Frege: Value-Ranges

The notion of the extension of a concept has been used in logic for a long time. It is usually considered to be closely connected to the intuitive notion of a set and thus seems as though it should be embedded into set theory. However, there are significant differences between this “logical” concept of set and the notion of set (class) as defined via standard axiomatic systems of set theory; it may, therefore, be quite misleading to consider the two concepts (...) as being continuous with each other. When we look at the writings of Gottlob Frege and consider the development of his attitude to extensions, we can see what the differences consist in and which of the two notions is more apt to be used in foundations of logic. Frege himself eventually rejected sets entirely. (shrink)

In this article, I first analyze and assess the epistemological and semantic status of canonical value-range equations in the formal language of Frege’s Grundgesetze der Arithmetik. I subsequently scrutinize the relation between (a) his informal, metalinguistic stipulation in Grundgesetze I, Section 3, and (b) its formal counterpart, which is Basic Law V. One point I argue for is that the stipulation in Section 3 was designed not only to fix the references of value-range names, but that it was probably also (...) intended to play a regulative role with an eye to Basic Law V. I further intend to shed new light on the status of Frege’s identification of the truth-values with their unit classes in Grundgesetze I, Section 10 and determine its place and importance in his overall logicist project. In a subsequent section, I first discuss the hypothetical extendability of the logical system of Grundgesetze. In what follows, I analyze the incompleteness of that system—were it not inconsistent and hence trivially complete—by focusing on the role of the definite description operator and the axiom governing it (= Basic Law VI) as well as on the role of the application operator. The latter is defined by means of the former. I further comment on the redundancy of an axiom that could be designed to supplement Basic Law VI by incorporating the second clause of Frege’s two-part elucidation of the description operator. I argue that although it seems likely that Basic Law VI could have been shown to be dispensable in Grundgesetze, Frege would probably have been reluctant to dispense with it, despite his commitment to axiomatic parsimony. In my view, the reason is that he considers it essential that every primitive function-name of his formal language be governed and grounded by a basic law that is tailored to the nature and the role of the primitive name occurring at a key point in the law. Toward the end of the article, I propose solutions to two problems with Frege’s use of “=” in the concept-script sentences that result from his definitions involving the replacement of the double stroke of definition with the judgment-stroke. I conclude the article with a summary of the main results that I have achieved. (shrink)

The aim of this article is to examine Frege’s view of the context principle in his mature philosophical doctrine. Here, the author argues that the context principle is embodied in the contextual explanation of value-ranges presented in Basic Laws of Arithmetic. The contextual explanation of value-ranges plays essentially the same role as the context principle in The Foundations of Arithmetic. It is supposed to show how a reference to natural numbers is possible. Moreover, the author argues against the view that (...) the context principle should be separated from the recarving thesis. This aspect of Frege’s view leads the author to the rejection of the purely referential view of the context principle, which is fairly widely accepted in the current literature. (shrink)

Frege's Grundgesetze der Arithmetik offers a conception of cpLogic as the study of functions. Among functions are included those that are concepts, i.e. characteristic functions whose values are the logical objects that are the True/the False. What, in Frege's view, are the objects the True/the False? Frege's stroke functions are themselves concepts. His stipulation introducing his negation stroke mentions that it yields [...]. But curiously no accommodating axiom is given, and there is no such theorem. Why is it that some (...) of Frege's informal stipulations never made appearances as axioms? I offer an explanation that sheds new light on the Grundgesetze. No axioms should over-determination the True as a logical object. Perhaps the True = 0, as would be common in the mathematics of characteristic functions. But the logical objects that are cardinal numbers are value ranges correlated with second-level numerical concepts by a non-homogeneous second-level value-range function [...]. The existence of concepts would be ontologically circular if the True is itself a number. We find this circularity perfectly agreeable to Frege, and suggest that he had accepted that the existence of functions that are concepts in his cpLogic may well be ontologically inseparable from the existence of his value-range function. His cpLogic itself stands or falls with the viability of some value-range function. (shrink)

