Frege: Functions and Concepts

Gottlob Frege is a significant figure in the philosophy of mathematics and logic insofar as he is the founder of modern symbolic logic and the articulator of the key distinctions and problems that have come to define much of contemporary analytic philosophy. Frege's concept-object distinction plays a major role in underwriting his thesis that numbers must be objects of a particular kind. Fregean numbers are generally interpreted as archetypal abstract objects, demanding some explanation as to how we come to know (...) them. Because Frege's distinction does not seem to deliver such an explanation, commentators have tended to dismiss it as a mistake. I argue that, despite its treatment by most scholars, Frege's distinction should not be read as an ontological claim, but as a methodological principle designed to prevent the errors being committed by his mathematical contemporaries. Analyzed as a norm of mathematical reasoning, Frege's concept-object distinction reveals itself as a neo-Kantian variant of Kant's distinction between a concept and an intuition. A key part of my argument involves delineating the different neo-Kantian camps that arose as a result of the clash between non-Euclidean geometries and Kant's epistemology. Frege is then shown to be aligned with the neo-Kantian camp led by Hermann Cohen and Paul Natorp. Seeing Frege as influenced by Cohen/Natorp's interpretation of Kantian epistemology not only supports my thesis that the C&O distinction is a methodological claim, but also explains why Frege defended the claim that Euclidean geometry was a body of necessary and universal truths grounded on pure intuition and continued to do so well into the 1900's. (shrink)

The paper presents a historical account of the primacy of concepts in Frege's conception of logic. Moreover, it argues that Frege's priority-thesis (i.e., the assumption that judgeable contents are prior to concepts) does not imply that sentential logic is more basic than the logic of concepts in his thought.

I offer an analysis of the sentence "the concept horse is a concept". It will be argued that the grammatical subject of this sentence, "the concept horse", indeed refers to a concept, and not to an object, as Frege once held. The argument is based on a criterion of proper-namehood according to which an expression is a proper name if it is so rendered in Frege's ideography. The predicate "is a concept", on the other hand, should not be thought of (...) as referring to a function. It will be argued that the analysis of sentences of the form "C is a concept" requires the introduction of a new form of statement. Such statements are not to be thought of as having function--argument form, but rather the structure subject--copula--predicate. (shrink)

It is a remarkable fact about the early history of the analytic tradition that its three most important protagonists all held, at least during significant intervals of their respective careers, that there are entities that cannot be named. This shared commitment on the part of Frege, Russell and the early Wittgenstein is the topic of this thesis. I first clarify the particular form this commitment takes in the work of these three authors. I also illustrate a distinctive cluster of philosophical (...) difficulties attending the view that there are unnameable entities, and explore the relationship between unnameability and inexpressibility. I then investigate what grounds there are for countenancing the unnameable, focussing in particular on the thesis that concepts cannot be named. I give a detailed hearing to four arguments for the unnameability of concepts discernible in, or suggested by, early analytic writings. The first and second arguments (chapters 3 and 4) are distinguishable in the locus classicus, Frege's 'On Concept and Object'. The first concerns the relationship between co-reference and intersubstitutability; the second concerns the unity of thought. The third argument (chapter 4) appeals to the alleged impossibility of expressing identities between objects and concepts, while the fourth (chapter 5) draws on considerations pertaining to diagonalization and Russell's paradox. I make the case that all four arguments fail to provide grounds for accepting the unnameable, contending that each argument can and should be resisted in defence of singular reference to concepts. In doing so I develop a novel defence of the view that absolutely any entity can be referred to with a singular term. (shrink)

In this paper I offer a conceptually tighter, quasi-Fregean solution to the concept horse paradox based on the idea that the unterfallen relation is asymmetrical. The solution is conceptually tighter in the sense that it retains the Fregean principle of separating sharply between concepts and objects, it retains Frege’s conclusion that the sentence ‘the concept horse is not a concept’ is true, but does not violate our intuitions on the matter. The solution is only ‘quasi’- Fregean in the sense that (...) it rejects Frege’s claims about the ontological import of natural language and his analysis thereof. (shrink)

In this essay, I respond to A. W. Moore’s instructive chapter on Frege. I respond by asking various questions, and I question particularly Moore’s claim that Frege, in reacting to Benno Kerry, falls into Hegelian excess. I toy with responding to my question by regarding Frege as anticipating a Wittgensteinian-Heideggerian exaction. It remains unclear whether this constitutes progress.

