The one-variable fragment of a first-order logic may be viewed as an “S5-like” modal logic, where the universal and existential quantifiers are replaced by box and diamond modalities, respectively. Axiomatizations of these modal logics have been obtained for special cases—notably, the modal counterparts $\mathrm {S5}$ and $\mathrm {MIPC}$ of the one-variable fragments of first-order classical logic and first-order intuitionistic logic, respectively—but a general approach, extending beyond first-order intermediate logics, has been lacking. To this end, a sufficient criterion is given in this paper for the one-variable fragment of a semantically defined first-order logic—spanning families of intermediate, substructural, many-valued, and modal logics—to admit a certain natural axiomatization. More precisely, an axiomatization is obtained for the one-variable fragment of any first-order logic based on a variety of algebraic structures with a lattice reduct that has the superamalgamation property, using a generalized version of a functional representation theorem for monadic Heyting algebras due to Bezhanishvili and Harding. An alternative proof-theoretic strategy for obtaining such axiomatization results is also developed for first-order substructural logics that have a cut-free sequent calculus and admit a certain interpolation property.

We show that Kozen and Tiuryn’s substructural logic of partial correctness $\mathsf{S}$ embeds into the equational theory of Kleene algebra with domain, $\mathsf{KAD}$. We provide an implicational formulation of $\mathsf{KAD}$ which sets $\mathsf{S}$ in the context of implicational extensions of Kleene algebra.

We introduce a sequent system which is Gentzen algebraisable with orthomodular lattices as equivalent algebraic semantics, and therefore can be viewed as a calculus for orthomodular quantum logic. Its sequents are pairs of non-associative structures, formed via a structural connective whose algebraic interpretation is the Sasaki product on the left-hand side and its De Morgan dual on the right-hand side. It is a substructural calculus, because some of the standard structural sequent rules are restricted—by lifting all such restrictions, one recovers a calculus for classical logic.

Lawvere hyperdoctrines give categorical algebraic semantics for intuitionistic predicate logic. Here we extend the hyperdoctrinal semantics to a broad variety of substructural predicate logics over the Typed Full Lambek Calculus, verifying their completeness with respect to the extended hyperdoctrinal semantics. This yields uniform hyperdoctrinal completeness results for numerous logics such as different types of relevant predicate logics and beyond, which are new results on their own; i.e., we give uniform categorical semantics for a broad variety of non-classical predicate logics. And we introduce an analogue of Lawvere–Tierney topology and cotopology in the hyperdoctrinal setting, which gives a unifying perspective on different logical translations, in particular allowing for a uniform treatment of Girard’s exponential translation between linear and intuitionistic logics and of Kolmogorov’s double negation translation between intuitionistic and classical logics. In the hyerdoctrinal conception, type theories are categories, logics over type theories are functors, and logical translations between them, then, are natural transformations, in particular Lawvere–Tierney topologies and cotopologies on hyperdoctrines. The view of logical translations as hyperdoctrinal Lawvere–Tierney topologies and cotopologies has not been elucidated before, and may be seen as a novel contribution of the present work. From a broader perspective, this work may be regarded as taking first steps towards interplay between algebraic and categorical logics; it is, technically, a combination of substructural (or Lambekian) algebraic logic and hyperdoctrinal (or Lawverian) categorical logic, as the hyperdoctrinal completeness theorem is shown via the integration of the Lindenbaum–Tarski algebra construction with the syntactic category construction. As such this work lays a foundation for further interactions between algebraic and categorical logics.

All known structural extensions of the substructural logic $\textbf{FL}_{\textbf{e}}$ , the Full Lambek calculus with exchange/commutativity (corresponding to subvarieties of commutative residuated lattices axiomatized by $\{\vee , \cdot , 1\}$ -equations), have decidable theoremhood; in particular all the ones defined by knotted axioms enjoy strong decidability properties (such as the finite embeddability property). We provide infinitely many such extensions that have undecidable theoremhood, by encoding machines with undecidable halting problem. An even bigger class of extensions is shown to have undecidable deducibility problem (the corresponding varieties of residuated lattices have undecidable word problem); actually with very few exceptions, such as the knotted axioms and the other prespinal axioms, we prove that undecidability is ubiquitous. Known undecidability results for non-commutative extensions use an encoding that fails in the presence of commutativity, so and-branching counter machines are employed. Even these machines provide encodings that fail to capture proper extensions of commutativity, therefore we introduce a new variant that works on an exponential scale. The correctness of the encoding is established by employing the theory of residuated frames.

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.

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.

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.

The seventeenth century can be viewed as an era of (closely related) innovation in the formal and natural sciences and of paradigmatic diversity in philosophy (due to the coexistence of at least the humanist, the late scholastic, and the early modern tradition). Within this environment, the present study focuses on scholastic logic and, in particular, syllogistic. In seventeenth-century scholastic logic two different approaches to logic can be identified, one represented by the Dominicans Báñez, Poinsot, and Comas del Brugar, the other represented by the Jesuits Hurtado, Arriaga, Oviedo, and Compton. These two groups of authors can be contrasted in three prominent features. First, in the role of the theory of validity, which is either a common basis for all particular theories (in this case, sentential logic and syllogistic), or a set of observations regarding a particular theory (in this case, syllogistic). Second, in the view of syllogistic, which is either an implication of a general theory of validity and a semantics of terms, or an algebra of structured objects. Third, in the role of the scholastic analysis of language in terms of suppositio, which either is a semantic underpinning of syllogistic, or it is replaced by a semantics of propositions.

Graph normal form, GNF, [1], was used in [2, 3] for analyzing paradoxes in propositional discourses, with the semantics—equivalent to the classical one—defined by kernels of digraphs. The paper presents infinitary, resolution-based reasoning with GNF theories, which is refutationally complete for the classical semantics. Used for direct (not refutational) deduction it is not explosive and allows to identify in an inconsistent discourse, a maximal consistent subdiscourse with its classical consequences. Semikernels, generalizing kernels, provide the semantic interpretation.

We prove that the set of formulae provable in the full Lambek calculus with the structural rule of contraction is undecidable. In fact, we show that the positive fragment of this logic is undecidable.

It is well known that classical propositional logic can be interpreted in intuitionistic prepositional logic. In particular Glivenko's theorem states that a formula is provable in the former iff its double negation is provable in the latter. We extend Glivenko's theorem and show that for every involutive substructural logic there exists a minimum substructural logic that contains the first via a double negation interpretation. Our presentation is algebraic and is formulated in the context of residuated lattices. In the last part of the paper, we also discuss some extended forms of the Koltnogorov translation and we compare it to the Glivenko translation.