N. Goodman and W. V. Quine, "Steps Toward a Constructive Nominalism", Journal of Symbolic Logic, 12 (1947).

7. Variables and Quantification

A variable of our object language is a vee, or a vee together with a string of one or more accents following it. We first define a string of accents as any inscription of which every part that is a character is an accent.

AcString x =. Insc x & (z)(Seg zx & Char z .--> Ac z).

The definition of a variable is then readily formulated.

Vbl x =. Vee x V (]y)(]z)(Vee y & AcString z & Cxyz) .

A variable is a vee or the result of concatenating a vee with a string of accents.

A quantifier will be simply a variable in parentheses. But it is more useful to define a string of (one or more) quantifiers directly. A method for doing this becomes evident when we reflect that any inscription will be a string of quantifiers if it begins and ends with facing parentheses and is such that every pair of facing parentheses within it frames an inscription that is either a variable or contains parentheses back to back.

QfrString x = (]y)(]z){LPar y & RPar z & (]w)Cxywz & (s)(t)(u)(k)[LPar t & RPar k & Seg sx .-->: ~Cstk : Cstuk .-->. Vbl u V (]p)(]q)(]r)(RPar q & LPar r & Cpqr & Seg pu)]}.

Then let us call x a quantification of y if x consists of a string of quantifiers followed by y.

Qfn xy = (]z)(QfrString z & Cxzy) .

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