In this paper, I seek to clarify an aspect of Frege's thought that has been only insufficiently explained in the literature, namely, his notion of logical objects. I adduce some elements of Frege's philosophy that elucidate why he saw extensions as natural candidates for paradigmatic cases of logical objects. Moreover, I argue (against the suggestion of some contemporary scholars, in particular, Wright and Boolos) that Frege could not have taken Hume's Principle instead of Axiom V as a fundamental law of (...) arithmetic. This would be inconsistent with his views on logical objects. Finally, I shall argue that there is a connection between Frege's view on this topic and the famous thesis first formulated in ‘Über Begriff und Gegenstand’ that ‘the concept horse is not a concept’. As far as I know, no due attention has been given to this connection in the scholarly literature so far. (shrink)

Frege’s rejection of singular reference to concepts is centrally implicated in his notorious paradox of the concept horse. I distinguish a number of claims in which that rejection might consist and detail the dialectical difficulties confronting the defense of several such claims. Arguably the least problematic such claim—that it is simply nonsense to say that a concept can be referred to with a singular term—has recently received a novel defense due to Robert Trueman. I set out Trueman’s argument for this (...) claim, identifying and remedying some omissions and errors of formulation therein. I then develop a response to the argument by showing, pace Trueman, that it is possible—and how it is possible—to express identities between objects and concepts. (shrink)

In this paper I argue that Frege’s concept horse paradox is not easily avoided. I do so without appealing to Wright’s Reference Principle. I then use this result to show that Hale and Wright’s recent attempts to avoid this paradox by rejecting or otherwise defanging the Reference Principle are unsuccessful.

What Frege’s paradox on concept and object (FP) consists in and the manner in which Frege coped with it (the ladder strategy) are briefly reviewed (§ 1). An idea for solving FP inspired by Husserl’s semantics is presented; it results in failure, for it leads to a version of Russell’s paradox, the usual solution of which implies something like a resurgence of FP (§ 2). A generalized version of Frege’s paradox (GFP) and an idea for solving it inspired by Davidson’s (...) semantics are presented; three theorems about recursive definability of truth are put forward and used to determine whether this idea can be successfully applied to certain putative forms of the Language of Science (§ 3). Proofs of these three theorems, in particular of the third, which answers a question that does not seem to have drawn logicians’ attention, are then given (§ 4). Finally, it turns out that there is a tension between the proposed solution of GFP and the idea of Language of Science assumed so far in this paper, and a way of solving it is proposed (§ 5). (shrink)

The „Paradox of the Concept Horse" arises on the assumption of the Reference Principle: that co-referential expressions should be cross-substitutable salva veritate in extensional contexts and salva congruitate in all. Accordingly no singular term can co-refer with an unsaturated expression. The paper outlines a number of desiderata for a satisfactory response to the problem and argues that recent treatments by Dummett and Wiggins fall short by their lights. It is then pointed out that a more consistent perception of the requirements (...) of the Reference Principle leads not to the Paradox but to the result that Frege had no business extending the notion of Bedeutung to unsaturated expressions in the first place. Rather the relation between, e.g., predicates and the entities that comprise the range of higher-order logical variables must be logically unlike that between singular terms and their referents; the way is therefore opened for singular terms to refer to entities of the former kind after all. The Concept Horse is a concept (and a Fregean object too.). (shrink)

I argue that Frege's so-called "concept 'horse' problem" is not one problem but many. When these separate sub-problems are distinguished, some are revealed to be more tractable than others. I further argue that there is, contrary to a widespread scholarly assumption originating with Peter Geach, little evidence that Frege was concerned with the general problem of the inexpressibility of logical category distinctions in writings available to Wittgenstein. In consequence, Geach is mistaken in thinking that in the Tractatus Wittgenstein simply accepts (...) from Frege certain lessons about the inexpressibility of logical category distinctions and the say-show distinction. In truth, Wittgenstein drew his own morals about these matters, quite possibly as the result of reflecting on how the general problem of the inexpressibility of logical category distinctions arises in Frege's writings , but also, quite possibly, by seeing certain glimmerings of these doctrines in the writings of Russell. (shrink)

