http://hdl.handle.net/11089/9983 2026-07-07T19:10:13Z 2026-07-07T19:10:13Z Drozdowska, Ela http://hdl.handle.net/11089/58695 2026-07-03T01:20:03Z 2026-06-10T00:00:00Z

Matrix Semantics for Classical Logic: The Case of the Lattice O6 Drozdowska, Ela It is well established that classical propositional logic is Boolean. However, this view has recently been challenged. In their paper Non-Orthomodular Models for Both Standard Quantum Logic and Standard Classical Logic: Repercussions for Quantum Computers, Mladen Pavic̆ić and Norman Megill present a non-distributive, non-orthomodular model for both classical and quantum logic based on lattice O6, and argue that classical propositional logic is non-distributive.In this paper, we examine this claim. Pavic̆ić and Megill’s model is formulated within unital matrix semantics rather than as an algebraic model in the sense of Abstract Algebraic Logic. An analysis of the lattice O6 in the framework of matrix semantics reveals that the matrix (O6,{1,a,b}) is adequate for CL, but not reduced, and induces the same consequence relation as the two-element Boolean matrix B2. Similarly, the unital matrix (O6,{1}) is adequate for CL through reduction to the four-element Boolean matrix B4. Furthermore, we present two lattice constructions that yield matrix models for CL lacking nontrivial lattice-theoretic properties.These results show that the adequacy of O6 is not intrinsic to its algebraic structure, but is inherited from its reducibility to Boolean matrices, and more generally that classical logic admits models with highly unconstrained lattice structure. Consequently, the existence of such non-distributive models does not undermine the distributive character of classical propositional logic.

2026-06-10T00:00:00Z Hałapacz, Maciej A. http://hdl.handle.net/11089/58696 2026-07-03T01:20:01Z 2026-06-10T00:00:00Z

Modal Logic of Lattices Hałapacz, Maciej A. We prove that the modal logic of lattices with the accessibility relation of being isomorphic to a sublattice is S4.2. The same is proven for modular and distributive lattices.

2026-06-10T00:00:00Z Galmiche, Didier Hornbeck, Brandon Méry, Daniel http://hdl.handle.net/11089/58697 2026-07-03T01:20:00Z 2026-06-29T00:00:00Z

Proof Translations between Label-free and Labeled Sequent Calculi in ISCI Galmiche, Didier; Hornbeck, Brandon; Méry, Daniel In this paper we consider the Intuitionistic Sentential Calculus with Identity (ISCI). We study two main families of sequent calculi. The first one, called G3ISCI, is based on a label-free multi-succedent sequent calculus that is sound and complete w.r.t. Kripke models and the second, called L3ISCI, is based on a multi-succedent labeled sequent calculus that is sound and complete w.r.t. Beth models. Our goal is to investigate how the calculi, that capture distinct semantics of the logic, relate to each other through proof translations. Proof translations from G3ISCI to L3ISCI provide new results about the soundness and (cut-free) completeness of G3ISCI w.r.t. Beth models. Proof translations from L3ISCI to G3ISCI are more difficult and require the definition of new calculi for ISCIthat provide intermediate steps in the translation process.

2026-06-29T00:00:00Z Pérez-Gaspar, Miguel Ramírez-Contreras, Juan Manuel Slagter, Juan Sebastián http://hdl.handle.net/11089/58694 2026-07-03T01:20:04Z 2026-06-10T00:00:00Z

Revisiting the Adequacy Theorem for Fragments of Łukasiewicz Logic Pérez-Gaspar, Miguel; Ramírez-Contreras, Juan Manuel; Slagter, Juan Sebastián A. V. Figallo introduced the 3-valued Super Łukasiewicz logic expanded with the Δ operator, denoted as C3↣,Δ, in 1990. This operator is used in the definition of 3-valued Łukasiewicz algebras, and it is not possible to recover Δ through implication and top in Super Łukasiewicz logic. On the other hand, Baaz introduced the Δ operator in Gödel logic, both in its propositional and quantified versions. Subsequently, this operator was extensively studied in the field of fuzzy logic.In this paper, we prove a strong version of the Adequacy Theorem for C3↣,Δ3. As a consequence, we demonstrate that the Deduction Theorem does not hold in this calculus. Furthermore, we introduce the first-order version of  C3↣,Δ3 and establish soundness and completeness results by adapting a recently developed algebraic technique. In this context, our presentation differs from others in the literature because we need to construct a special homomorphism, brought from the algebraic study of  C3↣,Δ3, in the syntactic setting. This homomorphism is also necessary to determine the generating algebras. While we can ascertain that the logical system is algebraizable by a (quasi-)variety of algebras, we cannot know a priori which are the subdirectly irreducible algebras.

2026-06-10T00:00:00Z