Konspekt wykładów z logiki wygłoszonych na Uniwersytecie Jana Kazimierza we Lwowie w roku akademickim 1924/1925 sporządzony przez Stefana Ołeksiuka

Author(s): Kazimierz Ajdukiewicz / Language(s): Polish Issue: 2 (98)/2017

We present the oldest known synopsis of Kazimierz Ajdukiewicz’s lectures. The lectures recorded by Stefan Oleksiuk mainly concern the part of logic called, in Polish terminology, “logical semiotics” (application of formal logic to matters of natural language). Ajdukiewicz discusses several important philosophical issues, such as the problem of meaning, representation, the status of general names, the truth-value of sentences or propositions, and identity.

Tipuri de argumentare greşită în perioada Covid-19

Author(s): Marius Dobre / Language(s): Romanian Issue: 4/2021

In the following, we propose an analysis of some of the most common types of argumentation (i.e. arguments that could be generic examples, which account for certain categories of arguments) coming from two currents of thought, called here radical skepticism and protectionist propaganda. The arguments presented below were mainly collected from the Romanian press (in the form of advertisements, propaganda spots, etc.) and from private discussions with various people from both camps.

Omitting Types in Fragments and Extensions of First Order Logic

Author(s): Tarek Sayed Ahmed / Language(s): English Issue: 3/2021

Fix 2< n <w. Let Ln denote first order logic restricted to the first n variables. Using the machinery of algebraic logic, positive and negative results on omitting types are obtained for Ln and for infinitary variants and extensions of Lw,w.

A Note on 3×3-valued Łukasiewicz Algebras with Negation

Author(s): Carlos Gallardo,Alicia Ziliani / Language(s): English Issue: 3/2021

Tense Operators on BL-algebras and Their Applications

Author(s): Akbar Paad / Language(s): English Issue: 3/2021

In this paper, the notions of tense operators and tense lters in BL-algebras are introduced and several characterizations of them are obtained. Also, the relation among tense BL-algebras, tense MV -algebras and tense Boolean algebras are investigated. Moreover, it is shown that the set of all tense lters of a BL-algebra is complete sublattice of F(L) of all filters of BL-algebra L. Also, maximal tense filters and simple tense BL-algebras and the relation between them are studied. Finally, the notions of tense congruence relations in tense BL-algebras and strict tense BL-algebras are introduced and an one-to-one correspondence between tense filters and tense congruences relations induced by tense filters are provided.

A Note on Gödel-Dummet Logic LC

Author(s): Gemma Robles ,José M. Méndez / Language(s): English Issue: 3/2021

Falling Shadow Theory with Applications in Hoops

Author(s): Rajab Ali Borzooei,Gholam Reza Rezaei,Mona Aaly Kologhani,Young Bae Jun / Language(s): English Issue: 3/2021

The falling shadow theory is applied to subhoops and filters in hoops. The notions of falling fuzzy subhoops and falling fuzzy filters in hoops are introduced, and several properties are investigated. Relationship between falling fuzzy subhoops and falling fuzzy filters are discussed, and conditions for a falling fuzzy subhoop to be a falling fuzzy filter are provided. Also conditions for a falling shadow of a random set to be a falling fuzzy filter are displayed.

A General Model of Neutrosophic Ideals in BCK/BCI-algebras Based on Neutrosophic Points

Author(s): Rajab Ali Borzooei,Hashem Bordbar,Florentin Smarandache,Young Bae Jun / Language(s): English Issue: 3/2021

More general form of (∈, ∈ ∨q)-neutrosophic ideal is introduced, and their properties are investigated. Relations between (∈, ∈)-neutrosophic ideal and (∈, ∈ ∨q(kT ,kI ,kF ))-neutrosophic ideal are discussed. Characterizations of (∈, ∈∨q(kT ,kI,kF ))-neutrosophic ideal are discussed, and conditions for a neutrosophic set to be an (∈, ∈∨q(kT ,kI ,kF ))-neutrosophic ideal are displayed.

