AUTOESTIMA Y RENDIMIENTO ACADÉMICO

However, Nolan’s reasoning is deeply mistaken. He is right that for a Lewis-world to represent a claim as true-at-it simply is for that world to instantiate that claim, i.e. for that claim to literally describe, or be true about,that world. (c.f. Nolan 1997: 541) But he is wrong to think that for a world to thus instantiate a claim, somehow means for that world to make that very claim literally – i.e. presumably actually, or simpliciter – true. This clearly isn’t so. For instance, it might be true at some world w that grass is red, but this does not mean that it is literally true that grass is red: it is certainly not thereby actually true that grass is red; nor is it true simpliciter. What is simpliciter true instead is some further proposition, namely that grass is red in worldw. So it is simply not the case that the truth-at-w of a proposition for GR amounts to literal truth, in any objectionable sense, of the proposition in question.

Another way to put the point is by focusing on what truth-at-w amounts to in GR- theory. According to Lewis, the role of the modifier ‘at w’ in ‘truth-at-w’ claims is simply to restrict all domains of explicit and implicit quantification to a particular world w (Lewis 1986a: 5), so that the relevant claim is really a truth about w properly speaking. But if so, then all it takes for Nolan’s exotic impossibilities to be true at a world, w,is for them to be true when restrict our quantifiers to all things in w. And once we restrict our quantifiers to

w, the expression ‘all worlds’ will simply refer to all worldsin w. Similarly atomic claims about the pluriverse will be true at a world, w, (vicariously) just when they are true when we quantify only over things at w. If so, then it may be a literal truth about w, i.e. when we restrict our quantifiers to w’s domain, that Anselm’s God exists at all worlds, but it certainly won’t be true simpliciter, or actually true that Anselm’s God exists at all worlds. Similarly, it might be true, when we quantify over everything in some world, w, that all disjunctions are false there, or even that all disjunctions are false at whatever represents the actual world at w, but it certainly will not be true simpliciter, or actually,

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that all disjunctions are false. (Moreover, I will take it as read that if any of the worlds that are required to represent these claims have to have inconsistencies true at them to do so, we can further apply the techniques we developed in response to Lewis in Ch IV to accommodate such worlds.)

In short, the crucial mistake that Nolan makes is in assuming that, just because GR’s notion of truth at some world really amounts to literal truth about some domain or other, this means that for something to be true at a world it has to be literally true in an

objectionable sense, whether simpliciter or actually true. All that is needed, according to traditional GR-theory, for a proposition to be true at a world w is for it to be literally true when we quantify over all things at w, not for it to be literally true, full-stop. So Nolan’s conclusion that IGR gets into trouble when faced with these claims is a non-sequitor.

What follows shows how we can represent the two sample impossibilities brought up by Nolan. Nolan’s first example, concerns the existence of an impossibilium (Anselm’s God) such that it exists at all worlds. We can represent the existence of such a thing simply by having a world that fits the following description in the home-language of GR:

wx(Ixw & Ax &y(Wy & Iyw  z(Izy & Cxz)))211

This tells us that there is a world w and an x in it such that x is Anselm’s God and such that all worlds in w contain a z which is a counterpart of x.212 Of course, when we quantify over all worlds ina world, we are simply quantifying over that world itself. And since the world contains Anselm’s God, it also thereby contains an item that is a counterpart of Anselm’s God, namely, Anselm’s God; for nothing is more similar to an individual than that individual itself. So we can have a world w that renders the relevant claim true at it, without causing us any problems.

Notably, we could understand Nolan’s example of the impossible God as involving multiple modal operators, instead. Maybe what he means to say is that while Anselm’s God doesn’t possibly exist, if we take him to impossibly exist then he exists necessarily. Then, the claim to be represented is not really a claim about what goes on at all worlds (that is jumping the gun); it is really a claim about a necessary existent that cannot possibly exist. Namely it is an impossibility, embedding a modal operator, of the form:

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I here use the notation w(Pw) to indicate that P is true at w, as indicated in Ch I.

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The fact that all these worlds are impossible is not directly relevant here. We can assume that these worlds are in some sense inaccessible from ours, as per CH V, although it is a good further question precisely what kind of accessibility relation applies.

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x(Ax)

Now, as we saw, under IGR, claims that are necessarily true relative to some world w are claims that are true at all worldsaccessible from w. And so the above claim is not true at any world that is (metaphysically) accessible from our world. But it can still be true at some inaccessible world. All we need in order to represent the relevant impossibility, is simply a world w such that w contains Anselm’s God and such that all worlds accessible from w contain counterparts of Anselm’s God. Then, the representation of the relevant claim merely commits the theory to the existence of a world, w1,that fits the following

description:

w1x(Ixw1 & Ax & w2(Rw1w2y(Iyw2 & Cyx))

Moreover, since we assume that the actual world is not accessible from w1, this

formulation accommodates the intuition that the impossibilium Anselm’s God is a necessary existent (at that world), without entailing the unacceptable consequence that Anselm’s God actually exists.

