Wydawnictwo Uniwersytetu Łódzkiego,
Rohan French,
Lloyd Humberstone,
Monash University Department of Philosophy

It is well known that no consistent normal modal logic contains (as theorems) both ◊A and ◊¬A (for any formula A). Here we observe that this claim can be strengthened to the following: for any formula A, either no consistent normal modal logic contains ◊A, or else no consistent normal modal logic contains ◊¬A.

Wydawnictwo Uniwersytetu Łódzkiego,
Calyampudi Radhakrishna Rao,
Venugopalam Undurthi,
Andhra University Department of Mathematics,
more

In this paper, we prove that the lattice of all closure operators of a complete Almost Distributive Lattice L with fixed maximal element m is dual atomistic. We define the concept of a completely meet-irreducible element in a complete ADL and derive a necessary and sufficient condition for a dual atom of Φ(L) to be complemented.

Wydawnictwo Uniwersytetu Łódzkiego,
George Voutsadakis,
Lake Superior State University School of Mathematics and Computer Science,
gvoutsad@lssu.edu

Wójcicki introduced in the late 1970s the concept of a referential semantics for propositional logics. Referential semantics incorporate features of the Kripke possible world semantics for modal logics into the realm of algebraic and matrix semantics of arbitrary sentential logics. A well-known theorem of Wójcicki asserts that a logic has a referential semantics if and only if it is selfextensional...

Wydawnictwo Uniwersytetu Łódzkiego,
Andrzej Biłat,
Faculty of Adminitration and Social Sciences Department of Philosophy,
a.bilat@ans.pw.edu.pl

The identity connective is usually interpreted in non-Fregean logic as an operator representing the identity of situations. This interpretation is related to the modal criterion of the identity of sentence correlates, characteristic of the WT system and some stronger systems. However, this connective can also be interpreted in a diﬀerent way – as an operator representing the identity of propositions...

Wydawnictwo Uniwersytetu Łódzkiego,
Vladimir Shalack,
Institute of Philosophy Section of Logic,
shalack@gmail.com

In the paper we formulate a sufficient criterion in order for the first order theory with finite set of axioms to be represented by definitions in predicate calculus. We prove the corresponding theorem. According to this criterion such theories as the theory of equivalence relation, the theory of partial order and many theories based on the equality relation with finite set of functional and predicate...

Wydawnictwo Uniwersytetu Łódzkiego,
Andrzej Biłat,
Faculty of Administration and Social Sciences Department of Philosophy,
a.bilat@ans.pw.edu.pl

This paper presents the main assumptions of Andrzej Grzegorczyk’s last research project concerning the logic of synonymity. It shows that the basis of logic of analytic equivalence, presented in the ﬁrst part of the work, fully corresponds with these assumptions.

Wydawnictwo Uniwersytetu Łódzkiego,
Wojciech Dzik,
Beniamin Wróbel,
Bankowa 12 University of Silesia Institute of Mathematics,
more

Unifiability of terms (and formulas) and structural completeness in the variety of relation algebras RA and in the products of modal logic S5 is investigated. Nonunifiable terms (formulas) which are satisfiable in varieties (in logics) are exhibited. Consequently, RA and products of S5 as well as representable diagonal-free n-dimensional cylindric algebras, RDfn, are almost structurally complete but...

Wydawnictwo Uniwersytetu Łódzkiego,
Marcin Łazarz,
Krzysztof Siemieńczuk,
Poland Department of Logic and Methodology of Sciences University of Wrocław,
more

Using known facts we give a simple characterization of the distributivity of lattices of finite length.

Wydawnictwo Uniwersytetu Łódzkiego,
Ja̅nis Cı̅rulis,
University of Latvia Institute of Mathematics and Computer Science,
janis.cirulis@lu.lv

An MV-algebra is an algebra (A, ⊕, ¬, 0), where (A, ⊕, 0) is a commutative monoid and ¬ is an idempotent operation on A satisfying also some additional axioms. Basic algebras are similar algebras that can roughly be characterised as nonassociative (hence, also non-commutative) generalizations of MV-algebras. Basic algebras and commutative basic algebras provide an equivalent algebraic semantics in...

