Bulletin of the Section of Logic | Tom 51 Numer 3

Opublikowano: 15 listopada 2022
BSL

Bulletin of the Section of Logic (BSL) jest kwartalnikiem naukowym Wydawnictwa Uniwersytetu Łódzkiego. Założone w 1972 przez Ryszarda Wójcickiego (przewodniczącego sekcji logiki Polskiej Akademii Nauk), czasopismo stanowi forum służące rozpowszechnianiu wyników badań dotyczących kalkulacji logicznych; ich metodologii, zastosowań i interpretacji algebraicznych – zawsze w krótkiej i zwięzłej formie.

Redaktorem naczelnym BSL jest Kierownik Katedry Logiki i Metodologii Nauk w Instytucie Filozofii na Wydziale Filozoficzno-Historycznym Uniwersytetu Łódzkiego – Profesor Andrzej Indrzejczak. Uznany autor wielu prac dotyczących teorii dowodu i logik nieklasycznych pozyskał również w tym roku prawie dwa miliony euro dofinansowania z grantu Europejskiej Rady Ds. Badań Naukowych. Realizowany w Centrum Filozofii Przyrody UŁ Advanced Grant ERC, Coming to Terms: Proof Theory Extended to Definite Descriptions and other Terms będzie miał niebagatelne znaczenie w badaniach sztucznej inteligencji (AI). Celem projektu jest dostosowanie narzędzi logiki formalnej do struktury języków naturalnych, w których złożone wyrażenia nazwowe odgrywają kluczową rolę jako nośniki informacji.

Rezultaty badań prowadzonych w projekcie poznamy za pięć lat, jednak najnowszy – trzeci numer Bulletin of the Section of Logic z roku 2022 dostępny jest już dziś na platformie czasopism Wydawnictwa Uniwersytetu Łódzkiego.

W numerze:

Interpolation Property on Visser’s Formal Propositional Logic

Majid Alizadeh, Masoud Memarzadeh

 In this paper by using a model-theoretic approach, we prove Craig interpolation property for Formal Propositional Logic, FPL, Basic propositional logic, BPL and the uniform left-interpolation property for FPL. We also show that there are countably infinite extensions of FPL with the uniform interpolation property.

First-Order Modal Semantics and Existence Predicate

Patryk Michalczenia

In the article we study the existence predicate ε in the context of semantics for first-order modal logic. For a formula φ we define φε – the so called existence relativization. We point to a gap in the work of Fitting and Mendelsohn concerning the relationship between the truth of φ and φε in classes of varying- and constant-domain models. We introduce operations on models which allow us to fill the gap and provide a more general perspective on the issue. As a result we obtain a series of theorems describing the logical connection between the notion of truth of a formula with the existence predicate in constant-domain models and the notion of truth of a formula without the existence predicate in varying-domain models.

Categorical Dualities for Some Two Categories of Lattices: An Extended Abstract

Wiesław Dziobiak, Marina Schwidefsky

The categorical dualities presented are: (first) for the category of bi-algebraic lattices that belong to the variety generated by the smallest non-modular lattice with complete (0,1)-lattice homomorphisms as morphisms, and (second) for the category of non-trivial (0,1)-lattices belonging to the same variety with (0,1)-lattice homomorphisms as morphisms. Although the two categories coincide on their finite objects, the presented dualities essentially differ mostly but not only by the fact that the duality for the second category uses topology. Using the presented dualities and some known in the literature results we prove that the Q-lattice of any non-trivial variety of (0,1)-lattices is either a 2-element chain or is uncountable and non-distributive.

Unification and Finite Model Property for Linear Step-Like Temporal Multi-Agent Logic with the Universal Modality

Stepan I. Bashmakov, Tatyana Yu. Zvereva

This paper proposes a semantic description of the linear step-like temporal multi-agent logic with the universal modality LTK.slU based on the idea of non-reflexive non-transitive nature of time. We proved a finite model property and projective unification for this logic.

Constructing a Hoop Using Rough Filters

Rajab Ali Borzooei, Elham Babaei

When it comes to making decisions in vague problems, rough is one of the best tools to help analyzers. So based on rough and hoop concepts, two kinds of approximations (Lower and Upper) for filters in hoops are defined, and then some properties of them are investigated by us. We prove that these approximations- lower and upper- are interior and closure operators, respectively. Also after defining a hyper operation in hoops, we show that by using this hyper operation, set of all rough filters is monoid. For more study, we define the implicative operation on the set of all rough filters and prove that this set with implication and intersection is made a hoop.

An (α,β)-Hesitant Fuzzy Set Approach to Ideal Theory in Semigroups

Pairote Yiarayong

The aim of this manuscript is to introduce the (α,β)-hesitant fuzzy set and apply it to semigroups. In this paper, as a generalization of the concept of hesitant fuzzy sets to semigroup theory, the concept of (α,β)-hesitant fuzzy subsemigroups of semigroups is introduced, and related properties are discussed. Furthermore, we define and study (α,β)-hesitant fuzzy ideals on semigroups. In particular, we investigate the structure of (α,β)-hesitant fuzzy ideal generated by a hesitant fuzzy ideal in a semigroup. In addition, we also introduce the concepts of (α,β)-hesitant fuzzy semiprime sets of semigroups, and characterize regular semigroups in terms of (α,β)-hesitant fuzzy left ideals and (α,β)-hesitant fuzzy right ideals. Finally, several characterizations of regular and intra-regular semigroups by the properties of (α,β)-hesitant ideals are given.

Complete Representations and Neat Embeddings

Tarek Sayed Ahmed

Let 2<n<ω. Then CAn denotes the class of cylindric algebras of dimension n, RCAn denotes the class of representable CAns, CRCAn denotes the class of completely representable CAns, and NrnCAω(⊆CAn) denotes the class of n-neat reducts of CAωs. The elementary closure of the class CRCAns (Kn) and the non-elementary class At(NrnCAω) are characterized using two-player zero-sum games, where At is the operator of forming atom structures. It is shown that Kn is not finitely axiomatizable and that it coincides with the class of atomic algebras in the elementary closure of ScNrnCAω where Sc is the operation of forming complete subalgebras. For any class L such that AtNrnCAω⊆L⊆AtKn, it is proved that SPCmL=RCAn, where Cm is the dual operator to At; that of forming complex algebras. It is also shown that any class K between CRCAn∩SdNrnCAω and ScNrnCAn+3 is not first order definable, where Sd is the operation of forming dense subalgebras, and that for any 2<n<m, any l≥n+3 any any class K (such that At(NrnCAm)∩CRCAn⊆K⊆AtScNrnCAl, K is not not first order definable either.

 

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