Algorithms for Relevant Logic

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

A star-free relational semantics for relevant logic is presented together with a sound and complete sequent proof theory (display calculus). It is an extension of the dualist approach to negation regarded as modality, according to which de Morgan negation in relevant logic is better understood as the confusion of two negative modalities. The present work shows a way to define them in terms of implication and a new connective, co-implication, which is modeled by respective ternary relations. The defined negations are confused by a special constraint on ternary relation, called the generalized star postulate, which implies definability of the Routley star in the frame. The resultant logic is shown to be equivalent to the well-known relevant logic R. Thus it can be seen as a reconstruction of R in the dualist framework.

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.

Over the last forty-odd years, Kit Fine has been one of the most influential and original analytic philosophers. He has made provocative and innovative contributions to several areas of systematic philosophy, including philosophy of language, metaphysics, and the philosophy of mathematics, as well as a number of topics in philosophical logic, such as modal logic, relevance logic, the logic of essence, and the logic of vagueness. These contributions have helped reshape the agendas of those fields and have given fresh impetus to a number of perennial debates. Fine’s work is distinguished by its great technical sophistication, philosophical breadth, and independence from current orthodoxy. A blend of philosophically sound common sense combined with a virtuosity of philosophical argumentation and construction, meant to back up the former, this seems to me to be the nature of Kit Fine’s lasting contributions to the current trends in analytic philosophy. The chapter gives a general overview of the groundbreaking work of Kit Fine and connects the contributed essays with Kit Fine’s work.

We introduce a general recipe for presenting non-classical logics in a modular and uniform way as labelled deduction systems. Our recipe is based on a labelling mechanism where labels are general entities that are present, in one way or another, in all logics, namely truth-values. More specifically, the main idea underlying our approach is the use of algebras of truth-values, whose operators reflect the semantics we have in mind, as the labelling algebras of our labelled deduction systems. The “truth-values as labels” approach allows us to give generalized systems for multiple-valued logics within the same formalism: since we can take multiple-valued logics as meaning not only finitely or infinitely many-valued logics but also power-set logics, i.e. logics for which the denotation of a formula can be seen as a set of worlds, our recipe allows us to capture also logics such as modal, intuitionistic and relevance logics, thus providing a first step towards the fibring of these logics with many-valued ones.


 
 
 In this article, we perform a detailed proof theoretic investigation of a wide number of relevant logics by employing the well-established methodology of labelled sequent calculi to build our intended systems. At the semantic level, we will characterise relevant logics by employing reduced Routley-Meyer models, namely, relational structures with a ternary relation between worlds along with a unique distinct element considered as the real (or actual) world. This paper realizes the idea of building a variety of modular labelled calculi by reflecting, at the syntactic level, semantic informations taken from reduced Routley-Meyer models. Central results include proofs of soundness and completeness, as well as a proof of cut- admissibility.
 
 

This article opposes a view widely accepted in studies concerning the history of modal logic, according to which (i) the approach of C. I. Lewis towards constructing modern modal logic was purely syntactical (i.e. limited to the construction of axiomatic systems S1–S5 of propositional modal logic), and (ii) the notion of a possible world was incorporated into modern logic and philosophy mainly by authors such as Rudolf Carnap and Saul Kripke. The article presents Lewis' definition of a possible world, and his formulation of the truth-conditions of statements containing strict implication as their main connective in terms of possible worlds. The main question of the article is whether it is possible to consider Lewis' work in this area as an early stage of the development of possible world semantics, and if so, in what sense? The article concludes by answering affirmatively, due to soundness and completeness proofs with respect to S5 using Lewis' semantics.

The Semantics of Entailment

The present paper studies two approaches to the expressiveness of propositional modal logics based on first-degree entailment logic, FDE. We first consider the basic FDE-based modal logic BK and certain systems in its vicinity, and then turn to some FDE-based modal logics in a richer vocabulary, including modal bilattice logic, MBL. On the one hand, model-theoretic proofs of the definability of connectives along the lines of [McCullough, “Logical connectives for intuitionistic propositional logic”, Journal of Symbolic Logic 36, 1 (1971): 15–20. DOI: 10.2307/2271511] and [[17] Wansing, “Logical connectives for constructive modal logic”, Synthese 150, 3 (2006): 459–482. DOI: 10.1007/s11229-005-5518-5] are given for various FDE-based modal logics. On the other hand, building on [Odintsov and Wansing, “Disentangling FDE-based paraconsistent modal logics, Studia Logica 105, 6 (2017): 1221–1254. DOI: 10.1007/s11225-017-9753-9], expressibility is considered in terms of mutual faithful embeddability of one logic into another logic. A distinction is drawn between definitional equivalence, which is defined with respect to a pair of structural translations between two languages, and weak definitional equivalence, which is defined with respect to a weaker notion of translations. Moreover, the definitional equivalence of some FDE-based modal logics is proven, especially the definitional equivalence of MBL and a conservative extension of the logic BK□×BK□, which underlines the central role played by BK among FDE-based modal logics.

This paper introduces a two-dimensional modal logic to represent agents’ concurrent common knowledge in distributed systems. Unlike common knowledge, concurrent common knowledge is a kind of agreement reachable in asynchronous environments. As a proper semantics to concurrent common knowledge we present the closed sub-product of modal logics. We axiomatize the presented logic issuing an idea of the soundness and completeness proofs.

A Hilbert-style version of an implicational logic can be represented by a set of axiom schemes and modus ponens or by the corresponding axioms, modus ponens and substitution. Certain logics, for example the intuitionistic implicational logic, can also be represented by axioms and the rule of condensed detachment, which combines modus ponens with a minimal form of substitution. Such logics, for example intuitionistic implicational logic, are said to be D-complete. For certain weaker logics, the version based on condensed detachment and axioms (the condensed version of the logic) is weaker than the original. In this paper we prove that the relevant logic T→, and any logic of which this is a sublogic, is D-complete.

According to Beall and Restall’s logical pluralism, classical logic, relevant logic, and intuitionistic logic are all correct. On this version of logical pluralism, logic is considered to be normative, in the sense that someone who accepts the truth of the premises of a valid argument, is bound to accept the conclusion. So-called collapse arguments are designed to show the incompatibility of the simultaneous acceptance of logical pluralism and the normativity of logic. Caret, however, by proposing logical contextualism, and Blake-Turner and Russell by proposing telic pluralism, have sought to nullify the collapse problem. In the present article, after setting out these two approaches to the collapse problem, we argue that by using the concept of the ‘rationality of beliefs’ in order to frame the canonical purpose of logic, it can be demonstrated that if logical contextualism and telic pluralism are considered as philosophically significant logical pluralisms, a refined version of the collapse argument is still a threat for both of these kinds of logical pluralism.

This paper deals with the axiomatizability problem for the matrix-based logics RMQ� and RMQ*. We present a Hilbert-style axiom system for RMQ�, and a quasi-axiomatization based on it for RMQ*. We further compare these logics to different well-known modal logics, and assess its status as relevance logics.

Ternary relations and relevant semantics