Quasi-Nelson algebras and fragments

The variety of quasi-Nelson algebras (QNAs) has been recently introduced and characterised in several equivalent ways: among others, as (1) the class of bounded commutative integral (but non-necessarily involutive) residuated lattices satisfying the Nelson identity, as well as (2) the class of (0, 1)-congruence orderable commutative integral residuated lattices. Logically, QNAs are the algebraic counterpart of quasi-Nelson logic, which is the (algebraisable) extension of the substructural logic ℱℒ ew (Full Lambek calculus with Exchange and Weakening) by the Nelson axiom. In the present paper, we collect virtually all the results that are currently known on QNAs, including solutions to certain questions left open in earlier publications. Furthermore, we extend our study to some subreducts of QNAs, that is, classes of algebras corresponding to fragments of the algebraic language obtained by eliding either the implication or the lattice operations.

HOW MUCH PROPOSITIONAL LOGIC SUFFICES FOR ROSSER’S ESSENTIAL UNDECIDABILITY THEOREM?

In this paper we explore the following question: how weak can a logic be for Rosser’s essential undecidability result to be provable for a weak arithmetical theory? It is well known that Robinson’s Q is essentially undecidable in intuitionistic logic, and P. Hájek proved it in the fuzzy logic BL for Grzegorczyk’s variant of Q which interprets the arithmetic operations as nontotal nonfunctional relations. We present a proof of essential undecidability in a much weaker substructural logic and for a much weaker arithmetic theory, a version of Robinson’s R (with arithmetic operations also interpreted as mere relations). Our result is based on a structural version of the undecidability argument introduced by Kleene and we show that it goes well beyond the scope of the Boolean, intuitionistic, or fuzzy logic.

Categorial Grammar

The term “categorial grammar” refers to a variety of approaches to syntax and semantics in which expressions are categorized by recursively defined types and in which grammatical structure is the projection of the properties of the lexical types of words. In the earliest forms of categorical grammar types are functional/implicational and interact by the logical rule of Modus Ponens. In categorial grammar there are two traditions: the logical tradition that grew out of the work of Joachim Lambek, and the combinatory tradition associated with the work of Mark Steedman. The logical approach employs methods from mathematical logic and situates categorial grammars in the context of substructural logic. The combinatory approach emphasizes practical applicability to natural language processing and situates categorial grammars within extended rewriting systems. The logical tradition interprets the history of categorial grammar as comprising evolution and generalization of basic functional/implicational types into a rich categorial logic suited to the characterization of the syntax and semantics of natural language which is at once logical, formal, computational, and mathematical, reaching a level of formal explicitness not achieved in other grammar formalisms. This is the interpretation of the field that is being made in this article. This research has been partially supported by MINICO project TIN2017–89244-R. Thanks to Stepan Kuznetsov, Oriol Valentín and Sylvain Salvati for comments and suggestions. All errors and shortcomings are the author’s own.

The Logic of Pseudo-Uninorms and their Residua

Our method for density elimination is generalized to the non-commutative substructural logic GpsUL∗. Then the standard completeness of HpsUL∗follows as a lemma by virtue of previous workbyMetcalfeandMontagna. This result shows that HpsUL∗ is the logic of pseudo-uninorms and their residua and answered the question posed by Prof. Metcalfe, Olivetti, Gabbay and Tsinakis.

MODAL LOGIC WITHOUT CONTRACTION IN A METATHEORY WITHOUT CONTRACTION

Standard reasoning about Kripke semantics for modal logic is almost always based on a background framework of classical logic. Can proofs for familiar definability theorems be carried out using a nonclassical substructural logic as the metatheory? This article presents a semantics for positive substructural modal logic and studies the connection between frame conditions and formulas, via definability theorems. The novelty is that all the proofs are carried out with a noncontractive logic in the background. This sheds light on which modal principles are invariant under changes of metalogic, and provides (further) evidence for the general viability of nonclassical mathematics.

SUBSTRUCTURAL INQUISITIVE LOGICS

This paper shows that any propositional logic that extends a basic substructural logic BSL (a weak, nondistributive, nonassociative, and noncommutative version of Full Lambek logic with a paraconsistent negation) can be enriched with questions in the style of inquisitive semantics and logic. We introduce a relational semantic framework for substructural logics that enables us to define the notion of an inquisitive extension of λ, denoted as ${\lambda ^?}$, for any logic λ that is at least as strong as BSL. A general theory of these “inquisitive extensions” is worked out. In particular, it is shown how to axiomatize ${\lambda ^?}$, given the axiomatization of λ. Furthermore, the general theory is applied to some prominent logical systems in the class: classical logic Cl, intuitionistic logic Int, and t-norm based fuzzy logics, including for example Łukasiewicz fuzzy logic Ł. For the inquisitive extensions of these logics, axiomatization is provided and a suitable semantics found.

Canonical formulas via locally finite reducts and generalized dualities

The method of canonical formulas is a powerful tool for investigating intuitionistic and modal logics. In this talk I will discuss an algebraic approach to this method. I will mostly concentrate on the case of intuitionistic logic. But I will also review the case of modal logic and possible generalizations to substructural logic.