Frege: Identity

The aim of this paper is to present a constructive solution to Frege's puzzle (largely limited to the mathematical context) based on type theory. Two ways in which an equality statement may be said to have cognitive significance are distinguished. One concerns the mode of presentation of the equality, the other its mode of proof. Frege's distinction between sense and reference, which emphasizes the former aspect, cannot adequately explain the cognitive significance of equality statements unless a clear identity criterion for (...) senses is provided. It is argued that providing a solution based on proofs is more satisfactory from the standpoint of constructive semantics. (shrink)

Many philosophers have argued or taken for granted that Frege's puzzle has little or nothing to do with identity statements. I show that this is wrong, arguing that the puzzle can only be motivated relative to a thinker's beliefs about the identity or distinctness of the relevant object. The result is important, as it suggests that the puzzle can be solved, not by a semantic theory of names or referring expressions as such, but simply by a theory of identity statements. (...) To show this, I sketch a framework for developing solutions of this sort. I also consider how this result could be implemented by two influential solutions to Frege's puzzle, Perry's referential-reflexivism and Fine's semantic relationism. (shrink)

In formulating the puzzle about cognitive significance in ‘Über Sinn und Bedeutung’, Frege rejects the approach he suggested in the Begriffsschrift on the ground that if the utterance of a sentence of the form a = b is understood as ‘a’ and ‘b’ referring to the same object we lose the subject matter. In this note, we will show how Frege’s concerns can be understood and circumvented.

The goal of the paper is to offer an explanation why Frege has changed his Begriffsschrift account of identity to the one presented in Über Sinn und Bedeutung. The main claim of the paper is that in order to better understand Frege’s motivation for the introduction of his distinction between sense and reference, which marks his change of views, one should place this change in its original setting, namely the broader framework of Frege’s fundamental preoccupations with the foundations of arithmetic (...) and logic. The Fregean thesis that mathematics is contentful, and its defense against formalism and psychologism, provides us an valuable interpretative key. Thus, Fregean senses are not just the mere outcome of some profound reflections on language, rather they play an important role in the articulation of Frege’s program in the foundations of arithmetic. (shrink)

The goal of the paper is to offer an explanation why Frege has changed his Begriffsschrift account of identity to the one presented in Über Sinn und Bedeutung.The main claim of the paper is that in order to better understand Frege’s motivation for the introduction of his distinction between sense and reference, which marks his change of views, one should place this change in its original setting, namely the broader framework of Frege’s fundamental preoccupations with the foundations of arithmetic and (...) logic. The Fregean thesis that mathematics is contentful, and its defense against formalism and psychologism, provides us an valuable interpretative key. Thus, Fregean senses are not just the mere outcome of some profound reflections on language, rather they play an important role in the articulation of Frege’s program in the foundations of arithmetic. (shrink)

The analysis of identity as coreference is strongly associated with Frege ; it is the view in Begriffsschrift, and, some have argued, henceforth throughout his work. This thesis is incorrect: Frege never held that identity is coreference. The case is made not by interpretation of “proof-quotes”, but rather by exploring how Frege actually deploys the concept. Two cases are considered. The first, from Grundgesetze, are the definitions of the core concepts, zero and truth; the second, from Begriffsschrift, is the validity (...) of Leibniz's Law. In both cases, if identity is coreference, results ensue that would be unacceptable to Frege. (shrink)

This essay offers a detailed philosophical criticism of Frege’s popular thesis that identity is a relation of names. I consider Frege’s position as articulated both in ‘On Sense and Reference’, and in the Grundgesetze, where he appears to take an objectual view of identity, arguing that in both cases Frege is clearly committed to the proposition that identity is a relation holding between names, on the grounds that two different things can never be identical. A counterexample to Frege’s thesis is (...) considered, and a positive thesis is developed according to which, in contradistinction to the Fregean position, identity is a reflexive, symmetric, and transitive relation holding only between a thing and itself which can be expressed as a relation between names. (shrink)

Wittgensteinian predicate logic (W-logic) is characterized by the requirement that the objects mentioned within the scope of a quantifier be excluded from the range of the associated bound variable. I present a sound and complete tableaux calculus for this logic and discuss issues of translatability between Wittgensteinian and standard predicate logic in languages with and without individual constants. A metalinguistic co-denotation predicate, akin to Frege’s triple bar of the Begriffsschrift, is introduced and used to bestow the full expressive power of (...) first-order logic with identity on W-logic in the presence of constants. (shrink)

Frege’s views on identity continue to provoke scholars, and rightly so. In particular his view in Begriffsschrift of 1879, and its relation to his view in ‘Über Sinn und Bedeutung’ of 1892 deserve careful attention. The issues involved have a wider significance than Frege’s specific views on identity in different periods, though these are important enough. They concern also the move from what I call below ‘thin’ semantics, which is exhausted in signs being assigned content, to a ‘thick’ semantics, in (...) which ‘modes of determining their content’, or Sinne are also concerned. (shrink)

Frege’s views on identity continue to provoke scholars, and rightly so. In particular his view in Begriffsschrift of 1879, and its relation to his view in ‘Über Sinn und Bedeutung’ of 1892 deserve careful attention. The issues involved have a wider significance than Frege’s specific views on identity in different periods, though these are important enough. They concern also the move from what I call below ‘thin’ semantics, which is exhausted in signs being assigned content, to a ‘thick’ semantics, in (...) which ‘modes of determining their content’, or Sinne are also concerned. (shrink)

The paper argues, as against Thau and Caplan, that the traditional interpretation that Frege abandoned his earlier views about identity and identity--statements is correct.

