In the late 1960s and early 1970s, Dana Scott introduced a kind of generalization (or perhaps simplification would be a better description) of the notion of inference, familiar from Gentzen, in which one may consider multiple conclusions rather than single formulas. Scott used this idea to good effect in a number of projects including the axiomatization of many-valued logics (of various kinds) and a reconsideration of the motivation of C.I. Lewis. Since he left the subject it has been vigorously prosecuted (...) by a number of authors under the heading of abstract entailment relations where it has found an important role in both algebra and theoretical computer science. In this essay we go back to the beginnings, as presented by Scott, in order to make some comments about Scott’s cut rule, and show how much of Scott’s main result may be applied to the case of single-conclusion logic. (shrink)

We provide a new proof of the following Pałasińska's theorem: Every finitely generated protoalgebraic relation distributive equality free quasivariety is finitely axiomatizable. The main tool we use are ${\mathcal{Q}}$ Q -relation formulas for a protoalgebraic equality free quasivariety ${\mathcal{Q}}$ Q . They are the counterparts of the congruence formulas used for describing the generation of congruences in algebras. Having this tool in hand, we prove a finite axiomatization theorem for ${\mathcal{Q}}$ Q when it has definable principal ${\mathcal{Q}}$ Q -subrelations. This (...) is a property obtained by carrying over the definability of principal subcongruences, invented by Baker and Wang for varieties, and which holds for finitely generated protoalgebraic relation distributive equality free quasivarieties. (shrink)

The notion of local deduction theorem (which generalizes on the known instances of indeterminate deduction theorems, e.g. for the infinitely-valued ukasiewicz logic C ) is defined. It is then shown that a given finitary non-pathological logic C admits the local deduction theorem iff the class Matr(C) of all matrices validating C has the C-filter extension property (Theorem II.1).

There exist important deductive systems, such as the non-normal modal logics, that are not proper subjects of classical algebraic logic in the sense that their metatheory cannot be reduced to the equational metatheory of any particular class of algebras. Nevertheless, most of these systems are amenable to the methods of universal algebra when applied to the matrix models of the system. In the present paper we consider a wide class of deductive systems of this kind called protoalgebraic logics. These include (...) almost all (non-pathological) systems of prepositional logic that have occurred in the literature. The relationship between the metatheory of a protoalgebraic logic and its matrix models is studied. The following results are obtained for any finite matrix model U of a filter-distributive protoalgebraic logic : (I) The extension U of is finitely axiomatized (provided has only finitely many inference rules); (II) U has only finitely many extensions. (shrink)

This paper is a continuation of [27], where we provide the background and the basic tools for studying the structural properties of classes of models over languages without equality. In the context of such languages, it is natural to make distinction between two kinds of classes, the so-called abstract classes, which correspond to those closed under isomorphic copies in the presence of equality, and the reduced classes, i.e., those obtained by factoring structures by their largest congruences. The generic problem described (...) in [27] is to investigate under what conditions this reduction process does not alter the metatheory of a class. Here we focus our attention on a concrete aspect of this generic problem that we import from universal algebra, namely the existence and description of free models. As in [27], we can find here again the basic notion of protoalgebraicity, which was originally introduced in [7] as the weakest condition to guarantee that the reduction process behaves reasonably well from an algebraic point of view. Our concern, however, takes us to handle a further notion, that of semialgebraicity, which corresponds to the notion of equivalential logic of [18]; semialgebraicity turns out to be the property which ensures that freeness is fully preserved by the reduction process. (shrink)

This paper is a continuation of [27], where we provide the background and the basic tools for studying the structural properties of classes of models over languages without equality. In the context of such languages, it is natural to make distinction between two kinds of classes, the so-calledabstruct classes, which correspond to those closed under isomorphic copies in the presence of equality, and thereduced classes, i.e., those obtained by factoring structures by their largest congruences. The generic problem described in [27] (...) is to investigate under what conditions thisreduction processdoes not alter the metatheory of a class.Here we focus our attention on a concrete aspect of this generic problem that we import from universal algebra, namely the existence and description of free models. As in [27], we can find here again the basic notion ofprotoalgebraicity, which was originally introduced in [7] as the weakest condition to guarantee that the reduction process behaves reasonably well from an algebraic point of view. Our concern, however, takes us to handle a further notion, that ofsemialgebraicity, which corresponds to the notion ofequivalential logicof [18]; semialgebraicity turns out to be the property which ensures that freeness is fully preserved by the reduction process. (shrink)

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 Wroski'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). (shrink)

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 C-filter, playing the role analogous to that of a congruence in universal algebra. A sentential logic C is called filter distributive if the lattice of C-filters 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 C-prime 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 well-known theorem from universal algebra [9, § 62, Theorem 2]. (shrink)

The first known statements of the deduction theorems for the first-order 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 and , C(X{{a}}) iff P(, ) AC(X). [P(, ) denotes the set of formulas which result by the simultaneous substitution of for p and 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 join-semilattice of finitely generated (= compact) deductive filters on each algebra A is dually Brouwerian. (shrink)

This paper is a contribution to the study of the rôle of disjunction inAlgebraic Logic. Several kinds of (generalized) disjunctions, usually defined using a suitable variant of the proof by cases property, were introduced and extensively studied in the literature mainly in the context of finitary logics. The goals of this paper are to extend these results to all logics, to systematize the multitude of notions of disjunction (both those already considered in the literature and those introduced in this paper), (...) and to show several interesting applications allowed by the presence of a suitable disjunction in a given logic. (shrink)