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By
Czelakowski, Janusz
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9 Citations
We examine the notion of primitive satisfaction in logical matrices. Theorem II. 1, being the matrix counterpart of Baker's wellknown result for congruently distributive varieties of algebras (cf [1], Thm. 1.5), links the notions of primitive and standard satisfaction. As a corollary we give the matrix version of Jónsson's Lemma, proved earlier in [4]. Then we investigate propositional logics with disjunction. The main result, Theorem III. 2, states a necessary and sufficient condition for such logics to be finitely based.
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By
Czelakowski, Janusz; Dziobiak, Wiesław
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4 Citations
We propose a new schema for the deduction theorem and prove that the deductive system S of a prepositional logic L fulfills the proposed schema if and only if there exists a finite set A(p, q) of propositional formulae involving only prepositional letters p and q such that A(p, p) ⊑ L and p, A(p, q) ⊢_{s}q.
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By
Czelakowski, Janusz; Dziobiak, Wiesław
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1 Citations
Let q(K) denote the least quasivariety containing a given class K of algebraic structures. Mal'cev [3] has proved that q(K) = ISP_{r}(K)^{(1)}. Another description of q(K) is given in Grätzer and Lakser [2], that is, q(K) = ISPP_{u}(K)^{2}. We give here other proofs of these results. The method which enables us to do that is borrowed from prepositional logics (cf. [1]).
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By
Czelakowski, Janusz
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20 Citations
The notion of local deduction theorem (which generalizes on the known instances of indeterminate deduction theorems, e.g. for the infinitelyvalued Łukasiewicz logic C_{∞}) is defined. It is then shown that a given finitary nonpathological logic C admits the local deduction theorem iff the class Matr(C) of all matrices validating C has the Cfilter extension property (Theorem II.1).
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By
Czelakowski, Janusz
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19 Citations
The main result of the present paper — Theorem 3 — establishes the equivalence of the interpolation and amalgamation properties for a large family of logics and their associated classes of matrices.
By
Czelakowski, Janusz
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31 Citations
The class of equivalential logics comprises all implicative logics in the sense of Rasiowa [9], Suszko's logicSCI and many Others. Roughly speaking, a logic is equivalential iff the greatest strict congruences in its matrices (models) are determined by polynomials. The present paper is the first part of the survey in which systematic investigations into this class of logics are undertaken. Using results given in [3] and general theorems from the theory of quasivarieties of models [5] we give a characterization of all simpleCmatrices for any equivalential logicC (Theorem I.14). In corollaries we give necessary and sufficient conditions for the class of all simple models for a given equivalential logic to be closed under free products (Theorem I.18).
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By
Czelakowski, Janusz
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6 Citations
The paper is conceived as a first study on the Suszko operator. The purpose of this paper is to indicate the existence of close relations holding between the properties of the Suszko operator and the structural properties of the model class for various sentential logics. The emphasis is put on generality both of the results and methods of tackling the problems that arise in the theory of this operator. The attempt is made here to develop the theory for nonprotoalgebraic logics.
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By
Czelakowski, Janusz
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1 Citations
The classesMatr(
$$ \subseteq $$
) of all matrices (models) for structural finitistic entailments
$$ \subseteq $$
are investigated. The purpose of the paper is to prove three theorems: Theorem I.7, being the counterpart of the main theorem from Czelakowski [3], and Theorems II.2 and III.2 being the entailment counterparts of Bloom's results [1]. Theorem I.7 states that if a classK of matrices is adequate for
$$ \subseteq $$
, thenMatr(
$$ \subseteq $$
) is the least class of matrices containingK and closed under the formation of ultraproducts, submatrices, strict homomorphisms and strict homomorphic preimages. Theorem II.2 in Section II gives sufficient and necessary conditions for a structural entailment to be finitistic. Section III contains theorems which characterize finitely based entailments.
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By
Czelakowski, Janusz
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7 Citations
With each sentential logic C, identified with a structural consequence operation in a sentential language, the class Matr^{*} (C) of factorial matrices which validate C is associated. The paper, which is a continuation of [2], concerns the connection between the purely syntactic property imposed on C, referred to as Maehara Interpolation Property (MIP), and three diagrammatic properties of the class Matr* (C): the Amalgamation Property (AP), the (deductive) Filter Extension Property (FEP) and Injections Transferable (IT). The main theorem of the paper (Theorem 2.2) is analogous to the Wroński's result for equational classes of algebras [13]. It reads that for a large class of logics the conjunction of (AP) and (FEP) is equivalent to (IT) and that the latter property is equivalent to (MIP).
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By
Czelakowski, Janusz
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9 Citations
In the first section logics with an algebraic semantics are investigated. Section 2 is devoted to subdirect products of matrices. There, among others we give the matrix counterpart of a theorem of Jónsson from universal algebra. Some positive results concerning logics with, finite degrees of maximality are presented in Section 3.