In ‘What’s Puzzling Gottlob Frege?’ Michael Thau and Ben Caplan argue that, contrary to the common wisdom, Frege never abandoned his early view that, as he puts it in Begriffsschrift, a statement of identity ‘expresses the circumstance that two names have the same content’ and thus asserts the existence of a relation between names rather than a relation between objects. The arguments at the beginning of ‘On Sense and Reference’ do, they agree, raise a problem for that view, but, they (...) insist, Frege does not, as the ‘standard’ interpretation has it, take these arguments to refute it. Rather, they claim, Frege is out to defend his earlier view against these objections: indeed, the defense he there offers is pretty much the same defense offered in Begriffsschrift against what are pretty much the same objections. (shrink)

*I am very pleased to be able to contribute this paper to a festschrift for Andrea Bonomi. This is not however, the paper I really wanted to write; I would have much rather have contributed a paper comparing the pianistic styles of Lennie Tristano and Bill Evans, which I think Andrea would have found much more fascinating than an essay devoted to an understanding of Frege’s thinking. But I do not totally despair. Andrea’s first paper published in English was entitled (...) “On the Concept of Logical Form in Frege,” so perhaps I can maintain some hope that this paper will appeal to lingering interests that Andrea wrote of in the past. I would like to thank Johannes Brandl, Ben Caplan, Bill Demopoulos, Bob Fiengo, Mark Kalderon, Patricia Marino, Gila Sher, Michael Thau, Dan Vest and especially Aldo Antonelli for very helpful discussion. (shrink)

In his paper Über Sinn und Bedeutung Frege discusses different views of identity but it is left open which of these views he ultimately holds. The theses that identity is a relation between objects, that it is a relation between names, and that it is no relation at all, seem to contradict each other. Remarkably, each of these three theses have been proposed as describing Frege's view. It is the aim of the paper to explain why there are such divergent (...) interpretations. The result is that all three theses are inadequately formulated. If one modifies them, one obtains three theses which do not really contradict each other, but describe the same fact in different manners. All three modified theses adequately characterize the core of Frege's view of identity. (shrink)

The idea underlying the Begriffsschrift account of identities was that the content of a sentence is a function of the things it is about. If so, then if an identity a=b is about the content of its contained terms and is true, then a=a and a=b have the same content. But they do not have the same content; so, Frege concluded, identities are not about the contents of their contained terms. The way Frege regarded the matter is that in an (...) identity the terms flanking the symbol for identity do not have their ordinary contents, but instead have themselves as their contents. In ?Uber Sinn und Bedeutung? Frege became convinced that if an identity a=bis about the signs aand b, then it expresses no proper knowledge. So, since it is evident that many such identities do express proper knowledge, Frege concluded that identities are not about their contained signs. So he became convinced that his Begriffsschrift account was incorrect. What was the error in the argument that led Frege to that account? It was thinking that the content of a sentence is a function of the contents of its constituent signs, that is, the things it is about. (shrink)

P. T. Geach, notoriously, holds the Relative Identity Thesis, according to which a meaningful judgment of identity is always, implicitly or explicitly, relative to some general term. ‘The same’ is a fragmentary expression, and has no significance unless we say or mean ‘the same X’, where ‘X’ represents a general term (what Frege calls a Begriffswort or Begriffsausdruck). (P. T. Geach, Mental Acts (London: Routledge and Kegan Paul, 1957), p. 69. I maintain that it makes no sense to judge whether (...) things are ‘the same’, or remain ‘the same’, unless we add or understand some general term - ‘the same F’. (P. T. Geach, Reference and Generality, third Edition (Ithaca, N.Y.: Cornell University Press, 1980), pp. 63f. I am arguing for the thesis that identity is relative. When one says ‘x is identical with y’, this, I hold, is an incomplete expression; it is short for ‘x is the same A as y’, where ‘A’ represents some count noun understood from the context of utterance - or else, it is just a vague expression of a half-formed thought. (P. T. Geach, ‘Identity,’ Review of Metaphysics 21 (1967-8), p. 3.) One of the ways Geach seeks to support this is by tying it to the well nigh universally admired Fregean thesis about cardinality. (shrink)

Interpretations from the past 30 years of frege's explanation of the cognitive value of identity sentences are considered. Frege's explanation, Which the author finds superior to any of these interpretations, Is that 'a=b' has greater cognitive value than 'a=a' because, Given that 'a' and 'b' are different sign-Types, The sense of 'a' "may" (though it need not) differ from that of 'b'. It is pointed out that this interpretation of frege shows that his problem with identity sentences can be resolved (...) prior to the larger problem of explicating the notion of sense. (shrink)

Frege continues by explaining what bothered him in the Begriffsschrift, and motivated his treatment of identity in that work.2 He goes on to criticize that account. By the end of the paragraph, he has introduced his key concept of sinn, abandonning not only the Begriffsschrift account of identity, but its basical semantical framework. In the Begriffsschrift Frege’s main semantic concept was content [Inhalt ]. Already in the Begriffsschrift, he is struggling with this concept. In §3 he..