A theory of restricted quantification I

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Abstract
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.Ifxis an individual-variable andQa formula of the predicate calculus, then for the idea ‘xsuch thatQ’ the notation ‘vxQ’ is proposed as one having the requisite degree of notational multiplicity – the ‘vx’ acting as a binding operator for free occurrences ofxinQ. The expression ‘vxQ’ is referred to as arestricted-variableand 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.

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