A Cut-Free Sequent Calculus for Defeasible Erotetic Inferences Abstract In recent years, the effort to formalize erotetic inferences—i.e., inferences to and from questions—has become a central concern for those working in erotetic logic. However, few have sought to formulate a proof theory for these inferences. To fill this lacuna, we construct a calculus for (classes of) sequents that are sound and complete for two species of erotetic inferences studied by Inferential Erotetic Logic (IEL): erotetic evocation and erotetic implication. While an effort has been made to axiomatize the former in a sequent system, there is currently no proof theory for the latter. Moreover, the extant axiomatization of erotetic evocation fails to capture its defeasible character and provides no rules for introducing or eliminating question-forming operators. In contrast, our calculus encodes defeasibility conditions on sequents and provides rules governing the introduction and elimination of erotetic formulas. We demonstrate that an elimination theorem holds for a version of the cut rule that applies to both declarative and erotetic formulas and that the rules for the axiomatic account of question evocation in IEL are admissible in our system.

Valuations: Bi, Tri, and Tetra Abstract This paper considers some issues to do with valuational presentations of consequence relations, and the Galois connections between spaces of valuations and spaces of consequence relations. Some of what we present is known, and some even well-known; but much is new. The aim is a systematic overview of a range of results applicable to nonreflexive and nontransitive logics, as well as more familiar logics. We conclude by considering some connectives suggested by this approach.

Categorical Equivalence Between $$\varvec{PMV}_{\varvec{f}}$$ PMV f -Product Algebras and Semi-Low $$\varvec{f}_{\varvec{u}}$$ f u -Rings Abstract An explicit categorical equivalence is defined between a proper subvariety of the class of \({ PMV}\) -algebras, as defined by Di Nola and Dvurečenskij, to be called \({ PMV}_{f}\) -algebras, and the category of semi-low \(f_u\) -rings. This categorical representation is done using the prime spectrum of the \({ MV}\) -algebras, through the equivalence between \({ MV}\) -algebras and \(l_u\) -groups established by Mundici, from the perspective of the Dubuc–Poveda approach, that extends the construction defined by Chang on chains. As a particular case, semi-low \(f_u\) -rings associated to Boolean algebras are characterized.

Translations Between Gentzen–Prawitz and Jaśkowski–Fitch Natural Deduction Proofs Abstract Two common forms of natural deduction proof systems are found in the Gentzen–Prawitz and Jaśkowski–Fitch systems. In this paper, I provide translations between proofs in these systems, pointing out the ways in which the translations highlight the structural rules implicit in the systems. These translations work for classical, intuitionistic, and minimal logic. I then provide translations for classical S4 proofs.

Disjunction and Existence Properties in Inquisitive First-Order Logic Abstract Classical first-order logic \(\texttt {FO}\) is commonly used to study logical connections between statements, that is sentences that in every context have an associated truth-value. Inquisitive first-order logic \(\texttt {InqBQ}\) is a conservative extension of \(\texttt {FO}\) which captures not only connections between statements, but also between questions. In this paper we prove the disjunction and existence properties for \(\texttt {InqBQ}\) relative to inquisitive disjunction and inquisitive existential quantifier \(\overline{\exists }\) . Moreover we extend these results to several families of theories, among which the one in the language of \(\texttt {FO}\) . To this end, we initiate a model-theoretic approach to the study of \(\texttt {InqBQ}\) . In particular, we develop a toolkit of basic constructions in order to transform and combine models of \(\texttt {InqBQ}\) .

On Tarski’s Axiomatization of Mereology Abstract It is shown how Tarski’s 1929 axiomatization of mereology secures the reflexivity of the ‘part of’ relation. This is done with a fusion-abstraction principle that is constructively weaker than that of Tarski; and by means of constructive and relevant reasoning throughout. We place a premium on complete formal rigor of proof. Every step of reasoning is an application of a primitive rule; and the natural deductions themselves can be checked effectively for formal correctness.

The Balanced Pseudocomplemented Ockham Algebras with the Strong Endomorphism Kernel Property Abstract An endomorphism on an algebra \({\mathcal {A}}\) is said to be strong if it is compatible with every congruence on \({\mathcal {A}}\) ; and \({\mathcal {A}}\) is said to have the strong endomorphism kernel property if every congruence on \({\mathcal {A}}\) , other than the universal congruence, is the kernel of a strong endomorphism on \({\mathcal {A}}\) . Here we characterise the structure of Ockham algebras with balanced pseudocomplementation those that have this property via Priestley duality.

Completeness in Equational Hybrid Propositional Type Theory Abstract Equational hybrid propositional type theory ( \(\mathsf {EHPTT}\) ) is a combination of propositional type theory, equational logic and hybrid modal logic. The structures used to interpret the language contain a hierarchy of propositional types, an algebra (a nonempty set with functions) and a Kripke frame. The main result in this paper is the proof of completeness of a calculus specifically defined for this logic. The completeness proof is based on the three proofs Henkin published last century: (i) Completeness in type theory, (ii) The completeness of the first-order functional calculus and (iii) Completeness in propositional type theory. More precisely, from (i) and (ii) we take the idea of building the model described by the maximal consistent set; in our case the maximal consistent set has to be named, \(\Diamond \) -saturated and extensionally algebraic-saturated due to the hybrid and equational nature of \(\mathsf {EHPTT}\) . From (iii), we use the result that any element in the hierarchy has a name. The challenge was to deal with all the heterogeneous components in an integrated system.

Varieties of BL-Algebras III: Splitting Algebras Abstract In this paper we investigate splitting algebras in varieties of logics, with special consideration for varieties of BL-algebras and similar structures. In the case of the variety of all BL-algebras a complete characterization of the splitting algebras is obtained.

Subminimal Logics in Light of Vakarelov’s Logic Abstract We investigate a subsystem of minimal logic related to D. Vakarelov’s logic \(\mathbf {SUBMIN}\) , using the framework of subminimal logics by A. Colacito, D. de Jongh and A. L. Vargas. In the course of it, the relationship between the two semantics in the respective frameworks is clarified. In addition, we introduce a sequent calculus for the investigated subsystem, and some proof-theoretic properties are established. Lastly, we formulate a new infinite class of subsystems of minimal logics.