Similar techniques apply to Nolan’s second case. The impossibility that there is something that renders all disjunctions false by its mere existence can be easily represented by a world where all disjunctions fail. Such a world might fit the following description:

wx(Ixw & y(Iyw & x=yD(~Dw))213

Of course it is a good question exactly how to formulate the claim ‘all disjunctions are false’, since it seems to involve second order quantification. But whether the formulation I choose here works well enough is not the point. The point is rather that disjunctions, too, can be false at some world (a world where, under the usual truth-conditions for

disjunction, presumably nothing is true), yet true at another. For instance it is true at this world that ‘either St Andrews is in Fife or pigs fly’, but it might well be false when we quantify over some other world w, where both disjuncts are false. Similarly with all other disjunctions, and given impossible worlds, even with disjunctions of the form Av~A. So

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This is equivalent, of course to: wx(Ixw & D(~Dw)). Further, using the amended truth-conditions for negation from Ch IV, section 4.4, we get wx(Ixw &D(D*w)).

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all we need in order to represent Nolan’s claim is to have a world where all disjunctions are false, not to have all disjunctions be literally false everywhere or about everything.

Finally, what if we want to interpret Nolan’s second example as involving the operator ‘actually’? There is a sense in which this interpretation involves no special challenges, or at least no special challenges along the lines outlined by Nolan. All we need, once more, is a world, w,such that when we quantify only over all things at w, the claim ‘all

disjunctions are false in @’ is trueat it. And then, as before, it can be true at w that all disjunctions are false in @’but this latter won’t be true simpliciter. What will be true

simpliciter at most is that all disjunctions are false in @ at w (or all disjunctions are

actually false at w). So whatever may be true-at-w will not affect what goes on at the actual world. In that sense, Nolan’s objection fails no matter what interpretation we give to his claim. For it is not the case that for a world to instantiate the claim ‘actually P’, for some actual falsehood P, the relevant falsehood must really hold of the actual world.

One may ask, exactly how can a GR-world represent claims of that nature? How can a world render true a proposition of the general form P-at-a, about some other world a?214

One might think that, given that there are no free variables in a sentence such as P-at-a, (or simply Pa), there is nothing for us to bind in the domain of quantification of the world, w, that is supposed to do the representing. So, one may think that for there to be a world where some theoretical falsehood Pa holds simply means for Pa to be true simpliciter, as per the equivalence:

(E) w(Pa) if and only if Pa

If so, then Nolan would be right that IGR must embrace contradictions, since, if worlds represent atomic claims as per (E), for there to be a world where some theoretical falsehood Pa holds simply means for Pa to be true simpliciter.

But, while (E) is clearly true, ‘w(Pa)’ is not how one would express the claim that Pa

is true at some worldw. Instead, one would translate such claims by recourse to

counterparts. It is as much of a mistake to think that for a world to represent the claim Pa

as true at it, it must be a world that fits the description ‘w(Pa)’, as it is to think that for a world to represent any atomic possibility of the form Fa as true at it simply is for that world to fit the description w(Fa). The possibility, Fa, is after all a proposition true at a

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set of worlds. And the members of that set do not just consist in the particular world that contains the individual a; (for let us not forget that all GR-individuals are world-bound). If it were so, then a possible proposition about a, Ga, involving some property G that a

contingently doesn’t but could have, would be true at no worlds, (and a fortiori no accessible worlds), and so would not possibly be true. But ex hypothesi Ga is possibly true. So, there is some world, other than the one that contains a (where Ga is false) that renders Ga true. How does it do so? Vicariously, by having something other than a – a counterpart of a – be G at that world.215 So the representation of ‘Ga’ as true at a world,

w,which doesn’t contain a,does not involve w in the simple (vacuous) description

w(Ga). Instead, what it means for Ga to be true at w is for w to fit this description, involving counterparts:

wx(Ixw & Cxa & Ga)

In short, while it is true that the relevant sentences that require representation here involve no (explicit) variables, this doesn’t mean that they are insensitive to domain-shifts. And so, this interpretation of Nolan’s argument also fails. What it takes for any claim ‘Ga’ to be true at a world w is for w to contain some other individual b,which stands in for a, and which instantiates G. Similarly what it takes for claims Pa to be true-at-w (where a is a world and P a proposition that fails to hold at it) is for w to contain some other individual

b, which stands in for a, and which instantiates P.

