Nebraska Academy of Sciences


Date of this Version



Published in Transactions of the Nebraska Academy of Sciences, Volume 4 (1977).


Copyright 1977 by the Authors; used by permission of the NAS


In this paper I discuss the distinction between the referential and substitutional interpretations of one of the basic concepts of modern logic-the concept expressed by "for all x." I try to bring out what is at stake in choosing between the two. Finally, I argue for the referential interpretation on the grounds that the substitutional interpretation allows defective formulation of sound principles.

One remarkable thing about modem logic is that the entire apparatus rests on just three basic concepts: the statement connective "not both ... and __ " "for all x" ("(x)" for short) and "=". There is controversy regarding the second of these concepts. One view has it that

(1) A universally quantified statement (x) A, is true just in case every object satisfies A.

This is the referential view. Another view has it that

(2) A universally quantified statement, (x) A, is true just in case every substitutive instance of A is true.

This is the substitutional view.