Relevance Logic: Problems Open and Closed

I discuss a collection of problems in relevance logic. The main problems discussed are: the decidability of the positive semilattice system, decidability of the fragments of R in a restricted number of variables, and the complexity of the decision problem for the implicational fragment of R. Some related problems are discussed along the way

A Note on the Relevance of Semilattice Relevance Logic

A propositional logic has the variable sharing property if φ →’ ψ is a theorem only if φ and ψ share some propositional variable(s). In this note, I prove that positive semilattice relevance logic (R+u) and its extension with an involution negation (R¬u) have the variable sharing property (as these systems are not subsystems of R, these results are not automatically entailed by the fact that R satisfies the variable sharing property). Typical proofs of the variable sharing property rely on ad hoc, if clever, matrices. However, in this note, I exploit the properties of rather more intuitive arithmetical structures to establish the variable sharing property for the systems discussed

A Note on the Relevance of Semilattice Relevance Logic

A propositional logic has the variable sharing property if φ →’ ψ is a theorem only if φ and ψ share some propositional variable(s). In this note, I prove that positive semilattice relevance logic (R+u) and its extension with an involution negation (R¬u) have the variable sharing property (as these systems are not subsystems of R, these results are not automatically entailed by the fact that R satisfies the variable sharing property). Typical proofs of the variable sharing property rely on ad hoc, if clever, matrices. However, in this note, I exploit the properties of rather more intuitive arithmetical structures to establish the variable sharing property for the systems discussed

Relevance Logic: Problems Open and Closed

I discuss a collection of problems in relevance logic. The main problems discussed are: the decidability of the positive semilattice system, decidability of the fragments of R in a restricted number of variables, and the complexity of the decision problem for the implicational fragment of R. Some related problems are discussed along the way

Modular labelled calculi for relevant logics

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

The algebraic significance of weak excluded middle laws

Please read abstract in the article.National Research Foundation of South Africa; Ministry of Science and Innovation of Spain; Agència de Gestió d'Ajuts Universitaris i de Recerca.https://onlinelibrary.wiley.com/journal/15213870hj2023Mathematics and Applied Mathematic

Inconsistency lemmas in algebraic logic

In this paper, the inconsistency lemmas of intuitionistic and classical propositional logic are formulated abstractly. We prove that, when a (finitary) deductive system is algebraized by a variety K, then has an inconsistency lemma—in the abstract sense—iff every algebra in K has a dually pseudo-complemented join semilattice of compact congruences. In this case, the following are shown to be equivalent: (1) has a classical inconsistency lemma; (2) has a greatest compact theory and K is filtral, i.e., semisimple with EDPC; (3) the compact congruences of any algebra in K form a Boolean lattice; (4) the compact congruences of any A ∈ K constitute a Boolean sublattice of the full congruence lattice of A. These results extend to quasivarieties and relative congruences. Except for (2), they extend even to protoalgebraic logics, with deductive filters in the role of congruences. A protoalgebraic system with a classical inconsistency lemma always has a deduction-detachment theorem (DDT), while a system with a DDT and a greatest compact theory has an inconsistency lemma. The converses are false.http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1521-3870hb201

Healthiness from Duality

Healthiness is a good old question in program logics that dates back to Dijkstra. It asks for an intrinsic characterization of those predicate transformers which arise as the (backward) interpretation of a certain class of programs. There are several results known for healthiness conditions: for deterministic programs, nondeterministic ones, probabilistic ones, etc. Building upon our previous works on so-called state-and-effect triangles, we contribute a unified categorical framework for investigating healthiness conditions. We find the framework to be centered around a dual adjunction induced by a dualizing object, together with our notion of relative Eilenberg-Moore algebra playing fundamental roles too. The latter notion seems interesting in its own right in the context of monads, Lawvere theories and enriched categories.Comment: 13 pages, Extended version with appendices of a paper accepted to LICS 201

Revisiting Constructive Mingle: Algebraic and Operational Semantics

Among Dunn’s many important contributions to relevance logic was his work on the system RM (R-mingle). Although RM is an interesting system in its own right, it is widely considered to be too strong. In this chapter, I revisit a closely related system, RM0 (sometimes known as ‘constructive mingle’), which includes the mingle axiom while not degenerating in the way that RM itself does. My main interest will be in examining this logic from two related semantical perspectives. First, I give a purely operational bisemilattice semantics for it by adapting previous work of Humberstone. Second, I examine a more conventional algebraic semantics for it and discuss how this relates to the operational semantics. A novel operational semantics for J (intuitionistic logic) as well as its conventional Heyting algebraic semantics emerge as special cases of the corresponding semantics for RM0. The results of this chapter suggest that RM0 is a more interesting logic than has been appreciated and that Humberstone’s operational semantic framework similarly deserves more attention than it has received