Patricia Blanchette

This chapter discusses the question of how modern model-theoretic methods should be assessed from a Fregean point of view. It is argued that while the central features of mathematical theories, from a post-Fregean point of view, have essentially to do with features of those theories’ classes of models, this is not the case from Frege’s point of view. Given a Fregean starting-point, from which logical relations are borne not by sentences but by thoughts, and from which conceptual analysis is relevant to the discovery of logical relations, classes of models are not sensitive to the crucial logically-significant features of theories. Hence the important logical properties of theories as Frege understands them cannot be cashed out in standard modern terms. For similar reasons, model-theoretic entailment in a sufficiently rich language will diverge from logical entailment as understood by Frege.

This chapter examines Frege’s dispute with David Hilbert over consistency- and independence-proofs in the foundations of geometry. It is argued that if one understands logical entailment in the way that Frege does, as a relation that can be partly revealed by conceptual analysis, then one can see that Frege’s much-misunderstood complaints against Hilbert are in fact decisive. Hilbert’s method of proving consistency does not demonstrate what Frege understood as “consistency.” It is argued that we see here an important distinction between two ways of understanding entailment and consistency, and that the now-unfamiliar Fregean understanding is a rich and valuable one in its own right.

This chapter treats the question of how Frege understands the connection between the familiar truths of ordinary arithmetic and the unfamiliar and complex truths with which his formal proofs conclude. It is argued that although Frege does not intend ordinary and formal sentences to express the same thoughts or to be “about” the same objects, nevertheless the relationship between the two kinds of sentence is intended to be sufficiently close that the success of the project would, in fact, have demonstrated the purely-logical status of ordinary arithmetic. It is argued, as against Joan Weiner, that the semantic distance between ordinary and derived sentences does not undermine this straightforward reading of Frege’s project.

It is often claimed that Frege requires all function-expressions to be everywhere defined. This, if true, poses a difficulty for Frege’s idea (as claimed in this book) that arithmetical function-expressions, both ordinary and formal, are not so defined. The purpose of this chapter is to argue that Frege does not require total definition of function-expressions, but instead that he imposes a much weaker requirement, here called that of “linguistic completeness,” and that the requirement is imposed just on formal languages. The importance of this point is twofold. First: only under such an understanding can Frege’s logicist project be understood, as urged here, as an attempt to establish the purely-logical status of ordinary arithmetical truths. Secondly: only under such an understanding can we take Frege at his word when he says that ordinary arithmetical sentences express truths and falsehoods.

The purpose of this chapter is to explain Frege’s understanding of _thoughts_, the non-linguistic propositions expressed by declarative sentences. It is stressed that thoughts form the contents of judgment and of theories, and that they are the things that logically entail one another. Difficulties in pinning down Frege’s view of the conditions under which two sentences express the same thought are discussed, along with tensions between the different roles he assigned to thoughts. Frege’s view that a given thought can be decomposed in multiple ways into simpler components is explained and defended (as against Dummett). Finally, the role of Frege’s formal languages in expressing thoughts, and some important differences between formal and natural languages, are laid out.

This chapter explains the role of conceptual analysis in Frege’s logicist project. It is explained that Frege employs conceptual analyses in order to reduce arithmetical truths to complexes of relatively simple components in order to facilitate the proof of those truths. The importance of the adequacy of the conceptual analyses is emphasized, and Frege’s view of that adequacy is explored in a preliminary way by looking carefully at a selection of Frege’s analyses in _Begriffsschrift_, _Grundlagen_, and _Grundgesetze_. Some of the difficulties surrounding Frege’s conception of analysis are explained, and it is pointed out that a clear conception of Frege’s understanding of analysis requires a clear understanding of his notion of _thought_, the subject of Chapter 2.