Wydawnictwo Uniwersytetu Łódzkiego,
Feng Gao,
George Tourlakis

A well established technique toward developing the proof theory of a Hilbert-style modal logic is to introduce a Gentzen-style equivalent (a Gentzenisation), then develop the proof theory of the latter, and finally transfer the metatheoretical results to the original logic (e.g., [1, 6, 8, 18, 10, 12]). In the first-order modal case, on one hand we know that the Gentzenisation of the straightforward...

Wydawnictwo Uniwersytetu Łódzkiego,
Alexej P Pynko,
V.M. Glushkov Institute of Cybernetics Academician Glushkov Department of Digital Automata Theory (100),
pynko@voliacable.com

The primary objective of this paper, which is an addendum to the author’s [8], is to apply the general study of the latter to Łukasiewicz’s n-valued logics [4]. The paper provides an analytical expression of a 2(n−1)-place sequent calculus (in the sense of [10, 9]) with the cut-elimination property and a strong completeness with respect to the logic involved which is most compact among similar calculi...

Wydawnictwo Uniwersytetu Łódzkiego,
Aldo V Figallo,
Gustavo Pelaitay,
Universidad Nacional de San Juan Instituto de Ciencias Básicas,
more

In 2015, A.V. Figallo and G. Pelaitay introduced tense n×m-valued Łukasiewicz–Moisil algebras, as a common generalization of tense Boolean algebras and tense n-valued Łukasiewicz–Moisil algebras. Here we initiate an investigation into the class tpLMn×m of tense polyadic n × m-valued Łukasiewicz–Moisil algebras. These algebras constitute a generalization of tense polyadic Boolean algebras introduced...

Wydawnictwo Uniwersytetu Łódzkiego,
Zofia Kostrzycka,
ul. Luboszycka 3 University of Technology,
z.kostrzycka@po.opole.pl

Halldén complete modal logics are defined semantically. They have a nice characterization as they are determined by homogeneous Kripke frames.

We give an elementary proof (in the sense that it is formalizable in Peano arithmetic) of the strong normalization of the atomic polymorphic calculus Fat (a predicative restriction of Girard’s system F).

Reference [12] introduced a novel formula to formula translation tool (“formula-tors”) that enables syntactic metatheoretical investigations of first-order modallogics, bypassing a need to convert them first into Gentzen style logics in order torely on cut elimination and the subformula property. In fact, the formulator tool,as was already demonstrated in loc. cit., is applicable even to the metatheoreticalstudy...

A natural deduction system NI, for the full propositional intuitionistic logic, is proposed. The operational rules of NI are obtained by the translation from Gentzen’s calculus LJ and the normalization is proved, via translations from sequent calculus derivations to natural deduction derivations and back.

The notions of a C-energetic subset and (anti) permeable C-value in BCK-algebras are introduced, and related properties are investigated. Conditions for an element t in [0, 1] to be an (anti) permeable C-value are provided. Also conditions for a subset to be a C-energetic subset are discussed. We decompose BCK-algebra by a partition which consists of a C-energetic subset and a commutative ideal.

In the paper a decision procedure for S5 is presented which uses a cut-free sequent calculus with additional rules allowing a reduction to normal modal forms. It utilizes the fact that in S5 every formula is equivalent to some 1-degree formula, i.e. a modally-flat formula with modal functors having only boolean formulas in its scope. In contrast to many sequent calculi (SC) for S5 the presented system...

Babenyshev and Martins proved that two hidden multi-sorted deductive systems are deductively equivalent if and only if there exists an isomorphism between their corresponding lattices of theories that commutes with substitutions. We show that the π-institutions corresponding to the hidden multi-sorted deductive systems studied by Babenyshev and Martins satisfy the multi-term condition of Gil-F´erez...

The logic BN4 can be considered as the 4-valued logic of the relevant conditional and the logic E4, as the 4-valued logic of (relevant) entailment. The aim of this paper is to endow E4 with a 2-set-up Routley-Meyer semantics. It is proved that E4 is strongly sound and complete w.r.t. this semantics.