Frege Explained by Weiner Joan;

Author:Weiner, Joan;
Language: eng
Format: epub
Publisher: Open Court
Published: 2011-04-18T16:00:00+00:00

is a perfectly good sentence (although not one anyone is likely to endorse). In contrast, the string of words,The Earth is a planet is identical to the number one

is not a sentence at all. But, given the view of concepts introduced in “Function and Concept,” such a distinction can no longer be maintained. As we have seen, ‘the Earth is a planet’ is now regarded as designating an object, namely the True. Since each concept holds or not of each object, it holds or not, in particular, of this object. Presuming we accept this, it makes perfect sense to say, ‘The Earth is a planet is identical to the number one’. Frege does not, of course, suggest that we introduce such sentences into natural language. But such sentences do appear in his logical notation.

Is this a difficulty with the new logical notation? It surely would be if Frege wanted the logical notation to represent the function of sentences in natural language. But the logical notation is meant to deviate from natural language in a number of ways. And there is no evidence that the tripartite distinction that is operative in our use of natural language has a role to play in Frege’s logic, given the purposes it is meant to serve. Moreover, given his purposes, it is essential that sentential expressions be regarded as object-names. To see this, let us consider Frege’s use of identity.

Although Frege distinguishes sentential from non-sentential expressions in Begriffsschrift, both sorts of expressions can appear flanking the identity sign. The reason is that identity, in the logic of Begriffsschrift, is simply sameness of conceptual content. A sentence, as much as any other expression, has conceptual content. Thus the identity sign is (and ought to be) usable to express the claim that two sentences have the same conceptual content. As long as identity is understood as sameness of conceptual content, the occurrence of sentences on either side of the identity sign does not force us to regard sentences as object names. That we are forced to do so is a consequence of Frege’s new understanding of identity. On the new view of identity, expressions that can flank the identity sign must be object-names. One might think that, once Frege modified his view of identity, he should also have modified his rules. One might think that he should, in particular, have prohibited such identity statements as: 1 = the Earth is a planet. In order to see why Frege did not choose to modify his rules in this way, we need to consider the role played by functions and concepts in Frege’s project.

One of Frege’s tasks is to define concepts. A concept is a function and, consequently, is defined by indicating what its value is for each object. Suppose we use the expression ‘E’ for the concept to be defined, and suppose we define it by saying that the property holds of the numbers that are divisible by 2. Our definition will look something like this: E(x) = x is a number that is divisible by 2.

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