Creencias y mitos sobre medidas de prevención a) Higiene Oral

Taking a cue from the foregoing developments in response to Nolan, I think that we can represent all GR-theoretical impossibility claims, whether they involve atomic

simpliciter falsehoods or such falsehoods involving unrestricted quantification over the entire plurality of worlds by simply allowing them to be true at some world, w,just when we quantify over w.

Let us begin with the case of universal claims. For any absolute universal claim

x(Fx)

to be true at a world, w,whether x here stands for a world or an individual, is for that claim to be true just when we quantify over w, namely for the following to be true simpliciter:

wx(IxwFx)

In that case, we can offer the following simple translations to some examples of such universal GR-theoretical impossibilities mentioned earlier:

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1. There are no worlds that contain donkeys:

w~xy(Wx & Ixw & Iyx & Dy)

This description picks out a world, in which no world contains donkeys; such a world is one which, itself, contains no donkeys.

2. There is no plurality of worlds/there is only one world:

wxy(Wx & Ixw & Iyw & (Wyx=y))

This description picks out a world, in which there is only one world. Clearly, if we restrict our quantifiers to a single world, it is true at that world that there is no plurality of worlds.

3. There are only two worlds:

wxyz(Ixw & Iyw & Izw & Wx & Wy & xy & (Wz (z=x v z=y))).

This description picks out a world, which contains only two distinct worlds. It picks out a world, which fails to be identical with itself.

4. There are more worlds than there are:

wx(Wx & Ixw & y(Wy & Iywyx))

This description, too, can be satisfied by a world which is not identical with itself.

5. There is an x such that x is part of all worlds:

wxy (Ixw & Iyw & (WyIxy)),

This, too, is quite uncontroversial. All we need is a world that contains some x such that for all worlds in that world, namely that world itself, x is part of those worlds.

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In short, there are no grounds for thinking that the representation of absolute universal false claims about the plurality as true at some impossible world creates unwelcome consequences for the theory. Whatever the theoretical claim, once put in the object- language as an impossibility eligible for truth at a world, it becomes subject to the usual world-restrictions. This allows the claim to be true at w when we restrict our quantifiers to

w, and yet be false simpliciter.

Additionally, if the need should arise, drawing from the theory’s truth-at-w conditions for necessity claims, we can develop a system whereby multi-world impossibility claims are represented as true at some impossible world w by being true at all worlds accessible (under some accessibility relation) from w. Under such a proposal any transworld universal claim like

x(Fx)

can be represented as if it had a box in front, namely as true at some impossible world, w1,

which fits the description:

w1w2(Rw1w2x(Ixw2Fx))

We can simplify notation for the case where x is a world(-variable) to read:

w1w2(Rw1w2Fw2)

Under this option, we could take the case where our quantifiers are bound within a single world, as we saw before, to be the special case, where a world only accesses itself. This more elaborate proposal for the representation of universal transworld impossibilities would make little practical difference in some cases, for instance:

1. There are no worlds that contain donkeys:

w1~w2x (Rw1w2 & Ixw2 & Dx)

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of worlds, accessible (under some contextually determined accessibility relation) from w1,

which contain no donkeys. The proposed alteration would have no effect at all on claims like:

2. There is no plurality of worlds/there is only one world:

w1w2(Rw1w2w1=w2)

And

5. There is an x such that x is part of all worlds:

w1x(Ixw & w2(Rw1w2Ixw2))

For, both are rendered true by a world that only accesses itself. However this alternative proposal might make a real difference in affording a consistent representation of claims like

3. There are only two worlds:

w1w2(Rw1w2 & w1w2 & w3 (Rw3w1 (w1=w3 v w2=w3))).

by letting this be made true by a world which only accesses one other.

It might be that there is no need for such a more complex proposal. The reason I put it forth is to showcase the riches from which IGR can draw, if need be, to accommodate impossibilities of an absolutely universal nature by merely redeploying existing resources. Whichever proposal works best, IGR can simply fall back on its own resources to

represent the relevant claims.

