The paper challenges a widely held interpretation of Frege's conception of logic on which the constituent clauses of basic law V have the same sense. I argue against this interpretation by first carefully looking at the development of Frege's thoughts in Grundlagen with respect to the status of abstraction principles. In doing so, I put forth a new interpretation of Grundlagen §64 and Frege's idea of ‘recarving of content’. I then argue that there is strong evidence in Grundgesetze that Frege (...) did not hold the relevant sense-identity claim regarding basic law V. (shrink)

The paper discusses the emergence of Frege's puzzle and the introduction of the celebrated distinction between sense and reference in the context of Frege's logicist project. The main aim of the paper is to show that not logicism per se is mainly responsible for this introduction, but Frege's constant struggle against formalism. Thus, the paper enlarges the historical context, and provides a reconstruction of Frege's philosophical development from this broader perspective.

Neo-logicism is, not least in the light of Frege’s logicist programme, an important topic in the current philosophy of mathematics. In this essay, I critically discuss a number of issues that I consider to be relevant for both Frege’s logicism and neo-logicism. I begin with a brief introduction into Wright’s neo-Fregean project and mention the main objections that he faces. In Sect. 2, I discuss the Julius Caesar problem and its possible Fregean and neo-Fregean solution. In Sect. 3, I raise (...) what I take to be a central objection to the position of neo-logicism. In Sect. 4, I attempt to clarify how we should understand Frege’s stipulation that the two sides of an abstraction principle qua contextual definition of a term-forming operator shall be “gleichbedeutend”. In Sect. 5, I consider the options that Frege might have had to establish the analyticity of Hume’s Principle: The number that belongs to the concept F is equal to the number that belongs to the concept G if and only if F and G are equinumerous. Section 6 is devoted to Frege’s two criteria of thought identity. In Sects. 7 and 8, I defend the position of the neo-logicist against an alleged “knock-down argument”. In Sect. 9, I comment on Frege’s description of abstraction in Grundlagen, §64 and the use of the terms “recarving” and “reconceptualization” in the relevant literature on Fregean abstraction and neo-logicism. I argue that Fregean abstraction has nothing to do with the recarving of a sentence content or its decomposition in different ways. I conclude with remarks on global logicism versus local logicisms. (shrink)

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)

One of the important discussions in the philosophy of mathematics, is that centered on Benacerraf’s Dilemma. Benacerraf’s dilemma challenges theorists to provide an epistemology and semantics for mathematics, based on their favourite ontology. This challenge is the point on which all philosophies of mathematics are judged, and clarifying how we might acquire mathematical knowledge is one of the main occupations of philosophers of mathematics. In this thesis I argue that this discussion has overlooked an important part of mathematics, namely mathematics (...) as it is exercised by ordinary people. I do so by looking at the different theories that have been put forward in the recent debate, and showing for each of these that they are unable to account for the mathematical practices of ordinary people. In order to show that these practices do need to be accounted for, I also argue that ordinary people are doing mathematics, i.e. that they engage in properly mathematical practices. Because these practices are properly mathematical, they should be accounted for by any philosophy of mathematics. The conclusion of my thesis, then, is that current theories fail to do something that they should do, while remaining neutral on how well they perform when it comes to accounting for the practices of professional mathematicians. (shrink)

In his logical foundation of arithmetic, Frege faced the problem that the semantic interpretation of his system does not determine the reference of the abstract terms completely. The contextual definition of number, for instance, does not decide whether the number 5 is identical to Julius Caesar. In a late writing, Quine claimed that the indeterminacy of reference established by Frege’s Caesar problem is a special case of the indeterminacy established by his proxy-function argument. The present paper aims to show that (...) Frege’s Caesar problem does not really support the conclusions that Quine draws from the proxy-function argument. On the contrary, it reveals that Quine’s argument is a non sequitur: it does not establish that there are alternative interpretations of our terms that are equally correct, but only that these terms are ambiguous. The latter kind of referential indeterminacy implies that almost all sentences of our overall theory of the world are either false or neither true nor false, because they contain definite descriptions whose uniqueness presupposition is not fulfilled. The proxy-function argument must therefore be regarded as a reductio ad absurdum of Quine’s behaviorist premise that the reference of terms is determined only by our linguistic behavior. (shrink)

This paper presents a new approach to the class-theoretic paradoxes. In the first part of the paper, I will distinguish classes from sets, describe the function of class talk, and present several reasons for postulating type-free classes. This involves applications to the problem of unrestricted quantification, reduction of properties, natural language semantics, and the epistemology of mathematics. In the second part of the paper, I will present some axioms for type-free classes. My approach is loosely based on the Gödel–Russell idea (...) of limited ranges of significance. It is shown how to derive the second-order Dedekind–Peano axioms within that theory. I conclude by discussing whether the theory can be used as a solution to the problem of unrestricted quantification. In an appendix, I prove the consistency of the class theory relative to Zermelo–Fraenkel set theory. (shrink)

In this article, I try to shed some new light onGrundgesetze§10, §29–§31 with special emphasis on Frege’s criteria and proof of referentiality and his treatment of the semantics of canonical value-range names. I begin by arguing against the claim, recently defended by several Frege scholars, that the first-order domain inGrundgesetzeis restricted to value-ranges, but conclude that there is an irresolvable tension in Frege’s view. The tension has a direct impact on the semantics of the concept-script, not least on the semantics (...) of value-range names. I further argue that despite first appearances truth-value names play a distinguished role as semantic “target names” for “test names” in the criteria of referentiality and do not figure themselves as “test names” regarding referentiality. Accordingly, I show in detail that Frege’s attempt to demonstrate that by virtue of his stipulations “regular” value-range names have indeed been endowed with a unique reference, can plausibly be regarded as a direct application of the context principle. In a subsequent section, I turn to some special issues involved in §10. §10 is closely intertwined with §31 and in my and Richard Heck’s view would have been better positioned between §30 and §31. In a first step, I discuss the piecemeal strategy which Frege applies when he attempts to bestow a unique reference on value-range names in §3, §10–§12. In a second step, I critically analyze his tentative, but predictably unsuccessful proposal to identify all objects whatsoever, including those already clad in the garb of value-ranges, with their unit classes. In conclusion, I present two arguments for my claim that Frege’s identification of the True and the False with their unit classes in §10 is illicit even if both the permutation argument and the identifiability thesis that he states in §10 are regarded as formally sound. The first argument is set out from the point of view of the syntax of his formal language. It suggests though that a reorganization of the exposition of the concept-script would have solved at least one of the problems to which the twin stipulations in §10 give rise. The second argument rests on semantic considerations. If it is sound, it may call into question, if not undermine the legitimacy of the twin stipulations. (shrink)