Neighbourhood Semantics for Graded Modal Logic

Author(s): Hans van Ditmarsch,Jinsheng Chen,Giuseppe Greco,Apostolos Tzimoulis / Language(s): English Issue: 3/2021

We introduce a class of neighbourhood frames for graded modal logic embedding Kripke frames into neighbourhood frames. This class of neighbourhood frames is shown to be first-order definable but not modally definable. We also obtain a new definition of graded bisimulation with respect to Kripke frames by modifying the definition of monotonic bisimulation.

Some cardinal characteristics related to the covering number and the uniformity of the meagre ideal

Author(s): Nattapon Sonpanow,Pimpen Vejjajiva / Language(s): English Issue: 56/2021

We extend the concepts of splitting, reaping, and independent families to families of functions and permutations on ω and define associated cardinal characteristics sf , sp, rf , rp, if , and ip. We study relationships among cov(M), non(M), and these cardinals. In this paper, we show that sf = non(M) = sp, rf = cov(M) ≤ rp, and cov(M) ≤ if , ip.

On the relation of negations in Nelson algebras

Author(s): Conrado Gomez,Miguel Andres Marcos,Hernan J. San Martin / Language(s): English Issue: 56/2021

The aim of this paper is to investigate the relation between the strong and the "weak" or intuitionistic negation in Nelson algebras. To do this, we define the variety of Kleene algebras with intuitionistic negation and explore the Kalman's construction for pseudocomplemented distributive lattices. We also study the centered algebras of this variety.

Tableau-based translation from first-order logic to modal logic

Author(s): Tin Perkov,Luka Mikec / Language(s): English Issue: 56/2021

We define a procedure for translating a given first-order formula to an equivalent modal formula, if one exists, by using tableau-based bisimulation invariance test. A previously developed tableau procedure tests bisimulation invariance of a given first-order formula, and therefore tests whether that formula is equivalent to the standard translation of some modal formula. Using a closed tableau as the starting point, we show how an equivalent modal formula can be effectively obtained.

A note on Humberstone's constant Ω

Author(s): Satoru Niki,Hitoshi Omori / Language(s): English Issue: 56/2021

We investigate an expansion of positive intuitionistic logic obtained by adding a constant Ω introduced by Lloyd Humberstone. Our main results include a sound and strongly complete axiomatization, some comparisons to other expansions of intuitionistic logic obtained by adding actuality and empirical negation, and an algebraic semantics. We also brie y discuss its connection to classical logic.

Embeddability Between Orderings and GCH

Author(s): Rodrigo A. Freire / Language(s): English Issue: 56/2021

We provide some statements equivalent in ZFC to GCH, and also to GCH above a given cardinal. These statements express the validity of the notions of replete and well-replete car- dinals, which are introduced and proved to be specially relevant to the study of cardinal exponentiation. As a byproduct, a structure theorem for linear orderings is proved to be equivalent to GCH: for every linear ordering L, at least one of L and its converse is universal for the smaller well-orderings.

Corrigendum to "On Definable Completeness for Ordered Fields" RML, 54 (2019), 95 - 100

Author(s): Mojtaba Moniri / Language(s): English Issue: 56/2021

The (Greatest) Fragment of Classical Logic that Respects the Variable-Sharing Principle (in the FMLA-FMLA Framework)

Author(s): Damian E. Szmuc / Language(s): English Issue: 4/2021

We examine the set of formula-to-formula valid inferences of Classical Logic, where the premise and the conclusion share at least a propositional variable in common. We review the fact, already proved in the literature, that such a system is identical to the first-degree entailment fragment of R. Epstein's Relatedness Logic, and that it is a non-transitive logic of the sort investigated by S. Frankowski and others. Furthermore, we provide a semantics and a calculus for this logic. The semantics is defined in terms of a p-matrix built on top of a 5-valued extension of the 3-element weak Kleene algebra, whereas the calculus is defined in terms of a Gentzen-style sequent system where the left and right negation rules are subject to linguistic constraints.

