects cannot have properties at worlds where they do not exist (see note 17), we will want to ensure that the only predications with non-referring terms that ever turn out true are of the form ‘‘p is necessary’’ or ‘‘p is possible.’’ We can do this by ensuring the interpretative rules conform with the following second-order axiom: ðAÞ

8x8Fðð E!ðxÞ f& FðxÞÞ ððF ¼ is necessaryÞ _ ðF ¼ is possibleÞÞÞ:

Thus no non-propositional objects have properties at worlds in which they do not exist (i.e., there are no true propositions that predicate properties of them at those worlds), but propositions can still be possible or impossible at worlds where they do not exist.23 Ontologically innocent semantics for positive free logics are possible (see Antonelli, 2000). Development of a modal positive free logic with the required expressive ability (including the required axiom) is beyond the scope of this paper. Nonetheless, barring the impossibility of such a logic, we have reason to believe (C4) can successfully account for the truth of modal propositions at worlds where they do not exist. 7.

PARTING REMARKS

One ﬁnal point is in order. After all the foregoing, someone may still insist that they do not understand what this notion of weak possibility is, or what it could mean for a proposition to be true at a world without being true in that world. While I think my account is intuitively plausible, I do not think that it is

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intuitively obvious; someone may, without ﬁnding fault with the speciﬁcs of my analysis, claim that they still ﬁnd truth at a world unintelligible. But what does such an objection amount to? We have given a clear deﬁnition of a relation, R, from propositions to worlds: a proposition p bears R to a world W if and only if it satisﬁes one of (C1), (C2), (C3), or (C4).24 Furthermore, the deﬁnition makes neither explicit nor covert appeal to a prior notion of weak possibility. I do not know what more could be asked of such an account. If anyone remains confused about truth at a world, I would simply say that a proposition is true at a world if and only if it bears R to that world. The objection, however, may be somewhat diﬀerent; perhaps one understands R, but does not see what a proposition’s bearing R to a word has to do with its being true at that world. If this is the complaint, I wonder what conditions a relation from propositions to worlds must satisfy in order to stand a chance of successfully playing the truth-at-a-world role. If the answer is ‘‘none – truth at a world simply makes no sense, so no relation can ever serve,’’ the objection seems little more than a dogmatic assertion that truth at a world cannot make sense, and no proposed analysis will be countenanced. On the other hand, if some list of desiderata is given, then I suggest that my interlocutor has at least a minimal understanding of truth at a world. What we must then do is ﬁnd a relation which satisﬁes all these conditions. Of course, it is possible that, given some (hitherto unspeciﬁed) set of conditions, nothing will be able to satisfy them all. If the conditions truly do capture the notion that we’re trying to get at, and they cannot be jointly satisﬁed, then truth at a world is indeed doomed. But if this is so, I should like to see why. Until then, I can see no reason for rejecting the general notion of truth at a world or the account of it given here.25 APPENDIX A

Claim: If p is a non-atomic, non-modal proposition which does not exist in W, then either p or ~p is true at W by (C1), (C2), or (C3).

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Assume (for the sake of simplicity) that the sentences of a ﬁrst-order negative free logic capture the syntactic structure of propositions. Then we need only concern ourselves to a set of primitive operators in the language, say, ‘‘~,’’ ‘‘&,’’ ‘‘",’’ and ‘‘=.’’ If p contains deﬁned operators, then there will be a proposition q logically equivalent to p that contains only primitive operators; by (C3), p will have the same at-W truth value as q. So without loss of generality suppose p has only primitive operators. Assume further that propositional constituents that play the logical operator role (if such there be) exist necessarily and that constituents of p are constituents of every logical construct involving p. I will prove the claim by induction on the number n of logical operators in p: Case n ¼ 1: Case 1: p ¼ ~q for some q. Since p has only one operator (n ¼ 1), q is atomic. Since p does not exist in W, some constituent a of p does not exist in W; hence, some constituent a of the atomic q does not exist in W, and so by (C2) and (C3), ~q is true in W. Case 2: p ¼ q & r for some q and r. Since n ¼ 1, q and r are atomic. Since p does not exist in W, some constituent a of either q or r does not exist in W. Suppose (without loss of generality) that a is a constituent of q; then ~q is true at W by (C2) and (C3). But, if ~q is true at W, then ~(q & r) is true at W by (C3); hence, ~p is true at W. Case 3: p ¼ ‘‘"xF(…x…)’’ for some formula ‘F.’ Since n ¼ 1, if b is any object that exists in W, ‘‘F(…b…)’’ is an atomic proposition. Since p does not exist in W, some constituent a of ‘‘F(…b…)’’ does not exist in W. So, ‘‘F(…b…)’’ has a constituent, a, which does not exist in W. Hence, ‘‘~F(…b…)’’ is true at W by (C2) and (C3). Therefore, ‘‘~"xF(…x…)’’ ¼ ~p is true at W by (C3). Case 4: p ¼ ‘‘a ¼ b’’ for some objects a and b. Since p does not exist in W, either a or b does not exist in W. Hence, by (C2) and (C3), ~p is true at W. Case n > 1. Inductive hypothesis: Suppose that, if q is a proposition with m < n operators, either q or ~q is true at W.

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Case 1: p ¼ ~q for some q. Since p has n operators, q has n ) 1 (

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