
Authors:
Andrzej INDRZEJCZAK, Janusz CIUCIURA
Title:Preface. Professor Grzegorz Malinowski in Honorem
Pages:110
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Abstract ( + )
It is both an honor and a privilege for us to serve as Editors of this special issue of the Bulletin of the Section of Logic in honor of Professor Grzegorz Malinowski. Additionally, this is also Anniversary Issue of the Bulletin; it has been 45 years since the first issue of this Journal appeared. The anniversary provides an opportunity to look back and reflect on the past years. It also coincides with the 70th birthday of Professor Grzegorz Malinowski, EditorinChief of the journal. On this occasion we would like to express our sincere gratitude for his efforts. The present volume, and the next one, are dedicated to Grzegorz Malinowski with best wishes of every success in his life.

Authors:
Ryszard WOJCICKI
Title:Three Doctrines of the Nature of Mathematics (Some Comments of a Knowledge Theorist)
Pages:1119
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Abstract ( + )
In this note I am reflecting on interrelations between three concepts of truth: (1) that employed by Hilbert arguing his formalist view on the nature of mathematics, (2) Freges idea of truth supported by mathematical intuition, and (3) known as Aristotelian correspondence idea of truth concerning any propositions not merely mathematical.
Keywords: mathematics, formalism, realism, intuitionism, truth.

Authors:
Jan WOLENSKI
Title:Universality of Logic
Pages:2132
File:bibtex
Abstract ( + )
This paper deals with the problem of universality property of logic. At first, this property is analyzed in the context of firstorder logic. Three senses of the universality property are distinguished: universal applicability, topical neutrality and validity (truth in all models). All theses senses can be proved to be justified. The fourth understanding, namely the amount of expressive power, is connected with the criticism of the firstorder thesis: firstorder logic is the logic. The categorical approach to logic is presented as associated with the last understanding of universality. The author concludes that two senses of universality should be sharply discriminated and defends the firstorder thesis.
Keywords: universality, logica docents, logica utens, firstorder logic, consequence operation, model, syntax, semantics, expressive power.

Authors:
Alexander S. KARPENKO
Title:FourValued Logics BD and DM4: Expansions
Pages:3345
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Abstract ( + )
The paper discusses functional properties of some fourvalued logics which are the expansions of fourvalued Belnap's logic DM4. At first, we consider the logics with two designated values, and then logics defined by matrices having the same underlying algebra, but with a different choice of designated values, i.e. with one designated value. In the preceding literature both approaches were developed independently. Moreover, we present the lattices of the functional expansions of DM4.
Keywords: Belnap's fourvalued logic, expansions and functional properties, lattices.

Authors:
Janusz CZELAKOWSKI
Title:Infinite Valued Łukasiewicz Logic
Pages:4764
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Abstract ( + )
The paper concerns the algebraic structure of the set of cumulative distribution functions as well as the relationship between the resulting algebra and the infinitevalued Łukasiewicz algebra. The paper also discusses interrelations holding between the logical systems determined by the above algebras.
Keywords: probability, cumulative distribution function, the infinitevalued standard Łukasiewicz algebra, consequence relation.
AMS Subject Classification: 03G20, 06D30, 60A05.

Authors:
Jan von PLATO
Title:From Gentzen to Jaśkowski and Back: Algorithmic Translation of Derivations Between the Two Main Systems of Natural Deduction
Pages:6573
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Abstract ( + )
The way from linearly written derivations in natural deduction, introduced by Jaśkowski and often used in textbooks, is a straightforward rootfirst translation. The other direction, instead, is tricky, because of the partially ordered assumption formulas in a tree that can get closed by the end of a derivation. An algorithm is defined that operates alternatively from the leaves and root of a derivation and solves the problem.
Keywords: proof systems, linear natural deduction, Gentzen, Jaśkowski.

Authors:
Wojciech BUSZKOWSKI
Title:Involutive Nonassociative Lambek Calculus: Sequent Systems and Complexity
Pages:7591
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Abstract ( + )
In [5] we study Nonassociative Lambek Calculus (NL) augmented with De Morgan negation, satisfying the double negation and contraposition laws. This logic, introduced by de Grooté and Lamarche [10], is called Classical NonAssociative Lambek Calculus (CNL). Here we study a weaker logic InNL, i.e. NL with two involutive negations. We present a onesided sequent system for InNL, admitting cut elimination. We also prove that InNL is PTIME.
Keywords: nonassociative Lambek calculus, linear logic, sequent system, cut elimination, PTIME complexity.

Authors:
NobuYuki SUZUKI
Title:Some Weak Variants of the Existence and Disjunction Properties in Intermediate Predicate Logics
Pages:93109
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Abstract ( + )
We discuss relationships among the existence property, the disjunction property, and their weak variants in the setting of intermediate predicate logics. We deal with the weak and sentential existence properties, and the Znormality, which is a weak variant of the disjunction property. These weak variants were presented in the author's previous paper [16]. In the present paper, the Kripke sheaf semantics is used.
Keywords: intermediate predicate logics, existence property, disjunction property.

Authors:
Andrzej PIETRUSZCZAK
Title:On Theses Without Iterated Modal ities of Modal Logics Between C1 and S5. Part 1
Pages:111133
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Abstract ( + )
This is the first, out of two papers, in which we identify all logics between C1 and S5 having the same theses without iterated modalities. All these logics can be divided into certain groups. Each such group depends only on which of the following formulas are theses of all logics from this group: (N), (T), (D), ^{┏}(T) ∨ □q^{┓}, and for any n > 0 a formula ^{┏}(T) ∨ (alt_{n})^{┓}, where (T) has not the atom ‘q’, and (T) and (alt_{n}) have no common atom. We generalize Pollack's result from [12], where he proved that all modal logics between S1 and S5 have the same theses which does not involve iterated modalities (i.e., the same firstdegree theses).
Keywords: firstdegree theses of modal logics; theses without iterated modalities; Pollack's theory of Basic Modal Logic; basic theories for modal logics between C1 and S5.

Authors:
Andrzej INDRZEJCZAK
Title:Cut Elimination Theorem for NonCommutative Hypersequent Calculus
Pages:135149
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Abstract ( + )
Hypersequent calculi (HC) can formalize various nonclassical logics. In [9] we presented a noncommutative variant of HC for the weakest temporal logic of linear frames Kt4.3 and some its extensions for dense and serial flow of time. The system was proved to be cutfree HC formalization of respective temporal logics by means of Schütte/Hintikkastyle semantical argument using models built from saturated hypersequents. In this paper we present a variant of this calculus for Kt4.3 with a constructive syntactical proof of cut elimination.
Keywords: temporal logic, linear time, hypersequent calculus, cut elimination.