However, now a further question arises. We already know a lot about what it is for some individual in a world to represent a possibility for another individual, namely for the former to be a counterpart of the latter. (Lewis 1986a: 230-232) But here we are not considering possibilities, so whatever does the representing in this case ought not to be a counterpart, at least not under any of the usual counterpart relations. If anything, the relevant individual should be a counterpart under similarity relations, which we would think inappropriate for the evaluation of possibility. To avoid confusion, I propose we call such unorthodox counterparts stand-ins. We can define a stand-in of an individual x

as an individual y that does not constitute a counterpart of x under any of the usual similarity relations that we take to govern possibility.216 One notable feature of the

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C.f. Lewis (1986a: 10)

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The satisfaction conditions for something being a stand-in can be just as vague and utterly subject to contextual features as those that something has to satisfy to count as a counterpart.

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introduction of stand-ins here (which are, in essence, inaccessible or dissimilar individuals) is that counterpart and accessibility relations can vary independently: an object can have a counterpart in an impossible (i.e. inaccessible) world under some unrelated accessibility relation, or have no counterparts at an accessible world. For instance a logically

inaccessible world (as per Ch V) may contain an apple that is a counterpart of a this- worldly apple; equally there might be a world which is logically accessible from ours yet contains, say, only dragons, or entities we would normally not invoke in our usual

counterpart relations for entities in this world. What particular individual might fill the role of a stand-in in any particular case will, as usual, be decided on the basis of pragmatic considerations, for instance, on the basis of whether it satisfies the properties that it needs to satisfy to represent the relevant claim as true at the world of which it is part.217

Notably, such stand-ins are not only required for the representation of exotic atomic GR-theoretical impossibilities. They are equally needed in order to represent any ordinary atomic impossibility as true at a world. Consider the ordinary impossibility, for instance, that ‘Obama is a boiled egg’, or Bo. In the first instance, the usual truth-conditions for the relevant modal claim dictate that it is impossible that Obama is boiled egg if and only if there is no (accessible) world containing a counterpart of Obama that is a boiled egg:

~Bo ~wx(Rw@ & Ixw & Cxo & Bx)

It is also true, strictly speaking, that Obama has no boiled-egg-counterparts, unrestrictedly speaking, at least under none of the similarity relations that govern de re possibility. (Lewis 1986a: 230) But the impossibility that ‘Obama is a boiled egg’ still needs to be represented as true at some inaccessible world under the extended theory. This is where stand-ins come in. Is there anything in such a world (indeed in the pluriverse) that can (vicariously) represent Obama per impossibile being a boiled egg? I think there is; indeed I think that any boiled egg will do, and so any (inaccessible) world that contains such an

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I take Lewis to pull a somewhat similar trick when he distinguishes the doxastic alternatives of a person from his counterparts. (Lewis 1986a §1.4) He takes, for instance, a proposition (the content of the belief) believed by Rene, namely that he is immaterial, to be represented by the set of those worlds where Rene has immaterial doxastic alternatives, thus allowing us to suppose that even if Rene is necessarily material, there are worlds that contain things which allow us to represent him otherwise. (1986a: 32-33) Along very similar lines, we here require individuals to be able to represent the content of claims involving certain other individuals, yet without the former being counterparts of the latter.

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egg can represent the impossibility that Obama is a boiled egg.218 It would strictly not be correct to say that the representing egg is a counterpart of Obama. Presumably, it bears no similarity to Obama with respect to his origins. Yet match of origins is often considered an important similarity respect governing de re possibility and so counterpart relations. So, let the egg simply stand-in for Obama at that world, thus allowing the world to render true (by proxy) the claim that Obama is a boiled egg.

The point is that whatever works for ordinary atomic impossibilities, works for extraordinary ones. For instance, is there anything in a world that can represent that all disjunctions are false at the actual world? I think there is; indeed I think that anything at which all disjunctions are false will do, and so will any world which contains such a thing. And given that we presumably still want the relevant proposition to constitute an

impossibility for the actual world, we do not want it instantiated by a world that contains a

counterpart of the actual world at which all disjunctions are false – at least not any kind of an ordinary counterpart. (If we take the representing individual to be the inaccessible world itself, the accessibility-relation and the counterpart-relation coincide: the world at which it is true that all disjunctions are actually false is neither accessible from the actual world nor, here, a counterpart of the actual world. Given the analysis of accessibility as similarity proposed in Ch V, we can simply say that the world that represents the relevant impossibility is not similar to the actual world under any of the usual respects that we take to govern possibility de dicto or de re.) Using the notation Sxy to express the claim that x

stands-in for y then, a world that renders true-at-it the claim Pb is a world that fits the following description in the theory:

wx(Ixw & Sxb & Px)

And in particular, the contended reading of Nolan’s second claim as involving the actuality operator ‘@’ can be represented by a world that falls under the following description: