http://hdl.handle.net/11089/56940 2026-04-06T20:39:08Z http://hdl.handle.net/11089/56972 From Translations to Non-Collapsing Logic Combinations Rasga, João; Sernadas, Cristina 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. 2025-11-28T00:00:00Z http://hdl.handle.net/11089/56973 On Generalization of Modular Lattices Stocka, Agnieszka 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. 2025-11-28T00:00:00Z http://hdl.handle.net/11089/56971 On Involutive Weak Exchange Algebras Walendziak, Andrzej 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. 2025-11-28T00:00:00Z http://hdl.handle.net/11089/56969 Some Results Concerning Axioms for Equivalential Calculus Czakon, Marcin 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. 2025-07-02T00:00:00Z