Journal of Applied Logic 1 (2003) 309–392 www.elsevier.com/locate/jal Mathematical modal logic: A view of its evolution Robert Goldblatt Centre for Logic, Language and Computation, Victoria University, P.O. Box 600, Wellington, New Zealand Abstract This is a survey of the origins of mathematical interpretations of modal logics, and their devel- opment over the last century or so. It focuses on the interconnections between algebraic semantics using Boolean algebras with operators and relational semantics using structures often called Kripke models. It reviews the ideas of a number of people who independently contributed to the emergence of relational semantics, and compares them with the work of Kripke. It concludes with an account of several applications of modal model theory to mathematics and theoretical computer science. 2003 Elsevier B.V. All rights reserved. Keywords: Modal logic; History of logic; Boolean algebra; Kripke semantics 1. Introduction Modal logic was originally conceived as the logic of necessary and possible truths. It is now viewed more broadly as the study of many linguistic constructions that qualify the truth conditions of statements, including statements concerning knowledge, belief, tempo- ral discourse, and ethics. Most recently, modal symbolism and model theory have been put to use in computer science, to formalise reasoning about the way programs behave and to express dynamical properties of transitions between states. Over a period of three decades or so from the early 1930s there evolved two kinds of mathematical semantics for modal logic. Algebraic semantics interprets modal connectives as operators on Boolean algebras. Relational semantics uses relational structures, often called Kripke models, whose elements are thought of variously as being possible worlds, moments of time, evidential situations, or states of a computer. The two approaches are intimately related: the subsets of a relational structure form a modal algebra (Boolean algebra with operators), while conversely any modal algebra can be embedded into an E-mail address: [email protected] (R. Goldblatt). URL: http://www.mcs.vuw.ac.nz/~rob. 1570-8683/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/S1570-8683(03)00008-9 310 R. Goldblatt / Journal of Applied Logic 1 (2003) 309–392 algebra of subsets of a relational structure via extensions of Stone’s Boolean representation theory. Techniques from both kinds of semantics have been used to explore the nature of modal logic and to clarify its relationship to other formalisms, particularly first and second order monadic predicate logic. The aim of this article is to review these developments in a way that provides some insight into how the present came to be as it is. The pervading theme is the mathematics underlying modal logic, and this has at least three dimensions. To begin with there are the new mathematical ideas: when and why they were introduced, and how they interacted and evolved. Then there is the use of methods and results from other areas of mathematical logic, algebra and topology in the analysis of modal systems. Finally, there is the applica- tion of modal syntax and semantics to study notions of mathematical and computational interest. There has been some mild controversy about priorities in the origin of relational model theory, and space is devoted to this issue in Section 4. An attempt is made to record in one place a sufficiently full account of what was said and done by early contributors to allow readers to make their own assessment (although the author does give his). Despite its length, the article does not purport to give an encyclopaedic coverage of the field. For instance, there is much about temporal logic (see Gabbay et al. [77]) and logics of knowledge (see Fagin et al. [64]) that is not reported here, while the surface of modal predicate logic is barely scratched, and proof theory is not discussed at all. I have not attempted to survey the work of the present younger generation of modal logicians (see Chagrov and Zakharyaschev [34], Kracht [141], and Marx and Venema [171], for exam- ple). There has been little by way of historical review of work on intensional semantics over the last century, and no doubt there remains room for more. 2. Beginnings 2.1. What is a modality? Modal logic began with Aristotle’s analysis of statements containing the words “nec- essary” 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 to create 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 S may be true. Here are some examples, grouped according to the subject they are naturally associated with tense logic: henceforth, eventually, hitherto, previously, now, tomorrow, yesterday, since, until, inevitably, finally, ultimately, endlessly, it will have been, it is being . 1 For the early history of modal logic, including the work of Greek and medieval scholars, see Bochenski 1961 [20] and Kneale and Kneale 1962 [134]. The Historical Introduction to Lemmon [156] gives a brief but informative sketch. R. Goldblatt / Journal of Applied Logic 1 (2003) 309–392 311 deontic logic: it is obligatory/forbidden/permitted/unlawful that epistemic logic: it is known to X that, it is common knowledge that doxastic logic: it is believed that dynamic logic: after the program/computation/action finishes, the program enables, throughout the computation geometric logic: it is locally the case that metalogic: it is valid/satisfiable/provable/consistent that The key to understanding the relational modal semantics is that many modalities come in dual pairs, with one of the pair having an interpretation as a universal quantifier (“in all ...”)andtheotherasanexistentialquantifier(“insome...”).Thisis illustrated by the following interpretations, the first being famously attributed to Leibniz (see Section 4). necessarily in all possible worlds possibly in some possible world henceforth at all future times eventually at some future time it is valid that in all models it is satisfiable that in some model after the program finishes after all terminating executions the program enables there is a terminating execution such that It 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. Notation Rather than trying to accommodate all the different notations used for truth-functional connectives by different authors over the years, we will fix on the symbols ∧, ∨, ¬, → and ↔ for conjunction, disjunction, negation, (material) implication, and (material) equiv- alence. The symbol is used for a constant true formula, equivalent to any tautology, while ⊥ is a constant false formula, equivalent to ¬.Wealsouse and ⊥ as symbols for truth values. The standard syntax for propositional modal logic is based on a countably infinite list p0,p1,...of propositional variables, for which we typically use the letters p,q,r.Formu- las are generated from these variables by means of the above connectives and the symbols and ✸. There are of course a number of options about which of these to take as prim- itive symbols, and which to define in terms of primitives. When describing the work of different authors we will sometimes use their original symbols for modalities, such as M for possibly, L or N for necessarily, and other conventions for deontic and tense logics. The symbol n stands for a sequence ··· of n copies of , and likewise ✸n for ✸✸ ···✸ (n times). 312 R. Goldblatt / Journal of Applied Logic 1 (2003) 309–392 A systematic notation will also be employed for Boolean algebras: the symbols + , ·, − denote the operations of sum (join), product (meet), and complement in a Boolean algebra, and 0 and 1 are the greatest and least elements under the ordering given by x y iff x · y = x. The supremum (sum) and infimum (product) of a set X of elements will be denoted X and X (when they exist). 2.2. MacColl’s iterated modalities The first substantial algebraic analysis of modalised statements was carried out by Hugh MacColl, in a series of papers that appeared in Mind between 1880 and 1906 under the title Symbolical Reasoning,2 as well as in other papers and his book [163] of 1906. MacColl symbolised the conjunction of two statements a and b by their concatenation ab,useda +b for their disjunction, and wrote a : b for the statement “a implies b”, which he said could be read “if a is true, then b must be true”, or “whenever a is true, b is also true”. The equation a = b was used for the assertion that a and b are equivalent, meaning that each implies the other. Thus a = b is itself equivalent to the “compound implication” (a : b)(b : a),an observation that was rendered symbolically by the equation (a = b) = (a : b)(b : a). MacColl wrote a for the “denial” or “negative” of statement a, and stated that (a + b) is equivalent to ab . However, while a + b is a “necessary consequence” of a : b (written (a : b) : a + b ), he argued that the two formulas are not equivalent because their denials are not equivalent, claiming that the denial of a : b “only asserts the possibility of the com- bination ab ”, while the denial of a + b “asserts the certainty of the same combination”.3 Boole had written a = 1anda = 0for“a is true” and “a is false”, giving a tempo- ral reading of these as always true and always false respectively (see Boole 1854 [21], Chapter XI).