2006: Alethic Modal Logics and Semantics (original) (raw)

2006, A Companion to Philosophical Logic

The first axiomatic development of modal logic was untertaken by C. I. Lewis in 1912. Being anticipated by H. MacColl in 1880, Lewis tried to cure logic from the 'paradoxes' of extensional (i.e. truthfunctional) implication … (cf. Hughes and Cresswell 1968: 215). He introduced the stronger notion of strict implication <, which can be defined with help of a necessity operator ᮀ (for 'it is neessary that:') as follows: A < B iff ᮀ(A … B); in words, A strictly implies B iff A necessarily implies B (A, B,. .. for arbitrary sentences). The new primitive sentential operator ᮀ is intensional (non-truthfunctional): the truth value of A does not determine the truth-value of ᮀA. To demonstrate this it suffices to find two particular sentences p, q which agree in their truth value without that ᮀp and ᮀq agree in their truth-value. For example, it p = 'the sun is identical with itself,' and q = 'the sun has nine planets,' then p and q are both true, ᮀp is true, but ᮀq is false. The dual of the necessity-operator is the possibility operator ‡ (for 'it is possible that:') defined as follows: ‡A iff ÿᮀÿA; in words, A is possible iff A's negation is not necessary. Alternatively, one might introduce ‡ as new primitive operator (this was Lewis' choice in 1918) and define ᮀA as ÿ ‡ÿA and A < B as ÿ ‡(A Ÿ ÿB). Lewis' work cumulated in Lewis and Langford (1932), where the five axiomatic systems S1-S5 were introduced. S1-S3 are weaker that the standard systems of § 2.2, but S4 and S5 coincide with standard S4 and S5 (for details on Lewis' systems cf. Hughes and Cresswell 1968: ch. 12; Chellas and Segerberg 1996). Lewis' pioneer work was mainly syntactic-axiomatic, except for the modal matrix-semantics (for details in the 'algebraic' tradition, started by Lukasiewicz, cf. Bull and Segerberg 1984: 8ff). The philosophically central semantics for modal logic is possible world semantics. It goes back to ideas of Leibniz, was first developed by Carnap and received broadest acceptance through the later work of Kripke. The actual world, in which we happen to live, is merely one among a multitude of other possible worlds, each realizing a different but logically complete collection of facts. The basic idea of possible world semantics as expressed by Carnap (1947: 9f, 174f) is: (1) ᮀA is true in the actual world iff A is true in all possible worlds. ‡A is true in the actual world iff A is true in some possible world.