Identity, variables, and impredicative definitions

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Abstract
Recent discussion serves to bring out, amply and convincingly, the utility of observing the ordinary correct use of words and phrases for the purpose of clearing up philosophical problems. In this paper, I shall endeavour to show, by means of an example, that the reverse method may have its interest, too. An attempt will be made to cultivate a minor deviation from the accepted ways of using certain words and phrases in idiomatic English as well as in the formalized “languages” of the logicians. The words and phrases in question are those for the formalization of which a logician employs (free or bound) variables. Cases in point are the words customarily called quantifiers. The deviation I have in mind affects the relation of these words to the notion of identity. The deviation is illustrated by the following sentences: (1a) Any two points of a straight line completely determine that line; (2a) He is John's brother if he has the same parents as John; (3a) Mazzini did more for the emancipation of his country than any living man of his time. These examples may be contrasted with the following closely related sentences: (1b) Any two distinct points of a straight line completely determine that line; (2b) He is John's brother if he has the same parents as John and if he is not John himself; (3b) Mazzini did more for the emancipation of his country than any other living man of his time.