Frege held various views about language and its relation to non-linguistic things. These views led him to the paradoxical-sounding conclusion that "the concept horse is NOT a concept." A key assumption that led him to say this is the assumption that phrases beginning with the definite article "the" denote objects, not concepts. In sections I-III this issue is explained. In sections IV-V Frege's theory is articulated, and it is shown that he was incorrect in thinking that this theory led to (...) the conclusion that "the concept horse is not a concept." Section VI goes on to show that his strict theory about the functioning of ordinary language is inconsistent. Sections VII-VIII investigate Frege's reasons for thinking that "the concept horse" must denote an object; these reasons are not adequate on Frege's own grounds. Section IX sketches a systematic way to allow such phrases to denote concepts (not objects) within the framework of Frege's main views about language. Section X comments briefly on the consequences of this idea for his logistic program. (shrink)

After describing the philosophical background of Kerry's work, an account is given of the way Kerry proposed to supplement Bolzano's conception of logic with a psychological account of the mental acts underlying mathematical judgements.In his writings Kerry criticized Frege's work and Kerry's views were then attacked by Frege.The following two issues were central to this controversy: (a) the relation between the content of a concept and the object of a concept; (b) the logical roles of the definite article. Not only (...) did Frege in 1892 offer an unconvincing solution to Kerry's puzzle concerning 'the concept horse' but he also overlooked the many criticisms levelled by Kerry against the notion of an (indefinite) extension on which his own definition of number was based. (shrink)

The purpose of the article is to explain two curious doctrines maintained by frege and rejected by wittgenstein in the 'tractatus logico-philosophicus'. that a special assertion sign is necessary was maintained by frege because he wanted to apply his concept-writing to ordinary language, and it was rejected by wittgenstein because his concern in the 'tractatus' was with scientific assertions only. frege's paradoxical notion that 'the concept horse is not a concept' was a consequence of his symbolizing functions by 'unsaturated' expressions. (...) wittgenstein's picture theory eliminated expressions for relations and thereby avoided the fregean paradox. (shrink)

In this paper I discuss Frege's distinction between objects and concepts and suggest a solution of Frege's paradox of the concept horse. The expression ''the concept horse'' is not eliminated and the concept is not identified with its extension, but the concept is identified with the sense of the corresponding predicate. This solution fits better into a fregean ontology and philosophy of language than alternative solutions and allows for a general answer to the question why Frege's system is infected with (...) Russell's paradox. Russell's paradox is caused by the reification of a concept. Certain problems of modern set theory seem to have a similar cause.Eine weithin sichtbare Warnungstafel muss aufgerichtet. (shrink)

The purpose of this paper is to discuss Frege’s view on vagueness, and to draw some relevant consequences of it. By examining what exactly Frege has in mind each time he complains about vagueness and advocates the sharpness requirement, I argue that he shows preoccupation with different kinds of vagueness in different periods of his thought. I also discuss the scope of the sharpness requirement, and argue that it is intended as applying primarily to mathematics and logic. Finally, I try (...) and argue that some of Frege’s remarks on incomplete functions suggest a view that is close in spirit to the contemporary supervaluationist approach to vagueness. (shrink)

Frege, we are told, believes that vague predicates have no referent (Bedeutung). But given other things he evidently believes, such a position would seem to commit him to a suspect nihilism according to which assertoric sentences containing vague predicates are neither true nor false. I argue that we have good reason to resist ascribing to Frege the view that vague predicates have no Bedeutung and thus good reason to resist seeing him as committed to the suspect nihilism. In the process, (...) I call attention to several under-appreciated texts in which Frege suggests that a vague predicate, though lacking a Bedeutung of its own, can come to acquire a Bedeutung in certain contexts. The upshot of this suggestion is that vague predicates can serve the purposes of ordinary communication quite well, even if they are useless for logical purposes. (shrink)

