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In this paper I will explain Frege’s reasons for initially supporting the view that identity statements express a relation between signs, rather than expressing a relation between objects. Subsequently, I will reconstruct Frege’s argument against this view. Finally, I will explain how this argument supports the view that identity statements express a relation between objects.
Initially, Frege is attracted to the thesis that identity statements express a relation between signs. The first reason he presents is that statements of a=a and a=b are of “differing cognitive value” (35). There seems to be an important distinction to be made between saying “Batman is Batman” and “Batman is Bruce Wayne” — namely that they are of different cognitive value. The way that the value differs is that the former is true a priori and analytic, whereas the latter is not true a priori and serves to extend our knowledge of Batman, rather than just analyze it (35). But if identity statements expressed a relation between objects, then the statements of a=a and a=b would be of the same cognitive value (35). This is because the relation that is being drawn is one between objects referred to, rather than the signs. But since the relation is that of identity, it is going to be a relation between the object and itself and nothing else (35). And so a=a and a=b, if both true, will never mean anything over and above “an object is the same as itself”. As in this way mean they same thing, for one thought is the same as the other, so they are of the same cognitive value.
But the statements are of different cognitive value, for we could hold a=a true and a=b false. For example, a person can know that Hesperus is Hesperus without believing that it is also Phosphorous. And when we say a=b we take ourselves to be saying that the signs “a” and “b” designate the same thing (35), not that an object is the same as itself. Thus the relation concerns signs, and not the actual object. So, if the identity relation were between objects these statements would be of the same cognitive value. But these statements are not of the same cognitive value. Thus, Frege initially supports the thesis that identity statements are relations between signs.
But Frege comes to reject this thesis. Consider the fact that the identity relation between signs holds only insofar as the signs designate something (35). For if a sign did not designate something, then there would be no common element between the signs that allow them to be identified with each other or be connected (35). But the connection between a sign and its object is arbitrary, for anyone can use an arbitrary object as a sign for something (35). This creates a problem where identity statements do not “express genuine knowledge” (35), namely that identity statements become linguistic artifacts.
In this way, “Hesperus is Phosphorous” is no longer about either Hesperus or Phosphorous, but rather a relation between arbitrarily chosen names, and so cannot express genuine knowledge; for genuine knowledge should be about the object of inquiry. When we want to know if Hesperus is Phosphorous, we want to know whether that particular celestial body in the sky is the same as that other particular celestial body in the sky — we are concerned about the object. We are not asking about the name (though we are using it). The name serves to pick out the object we want to identify, but we do not identify the name. A name’s connection to its object is something arbitrary, but an object’s connection to itself is not. If the knowledge we obtain is about the equation of arbitrarily chosen names, then we have arrived at empty, not genuine, knowledge. For suppose a=b. Now I stipulate that a=c. I conclude that b=c. Now I stipulate that a=d. I conclude that c=d. The conclusion is empty. If I needed to find out something, or solve a problem about, a, the knowledge that c=d is of no use, for the cognitive value of a=b and c=d is the same. But this is exactly the sort of knowledge we get if statements of identity are about signs rather than objects. Because if an identity statement expresses a relation between signs, no genuine knowledge is expressed, Frege thus rejects the thesis that identity statements are a relation between signs.
Frege argues that in order for the cognitive value of a=a and a=b to differ, there must be a difference in the “mode of presentation” (35) of the object. This is like the way of thinking about an object and constitutes a sort of definite description. The sense of Batman differs from that of Bruce Wayne, even though they are the same. A person could believe that Bruce Wayne is Bruce Wayne without believing that Batman is Bruce Wayne, so each expression corresponds to a different thought. In this way, the cognitive value of Batman differs from that of Bruce Wayne.
We might ask whether Frege’s reasons for rejecting the thesis that identity statements express a relation between signs provide sufficient support for his final view that they express a relation between objects. There are two items that identity statements could express a relation between, viz. sign and object. In arguing that it cannot express a relation between signs, Frege leaves only one possibility, viz. object. If it can’t be sign, then it must be object.
But Frege had raised worries against identity being between objects, namely that the cognitive value of a=a and a=b is the same. How does this worry disappear? Frege’s notion of sense must come into play. A name is associated with a sense, not an object, though connection between name and sense will be arbitrary (as discussed in the problem between name and object). Sense, however, corresponds to an object in a nonarbitrary way. For instance “the smallest natural number” is the sense of 1 — 1 could never be associated with the sense “the greatest natural number”, for that does not describe 1, thus it is not arbitrary. This explains how a=a and a=b can be about the objects but still differ in cognitive value. Because sign is connected to sense, not object, the sense of a differs from b and so the cognitive value of the statements (the way of thinking about them) differs. But both senses still correspond to the same object (nonarbitrarily), so there is an identity with respect to the object (which is to say that the identity relation exists between objects). In this way, the statements are of different cognitive value while the identity between the objects is preserved, and so Frege’s initial worry is avoided and his argument provides support for the thesis that identity statements express a relation between objects.