Sommers posed the question 'Do We Need Identity?' and answered in the negative. According to Sommers, the need for a special identity relation resulted from an arbitrary distinction between concept and object introduced by Frege and retained in modern predicate logic (MPL). This is reflected in the syntactic distinction between predicate and individual constant. Traditional formal logic (TFL) does not respect this distinction and, as a consequence, has no need for a special identity relation. But Sommers' position has not gained general acceptance. On the contrary, it has received considerable criticism. While it is conceded that TFL can express the identity of individual constants, it is quickly pointed out that this falls far short of providing the expressiveness of the logical identity relation. But the precise extent of the deficit in expressiveness, if indeed there is any deficit, has not been determined. It appears that Sommers' position on identity has not been adequately formalized to permit such a determination. This paper formalizes and extends Sommers' position on identity. This formalization is compared with MPL to define precisely the difference in expressive power. The conclusion is that it has less expressive power than MPL, but nonetheless does provide essentially all the expressiveness of the logical identity relation. The formal language defined for this investigation is similar to the language of MPL. The similarity will not only facilitate comparison, but perhaps will also make this formal language more palatable to readers whose experience and/or predisposition favors MPL.
Purdy, William C., "On the Question ‘Do We Need Identity?’" (1991). Electrical Engineering and Computer Science - Technical Reports. 101.