modal logic (redirected from Actual propositions)

modal logic

Formal systems incorporating modalities such as necessity, possibility, impossibility, contingency, strict implication, and certain other closely related concepts. The most straightforward way of constructing a modal logic is to add to some standard nonmodal logical system a new primitive operator intended to represent one of the modalities, to define other modal operators in terms of it, and to add axioms and/or transformation rules involving those modal operators. For example, one may add the symbol L, which means “It is necessary that,” to classical propositional calculus; thus, Lp is read as “It is necessary that p.” The possibility operator M (“It is possible that”) may be defined in terms of L as Mp = ¬L¬p (where ¬ means “not”). In addition to the axioms and rules of inference of classical propositional logic, such a system might have two axioms and one rule of inference of its own. Some characteristic axioms of modal logic are: (A1) Lp ⊃ p and (A2) L(p ⊃ q) ⊃ (Lp ⊃ Lq). The new rule of inference in this system is the Rule of Necessitation: If p is a theorem of the system, then so is Lp. Stronger systems of modal logic can be obtained by adding additional axioms. Some add the axiom Lp ⊃ LLp; others add the axiom Mp ⊃ LMp.

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(logic)

modal logic - An extension of propositional calculus with operators that express various "modes" of truth. Examples of modes are: necessarily A, possibly A, probably A, it has always been true that A, it is permissible that A, it is believed that A. "It is necessarily true that A" means that things being as they are, A must be true, e.g. "It is necessarily true that x=x" is TRUE while "It is necessarily true that x=y" is FALSE even though "x=y" might be TRUE. Adding modal operators [F] and [P], meaning, respectively, henceforth and hitherto leads to a "temporal logic". Flavours of modal logics include: Propositional Dynamic Logic (PDL), Propositional Linear Temporal Logic (PLTL), Linear Temporal Logic (LTL), Computational Tree Logic (CTL), Hennessy-Milner Logic, S1-S5, T. C.I. Lewis, "A Survey of Symbolic Logic", 1918, initiated the modern analysis of modality. He developed the logical systems S1-S5. JCC McKinsey used algebraic methods (Boolean algebras with operators) to prove the decidability of Lewis' S2 and S4 in 1941. Saul Kripke developed the relational semantics for modal logics (1959, 1963). Vaughan Pratt introduced dynamic logic in 1976. Amir Pnuelli proposed the use of temporal logic to formalise the behaviour of continually operating concurrent programs in 1977. [Robert Goldblatt, "Logics of Time and Computation", CSLI Lecture Notes No. 7, Centre for the Study of Language and Information, Stanford University, Second Edition, 1992, (distributed by University of Chicago Press)]. [Robert Goldblatt, "Mathematics of Modality", CSLI Lecture Notes No. 43, Centre for the Study of Language and Information, Stanford University, 1993, (distributed by University of Chicago Press)]. [G.E. Hughes and M.J. Cresswell, "An Introduction to Modal Logic", Methuen, 1968]. [E.J. Lemmon (with Dana Scott), "An Introduction to Modal Logic", American Philosophical Quarterly Monograpph Series, no. 11 (ed. by Krister Segerberg), Basil Blackwell, Oxford, 1977].