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.

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. (shrink)

This paper examines the connection between model-theoretic truth and necessary truth. It is argued that though the model-theoretic truths of some standard languages are demonstrably ''''necessary'''' (in a precise sense), the widespread view of model-theoretic truth as providing a general guarantee of necessity is mistaken. Several arguments to the contrary are criticized.

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. (shrink)

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. (shrink)

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. (shrink)

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. (shrink)

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. (shrink)

All contributions included in the present issue were originally presented at an ‘Author Meets Critics’ session organised by Richard Zach at the Pacific Meeting of the American Philosophical Association in San Diego in the Spring of 2014.

Whenever one asserts a claim of any kind, one engages in a commitment not just to that claim itself, but to a variety of other claims that follow in its wake, claims that, as we tend to say, follow logically from the original claim. To say that Smith and Jones are both great basketball players is to say something from which it follows that Smith is a great basketball player, that someone is a great basketball player, that there is something (...) at which Smith is great, and so on. (shrink)

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. (shrink)

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. (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.