Algebraic aspects of deduction theorems

Studia Logica 44 (4):369 - 387 (1985)
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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.



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Citations of this work

Fregean Logics.J. Czelakowski & D. Pigozzi - 2004 - Annals of Pure and Applied Logic 127 (1-3):17-76.
A Study of Truth Predicates in Matrix Semantics.Tommaso Moraschini - 2018 - Review of Symbolic Logic 11 (4):780-804.
Local Deductions Theorems.Janusz Czelakowski - 1986 - Studia Logica 45 (4):377 - 391.

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References found in this work

Equivalential Logics (I).Janusz Czelakowski - 1981 - Studia Logica 40 (3):227 - 236.
Equivalential logics.Janusz Czelakowski - 1981 - Studia Logica 40 (3):227-236.
Reduced Products of Logical Matrices.Janusz Czelakowski - 1980 - Studia Logica 39 (1):19 - 43.

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