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By
Czelakowski, Janusz
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This paper, being a companion to the book [2] elaborates the deontology of sequential and compound actions based on relational models and formal constructs borrowed from formal linguistics. The semantic constructions presented in this paper emulate to some extent the content of [3] but are more involved. Although the present work should be regarded as a sequel of [3] it is selfcontained and may be read independently. The issue of permission and obligation of actions is presented in the form of a logical system
. This system is semantically defined by providing its intended models in which the role of actions of various types (atomic, sequential and compound ones) is accentuated. Since the consequence relation
is not finitary, other semantically defined variants of
are defined. The focus is on the finitary system
in which only finite compound actions are admissible. An adequate axiom system for
it is defined. The strong completeness theorem is the central result. The role of the canonical model in the proof of the completeness theorem is emphasized.
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By
Czelakowski, Janusz
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19 Citations
The first known statements of the deduction theorems for the firstorder predicate calculus and the classical sentential logic are due to Herbrand [8] and Tarski [14], respectively. The present paper contains an analysis of closure spaces associated with those sentential logics which admit various deduction theorems. For purely algebraic reasons it is convenient to view deduction theorems in a more general form: given a sentential logic C (identified with a structural consequence operation) in a sentential language I, a quite arbitrary set P of formulas of I built up with at most two distinct sentential variables p and q is called a uniform deduction theorem scheme for C if it satisfies the following condition: for every set X of formulas of I and for any formulas α and β, βεC(X∪{{a}}) iff P(α, β) AC(X). [P(α, β) denotes the set of formulas which result by the simultaneous substitution of α for p and β for q in all formulas in P]. The above definition encompasses many particular formulations of theorems considered in the literature to be deduction theorems. Theorem 1.3 gives necessary and sufficient conditions for a logic to have a uniform deduction theorem scheme. Then, given a sentential logic C with a uniform deduction theorem scheme, the lattices of deductive filters on the algebras A similar to the language of C are investigated. It is shown that the joinsemilattice of finitely generated (= compact) deductive filters on each algebra A is dually Brouwerian.
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By
Czelakowski, Janusz
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15 Citations
The class Matr(C) of all matrices for a prepositional logic (ℒ, C) is investigated. The paper contains general results with no special reference to particular logics. The main theorem (Th. (5.1)) which gives the algebraic characterization of the class Matr(C) states the following. Assume C to be the consequence operation on a prepositional language induced by a class K of matrices. Let m be a regular cardinal not less than the cardinality of C. Then Matr (C) is the least class of matrices containing K and closed under mreduced products, submatrices, matrix homomorphisms, and matrix homomorphic counterimages.
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By
Czelakowski, Janusz
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The paper is concerned with reflexive points of relations. The significance of reflexive points in the context of indeterminate recursion principles is shown.
By
Czelakowski, Janusz
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5 Citations
The present paper is thought as a formal study of distributive closure systems which arise in the domain of sentential logics. Special stress is laid on the notion of a Cfilter, playing the role analogous to that of a congruence in universal algebra. A sentential logic C is called filter distributive if the lattice of Cfilters in every algebra similar to the language of C is distributive. Theorem IV.2 in Section IV gives a method of axiomatization of those filter distributive logics for which the class Matr (C)_{prime}of Cprime matrices (models) is axiomatizable. In Section V, the attention is focused on axiomatic strengthenings of filter distributive logics. The theorems placed there may be regarded, to some extent, as the matrix counterparts of Baker's wellknown theorem from universal algebra [9, § 62, Theorem 2].
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By
Czelakowski, Janusz
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1 Citations
The purpose of this paper is to present in a uniform way the commutator theory for kdeductive system of arbitrary positive dimension k. We are interested in the logical perspective of the research — an emphasis is put on an analysis of the interconnections holding between the commutator and logic. This research thus qualifies as belonging to abstract algebraic logic, an area of universal algebra that explores to a large extent the methods provided by the general theory of deductive systems. In the paper the new term ‘commutator formula’ is introduced. The paper is concerned with the meanings of the above term in the models provided by the commutator theory and clarifies contexts in which these meanings occur. The work is presented in an abstracted form: main ideas are outlined but proofs are deferred to the second part of the paper.
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By
Czelakowski, Janusz
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2 Citations
The article deals with compatible families of Boolean algebras. We define the notion of a partial Boolean algebra in a broader sense (PBA(bs)) and then we show that there is a mutual correspondence between PBA(bs) and compatible families of Boolean algebras (Theorem (1.8)). We examine in detail the interdependence between PBA(bs) and the following classes: partial Boolean algebras in the sense of Kochen and Specker (§ 2), ortholattices (§ 3, § 5), and orthomodular posets (§ 4), respectively.
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By
Czelakowski, Janusz; Malinowski, Grzegorz
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5 Citations
The aim of the article is to outline the historical background and the present state of the methodology of deductive systems invented by Alfred Tarski in the thirties. Key notions of Tarski's methodology are presented and discussed through, the recent development of the original concepts and ideas.
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