Frege’s famous definition of number famously uses the concept of “extension”. Extensions, in the Fregean framework, are susceptible to bringing many difficulties, and, some say, even paradoxes. Therefore, neo-logicist programs want to avoid the problems and to replace the classical Fregean definition of number with Hume’s Principle. I argue that this move, even if it makes sense from a computational point of view, is at odds with Frege’s larger philosophical project. For Frege, I claim, extensions were an important part of (...) his philosophical program of logic-as-an-universal-language. This is why Frege places his project in line with Leibniz’ philosophical project of finding a lingua characterica universalis. (shrink)

Frege’s famous definition of number famously uses the concept of “extension”. Extensions, in the Fregean framework, are susceptible to bringing many difficulties, and, some say, even paradoxes. Therefore, neo-logicist programs want to avoid the problems and to replace the classical Fregean definition of number with Hume’s Principle. I argue that this move, even if it makes sense from a computational point of view, is at odds with Frege’s larger philosophical project. For Frege, I claim, extensions were an important part of (...) his philosophical program of logic-as-an-universal-language. This is why Frege places his project in line with Leibniz’ philosophical project of finding a lingua characterica universalis. (shrink)

Frege’s famous definition of number famously uses the concept of “extension”. Extensions, in the Fregean framework, are susceptible to bringing many difficulties, and, some say, even paradoxes. Therefore, neo-logicist programs want to avoid the problems and to replace the classical Fregean definition of number with Hume’s Principle. I argue that this move, even if it makes sense from a computational point of view, is at odds with Frege’s larger philosophical project. For Frege, I claim, extensions were an important part of (...) his philosophical program of logic-as-an-universal-language. This is why Frege places his project in line with Leibniz’ philosophical project of finding a lingua characterica universalis. (shrink)

Gottlob Frege defined cardinal numbers in terms of value-ranges governed by the inconsistent Basic Law V. Neo-logicists have revived something like Frege's original project by introducing cardinal numbers as primitive objects, governed by Hume's Principle. A neo-logicist foundation for set theory, however, requires a consistent theory of value-ranges of some sort. Thus, it is natural to ask whether we can reconstruct the cardinal numbers by retaining Frege's definition and adopting an alternative consistent principle governing value-ranges. Given some natural assumptions regarding (...) what an acceptable neo-logicistic theory of value-ranges might look like, successfully implementing this alternative approach is impossible. (shrink)

Although the notion of logical object plays a key role in Frege's foundational project, it has hardly been analyzed in depth so far. I argue that Marco Ruffino's attempt to fill this gap by establishing a close link between Frege's treatment of expressions of the form ‘the concept F’ and the privileged status Frege assigns to extensions of concepts as logical objects is bound to fail. I argue, in particular, that Frege's principal motive for introducing extensions into his logical theory (...) is not to be able to make in-direct statements about concepts, but rather to define all numbers as logical objects of a fundamental kind in order to ensure that we have the right cognitive access to them qua logical objects via Axiom V. Contrary to what Ruffino claims, reducibility to extensions cannot be the ‘ultimate criterion’ for Frege of what is to be regarded as a logical object. (shrink)

In Section 10 of Grundgesetze, Volume I, Frege advances a mathematical argument (known as the permutation argument), by means of which he intends to show that an arbitrary value-range may be identified with the True, and any other one with the False, without contradicting any stipulations previously introduced (we shall call this claim the identifiability thesis, following Schroeder-Heister (1987)). As far as we are aware, there is no consensus in the literature as to (i) the proper interpretation of the permutation (...) argument and the identifiability thesis, (ii) the validity of the permutation argument, and (iii) the truth of the identifiability thesis. In this paper, we undertake a detailed technical study of the two main lines of interpretation, and gather some evidence for favoring one interpretation over the other. (shrink)

Edmund Husserl was one of the very first to experience the direct impact of challenging problems in set theory and his phenomenology first began to take shape while he was struggling to solve such problems. Here I study three difficulties associated with Frege's use of sets that Husserl explicitly addressed: reference to non-existent, impossible, imaginary objects; the introduction of extensions; and 'Russell's paradox'.I do so within the context of Husserl's struggle to overcome the shortcomings of set theory and to develop (...) his own theory of manifolds. I define certain issues involved and discuss how Husserl's theory of manifolds might confront them. In so doing I hope to help bring Husserl's theories about sets and manifolds out of the realm of abstract theorizing and prompt further exploration of uncharted philosophical territory rich in philosophical implications. (shrink)

George Boolos, Crispin Wright, and others have demonstrated how most of Frege's treatment of arithmetic can be obtained from a second-order statement that Boolos dubbed ‘Hume's principle’. This note explores the historical evidence that Frege originally planned to develop a philosophical approach to numbers in which Hume's principle is central, but this strategy was abandoned midway through his Grundlagen.