In Frege's Conception of Logic Patricia A. Blanchette explores the relationship between Gottlob Frege's understanding of conceptual analysis and his understanding of logic. She argues that the fruitfulness of Frege's conception of logic, and the illuminating differences between that conception and those more modern views that have largely supplanted it, are best understood against the backdrop of a clear account of the role of conceptual analysis in logical investigation.The first part of the book locates the role of conceptual analysis in Frege's logicist project. Blanchette argues that despite a number of difficulties, Frege's use of analysis in the service of logicism is a powerful and coherent tool. As a result of coming to grips with his use of that tool, we can see that there is, despite appearances, no conflict between Frege's intention to demonstrate the grounds of ordinary arithmetic and the fact that the numerals of his derived sentences fail to co-refer with ordinary numerals. In the second part of the book, Blanchette explores the resulting conception of logic itself, and some of the straightforward ways in which Frege's conception differs from its now-familiar descendants. In particular, Blanchette argues that consistency, as Frege understands it, differs significantly from the kind of consistency demonstrable via the construction of models. To appreciate this difference is to appreciate the extent to which Frege was right in his debate with Hilbert over consistency- and independence-proofs in geometry. For similar reasons, modern results such as the completeness of formal systems and the categoricity of theories do not have for Frege the same importance they are commonly taken to have by his post-Tarskian descendants. These differences, together with the coherence of Frege's position, provide reason for caution with respect to the appeal to formal systems and their properties in the treatment of fundamental logical properties and relations.

Frege’s conception of axioms is an old-fashioned one. According to it, each axiom is a determinate non-linguistic proposition, one with a fixed subject-matter, and with respect to which the notion of a ‘model’ or an ‘interpretation’ makes no sense. As contrasted with the fruitful modern conception of mathematical axioms as collectively providing implicit definitions of structure-types, a conception on which the range of models of a set of axioms is of the essence of those axioms’ significance, Frege’s view is a dinosaur. This essay investigates some of the philosophically-important aspects of that dinosaur, in order to shed light on Frege’s understanding of the foundational role of axioms, and on some of the ways in which our current conception of such axiomatic virtues as independence and categoricity have (and in some cases have not) been informed by a move away from Frege’s understanding of the foundational role of axioms.

Frege claims that mathematical theories are collections of thoughts, and that scientific continuity turns on thought-identity. This essay explores the difficulties posed for this conception of mathematics by the conceptual development canonically involved in mathematical progress. The central difficulties are that mathematical development often involves sufficient conceptual progress that mature versions of theories do not involve easily-recognizable synonyms of their earlier versions, and that the introduction of new elements in the domains of mathematical theories would seem to conflict with Frege’s view that the original theories involved determinate reference. It is argued here that the difficulties apparently posed to Frege’s central views stem from an overly-simple view of Frege’s understanding of mathematical objects and of reference. The positive view recommended is one on which Frege’s view of mathematical theories is largely consistent with, and helps make sense of, the phenomenon of theoretical unity across conceptual development.

This paper defends the view that Frege’s reduction of arithmetic to logic would, if successful, have shown that arithmetical knowledge is analytic in essentially Kant’s sense. It is argued, as against Paul Benacerraf, that Frege’s apparent acceptance of multiple reductions is compatible with this epistemological thesis. The importance of this defense is that (a) it clarifies the role of proof, definition, and analysis in Frege’s logicist works; and (b) it demonstrates that the Fregean style of reduction is a valuable tool for those who would investigate the nature of arithmetical knowledge.

Gottlob Frege famously rejects the methodology for consistency and independence proofs offered by David Hilbert in the latter's Foundations of Geometry. The present essay defends against recent criticism the view that this rejection turns on Frege's understanding of logical entailment, on which the entailment relation is sensitive to the contents of non-logical terminology. The goals are (a) to clarify further Frege's understanding of logic and of the role of conceptual analysis in logical investigation, and (b) to point out the extent to which his understanding of logic differs importantly from that of the model-theoretic tradition that grows out of Hilbert's work.

