Remark on a Recent Paper by Hollcroft
نویسنده
چکیده
The method in mathematical logic invented by Schönfinkelf and developed in detail by Curry | is important in that it completely eliminates the variable from the formal presuppositions of logic and mathematics. Constructed according to Schönfinkel's scheme the primitive language of mathematical logic consists only of a few constants; variables, if wanted as a convenience, are introduced afterward through conventions of shorthand. Central to Schönfinkel's scheme is the device of construing relations as unitary operators. This is accomplished in the case of a dyadic relation by construing the proposition xy (x bears the relation to y) as (x)y, that is, as the proposition predicating of y an attribute x which, in its turn, is the result of applying an operator to x. (The present recourse to variables is expository only, and foreign to the formal system.) Thus dyadic relations are for Schönfinkel unitary operators yielding attributes, where attributes may, for uniformity, be regarded in turn as unitary operators yielding propositions. In general, any nadic relation is construed in corresponding fashion by taking the proposition XiX2 • • Xfi a s ( • • • ((<£xi)x2) • • • )xn] w-adic relations become unitary operators yielding unitary operators • • • yielding unitary operators yielding propositions. My present concern is to point out that Schönfinkers explanation of relations as unitary operators is gratuitous : inessential to the net interpretation of his formulas, and avoidable by a slight reinterpretation of his notation. The new interpretation is advanced not necessarily as an improvement in an intuitive
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تاریخ انتشار 2007