A survey of deduction theorems for the propositional calculi

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A survey of deduction theorems for the propositional calculi

     

A deduction theorem schema for deductive systems of propositional logics

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.     

A contribution to the history of propositional calculus

Summary The anonimous scholiumOn all forms of syllogism was copied in 1884 from the Paris Codex 2064 by E. Richter. In 1899 M. Wallies published it in the preface to Ammonius' commentary on the Prior Analytics of Aristotle. There appear in that scholium, apart from the complex figure of Galenos, other characteristic forms of inference.Among these forms I found five so-called non-demonstrable stoic syllogisms, three modifications of the law of transposition of which the third is not mentioned by the authors of Princ pia Mathematica, and a modification of the form of inference known as Euclid's law. This form of inference was applied by Euclid in mathematics and by Saccherius in syllogistics; it is mentioned for the first time by Cardan in a treatise of 1570 and later by Clavius in his commentary of 1574 on the Elements of Euclid and in the commentary on Theodosius'Sphaerica of the year 1586.In 1658 Erhard Weigel made the first attempt at refuting the logical law of Euclid as formulated by Cardan and Clavius and in 1686 James Bernoulli tried to prove it.     

Algebraic aspects of deduction theorems

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.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.     

Three contributions to the two-valued propositional calculus

Summary Three chapters contain the results independent of each other. In the first chapter I present a set of axioms for the propositional calculus which are shorter than the ones known so far, in the second one I give a method of defining all ternary connectives, in the third one, I prove that the probability of propositional functions is preserved under reversible substitutions. This paper appeard orginally under the title “Trois contributions an calcul des propositions bivalent” inStudia Societatis Scientiarum Torunensis, Toruń, Polonia, Sectio A, vol. I (1948), pp. 3–15.