Elsevier

Information Sciences

Volume 1, Issue 4, October 1969, Pages 381-402
Information Sciences

Natural deduction with few restrictions on variables☆

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Abstract

In this paper, systems of natural deduction are described in which the usually cumbersome quantifier rules (∀-introduction and ∃-elimination) are replaced by somewhat more convenient ones. These systems have deducibility from a set of assumption formulas as a primitive notion which is shown to have the compactness property. They are equivalent to the system obtained from a logistic system of Kleene by excluding variation of variables, admit the usual subsystems, have some separation properties for logical operators and, together with Kleene's system under exclusion of variation of variables, are complete with respect to logical consequence.

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☆

An abstract of parts of this paper was presented at the Rome Air Development Center—Hughes Aircraft Company Symposium on Logic, Computability, and Automata, Rome, New York, August 1965.

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