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In the fourth chapter of "Counterfactuals", David Lewis tries to justify his positions about modal realism. He claims that: "We might take them [modalities] as metalinguistic predicates analyzable in terms of consistency: 'Possibly P' means that P is a consistent sentence [...] If a consistent sentence is one whose denial is not a theorem of some specified deductive system, then the theory is incorrect: no falsehood of arithmetic is possibly true, but for any deductive system you care to specify either there are falsehoods among its theorems or there is some falsehood of arithmetic whose denial is not among its theorem"

"We might take [modalities] as quantifiers over possible worlds where these are treated as linguistic entities: maximal consistent sets of sentences (atomic sentences of sentences of a language enriched by the addition of names for all the things there are, that is diagrammed models) [...] But again the theory would be circular or incorrect, according as we explain consistency in modal terms or in deductive (or purely model-theoretic) terms"

Two things are not so clear to me: why for any deductive system we specify we either have false sentences in its theorems or there is some falsehood of arithmetic whose denial is not among its theorem?

Second, why would the theory be circular or incorrect if we were to consider possible worlds as sets of consistent sentences or set of diagrammed models?

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Lewis observes that our ordinary language of modality contains apparent quantifiers. We commonly say that there are ways the world could be, or ways the world could have been, but actually aren't. According to Lewis, the quantifier "there are" is most simply understood as saying that there exist possible worlds where such things are true, and these possible worlds are not actual. The only thing that distinguishes the actual world from other possible worlds is that it is the one where we find ourselves.

But Lewis is aware that this sounds implausible, so he is concerned to defend it against alternatives. One might try to explain modal language in terms of consistency. We could say that "possibly P" means "P is consistent", but we would then need to specify what kind of consistency we have in mind. If it means that the denial of P cannot be reconciled with some theory then this account is circular, because it appeals in turn to the modal 'cannot'. On the other hand, if it means that the denial of P is not deducible from some theory, then this is too weak and would accommodate necessarily false propositions as possible.

On the specific point about arithmetic: to say that any deductive system of arithmetic either contains falsehoods or has some falsehood whose denial is not a theorem, is Lewis' colloquial way of saying that any deductive (recursive) theory of arithmetic is either inconsistent or incomplete.

To claim that worlds are just sets of consistent sentences, or states of affairs, runs into the objection above that it is either circular because it appeals to what can or cannot be reconciled with a given theory, or it is false because it accommodates sentences that are impossible but whose denial is not deducible.

It is perhaps worth noting that Lewis is in effect assuming a kind of logical and mathematical platonism. Also, he does not allow that there are impossible worlds. Otherwise, we could say that it is not a problem for the range of possible worlds to include necessarily false propositions.

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