The Bulletin of the Section of Logic serves as a forum for disseminating research results on logical calculi—their methodology, applications, and algebraic interpretations. Although the published texts appear in a short and concise form, they represent some of the most demanding intellectual challenges available in our offer.
The following articles are included in Vol. 54 nr 2:
Cut-elimination and Normalization Theorems for Connexive Logics over Wansing’s C
Norihiro Kamide
Gentzen-style sequent calculi and Gentzen-style natural deduction systems are introduced for a family (C-family) of connexive logics over Wansing’s basic constructive connexive logic C. The C-family is derived from C by incorporating Peirce’s law, the law of excluded middle, and the generalized law of excluded middle. Natural deduction systems with general elimination rules are also introduced for the C-family. Theorems establishing the equivalence between the proposed sequent calculi and natural deduction systems are demonstrated. Cut-elimination and normalization theorems are established for the proposed sequent calculi and natural deduction systems, respectively. Additionally, similar results are obtained for a family (N-family) of paraconsistent logics over Nelson’s constructive four-valued logic N4.
Semantic Incompleteness of Liberman et al. (2020)’s Hilbert-style Systems for Term-modal Logics with Equality and Non-rigid Terms
Takahiro Sawasaki:
In this paper, we prove the semantic incompleteness of some expansions of the Hilbert-style system for the minimal normal term-modal logic with equality and non-rigid terms that were proposed in Liberman et al. (2020) “Dynamic Term-modal Logics for First-order Epistemic Planning.” Term-modal logic is a family of first-order modal logics having term-modal operators indexed with terms in the first-order language. While some first-order formula is valid over the corresponding class of frames in the involved Kripke semantics, it is not provable in those expansions. We show this fact by introducing a non-standard Kripke semantics which makes the meanings of constants and function symbols relative to the meanings of relation symbols combined with them. We also address an incorrect frame correspondence result given in Liberman et al. (2020).
Unified Sequent Calculi and Natural Deduction Systems for Until-free Linear-time Temporal Logics
Norihiro Kamide, Sara Negri:
A unified Gentzen-style proof-theoretic framework for until-free propositional linear-time temporal logic and its intuitionistic variant is introduced. The framework unifies Gentzen-style single-succedent sequent calculi and natural deduction systems for both the classical and intuitionistic versions of these temporal logics. Theorems establishing the equivalence between the proposed sequent calculi and natural deduction systems are proved. Furthermore, the cut-elimination theorems for the proposed sequent calculi and the normalization theorems for the proposed natural deduction systems are established.
Continua of Logics Related to Intuitionistic and Minimal Logics
Kaito Ichikura:
We analyze the relationship between logics around intuitionistic logic and minimal logic. We characterize the intersection of minimal logic and co-minimal logic introduced by Vakarelov, and reformulate logics given in the previous studies by Vakarelov, Bezhanishvili, Colacito, de Jongh, Vargas, and Niki in a uniform language. We also compare the new logic with other known logics in terms of the cardinalities of logics between them. Specifically, we apply Wronski’s algebraic semantics, instead of neighborhood semantics used in the previous studies, to show the existence of continua of logics between known logics and the new logic. This result is an extension of the conventional results, and the proof is given in a simpler way.
Preface: Non-Classical Logics. Theory and Applications (Part I)
Michał Zawidzki
Vol. 54 nr 3 features the following articles:
Some Results Concerning Axioms for Equivalential Calculus
Marcin Czakon
One of the most important questions in the area of the equivalential calculus (EC) currently is the issue of the single shortest axiom. We show some new a single organic and inorganic axioms for EC which are either D-complete or R-complete. We also present a number of two-element sets of axioms which posses some special properties. Two matrix are also discussed, which exclude two formulas from the set of potential 2MP-complete axioms.
Dialogical Ecumenism
Miguel Álvarez Lisboa
Ecumenical logics are systems where two logics can coexist, sharing vocabulary and avoiding collapses between them. The literature has focused mainly on ecumenism between classical and intuitionistic logic, and several calculi of Natural Deduction and Sequents have been proposed. In this paper I contribute to this project with a dialogical ecumenical system. This Game utilizes an extension of the intuitionistic structural rules that permits to handle classical disjunctions and conditionals. I show that this is indeed an ecumenical dialogical system, where classical formulas and intuitionistic formulas can be validated without collapses between them, and provide a philosophical defense of its design.
On Involutive Weak Exchange Algebras
Andrzej Walendziak
In this paper, involutive weak exchange algebras (for short, involutive WE algebras) are introduced and studied. Their properties and characterizations are investigated. Some important results and examples are given. In particular, it is proven that in involutive WE algebras, the properties (BB), (B), (*), (**) and (Tr) are equivalent. Moreover, involutive BE, involutive GE, involutive pre-BCK and involutive pre-Hilbert algebras are considered, their connections are established. It is shown that involutive WE algebras (respectively, involutive GE algebras) satisfying the commutative property are Wajsberg algebras (respectively, Boolean algebras). Finally, the interrelationships between the classes of involutive algebras considered here are presented.
From Translations to Non-Collapsing Logic Combinations
João Rasga, Cristina Sernadas
Prawitz suggested expanding a natural deduction system for intuitionistic logic to include rules for classical logic constructors, allowing both intuitionistic and classical elements to coexist without losing their inherent characteristics. Looking at the added rules from the point of view of the Gödel-Gentzen translation, led us to propose a general method for the coexistent combination of two logics when a conservative translation exists from one logic (the source) to another (the host). Then we prove that the combined logic is a conservative extension of the original logics, thereby preserving the unique characteristics of each component logic. In this way there is no collapse of one logic into the other in the combination. We also demonstrate that a Gentzen calculus for the combined logic can be induced from a Gentzen calculus for the host logic by considering the translation. This approach applies to semantics as well. We then establish a general sufficient condition for ensuring that the combined logic is both sound and complete. We apply these principles by combining classical and intuitionistic logics capitalizing on the Gödel-Gentzen conservative translation, intuitionistic and S4 modal logics relying on the Gödel-McKinsey-Tarski conservative translation, and classical and Jaśkowski’s paraconsistent logics taking into account the existence of a conservative translation.
On Generalization of Modular Lattices
Agnieszka Stocka
We introduce the concepts of dually balanced lattices and M-lattices and provide some basic properties of these classes of lattices. Both classes can be viewed as generalizations of the well-known class of modular lattices. In particular, we obtain analogues of the Kurosh-Ore theorem for dually balanced lattices and the Jordan-Hölder theorem for M-lattices. Furthermore, we investigate the behaviour of several invariants, including the hollow dimension and the Kurosh-Ore dimension in dually balanced lattices, as well as the maximal dimension in M-lattices.
Together, these two issues highlight the diversity and depth of contemporary research in non-classical logics—from proof theory and algebraic semantics to dialogical approaches and lattice theory. They showcase both technical advances and broader philosophical perspectives, offering readers a rich panorama of current developments in the field.
We invite you to explore both full issues. You can find the complete editions HERE and HERE.
