This paper aims to answer the question of whether or not Frege's solution limited to value-ranges and truth-values proposed to resolve the "problem of indeterminacy of reference" in section 10 of Grundgesetze is a violation of his principle of complete determination, which states that a predicate must be defined to apply for all objects in general. Closely related to this doubt is the common allegation that Frege was unable to solve a persistent version of the Caesar problem for value-ranges. It (...) is argued that, in Frege’s standards of reducing arithmetic to logic, his solution to the indeterminacy does not give rise to any sort of Caesar problem in the book. (shrink)

This paper has a two-fold objective: to provide a balanced, multi-faceted account of the origins of logicism; to rehabilitate Richard Dedekind as a main logicist. Logicism should be seen as more deeply rooted in the development of modern mathematics than typically assumed, and this becomes evident by reconsidering Dedekind's writings in relation to Frege's. Especially in its Dedekindian and Fregean versions, logicism constitutes the culmination of the rise of ?pure mathematics? in the nineteenth century; and this rise brought with it (...) an inter-weaving of methodological and epistemological considerations. The latter aspect illustrates how philosophical concerns can grow out of mathematical practice, as opposed to being imposed on it from outside. It also sheds new light on the legacy and the lasting significance of logicism today. (shrink)

Frege’s project has been characterized as an attempt to formulate a complete system of logic adequate to characterize mathematical theories such as arithmetic and set theory. As such, it was seen to fail by Gödel’s incompleteness theorem of 1931. It is argued, however, that this is to impose a later interpretation on the word ‘complete’ it is clear from Dedekind’s writings that at least as good as interpretation of completeness is categoricity. Whereas few interesting first-order mathematical theories are categorical or (...) complete, there are logical extensions of these theories into second-order and by the addition of generalized quantifiers which are categorical. Frege’s project really found success through Gödel’s completeness theorem of 1930 and the subsequent development of first- and higher-order model theory. (shrink)

I present a novel interpretation of Frege’s attempt at Grundgesetze I §§29-31 to prove that every expression of his language has a unique reference. I argue that Frege’s proof is based on a contextual account of reference, similar to but more sophisticated than that enshrined in his famous Context Principle. Although Frege’s proof is incorrect, I argue that the account of reference on which it is based is of potential philosophical value, and I analyze the class of cases to which (...) it may successfully be applied. (shrink)

Hourya Benis-Sinaceur, Marco Panza, and Gabriel Sandu. Functions and Generality of Logic: Reflections on Dedekind’s and Frege’s Logicisms. Logic, Epistemology, and the Unity of Science; 37. Springer, 2015. ISBN: 978-3-319-17108-1 ; 978-3-319-36782-8, 978-3-319-17109-8.. Pp. xxi + 125.

Both Frege's Grundgesetze, and Lagrange's treatises on analytical functions pursue a foundational purpose. Still, the former's program is not only crucially different from the latter's. It also depends on a different idea of what foundation of mathematics should be like . Despite this contrast, the notion of function plays similar roles in their respective programs. The purpose of my paper is emphasising this similarity. In doing it, I hope to contribute to a better understanding of Frege's logicism, especially in relation (...) to its crucial differences with a set-theoretic foundational perspective. This should also spread some light on a question arisen by J. Hintikka and G. Sandu in a widely discussed paper, namely whether Frege should or should not be credited with the notion of arbitrary function underlying our standard interpretation of second-order logic. In section 1, I account for Lagrange's notion of function. In section 2, I advance some remarks on connected historical matters. This will provide an appropriate framework for discussing the role played by the notion of function in Frege's Grundgesetze. Section 3 is devoted to this. Some concluding remarks will close the paper. (shrink)

Carnap's philosophy is examined from new viewpoints, including three important distinctions: (i) language as calculus vs language as universal medium; (ii) different senses of completeness: (iii) standard vs nonstandard interpretations of (higher-order) logic. (i) Carnap favored in 1930-34 the "formal mode of speech," a corollary to the universality assumption. He later gave it up partially but retained some of its ingredients, e.g., the one-domain assumption. (ii) Carnap's project of creating a universal self-referential language is encouraged by (ii) and by the (...) author's recent work. (iii) Carnap was aware of (iii) and occasionally used the standard interpretation, but was not entirely clear of the nature of the contrast. (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)

This explorative article is organized around a set of questions concerning the concept of a function. First, a summary of certain general facts about functions that are a common coin in contemporary logic is given. Then Frege's attempt at clarifying the nature of functions in his famous paper Function and Concept and in his Grundgesetze is discussed along with some questions which Freges' approach gave rise to in the literature. Finally, some characteristic uses of functional notions to be found in (...) the work of Bernard Bolzano and in Edmund Husserl's early work are presented and elucidated. (shrink)

Frege's docent's dissertation Rechnungsmethoden, die sich auf eine Erweiterung des Grössenbegriffes gründen(1874) contains indications of a bold attempt to extend arithmetic. According to it, arithmetic means the science of magnitude, and magnitude must be understood structurally without intuitive support. The main thing is insight into the formal structure of the operation of ?addition?. It turns out that a general ?magnitude domain? coincides with a (commutative) group. This is an interesting connection with simultaneous developments in abstract algebra. As his main application, (...) Frege studies iterations of functions. He does not yet pose the question of existence proofs. Measurement of magnitudes is also connected to numbers, but the discussion is here ambiguous in a way which calls for the systematic account of numbers in Grundgesetze. (shrink)