This essay examines Frege's reaction to Russell's Paradox and his views about the grounding of existence claims in mathematics. It is argued that Frege's strict requirements on existential proofs would rule out the attempt to ground arithmetic in. It is hoped that this discussion will help to clarify the ways in which Frege's position is both coherent and significantly different from the neo-logicist position on the issues of: what's required for proofs of existence; the connection between models, consistency, and existence; and the prospects for a logical grounding of arithmetic in the wake of the paradox.

In the early years of the twentieth century, Gottlob Frege and David Hilbert, two titans of mathematical logic, engaged in a controversy regarding the correct understanding of the role of axioms in mathematical theories, and the correct way to demonstrate consistency and independence results for such axioms. The controversy touches on a number of difficult questions in logic and the philosophy of logic, and marks an important turning-point in the development of modern logic. This entry gives an overview of that controversy and of its philosophical underpinnings.

Peter Geach famously holds that there is no such thing as absolute identity. There are rather, as Geach sees it, a variety of relative identity relations, each essentially connected with a particular monadic predicate. Though we can strictly and meaningfully say that an individual a is the same man as the individual b, or that a is the same statue as b, we cannot, on this view, strictly and meaningfully say that the individual a simply is b. It is difficult to find anything like a persuasive argument for this doctrine in Geach's work. But one claim made by Geach is that his account of identity is the account most naturally aligned with Frege's widely admired treatment of cardinality. And though this claim of an affinity between Frege's and Geach's doctrines has been challenged, the challenge has been resisted. William Alston and Jonathan Bennett, indeed, go further than Geach to argue that Frege's doctrine implies Geach's. If Alston and Bennett are right, then those who favor a broadly Fregean treatment of cardinality have no choice but to adopt Geach's doctrine of relative identity. And because contemporary set-theoretic accounts of cardinality are essentially Fregean in all respects relevant to this issue (see §11 below), it follows from Alston and Bennett's claim that virtually every standard modern treatment of cardinality requires a 'relativized' treatment of identity. The purpose of this paper is to argue against the implication claimed by Alston and Bennett. Despite some superficial similarities between Frege's treatment of cardinality and Geach's treatment of identity, the two doctrines are fundamentally opposed. Frege's account of cardinality, and its set-theoretic descendants, are all inconsistent with the claim that identity is 'relative' in Geach's sense.

This essay addresses the question of the effect of Russell's paradox on Frege's distinctive brand of arithmetical realism. It is argued that the effect is not just to undermine Frege's specific account of numbers as extensions (courses of value) but more importantly to undermine his general means of explaining the object-directedness of arithmetical discourse. It is argued that contemporary neo-Fregean attempts to revive that explanation do not successfully avoid the central problem brought to light by the paradox. Along the way, it is argued that the need to fend off an eliminative construal of arithmetic can help explain the so-called Caesar problem in the Grundlagen, and that the "syntactic priority thesis" is insufficient to establish the claim that numbers are objects

This thesis is an examination of Frege's logicism, and of a number of objections which are widely viewed as refutations of the logicist thesis. In the view offered here, logicism is designed to provide answers to two questions: that of the nature of arithmetical truth, and that of the source of arithmetical knowledge. ;The first objection dealt with here is the view that logicism is not an epistemologically significant thesis, due to the fact that the epistemological status of logic itself is not well understood. I argue to the contrary that on Frege's conception of logic, logicism is of clear epistemological importance. ;The second objection examined is the claim that Godel's first incompleteness theorem falsifies logicism. I argue that the incompleteness theorem has no impact on logicism unless the logicist is compelled to hold that logic is recursively enumerable. I argue, further, that there is no reason to impose this requirement on logicism. ;The third objection concerns Russell's paradox. I argue that the paradox is devastating to Frege's conception of numbers, but not to his logicist project. I suggest that the appropriate course for a post-Fregean logicist to follow is one which divorces itself from Frege's platonism. ;The conclusion of this thesis is that logicism has of late been too easily dismissed. Though several critical aspects of Frege's logicism must be altered in light of recent results, the central Fregean thesis is still an important and promising view about the nature of arithmetic and arithmetical knowledge.