# modal logic

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## modal logic

(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].

## Logic, Modal

the branch of logic devoted to the study of modalities, the construction of calculi in which modalities are applied to propositions, in addition to logical operations, and the comparative study of such calculi. Modal operators, such as “possible” and “necessary,” may refer to propositions or predicates and to words that express certain actions or acts. Interest in modal logic is chiefly due to the natural relation between modalities such as “necessary” and the concept of logical law (that is, an identically valid proposition of some logical system), on the one hand, and between modalities such as “possible” and such epistemological and general scientific concepts as “(effectively) realizable” and “calculable,” on the other.

In classical systems of modal logic (for which the law of the excluded middle A V ┐ A or the law of double negation ┐ ┐ AA is valid), duality relations—analogous to De Morgan’s laws ┐(A V B)↔(┐A & ┐B) and ┐(A & B)↔(┐A V ┐B) of the algebra of logic and to the corresponding equivalencies for quantifiers — relate the possibility operator ✧ and the necessity operator ☐ to negation ┐ obtain for modalities:

A ↔ ┐ ✧ ┐A and ✧ A ↔ ┐ ☐ ┐A

Therefore, one modal operation is usually introduced as the initial operation in axiomatic systems of modal logic (by using one of these equivalencies to define the other operation). Other modal operations— which are not logical operations and cannot be expressed in terms of them —also are introduced in a similar manner.

The systems of modal logic may be interpreted in terms of many-valued logic. The simplest systems may be interpreted as three-valued systems: “true,” “false,” “possible.” This fact, as well as the possibility of applying modal logic to the construction of a theory of “probable” conclusions, points to its strong kinship with probability logic.

In addition to the “absolute” modalities considered above, modal logic also deals with relative modalities— that is, modalities linked to certain conditions, such as “if B, then A is possible.” The formalization of the rules for dealing with such modalities does not create additional difficulties and is carried out by means of restricted quantifiers (using predicates expressing restrictions and the logical operations of material implication).

IU. A. GASTEV

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