Let us now turn to the case of atomic claims. For any absolute atomic claim

Pa

to be true at a world, w – whether a names a world or an individual – is for that claim to be true (by proxy) just when we quantify over everything in w, namely for the following to be

181 true simpliciter:

wx(Ixw & Sxa & Px)

For instance, we can offer the following translations for the series of atomic impossibility examples mentioned earlier. Samples like

6. actually St Andrews is in Australia

7. actually Brown is the prime minister of Timbuktu 8. actually Caesar killed Brutus

have pretty much the same form, so, let us translate just one of these here. A world which renders true, say (8), is a world that satisfies

wxyz(Ixw & Iyw & Izw & Sxc & Syb & Sz@ & Ixz & Iyz & Kxy)

It contains stand-ins x, y, z,for Caesar, Brutus and the actual world, such that z contains x

and y and such that the counterpart of Caesar kills the counterpart of Brutus. Again, it might be that w=z or it might equally be that the stand-in for the actual world is some proper part of w. Equally unproblematic are:

9. Obama is spatiotemporally related to a talking donkey

wxy(Ixw & Iyw & Ty & Sxo & Rxy)

Again, all we have is a world that contains a stand-in for Obama that is spatiotemporally related to a talking donkey.219 And

10.Obama is part of w17

wxy(Ixw & Iyw & Sxo & Cyw17& Ixy)

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If the claim is about some particular donkey d, then we offer the slightly different translation:

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This is a world that contains stand-ins for Obama and world seventeen such that the former is part of the latter. Any individual in w that is part of another, also in w, can serve. Of some interest might be the following:

11.Obama is not part of the actual world

wxy(Ixw & Iyw & Sxo & Sy@& ~Ixy)

This world contains a counterpart, x, of Obama, o, and one, y, of the actual world, @,and x

is not part of y. What is of interest here is that if w itself acts as a stand-in for the actual world, i.e. if y=w, then w is inconsistent in that it both contains and does not contain x. To the extent that if Obama was not part of the actual world, then it would not also be the case that he is part of the actual world, we might not want w to act as a stand-in for @ here. So, this gives us positive reason to want individuals other than worlds to stand-in for worlds in our representations.

Lastly, we have a couple of mixed cases:

12.Everything is actual

This is of interest insofar as it combines universal quantification and the employment of the rigid operator denoting the actual world. We might give it different readings:

(a) If we understand it to use the rigid actuality operator (so, read: x(Ix@)), then for a world, w, to represent this claim, it would satisfy:

wx(Ixw & Sx@y(IywIyx))

Namely, it would contain an x as a stand-in for @, such that everything in w is in x. Contrary to (11), for this translation to be consistent (and there seems to be no call for an logically inconsistent translation here), w itself would act as the stand-in for @, i.e. w=x.

(b) Alternatively, if we treat the expression ‘actually’ as an indexical, (12) simply states the GR-tautology that everything at w is part of w.

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(c) Lastly, echoing the discussion on the GR-notion of a world in section 6.3.2, we could read the relevant impossibility, as an actualist might often think of it: namely to imply that actuality and existence are not just coextensive, but that for something to exist just means for it to be actual. But, then we can simply evaluate this claim at a world where ‘actual’ means ‘exists’.

The second mixed case of interest is this:

13.w1 is accessible from w2

On a straight-forward understanding, (13) does not present any further challenges. All we need is a world that contains two things, which stand-in for w1 and w2, and which bear the

relevant relation, possibly by being similar to each other in some relevant sense (I will use non-italicised lettering below to indicate w1 and w2 as rigid names of worlds):

wxy(Ixw & Iyw & Sxw1 & Syw2 & Rxy)

(13) is of interest insofar as it might seem more intuitive to translate it employing both the stand-ins and accessibility relations of the above proposals:

w3w4w5(Rw3w4 & Rw3w5 & Sw4w1 & Sw5w2 & Rw4w5)

This translation renders the relevant claim true at world w3 in virtue of it accessing two

worlds w4 and w5 which stand-in for w1 and w2, such that Rw4w5.

My aim was to show that the representation of GR-theoretical impossibilities can be accommodated using nothing more than the existing GR-theoretical tools of quantification over worlds, accessibility and counterpart relations.220 I hope to have demonstrated that IGR can fall back on familiar resources in order to reply, in a systematic manner, to any representational challenges of this nature.

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I have not touched upon questions of how IGR can represent complex claims involving sets, here. Lewis comments that a set exists according to a world just when it exists from the standpoint of a world; moreover, he takes it that although some sets, numbers for instance, exist from the standpoint of all worlds, others, like singletons of concreta, exist only from the standpoint of some worlds. (Lewis 1983b: 40; c.f. also Lewis 1986a: 96 fn 61)) Needless to say, we need not take any sets to exist according to all worlds under

impossibilist GR. We could make use of the notion of the existence of sets ‘from the standpoint’ of worlds, here, according to our needs. For instance there may be worlds that fail to have any arithmetical truths hold at them, for failing to contain the relevant sets; or it might be that there are worlds according to which properties are not identical with sets, since there are none of the latter.

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