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A theory of restricted quantification II

Published online by Cambridge University Press:  12 March 2014

Theodore Hailperin
Theodore Hailperin*
Affiliation:
Lehigh University
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Extract

In Part I of this paper no official significance was given to free occurrences of restricted-variables. Indeed doubt has been expressed as to the desirability or feasibility of such usage. Thus, to quote from Rosser [4], p. 146:

“We now raise the question of the significance of F (α), in which the occurrences of α are free [α a variable subject to the restriction K(α)]. In deciding to take (x).K(x) ⊃ F(x) and (Ex).K(x).F(x) as the meanings of (α)F(α) and (Eα)F(α), we were guided by the intuitive meanings. In the case of F(α), the intuitive meaning does not furnish a satisfactory guide. In everyday mathematics, if it has been agreed that α stands for a quantity satisfying the restriction K(α), it is commonly the case that, if one is assuming F (α), then K(α)&F(α) is understood, but if one is trying to prove F (α), then K(α) ⊃ F (α) is understood. It seems that in symbolic logic perhaps it is best not to give any especial significance to α in F(α) when it occurs free.”

Despite this anomalous behaviour we shall, in this Part, show how one can have unhampered use of free restricted-variables in appropriate contexts.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 22 , Issue 2 , June 1957 , pp. 113 - 129
Copyright
Copyright © Association for Symbolic Logic 1957

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References

ADDITIONAL REFERENCES

[9]Quine, W. V., On natural deduction, this Journal, vol. 15 (1950), pp. 93–102.Google Scholar
[10]Wang, Hao, On denumerable bases of formal systems, Mathematical interpretation of formal systems. Studies in logic and the foundations of mathematics. North-Holland Publishing Company, Amsterdam, 1955, pp. 57–84.CrossRefGoogle Scholar
[11]Jaakko, K.Hintikka, J., Distributive normal forms in the calculus of predicates. Acta Philosophica Fenilica, no. 6 Helsinki 1953.Google Scholar
[12]Whitehead, A. N. and Russell, B., Principia mathematica, vols. I–III, second edition. Cambridge University Press, Cambridge, England, 1925–1927.Google Scholar