Pretendo usar o exemplo dos nomes de percursos de valores como prova de que, contrariamente ao que Michael Resnik e Michael Dummett sustentam, Frege nunca abandonou o seu princípio do contexto: “Apenas no contexto de uma sentenya tem uma palavra significado”. Em particular, pretendo mostrar que a prova da completude com relação ao significado, que Frege tentou introduzir na linguagem formal das Grundgesetze der Arithmetik, baseia-se em uma aplicação do principio do contexto, e que, em consequencia, tambem nomes de percursos (...) de valores tem significado apenas nocontexto de uma sentença. A teoria Fregeana do sentido e do significado somente pode ser entendida adequadamente sob o pano de fundo do princfpio do contexto.Taking course-of-values names as an example, I want to show that, contrary to what Michael Resnik and Michael Dummett claim, Frege never abandoned his context principle “Only in the context of a sentence do words have meaning”. In particular, I want to show that Frege’s attempted proof of referentiality for the formal language of Grundgesetze der Arithmetik rests on the context principle and that, consequently, course-of-values names have a reference only in the context of a sentence. It is only in the light of the context principle that Frege’s theory of sense and reference can be understood appropriately. (shrink)

Frege’s method of introducing abstract singular terms by transforming an equivalence statement into an identity statement suffers from one major defect: it is haunted by a pervasive indeterminacy of putative reference. In this paper, I. discuss mainly Frege’s introduction of courses-of-values in his magnum opus Grundgesetze der Arithmetik (Volume I, 1893, Volume 11, 1903). More specifically, I want to assesscritically, with respect to course-of-values names, what I call Frege’s indeterminacy problem. In the first part, I sketch the nature of this (...) problem in connection with the introduction of numerical singular terms in his book Die Grundlagen der Arithmetik of 1884. In the second part, I try to shed light on the analogies and differences concerning the introduction of numerical terms on the one hand and of course-of-values terms on the other. (shrink)

Frege’s method of introducing abstract singular terms by transforming an equivalence statement into an identity statement suffers from one major defect: it is haunted by a pervasive indeterminacy of putative reference. In this paper, I. discuss mainly Frege’s introduction of courses-of-values in his magnum opus Grundgesetze der Arithmetik. More specifically, I want to assesscritically, with respect to course-of-values names, what I call Frege’s indeterminacy problem. In the first part, I sketch the nature of this problem in connection with the introduction (...) of numerical singular terms in his book Die Grundlagen der Arithmetik of 1884. In the second part, I try to shed light on the analogies and differences concerning the introduction of numerical terms on the one hand and of course-of-values terms on the other. (shrink)

Frege’s method of introducing abstract singular terms by transforming an equivalence statement into an identity statement suffers from one major defect: it is haunted by a pervasive indeterminacy of putative reference. In this paper, I. discuss mainly Frege’s introduction of courses-of-values in his magnum opus Grundgesetze der Arithmetik (Volume I, 1893, Volume 11, 1903). More specifically, I want to assesscritically, with respect to course-of-values names, what I call Frege’s indeterminacy problem. In the first part, I sketch the nature of this (...) problem in connection with the introduction of numerical singular terms in his book Die Grundlagen der Arithmetik of 1884. In the second part, I try to shed light on the analogies and differences concerning the introduction of numerical terms on the one hand and of course-of-values terms on the other. (shrink)

This paper casts doubt on a recent criticism of Frege's theory of concepts and extensions by showing that it misses one of Frege's most important contributions: the derivation of the infinity of the natural numbers. We show how this result may be incorporated into the conceptual structure of Zermelo- Fraenkel Set Theory. The paper clarifies the bearing of the development of the notion of a real-valued function on Frege's theory of concepts; it concludes with a brief discussion of the claim (...) that the standard interpretation of second-order logic is necessary for the derivation of the Peano Postulates and the proof of their categoricity. (shrink)

This is a study of Frege's permutation argument in Part I, Section 10, of Frege's Basic Laws of Arithmetic.