Online research in philosophy

Kripke-completeness of every classical modal logic with Sahlqvist formulas is one of the basic general results on completeness of classical modal logics. This paper shows a Sahlqvist theorem for modal logic over the relevant logic Bin terms of Routley-Meyer semantics. It is shown that usual Sahlqvist theorem for classical modal logics can be obtained as a special case of our theorem.

This long-awaited book replaces not one but both of Hughes and Cresswell's two previous classic studies of modal logic: An Introduction to Modal Logic and A Companion to Modal Logic . A New Introduction to Modal Logic has been completely rewritten by the authors to incorporate all the developments that have taken place since 1968 both in modal propositional logical and modal predicate logic, but without sacrificing the clarity of exposition and approachability that were essential features of the earlier works. The book takes readers through the most basic systems of modal prepositional logic right up to systems of modal predicate with identity. It deals with both technical developments such as completeness and incompleteness, and finite and infinite models, and discusses philosophical applications, especially, in the area of modal predicate logic.

A general strategy for proving completeness theorems for quantified modal logics is provided. Starting from free quantified modal logic K, with or without identity, extensions obtained either by adding the principle of universal instantiation or the converse of the Barcan formula or the Barcan formula are considered and proved complete in a uniform way. Completeness theorems are also shown for systems with the extended Barcan rule as well as for some quantified extensions of the modal logic B. The incompleteness of Q°.B + BF is also proved.

We define a multi-modal version of Computation Tree Logic (CTL) by extending the language with path quantifiers $E^\delta $ and $E^\delta $ where δ denotes one of finitely many dimensions, interpreted over Kripke structures with one total relation for each dimension. As expected, the logic is axiomatised by taking a copy of a CTL axiomatisation for each dimension. Completeness is proved by employing the completeness result for CTL to obtain a model along each dimension in turn. We also show that the logic is decidable and that its satisfiability problem is no harder than the corresponding problem for CTL. We then demonstrate how Normative Systems can be conceived as a natural interpretation of such a multi-dimensional CTL logic.

The paper generalises Goldblatt's completeness proof for Lemmon–Scott formulas to various modal propositional logics without classical negation and without ex falso, up to positive modal logic, where conjunction and disjunction, andwhere necessity and possibility are respectively independent.Further the paper proves definability theorems for Lemmon–Scottformulas, which hold even in modal propositional languages without negation and without falsum. Both, the completeness theorem and the definability theoremmake use only of special constructions of relations,like relation products. No second order logic, no general frames are involved.

In this work we propose an encoding of Reiter’s Situation Calculus solution to the frame problem into the framework of a simple multimodal logic of actions. In particular we present the modal counterpart of the regression technique. This gives us a theorem proving method for a relevant fragment of our modal logic.

We define a multi-modal version of Computation Tree Logic ( ctl ) by extending the language with path quantifiers E δ and A δ where δ denotes one of finitely many dimensions, interpreted over Kripke structures with one total relation for each dimension. As expected, the logic is axiomatised by taking a copy of a ctl axiomatisation for each dimension. Completeness is proved by employing the completeness result for ctl to obtain a model along each dimension in turn. We also show that the logic is decidable and that its satisfiability problem is no harder than the corresponding problem for ctl. We then demonstrate how Normative Systems can be conceived as a natural interpretation of such a multi-dimensional ctl logic.

We show that if we interpret modal diamond as the derived set operator of a topological space, then the modal logic of Stone spaces is K4 and the modal logic of weakly scattered Stone spaces is K4G. As a corollary, we obtain that K4 is also the modal logic of compact Hausdorff spaces and K4G is the modal logic of weakly scattered compact Hausdorff spaces.

In this paper the PA-completeness of modal logic is studied by syntactical and constructive methods. The main results are theorems on the structure of the PA-proofs of suitable arithmetical interpretationsS of a modal sequentS, which allow the transformation of PA-proofs ofS into proof-trees similar to modal proof-trees. As an application of such theorems, a proof of Solovay's theorem on arithmetical completeness of the modal system G is presented for the class of modal sequents of Boolean combinations of formulas of the form p i,m i=0, 1, 2, ... The paper is the preliminary step for a forthcoming global syntactical resolution of the PA-completeness problem for modal logic.

This paper extends David Lewis result that all first degree modal logics are complete to weakly aggregative modal logic by providing a filtration-theoretic version of the canonical model construction of Apostoli and Brown. The completeness and decidability of all first-degree weakly aggregative modal logics is obtained, with Lewiss result for Kripkean logics recovered in the case k=1.