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The worlds of possibility. Modal realism and the semantics of modal logic. (English) Zbl 0901.03005

Oxford: Clarendon Press. xii, 342 p. (1998).
This book argues that a realistic conception of possible worlds is not necessary to provide a semantics for quantificational S5 with actualist quantifiers and an existence predicate. The first move is to replace the possible worlds in an S5 interpretation by pure sets. The next task is to connect up these pure sets with modal discourse in natural language. In essence it works like this: For every \(n\)-place predicate \(\theta\) assign to \(\theta\) as semantic value an \(n\)-place predicate of natural language. Then label all the ways the world might have been by pure sets, and say that \(\theta\) holds of an \(n\)-tuple at that set iff the natural language predicate which interprets \(\theta\) would have been true of those individuals if the world had been the way that set labels. The possibility operator \(\diamondsuit\) is interpreted by the following rule (p. 201):
\[ \begin{aligned} & [\text{T-d}] \text{ A sentence `} \diamondsuit \varphi \text{' is true just in case the}\\ & \text{world could have been such that } \varphi \text{ was true}. \end{aligned} \] \((\square\) is its dual.) With this definition it is straightforward to shew that the wff valid in every natural language interpretation are precisely the wff valid in the given version of S5, and so, in that sense, modal logic does not require possible worlds.

MSC:

03A05 Philosophical and critical aspects of logic and foundations
03B45 Modal logic (including the logic of norms)
03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations
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