An Essay in Classical Modal Logic

This work forms the author’s Ph.D. dissertation, submitted to Stanford University in 1971. The author’s overall purpose is to present in an organized fashion the theory of relational semantics (Kripke semantics) in modal propositional logic, as well as the more general neighbourhood semantics (Montague-Scott semantics), and then to apply these systematically to the examination of a wide range of individual modal logics. He restricts himself to propositional modal logics; quantified modal logics are not considered. The author brings together under one cover a great many results that were already known in scattered form in journals, as well as others from oral communications; he systematizes these results, relates them to each other, and refines them; he provides new proofs of many old theorems, constructing, for example, demonstrations via relational models for theorems previously known only by algebraic methods; and he also contributes an impressive number of new results to the field. These works established some notational and terminological conventions that have been lasting. For instance, the term frame was used in place of model structure. In the first volume the author sets out some preliminary notions, introduces the idea of neighbourhood semantics, establishes several basic consistency and completeness theorems in terms of such semantics, introduces relational semantics and relates them to neighbourhood semantics, and begins a study of p-morphisms and filtrations of relational and neighbourhood models. In the second volume he applies these semantic techniques to a detailed study of transitive relational models and associated logics. In the third volume he adapts the notions and techniques developed in the first two so as to cover modal logics that are quasi-normal or quasi-regular, in the sense of including the least normal [regular] modal logic without necessarily being themselves normal [regular]. [From the review by David Makinson.] Filtration was used extensively by Segerberg to prove completeness theorems. This technique can be effective in dealing with logics whose canonical model does not satisfy some desired property, and comes into its own when seeking to axiomatise logics defined by some condition on finite frames. This method was applied in ``Essay'' to axiomatise a whole range of logics, including those characterised by the classes of finite partial orderings, finite linear orderings (both irreflexive and reflexive), and the modal and tense logics of the structures of N, Z, Q, R, with the relation "more", "less", or their reflexive counterparts. [Taken from R.Goldblatt, Mathematical modal logic: A view of its evolution, J. of Applied Logic, vol.1 (2003), 309-392.]