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A theory of restricted quantification I

Published online by Cambridge University Press:  12 March 2014

Theodore Hailperin
Theodore Hailperin*
Affiliation:
Lehigh University
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Extract

The notion of a variable restricted in range to some set of values, or to values having some property, is widespread in logic and mathematics. To avoid continual explicit mention of such a restriction the common device is to resort to a typographically different set of letters for such variables. But a notation of this kind is clearly inadequate when a general discussion, including more than one kind of restriction, is under consideration. For example, two restrictions may have some inner connection, as in the case of mutually exclusive restrictions, and this the usual symbolism doesn't depict; or, one may wish to study the dependence of such restrictions upon one or more parameters as in the case of bounded quantifiers of recursive function theory; or, one can even conceive of restrictions which may in turn depend upon restricted variables. In each of these cases the currently used notation is deficient.

If x is an individual-variable and Q a formula of the predicate calculus, then for the idea ‘x such that Q’ the notation ‘vxQ’ is proposed as one having the requisite degree of notational multiplicity – the ‘vx’ acting as a binding operator for free occurrences of x in Q. The expression ‘vxQ’ is referred to as a restricted-variable and it may occupy positions in formulas appropriate to individual-variables, in argument places as well as in quantifiers. In § 2 a first-level predicate calculus, called ‘‘, is presented in which the restricted-variable is an integral part.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 22 , Issue 1 , March 1957 , pp. 19 - 35
Copyright
Copyright © Association for Symbolic Logic 1957

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References

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