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
algebra}s 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].
(1995-02-15)