Showing papers in "Notre Dame Journal of Formal Logic in 1978"

A formal system for the non-theorems of the propositional calculus.

Abstract: Introduction The completeness of the classical propositional calculus allows us to give a deductive system consisting of finitely many axiom schemas and finitely many rules of inference, that permit us to pass from a formula or a pair of formulae to a syntactically related formula, in such a manner that the formulae obtained inductively from the axioms by repeated application of the rules are exactly the tautologies. In this paper we give an analogous deductive system (more concretely, a Hubert type system) such that the formulae deduced are exactly those that are not tautologies, the non-theorems of the propositional calculus. Obviously, this has to be the most non-standard of the non-classical logics. It is important to note that there are many other algorithms to generate recursively the nontheorems, since the propositional calculus is decidable. Usually they are based in the methodical search for a counterexample, but they lack the inductive character of a Hubert type system, where every formula involved in a deduction is itself deductible. In our system, unlike semantic tableaux or refutation trees, every formula introduced in a deduction is a nontautology, and it is introduced only if it is a non-tautological axiom, or it follows by one of the non-tautological rules of inference from nontautologies introduced earlier in the deduction.

On the consistency and independence of some set-theoretical axioms

Abstract: In this paper by means of simple models it is shown that the five set-theoretical axioms of Extensionalit y, Replacement, Power-Set, SumSet, and Choice are consistent and that each of the axioms of Extensionality, Replacement, and Power-Set is independent from the remaining four axioms. Although the above results are known and can be found in part in [l], it is believed that this paper has some expository merits. The abovementioned axioms are five of the six axioms of the ZermeloFraenkel Theory of Sets, the sixth being Axiom of Infinity. We accept that every element of a set is a set and (without borrowing "=" from Logic) we define two sets u and υ to be equal if and only if they possess the same elements, in which case we write u = v. With this in mind, the six axioms of the Zermelo-Fraenkel Theory of Sets can be stated as follows [2]:

The adequacy of material dialogue-games.

Abstract: Les jeux de dialogues materiels qui possedent certaines proprietes (finitude locale, regularite, compatibilite avec une theorie des modeles) sont adequats, du point de vue d'une quelconque theorie des modeles a 2valeurs de verite.

A note on the law of identity and the converse Parry property.

Abstract: Proof: 1.1. Every mef will be of the form p—> p in virtue of variablesharing. Eefs of the form p —> C and C -+ p, where C is an ef, are ruled out, the first because of the Ackermann property, the second because by modus ponens p would be a theorem. So every eef is a mef, and Lemma 1.1. follows. 1.2. Let A contain two or more propositional variables. In virtue of variable-sharing every ef of A containing two or more propositional variables will have on at least one side of its arrow a subformula containing at least two propositional variables (so this subformula will be an ef). (E.g., imagine an ef B containing two propositional variables, p and q; then it will contain p either on both sides of the arrow, in which case q is on at least one side, or on only one side, in which case q must be on both.)*