A binary modal logic for the intersection types of lambda-calculus

The chapter is an attempt to systematize and to provide consistency and inconsistency results for axiomatic modal theories that arise in an expressively rich context. The study is carried out in diagonal modal logic, that is, modal operator logic with fixed points for boxed formulas, as studied by Smorynski. In modal operator logic the interdependencies of the various modal logics can be exploited, which allows for a systematization of the consistencies and inconsistencies of the modal systems depending on the modal properties they assume the modal notion to have. Toward the end of the chapter the focus is on the consistency and inconsistency of multimodal theories. In particular, it is investigated whether multimodal paradoxes are genuinely new paradoxes or whether they are reducible in some way to the unimodal paradoxes. To this end a notion of reducibility is developed and it is shown that there are both reducible and irreducible paradoxes.

The famous van Benthem theorem states that modal logic corresponds exactly to the fragment of first-order logic that is invariant under bisimulation. In this article we prove an exact analogue of this theorem in the framework of modal dependence logic (MDL) and team semantics. We show that Modal Team Logic (MTL) extending MDL by classical negation captures exactly the FO-definable bisimulation invariant properties of Kripke structures and teams. We also compare the expressive power of MTL to most of the variants and extensions of MDL recently studied in the area.

Even if Leibniz didn’t have the opportunity to actually conceive an explicit modal logic system, remains the fact that he had worked out a modal metaphysics, of which the inaugural act, in his Elementa juris naturalis (c. 1671) was an obvious reference to the Apulean square of opposition. Later, scholars acknowledged in this passage probably one of the first sketch of deontic logic of norms. His modal metaphysics rather deals with the so-called alethic modalities, sometimes expounded through a language such as R.M. Adams wondered whether Leibniz could be “a sort of grandfather of possible worlds semantics for modal logic”. In the following study, the Apulean square is used as a hermeneutic tool: however, is the square really well-fitted to express some basic implications of the modal metaphysics? The most remarkable point is that Leibniz was well aware of the K-distributive axiom □(p⊃q)⊃(□p⊃□q) common to the main modal systems today. This awareness dissuaded him to trust the too easy solution of the necessitarianism issue grounded on the well-known distinction coming from Boethius between a “necessitas consequentiae” and a “necessitas consequentis”.

In this paper, we investigate what kind of modal logics can be constructed into the Gentzen-type formulation which enjoys the Cut elimination theorem. We treat normal, regular and monotonic modal logics. A normal modal logic is the modal logic with the Rule of Necessitation and the axiom K:□(A⊃B)⊃(□A⊃□B), a regular modal logic with the rule of inference RR :A,Г→B□A,□Г→□B and a monotonic modal logic with the rule of inference RM :A→B□A→□B. We provide Gentzen-type formulation and prove the cut elimination theorem for the following modal logics: RT4 (a regular modal logic with the axioms T:□A⊃A and 4:□A⊃□□A), MT4 (a monotonic modal logic with the axioms T and 4), MNT4 (a monotonic modal logic with the Rule of Necessitation and the axioms T and 4), RT (a regular modal logic with the axiom T), MT (a monotonic modal logic with the axiom T), MNT (a monotonic modal logic with the Rule of Necessitation and the axiom T), KL (a normal modal logic with the axiom L:□A⊃(A∨□B)) and RTA (a regular modal logic with axioms T and A:□A&B⊃□B).

The classical analytic tableau method has been extended successfully to modal logics (see e.g. Fitting 1983; Fitting 1993; Gore 1992) and also to relevant and paraconsistent logics Bloesch 1993a; Bloesch 1993b. The classical connection method has been extended to modal and intuitionistic logics Wallen 1990, and the purpose of this paper is to investigate whether a similar adaptation to relevant logic is possible. A hybrid method is developed for B+, with a specific solution to the “multiplicity problem”, as in the technique of modal semantic diagrams introduced in Hughes and Cresswell 1968. Proofs of soundness and completeness are also given.

Analytic proof calculi are introduced for box and diamond fragments of basic modal fuzzy logics that combine the Kripke semantics of modal logic K with the many-valued semantics of G\"odel logic. The calculi are used to establish completeness and complexity results for these fragments.

This book, written by one of the most distinguished of contemporary philosophers of mathematics, is a fully rewritten and updated successor to the author's earlier The Unprovability of Consistency (1979). Its subject is the relation between provability and modal logic, a branch of logic invented by Aristotle but much disparaged by philosophers and virtually ignored by mathematicians. Here it receives its first scientific application since its invention. Modal logic is concerned with the notions of necessity and possibility. What George Boolos does is to show how the concepts, techniques, and methods of modal logic shed brilliant light on the most important logical discovery of the twentieth century: the incompleteness theorems of Kurt Godel and the 'self-referential' sentences constructed in their proof. The book explores the effects of reinterpreting the notions of necessity and possibility to mean provability and consistency.

Ternary relations and relevant semantics

The Semantics of Entailment

Boolean contact algebras constitute a suitable algebraic theory for qualitative spatial reasoning. They are Boolean algebras with an additional contact relation grasping the topological aspect of spatial entities. In this paper we present a logic that combines propositional logic, relevance logic, and modal logic to reason about Boolean contact algebras. This is done in two steps. First, we use the relevance logic operators to obtain a logic suitable for Boolean algebras. Then we add modal operators that are based on the contact relation. In both cases we present axioms that are equivalent to requirement that every frame for the logic is indeed a Boolean algebra resp. Boolean contact algebra. We also provide a natural deduction system for this logic by defining introduction and eliminations rules for each logical operator. The system is shown to be sound. Furthermore, we sketch an implementation of the natural deduction system in the functional programming language and interactive theorem prover Coq.

We present a neighbourhood-style semantic framework for modal epistemic logic modelling agents who process information using relevant logic. The distinguishing feature of the framework in comparison to relevant modal logic is that the environment the agent is situated in is assumed to be a classical possible world. This framework generates two-layered logics combining classical logic on the propositional level with relevant logic in the scope of modal operators. Our main technical result is a general soundness and completeness theorem.

Decidability of order-based modal logics

Coalgebraic semantics of modal logics: An overview

CONCEPTUAL FOUNDATIONS OF QUANTUM MECHANICS: THE ROLE OF EVIDENCE THEORY, QUANTUM SETS, AND MODAL LOGIC

We develop a proof-theoretic semantics—in particular, a base-extension semantics—for multi-agent $S5$ modal logic (and hence also for the usual unindexed $S5$). Following the inferentialist interpretation of logic, this gives us a semantics in which validity is based on proof rather than truth. In base-extension semantics, the validity of formulae is generated by provability in a ‘base’ of atomic rules and an inductive definition of the validity of the connectives. Base-extension semantics for many interesting logics has been explored by several authors and, in particular, a base-extension semantics for the modal logics $K$, $KT$, $K4$, and $S4$ has been developed by the present authors. Here, we give a base-extension semantics for multi-agent $S5$ with $\square _{a}$, for an agent $a$, as our primary operators, framed as the knowledge operator $K_{a}$. Similarly to Kripke semantics, we make use of relational structure between bases, allowing us to establish a correspondence between certain bases and worlds. We use this to establish the appropriate soundness and completeness results. We conclude by discussing how this semantics can be extended to Dynamic Epistemic Logics (DEL) starting with Public Announcement Logic (PAL). This note directly supplements [4] and does not repeat the discussion in it.