Controversy exists concerning the consequences of Frege's sharpness requirement for concepts and functions. Some say that the sharpness requirement, if taken to be a necessary condition for truth functional language use, renders most of our natural language discourse meaningless. This is because most if not all natural language concepts and predicates are not sharp. In this essay I argue first that Frege does indeed see the sharpness requirement as a necessary condition on a language's truth- functionality in all contexts in (...) which language is used, and that the attempt to eschew the difficulty that this requirement presents by stipulating within a metalanguage what the extensions of our natural language concepts and predicates shall be is fundamentally at odds with Frege's conception of logic. I then turn to a possible application of Frege's notion of sharpness as a set of metaphysical presuppositions underlying the everyday use of concepts in contexts in which truth is being addressed. (shrink)

Vagueness provides the first comprehensive examination of a topic of increasing importance in metaphysics and the philosophy of logic and language. Timothy Williamson traces the history of this philosophical problem from discussions of the heap paradox in classical Greece to modern formal approaches such as fuzzy logic. He illustrates the problems with views which have taken the position that standard logic and formal semantics do not apply to vague language, and defends the controversial realistic view that vagueness is a kind (...) of ignorance--that there really is a grain of sand whose removal turns a heap into a non-heap, but we cannot know which one it is. (shrink)

_ Source: _Volume 95, Issue 3, pp 368 - 413 Frege famously maintained that concepts are not objects. A key argument of Frege’s for this view is, in outline, as follows: if we are to account for the unity of thought, concepts must be deemed _unsaturated_; since objects are, by contrast, saturated entities, concepts cannot be objects. The author investigates what can be made of this argument and, in particular, of the unsaturated/saturated distinction it invokes. Systematically exploring a range of (...) reconstructions suggested by Frege’s writings, and drawing on contemporary work, the author illustrates that no plausible reconstruction is forthcoming. In essence, it is altogether unclear how to simultaneously substantiate, on the one hand, the claim that unsaturated entities must be recognized in order to account for unity and, on the other, the claim that unsaturatedness is incompatible with objecthood. (shrink)

ABSTRACTFregeans hold that predicates denote things, albeit things different in kind from what singular terms denote. This leads to a familiar problem: it seems impossible to say what any given predicate denotes. One strategy for avoiding this problem reduces the Fregean position to form of nominalism. I develop an alternative strategy that lets the Fregean hold on to the view that predicate denote things by reconceiving the nature of singular denotation and of Fregean objects.

This paper identifies a tension in Frege’s philosophy and offers a diagnosis of its origins. Frege’s Context Principle can be used to dissolve the problem of propositional unity. However, Frege’s official response to the problem does not invoke the Context Principle, but the distinction between ‘saturated’ and ‘unsaturated’ propositional constituents. I argue that such a response involves assumptions that clash with the Context Principle. I suggest, however, that this tension is not generated by deep-seated philosophical commitments, but by Frege’s occasional (...) attempt to take a dubious shortcut in the justification of his conception of propositional structure. (shrink)

What Frege’s paradox on concept and object (FP) consists in and the manner in which Frege coped with it (the ladder strategy) are briefly reviewed (§ 1). An idea for solving FP inspired by Husserl’s semantics is presented; it results in failure, for it leads to a version of Russell’s paradox, the usual solution of which implies something like a resurgence of FP (§ 2). A generalized version of Frege’s paradox (GFP) and an idea for solving it inspired by Davidson’s (...) semantics are presented; three theorems about recursive definability of truth are put forward and used to determine whether this idea can be successfully applied to certain putative forms of the Language of Science (§ 3). Proofs of these three theorems, in particular of the third, which answers a question that does not seem to have drawn logicians’ attention, are then given (§ 4). Finally, it turns out that there is a tension between the proposed solution of GFP and the idea of Language of Science assumed so far in this paper, and a way of solving it is proposed (§ 5). (shrink)

A brief study of the logical, linguistic, psychological and ontological problem of ‘impersonalia’, which is to say of assertions such as ‘it’s raining’ or ‘es blitzt’ which seem to have no subject. Such assertions cause problems not only for defenders of traditional subject-predicate views of assertive sentences, but also for those, such as Frege, who defended a view in terms of functions and arguments.