Abstract
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 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 join-semilattice of finitely generated (= compact) deductive filters on each algebra A is dually Brouwerian.
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References
W. Blok, The lattice of modal logics. An algebraic investigation, Journal of Symbolic Logic, Vol. 45 (1980), pp. 221–236.
J. Czelakowski, Reduced products of logical matrices, Studia, Logica, Vol. 39, No. 1 (1980), pp. 19–43.
—, Eguivalential logics, Part I, Studia Logica, Vol. 40, No. 3 (1981), pp. 227–236, Part II, ibidem, Vol. 40, No. 4 (1981), pp. 355–372.
—, Logical matrices, primitive satisfaction and finitely based logics, Studia Logica, Vol. 42, No. 1 (1983), pp. 89–104.
—, Filter distributive sentential logics, Abstracts of the 7th International Congress of Logic, Methodology and Philosophy of Science, Salzburg, Vol. 2, pp. 39–42 (an extended version will appear in Studia Logica).
J. M. Dunn, Algabraic completeness results for R-mingle and its extensions, Journal of Symbolic Logic, Vol. 35 (1970), pp. 1–13.
G. Grätzer, Universal Algebra, Van Nostrand, Princeton, New Jersey, 1968.
J. Herbrand, Recherches sur la théorie de la démonstration (itthèse), Prace Towarzystwa Naukowego Warszawskiego, Wydział 3, Vol. 33 (1931).
P. Köhler and D. Pigozzi, Varieties with eguationally definable principal congruences, Algebra Universalis, Vol. 11 (1980), pp. 213–219.
J. Łoś and R. Suszko, Remarks on sentential logics, Indagationes Mathematicae, Vol. 20 (1958), pp. 177–183.
W. C. Nemitz, On the lattice of filters of an implicative semilattice, Journal of Mathematics and Mechanics, Vol. 18 (1969), pp. 683–688.
W. A. Pogorzelski, Przglqd twierdzeń o dedukcji dla rachunków zdaniowych (in Polish), Studia Logica, Vol. XV (1964), pp. 163–178.
J. Porte, Fifty years of deduction theorems, in: J. Stern (ed.), Proceedings of the Herbrand Symposium — Logic Colloquium '81, North-Holland, Amsterdam 1982.
A. Tarski, Über einige fundamentale Begriffe der Metamathematik, Sprawozdania z posiedzeń Towarzystwa Naukowego Warszawskiego, Wydział 3, Vol. 23 (1930), pp. 35–43.
M. Tokarz, Essays in Matrix Semantics of Relevant Logics, The Polish Academy of Sciences, Institute of Philosophy and Sociology, Warszawa 1980.
E. Wójcicki, Matrix approach in sentential calculi, Studia Logica, Vol. 32 (1973), pp. 7–37.
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A part of this paper was presented in abstracted form in Bulletin of the Section of Logic, Vol. 12, No. 3 (1983), pp. 111–116, and in The Journal of Symbolic Logic.
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Czelakowski, J. Algebraic aspects of deduction theorems. Stud Logica 44, 369–387 (1985). https://doi.org/10.1007/BF00370428
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DOI: https://doi.org/10.1007/BF00370428