CRONOGRAMA DE TRABAJO

Our question is: how do we add genuine impossible worlds into GR without compromising (P)? I propose the answer lies in drawing attention to the notion of accessibility. So, I propose we amend (P) to read:

(P’) Possibly A iff there is an accessible world w such that A at w.

This answer draws on the relatively common idea that modality usually amounts to restricted quantification over worlds. When Lewis rejects concrete impossible worlds, he rejects these conceived of as absolute rather than relative impossibilia. This opens up the option of preserving (P) by conceiving of impossible worlds as worlds that are

restrictedly impossible, worlds which are impossible in the sense that they are inaccessible

from the perspective of some world, usually ours. The conception of modality as restricted quantification is strongly present in Lewis, both in his Plurality of Worlds and his

Counterfactuals, for instance:

“More often than not, modality is restricted quantification; and restricted from the standpoint of a given world, perhaps ours, by means of so called ‘accessibility’ relations.” (Lewis 1986a: 7)

“A necessity operator in general, is an operator that acts like a restricted quantifier over possible worlds. ...we call these worlds accessible meaning thereby simply that they satisfy the restriction associated with the sort of necessity operator under consideration. Necessity is truth at all accessible worlds, and different sorts of necessity correspond to different accessibility restrictions. A possibility operator

likewise, is an operator that acts like a restricted existential quantifier over worlds. Possibility is truth at some accessible world, and the accessibility restriction imposed depends on the sort of possibility under consideration. If a necessity and a possibility

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operator correspond to the same accessibility restriction on the worlds quantified over, then they will be a dual interdefinable pair.” (Lewis 1973a: 4-5)178

But as Robert Pargetter points out, “[w]hile care is taken with these formal accounts of possibility and necessity to see them as relative notions, when they are viewed

metaphysically they seem to lose their relativity and take on an absoluteness.” (Pargetter 1984: 336) And so, the present proposal simply hopes to dispel this air of absoluteness. In more precise terms, the amended analysis reads as follows:

(PM) it is M-possible at w0 that A iff there is some world w1 which is accessible

from w0 under the accessibility relation RM, and at w1, A.

Then, for any family of modal notions M, M-possibility corresponds to the existence of some M-accessible world where the relevant claim holds, M-necessity corresponds to the relevant claim holding at all M-accessible worlds; and M-impossibility corresponds to it holding at none. The explicit introduction of accessibility relations thus allows us to preserve the accuracy of (P), in its more explicit form (PM). Of course, we can allow that the original reductive analysis, (P) with its quantifiers unrestricted, still constitutes a generic absolute analysis of possibility. After all it is simply the trivial case where the accessibility relation is universal. The difference is that it is unlikely that this absolute

unrestricted notion of possibility will, any longer, fit any of our common modal notions.179 Now all that remains is to specify the details. And as Stalnaker notes, herein lies the real challenge:

“...if you want to interpret possibilities and impossibilities... in terms of restricted quantification – that is fine with me. And if you do, I shouldn’t even accuse you of ruining good old words, since modal words...are most commonly interpreted in terms

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See also Lewis’ (1968: 37-38) and Divers (2002: 68). The importance of accessibility relations in the analysis of possibility is also highlighted in Stalnaker (1996), Perszyk (1993) and Salmon (1984). See also Lycan (1994: §8) and on a different note Barwise (1997), who also proposes that a world is impossible, not

simpliciter, but relative to a state of information. See also Smiley (1963) for a notion of necessity relative to a body of propositions.

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This is an intuitive idea. For instance, Barwise (1997), too, proposes that our common modal notions arise out of holding fixed a particular set of facts (about our world): in the case of physical possibility, for instance, the physical laws, in the case of logical or mathematical possibility, the logical laws and mathematical truths and in the case of metaphysical possibility, “those regularities that fall out of the way that humans individuate objects, properties and relations.” (Barwise 1997: 496)

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of some proper subset of the possibilities, a set defined by an accessibility relation between worlds... All I need to know is what the basis is for your restriction.” (Stalnaker: 1996: 200)

This request will be taken up at some length, here.

In the first place we need to say something about the nature of the relations that effect the relevant restrictions. While accessibility sounds very much like a modal term, it has a natural non-modal rendering given GR’s existing conceptual tools, namely in terms of similarity. (Lewis 1986a: 8, 234) Thus, relations of accessibility reduce to basic non- modal relations of similarity: The basis of the relevant restriction is a matter of similarity between the base world w and the accessed world v in respect to the relevant set of facts M

about w:180

(RM) a world w1 is accessible from another world w0 under the accessibility

relation RM if and only if w1is similar to w0with respect to a set of base

facts M about w0.

It is worth noting that the notion of an impossible world that results from this picture is doubly relative: a world counts as an impossible world only relative to (a) some other world w and (b) to the particular similarity relation we choose to employ. Hence:

(IW) A world w1 is impossible relative to another world w2 if and only if w1 is

inaccessible from w2 under some (specified) similarity relation R.

Intuitively this means that a world might be impossible relative to our world given some relation R1 yet possible under the same relation (say, nomological similarity) from some

other world w. Alternatively a world may be impossible relative to our world given some relation R1 (nomological similarity) yet possible under another relation R2(say, logical

similarity). Moreover, this understanding of accessibility as similarity between worlds paints a picture of de dicto modality very much along the lines on which GR accounts for modality de re, namely on the basis of similarity between individuals, this time worlds. If, as per Lewis (1986a: 53), propositions are properties of entire worlds, then the notion of

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accessibility operative here is that of similarity between entire worlds with respect to certain propositions that hold of them.

Now, the (reductive) success of this proposal will depend on whether the relevant propositions, M, can be specified without appeal to modal terms. So the second part of an answer to the challenge voiced here by Stalnaker will be to show that the base set of facts

M that fix the relevant similarity relation can be non-modally specified and their modal status, if any, reduced to truth at all M-accessible worlds. While I cannot address this latter for each particular family of modal notions here, I will try to show that the same arguments that show this desideratum to be met in the case of the nomological modalities can be employed to show that it can be met in the case of the (broadly) logical modalities.