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On weakened quantification

Published online by Cambridge University Press:  12 March 2014

A. Heyting
A. Heyting*
Affiliation:
University of Amsterdam
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Extract

In this note I give an account of the relations between different notions of weakened quantification, which can be defined in intuitionistic logic. The reason for publishing it is the fact that the last gap in our knowledge of these relations has been filled in an article by Kleene [5].

For the intuitionistic calculus of propositions I refer to my article [2]; besides the rules of substitution and of inference I need only a few formulas from this calculus, which I shall repeat below. The simplest way of formalizing the intuitionistic predicate calculus is obtained by adjoining to the calculus of propositions the symbols, rules, and axioms as given by Hilbert and Ackermann [4], Chapter 3, §§4-5, pp. 53-57. (Reference is made to this book as HA.) I repeat only the axioms (e) and (f) and the rule γ.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 11 , Issue 4 , December 1946 , pp. 119 - 121
Copyright
Copyright © Association for Symbolic Logic 1947

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References

1 In this article the rule 1.2 must be omitted, as it can be derived from the other rules and the axioms. Prof. Bernays made this remark soon after the publication of the paper.

2 We have proved (B) ⊃ (F) and (F) ⊃ (B), but not the corresponding formulas for (D) and (E). For our purpose this difference is of no importance.