MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

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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.
Mathematical Modal Logic: A View of its Evolution 3create a new statement that makes an assertion about the mode of truth of S:about when, where or how S is true, or about the circumstances under which Smay be true. Here are some examples, grouped according to the subject they arenaturally associated withtense logic:deontic logic:epistemic logic:doxastic logic:dynamic logic:geometric logic:metalogic:henceforth, eventually, hitherto, previously, now,tomorrow, yesterday, since, until, inevitably, finally,ultimately, endlessly, it will have been, it is being . . .it is obligatory/forbidden/permitted/unlawful thatit is known to X that, it is common knowledge thatit is believed thatafter the program/computation/action finishes,the program enables, throughout the computationit is locally the case thatit is valid/satisfiable/provable/consistent thatThe key to understanding the relational modal semantics is that many modalitiescome in dual pairs, with one of the pair having an interpretation as a universalquantifier (“in all. . . ”) and the other as an existential quantifier (“in some. . . ”).This is illustrated by the following interpretations, the first being famously attributedto Leibniz (see section 4).necessarilypossiblyhencefortheventuallyit is valid thatit is satisfiable thatafter the program finishesthe program enablesin all possible worldsin some possible worldat all future timesat some future timein all modelsin some modelafter all terminating executionsthere is a terminating execution such thatIt is now common to use the symbol ✷ for a modality of universal character, and✸ for its existential dual. In systems based on classical truth-functional logic, ✷is equivalent to ¬✸¬, and ✸ to ¬✷¬, where ¬ is the negation connective. Thus“necessarily” means “not possibly not”, “eventually” means “not henceforth not”,a statement is valid when its negation is not satisfiable, etc.NotationRather than trying to accommodate all the notations used for truth-functionalconnectives by different authors over the years, we will fix on the symbols ∧,∨, ¬, → and ↔ for conjunction, disjunction, negation, (material) implication,and (material) equivalence. The symbol ⊤ is used for a constant true formula,
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