Hyperintensional logics for everyone
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Hyperintensional logics for everyone Igor Sedlár1 Received: 1 March 2017 / Accepted: 26 December 2018 © Springer Nature B.V. 2019
Abstract We introduce a general representation of unary hyperintensional modalities and study various hyperintensional modal logics based on the representation. It is shown that the major approaches to hyperintensionality known from the literature, that is state-based, syntactic and structuralist approaches, all correspond to special cases of the general framework. Completeness results pertaining to our hyperintensional modal logics are established. Keywords Awareness logic · Hyperintensionality · Hyperintensional logic · Hyperintensional modalities · Impossible worlds · Modal logic · Non-Fregean logic · Structured propositions
1 Introduction The possible-worlds framework has provided semantics for various formal languages as well as large portions of natural language. The general strategy is to represent semantic contents of expressions by intensions, i.e. functions from possible worlds (usually taken as unanalysed indices) to extensions. The specific kind of extension depends on the kind of expression at hand. For instance, extensions are truth-values 0, 1 in the case of sentences, individuals (from some fixed domain) in the case of names and n-ary relations in the case of n-ary predicates. Modalities are seen as expressing properties of (or relations between) intensions; the corresponding intensions are functions from possible worlds to relations on intensions. In the case of unary sentential modalities, an equivalent approach is to take functions from possible worlds to sets of propositions (i.e. to sets of sets of possible worlds). A prominent example of this approach is the Montague–Scott semantics for modal logic (Montague 1968; Scott 1970; Segerberg 1971; Chellas 1980; Pacuit 2017). In
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Igor Sedlár [email protected] Institute of Computer Science, The Czech Academy of Sciences, Pod Vodárenskou vˇeží 2, 182 07 Prague 8, Czech Republic
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the MS semantics, a modal formula F says that F, the proposition expressed by the formula F, has the property expressed by . This property is represented by a function N from possible worlds to sets of propositions. So, F is true in a world w iff F ∈ N (w). The problem with this picture is that some natural-language modalities do not express properties of propositions. Take epistemic modalities, for example. Sentences ‘2 + 3 = 5’ and ‘110119 is a prime number’ express the same proposition (the set of all possible worlds), but mutual substitution of these sentences in the scope of an epistemic modality is not guaranteed to preserve truth value. Simply put, it is possible for some agent, say John, to believe that 2+3 = 5 without also believing that 110119 is a prime number. This would be impossible if ‘John believes that’ expressed a property of propositions. Modalities that express properties of sentential contents grained finer than sentential intensions are known as hyperintensional modalities. Semantic frameworks where
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