A Sound Interpretation of Leśniewski's Epsilon in Modal Logic KTB

Author(s): Takao Inoue / Language(s): English Issue: 4/2021

In this paper, we shall show that the following translation IM from the propositional fragment L1 of Leśniewski's ontology to modal logic KTB is sound: for any formula ϕ and ψ of L1, it is defined as(M1) IM(ϕ∨ψ)=IM(ϕ)∨IM(ψ),(M2) IM(¬ϕ)=¬IM(ϕ),(M3) IM(ϵab)=◊pa⊃pa.∧.◻pa⊃◻pb.∧.◊pb⊃pa,where pa and pb are propositional variables corresponding to the name variables a and b, respectively. In the last, we shall give some comments including some open problems and my conjectures.

On Complete Representations and Minimal Completions in Algebraic Logic, Both Positive and Negative Results

Author(s): Tarek Sayed Ahmed / Language(s): English Issue: 4/2021

Fix a finite ordinal n≥3 and let α be an arbitrary ordinal. Let CAn denote the class of cylindric algebras of dimension n and RA denote the class of relation algebras. Let PAα(PEAα) stand for the class of polyadic (equality) algebras of dimension α. We reprove that the class CRCAn of completely representable CAns, and the class CRRA of completely representable RAs are not elementary, a result of Hirsch and Hodkinson. We extend this result to any variety V between polyadic algebras of dimension n and diagonal free CAns. We show that that the class of completely and strongly representable algebras in V is not elementary either, reproving a result of Bulian and Hodkinson. For relation algebras, we can and will, go further. We show the class CRRA is not closed under ≡∞,ω. In contrast, we show that given α≥ω, and an atomic A∈PEAα, then for any n<ω, NrnA is a completely representable PEAn. We show that for any α≥ω, the class of completely representable algebras in certain reducts of PAαs, that happen to be varieties, is elementary. We show that for α≥ω, the the class of polyadic-cylindric algebras dimension α, introduced by Ferenczi, the completely representable algebras (slightly altering representing algebras) coincide with the atomic ones. In the last algebras cylindrifications commute only one way, in a sense weaker than full fledged commutativity of cylindrifications enjoyed by classical cylindric and polyadic algebras. Finally, we address closure under Dedekind-MacNeille completions for cylindric-like algebras of dimension n and PAαs for α an infinite ordinal, proving negative results for the first and positive ones for the second.

An Arithmetically Complete Predicate Modal Logic

Author(s): Yunge Hao,George Tourlakis / Language(s): English Issue: 4/2021

This paper investigates a first-order extension of GL called ML3. We outline briefly the history that led to ML3, its key properties and some of its toolbox: the \emph{conservation theorem}, its cut-free Gentzenisation, the ``formulators'' tool. Its semantic completeness (with respect to finite reverse well-founded Kripke models) is fully stated in the current paper and the proof is retold here. Applying the Solovay technique to those models the present paper establishes its main result, namely, that ML3 is arithmetically complete. As expanded below, ML3 is a first-order modal logic that along with its built-in ability to simulate general classical first-order provability―"◻" simulating the the informal classical "⊢"―is also arithmetically complete in the Solovay sense.

A Sequent Systems without Improper Derivations

Author(s): Katsumi Sasaki / Language(s): English Issue: 1/2022

In the natural deduction system for classical propositional logic given by G. Gentzen, there are some inference rules with assumptions discharged by the rule. D. Prawitz calls such inference rules improper, and others proper. Improper inference rules are more complicated and are often harder to understand than the proper ones. In the present paper, we distinguish between proper and improper derivations by using sequent systems. Specifically, we introduce a sequent system \(\vdash_{\bf Sc}\) for classical propositional logic with only structural rules, and prove that \(\vdash_{\bf Sc}\) does not allow improper derivations in general. For instance, the sequent \(\Rightarrow p \to q\) cannot be derived from the sequent \(p \Rightarrow q\) in \(\vdash_{\bf Sc}\). In order to prove the failure of improper derivations, we modify the usual notion of truth valuation, and using the modified valuation, we prove the completeness of \(\vdash_{\bf Sc}\). We also consider whether an improper derivation can be described generally by using \(\vdash_{\bf Sc}\).