MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

2 Robert Goldblattintroduced, and how they interacted and evolved. Then there is the use of methodsand results from other areas of mathematical logic, algebra and topology in theanalysis of modal systems. Finally, there is the application of modal syntax andsemantics to study notions of mathematical and computational interest.There has been some mild controversy about priorities in the origin of relationalmodel theory, and space is devoted to this issue in section 4. An attempt is madeto record in one place a sufficiently full account of what was said and done by earlycontributors to allow readers to make their own assessment (although the authordoes give his).Despite its length, the article does not purport to give an encyclopaedic coverageof the field. For instance, there is much about temporal logic (see [Gabbay et al.,1994]) and logics of knowledge (see [Fagin et al., 1995]) that is not reported here,while the surface of modal predicate logic is barely scratched, and proof theoryis not discussed at all. I have not attempted to survey the work of the presentyounger generation of modal logicians (see [Chagrov and Zakharyaschev, 1997],[Kracht, 1999], and [Marx and Venema, 1997], for example). There has been littleby way of historical review of work on intensional semantics over the last century,and no doubt there remains room for more.Several people have provided information, comments and corrections, both historicaland editorial. For such assistance I am grateful to Wim Blok, Max Cresswell,John Dawson, Allen Emerson, Saul Kripke, Neil Leslie, Ed Mares, RobinMilner, Hiroakira Ono, Amir Pnueli, Lawrence Pedersen, Vaughan Pratt, ColinStirling and Paul van Ulsen.This article originally appeared as [Goldblatt, 2003c]. As well as correctionsand minor adjustments, there are two significant additions to this version. Thelast part of section 6.6 has been rewritten in the light of the discovery in 2003 of asolution of what was described in the first version as a “perplexing open question”.This was the question of whether a logic validated by its canonical frame must becharacterised by a first-order definable class of frames. Also, a new section 7.7has been added to describe recent work in theoretical computer science on modallogics for “coalgebras”.2.1 What is a Modality?2 BEGINNINGSModal logic began with Aristotle’s analysis of statements containing the words“necessary” and “possible”. 1 These are but two of a wide range of modal connectives,or modalities that are abundant in natural and technical languages. Briefly,a modality is any word or phrase that can be applied to a given statement S to1 For the early history of modal logic, including the work of Greek and medieval scholars, see[Bochenski, 1961] and [Kneale and Kneale, 1962]. The Historical Introduction to [Lemmon andScott, 1966] gives a brief but informative sketch.