Informative Identities in the Begriffsschrift and 'On Sense and Reference'

This paper is about the relationship between Frege's discussions of informative identity statements in the Begriffsschrift and 'On Sense and Reference'. The question of how these discussions relate to one another has a more-or-less standard answer that goes like this. In the Begriffsschrift Frege proposes a metalinguistic solution to the puzzle about how an identity statement can be informative. He says that what you find out when you discover that, for example, Hesperus is identical with Phosphorus, is that the two names 'Hesperus' and 'Phosphorus' are names for the same thing. In 'On Sense and Reference' Frege rejects this solution on the ground that it treats identity statements as statements about names rather than statements about objects. His new solution is that if 'a = b' is potentially informative for co-referring 'a' and 'b' this is because 'a' and 'b', though they refer to the same object, are associated with different ways of being presented with the object: finding out that a = b is finding out that the objects presented in these ways are the same.¹

In this paper I shall argue that, though the standard view of the relationship between the two discussions of informative identity is right in all its skeletal features, there is an important interpretive difficulty that it overlooks. And I shall suggest that clearing up this difficulty sheds considerable light on how the explanatory role of Fregean senses should be understood.

The paper has three parts. The first summarises the two discussions of informative identities and sets out the generally overlooked problem raised by Frege's own account of the transition from the Begriffsschrift view to the 'On Sense and Reference' view. The second proposes a solution to this problem. The third relates the problem and solution to wider questions about the explanatory role of the notion of the sense of a name.

I The Begriffsschrift and 'On Sense and Reference' treatments of informative identities

At the start of 'On Sense and Reference' Frege gives a reconstruction of the argument for the Begriffsschrift view of identity statements that goes like this²:

1. i. 'a = a' and 'a = b' can differ in cognitive value even if a = b. ['a = a' is known a priori and cannot extend knowledge³; 'a = b' may not be knowable a priori and may contain a 'valuable extension of our knowledge']

2. ii. An identity statement states either a relation between the objects that the names flanking the identity sign stand for, or a relation between the names themselves.

3. iii. If an identity statement states a relation between the objects that the names flanking the identity sign stand for, 'a = a' and 'a = b' cannot differ in cognitive value if a = b. [This is because if 'a = a' and 'a = b' state relations of this kind and 'a' and 'b' co-refer, each statement states the same relation between a thing and itself.]

therefore

iv) An identity statement states a relation between the names flanking the identity sign.

Here is the Begriffsschrift statement of this argument's conclusion:

Equality of content differs from conditionality and negation by relating to names, not to contents. Elsewhere, signs are mere proxies for their content, and thus any phrase they occur in just expresses a relation between their various contents; but names at once appear in propria persona [as themselves] so soon as they are joined together by the symbol for equality of content, for this signifies the circumstance of two names' having the same content.

[Begriffsschrift sect. 8]⁴

So the suggestion is that 'a = b' is to be treated as short for '"a" and "b" stand for the same object'. The possible difference in cognitive significance between 'a = a' and 'a = b' is to be explained in terms of the fact that the first sentence says '"a" stands for what "a" stands for' (or '"a" and "a" are intersubstitutable') while the second says '"a" stands for what "b" stands for' (or '"a" and "b" are intersubstitutable'). For it is never informative to be told that if you substitute a name for itself...