Journal of the History of Philosophy

Frege's Begriffsschrift Theory of Identity RICHARD L. MENDELSOHN MY CONCERNin this paper is with Frege's Begriffsschrift theory of identity, on which identity is taken to relate expressions instead of the object(s) denoted by them. This nonstandard theory of identity is widely regarded as a youthful aberration, and it has been by and large ignored in the philosophical literature. As a result, there is insufficient understanding of Frege's reasons for adopting the theory and almost no appreciation of the continuity between his Begriffsschrift view and the later, more successful sense/reference account. My intention is to fill this gap in our knowledge of Frege's development . In Section 1, I examine the paradox of identity and show that the underlying problem lies not with identity but with a highly dubious substitution principle. In Section 2, I explore the Begriffsschrift semantic theory to determine why Frege had held such a principle. I argue that although Frege had recognized a distinction for singular terms comparable to the sense/reference distinction, confusion about the notion of the content of a sentence prevented him from drawing the corresponding distinction for them: the function/argument connection between the semantics of singular terms and sentences was therefore drawn incorrectly. Taking identity to relate expressions was Frege's attempt to make a coherent connection, without, I argue, much success. In Section 3, I discuss the traditional criticisms of the Begriffsschrift theory and then detail my own, based on the findings of Section 2. I At the beginning of "On Sense and Reference" (hereafter cited as SR), Frege observed that identity sentences (/-sentences differ in what he calls "cognitive value" (Erkenntniswerth): a=a and a=b are obviously statements of differing cognitive value; a=a holds a priori and, according to Kant, is to be labelled analytic, while statements of the form a=b often contain very valuable extensions of our knowledge and cannot alwaysbe [279] 280 HISTORY OF PHILOSOPHY established a priori. The discover), that the rising sun is not new every morning, but always the same, was one of the most fertile astronomical discoveries. Even to-day the identification of a small planet or a comet is not always a matter of course? Roughly, /-sentences of the form let=et] are true but trivial, whereas isentences of the form tt~=~l are, if true, often interesting and informative. ~ This observation is accurate, and examples come readily to mind: 'Mark Twain = Mark Twain' is a mere truism hardly worth remarking, but 'Mark Twain = Samuel Clemens' says something of considerable literary and historical significance. However, Frege continues, the fact that /-sentences differ in cognitive value might appear to conflict with the generally accepted view that "identity relates objects," the view, that is, on which let=13j is understood to express that a relation, being one and the same thing as, holds between the objects denoted by a and [3. For since the relation is said to hold between the objects themselves, he argues, then what is said or expressed by Ea=[3J--the cognitivie or informational content of the sentence--is simply that the objects stand in the given relation. Therefore tet=[3J and cy=61 (et, [3, y, 6 not necessarily distinct) would say the same thing--have the same cognitive or informational content--if the object denoted by et were one and the same as the object denoted by y and the object denoted by [3 were one and the same as the object denoted by 5; for the same relation would be said to hold between the same objects. So, Frege concludes, ra=al and true ta=131 could not differ in cognitive value: "if we were to regard equality as a relation between that which the names 'a' and 'b' designate, it would seem that a=b could not differ from a=a (i.e. provided a=b is true). A relation would thereby be expressed of a thing to itself, and indeed one in which each thing stands to itself but to no other thing. ''~ On this interpretation of '=', then, it appears that our two/-sentences, 'Mark Twain = Mark Twain' and 'Mark Twain = Samuel Clemens', both say the same thing...