Edited by Mark Jago (Nottingham University)
|Summary||Several kinds of semantics for modal logic have been proposed, the most popular of which is Kripke semantics. (This is sometimes called possible worlds semantics, although the formal semantics doesn’t require us to think of the entities in its domain as possible worlds.) In Kripke semantics, models comprise a domain of entities, variously called points, situations, scenarios or possible worlds, with one or more accessible relations between them. The truth-value of a sentence at a possible world w may depend on the truth-value of other sentences at worlds accessible from w. In modal logics, for example, ‘necessarily, A’ is true at a worldw iff A is true at all worlds u accessible from w, and ‘possibly, A’ is true at w iff A is true at some world u accessible from w. This approach can be applied to other kinds of modalities, including ‘agent a knows that’ in modal epistemic logics. One can also give algebraic, topological and categorical semantics for modal logics, in place of Kripke semantics. These approaches have intrinsic mathematical interest, but have received less attention in the philosophical literature (perhaps because they do not provide an analysis of modal concepts in the way that possible worlds semantics does).|
|Key works||Possible worlds semantics was first presented as a formal semantics for modal logic in Kripke 1959, 1963 and for intuitionistic logic in Kripke 1963. Carnap 1947 was an important precursor to possible worlds semantics. Hintikka 1962, 1967 develops the possible worlds semantics and applies it to epistemic concepts. van Benthem 1983 is the classic investigation of the relationship between accessibility relations in the semantics and modal axioms. Lewis 1968 develops counterpart theory in the context of (first-order) possible worlds semantics. Lewis 1986 argues for an extreme realist philosophical interpretation of the possible worlds semantics.|
|Introductions||Girle 2003 and Girle 2000 are introductory textbooks on possible worlds and modal logic. Priest 2001 is a general introduction to propositional modal, intuitionistic and relevant logics. Hughes & Cresswell 1996 is a classic textbook in modal logic. Sider 2010 includes a good presentation of quantified first-order logic.|
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