<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
"It is necessarily true that x=x" is TRUE
"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,
[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].
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Mode » mode