This issue features six articles addressing key topics in algebraic logic, classical and non-classical semantics, and relationships between different proof systems. The authors introduce new algebraic constructions and revisit classical theorems, opening new directions for research on logical structures.
In this issue you will find:
A Study of a Special Semi Maximal Filters in BL−algebras
Motahareh Atabaki, Mahta Bedrood, Fereshteh Forouzesh
In this article, a specific type of semi maximal filters is introduced, which form a lattice structure. These filters are called J- semi maximal and NJ-semi maximal, and their key properties in BL-algebras are analyzed. Additionally, these special filters are compared with other defined filters, particularly semi maximal and maximal filters. The purpose of this article is to provide a new analysis of filters in BL-algebras.
Topology on Equality Algebras
Mona Aaly Kologani, Sogol Niazian, Rajab Ali Borzooei
In this paper, by special upsets on equality algebras, we construct a topology on bounded equality algebras and investigate some of their topological properties, such as Hausdorff, T0-space, T1-space and disconnected. In addition, we express the relation between closed and compact sets in this topology. Moreover, by considering the binary operation ↠ and constructing a topology on the bounded equality algebra E, we introduce the notion of semi-topological algebra and prove that any involutive equality algebra is a right semi-topological algebra and by some conditions it can be a semi-⋏-topological algebra. Also, we show that it is not necessarily a left semi-topological algebra. Finally, we investigate converse image, product and quotient topology on equality algebra and show that under what condition we can make finer topology.
Revisiting the Adequacy Theorem for Fragments of Łukasiewicz Logic
Miguel Pérez-Gaspar, Juan Manuel Ramírez-Contreras, Juan Sebastián Slagter
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.
Matrix Semantics for Classical Logic: The Case of the Lattice O6
Ela Drozdowska
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.
Modal Logic of Lattices
Maciej A. Hałapacz
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.
Proof Translations between Label-free and Labeled Sequent Calculi in ISCI
Didier Galmiche, Brandon Hornbeck, Daniel Méry
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.
“Bulletin of the Section of Logic” remains one of the leading international journals devoted to formal logic, proof theory, and semantics, providing a long-standing platform for research in non-classical, algebraic, and modal logics.
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