Robert Goldblatt

Mathematical modal logic: a view of its evolution

This is a survey of the origins of mathematical interpretations of modal logics, and their development 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.

Diodorean modality in Minkowski spacetime

The Diodorean interpretation of modality reads the operator □ as “it is now and always will be the case that”. In this paper time is modelled by the four-dimensional Minkowskian geometry that forms the basis of Einsteins special theory of relativity, with “event” y coming after event x just in case a signal can be sent from x to y at a speed at most that of the speed of light (so that y is in the causal future of x).It is shown that the modal sentences valid in this structure are precisely the theorems of the well-known logic S4.2, and that this system axiomatises the logics of two and three dimensional spacetimes as well.Requiring signals to travel slower than light makes no difference to what is valid under the Diodorean interpretation. However if the “is now” part is deleted, so that the temporal ordering becomes irreflexive, then there are sentences that distinguish two and three dimensions, and sentences that can be falsified by approaching the future at the speed of light, but not otherwise.

First-Order Definability in Modal Logic

In the early days of the development of Kripke-style semantics for modal logic a great deal of effort was devoted to showing that particular axiom systems were characterised by a class of models describable by a first-order condition on a binary relation. For a time the approach seemed all encompassing, but recent work by Thomason [6] and Fine [2] has shown it to be somewhat limited—there are logics not determined by any class of Kripke models at all. In fact it now seems that modal logic is basically second-order in nature, in that any system may be analysed in terms of structures having a nominated class of second-order individuals (subsets) that serve as interpretations of propositional variables (cf. [7]). The question has thus arisen as to how much of modal logic can be handled in a first-order way, and precisely which modal sentences are determined by first-order conditions on their models. In this paper we present a model-theoretic characterisation of this class of sentences, and show that it does not include the much discussed LMp → MLp . Definition 1. A modal frame ℱ = 〈 W, R 〉 consists of a set W on which a binary relation R is defined. A valuation V on ℱ is a function that associates with each propositional variable p a subset V(p) of W (the set of points at which p is “true”).

The Mckinsey axiom is not canonical

The logic KM is the smallest normal modal logic that includes the McKinsey axiom It is shown here that this axiom is not valid in the canonical frame for KM , answering a question first posed in the Lemmon-Scott manuscript [Lemmon, 1966]. The result is not just an esoteric counterexample: apart from interest generated by the long delay in a solution being found, the problem has been of historical importance in the development of our understanding of intensional model theory, and is of some conceptual significance, as will now be explained. The relational semantics for normal modal logics first appeared in [Kripke, 1963], where a number of well-known systems were shown to be characterised by simple first-order conditions on binary relations (frames). This phenomenon was systematically investigated in [Lemmon, 1966], which introduced the technique of associating with each logic L a canonical frame which invalidates every nontheorem of L . If, in addition, each L -theorem is valid in , then L is said to be canonical . The problem of showing that L is determined by some validating condition C , meaning that the L -theorems are precisely those formulae valid in all frames satisfying C , can be solved by showing that satisfies C —in which case canonicity is also established. Numerous cases were studied, leading to the definition of a first-order condition C φ associated with each formula φ of the form where Ψ is a positive modal formula.

Parallel action: Concurrent dynamic logic with independent modalities

Regular dynamic logic is extended by the program constructα∩β, meaning “α andβ executed in parallel”. In a semantics due to Peleg, each commandα is interpreted as a set of pairs (s,T), withT being the set of states “reachable” froms by a single execution ofα, possibly involving several processes acting in parallel. The modalities A is true ats iff there existsT withsRβT andA true throughoutT, and[α]A is true ats iff for allT, ifsRβT thenA is true throughoutT, which make and [α] no longer interdefinable via negation, as they are in the regular case.We prove that the logic defined by this modelling is finitely axiomatisable and has the finite model property, hence is decidable. This requires the development a new theory of canonical models and filtrations for “reachability” relations.

Erdős graphs resolve fine's canonicity problem

¤ Abstract. We show that there exist 2 @0 equational classes of Boolean algebras with operators that are not generated by the complex algebras of any first-order definable class of relational structures. Using a variant of this construction, we resolve a long-standing question of Fine, by exhibiting a bimodal logic that is valid in its canonical frames, but is not sound and complete for any first-order definable class of Kripke frames (a monomodal example can then be obtained using simulation results of Thomason). The constructions use the result of Erdýos that there are finite graphs with arbitrarily large chromatic number and girth.

Deduction Systems for Coalgebras Over Measurable Spaces

A theory of infinitary deduction systems is developed for the modal logic of coalgebras for measurable polynomial functors on the category of measurable spaces. These functors have been shown by Moss and Viglizzo to have final coalgebras that represent certain universal type spaces in game-theoretic economics. A notable feature of the deductive machinery is an infinitary Countable Additivity Rule. A deductive construction of canonical spaces and coalgebras leads to completeness results. These give a proof-theoretic characterization of the semantic consequence relation for the logic of any measurable polynomial functor as the least deduction system satisfying Lindenbaums Lemma. It is also the only Lindenbaum system that is sound. The theory is additionally worked out for Kripke polynomial functors, on the category of sets, that have infinite constant sets in their formation.

Algebraic polymodal logic: a survey

This is a review of those aspects of the theory of varieties of Boolean algebras with operators (BAO’s) that emphasise connections with modal logic and structural properties that are related to natural properties of logical systems. It begins with a survey of the duality that exists between BAO’s and relational structures, focusing on the notions of bounded morphisms, inner substructures, disjoint and bounded unions, and canonical extensions of structures that originate in the study of validity-preserving operations on Kripke frames. This duality is then applied to polymodal propositional logics having finitary intensional connectives that generalise the Box and Diamond connectives of unary modal logic. Issues discussed include validity in canonical structures, completeness and incompleteness under the relational semantics, and characterisations of logics by elementary classes of structures and by finite structures. It turns out that a logic is strongly complete for the relational semantics iff the variety of algebras it defines is complex , which means that every algebra in the variety is embeddable into a full powerset algebra that is also in the variety. A hitherto unpublished formulation and proof of this is given (Theorem 5.6.1) that applies to quasi-varieties. This is followed by an algebraic demonstration that the temporal logic of Dedekind complete linear orderings defines a complex variety, adapting Gabbay’s model-theoretic proof that this logic is strongly complete.

Mathematical modal logic: A view of its evolution

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, temporal 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 1930’s 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 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

Final coalgebras and the Hennessy–Milner property

Abstract The existence of a final coalgebra is equivalent to the existence of a formal logic with a set (small class) of formulas that has the Hennessy–Milner property of distinguishing coalgebraic states up to bisimilarity. This applies to coalgebras of any functor on the category of sets for which the bisimilarity relation is transitive. There are cases of functors that do have logics with the Hennessy–Milner property, but the only such logics have a proper class of formulas. The main theorem gives a representation of states of the final coalgebra as certain satisfiable sets of formulas. The key technical fact used is that any function between coalgebras that is truth-preserving and has a simple codomain must be a coalgebraic morphism.