An investigation of Frege’s various contributions to the study of language, focusing on three of his most famous doctrines: that concepts are unsaturated, that sentences refer to truth-values, and that sense must be distinguished from reference.

This paper explores the relationship between Hume's Prinicple and Basic Law V, investigating the question whether we really do need to suppose that, already in Die Grundlagen, Frege intended that HP should be justified by its derivation from Law V.

The Caesar problem arises for abstractionist views, which seek to secure reference for terms such as ‘the number of Xs’ or #X by stipulating the content of ‘unmixed’ identity contexts like ‘#X = #Y’. Frege objects that this stipulation says nothing about ‘mixed’ contexts such as ‘# X = Julius Caesar’. This article defends a neglected response to the Caesar problem: the content of mixed contexts is just as open to stipulation as that of unmixed contexts.

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)

A running commentary is offered on the first half of Frege’s Grundlagen der Arithmetik, §17, and suggests that Frege anticipated the method of demonstration used by Paul Bernays for the Deduction Theorem.

This open access book is a superb collection of some fifteen chapters inspired by Schroeder-Heister's groundbreaking work, written by leading experts in the field, plus an extensive autobiography and comments on the various contributions by Schroeder-Heister himself. For several decades, Peter Schroeder-Heister has been a central figure in proof-theoretic semantics, a field of study situated at the interface of logic, theoretical computer science, natural-language semantics, and the philosophy of language. -/- The chapters of which this book is composed discuss the (...) subject from a rich variety of angles, including the history of logic, the proper interpretation of logical validity, natural deduction rules, the notions of harmony and of synonymy, the structure of proofs, the logical status of equality, intentional phenomena, and the proof theory of second-order arithmetic. All chapters relate directly to questions that have driven Schroeder-Heister's own research agenda and to which he has made seminal contributions. The extensive autobiographical chapter not only provides a fascinating overview of Schroeder-Heister's career and the evolution of his academic interests but also constitutes a contribution to the recent history of logic in its own right, painting an intriguing picture of the philosophical, logical, and mathematical institutional landscape in Germany and elsewhere since the early 1970s. The papers collected in this book are illuminatingly put into a unified perspective by Schroeder-Heister's comments at the end of the book. Both graduate students and established researchers in the field will find this book an excellent resource for future work in proof-theoretic semantics and related areas. (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)

Frege's concern in GGI §10 is neither with the epistemological issue of how we come to know about value-ranges, nor with the semantic-metaphysical issue of whether we have said enough about such objects in order to ensure that any kind of reference to them is possible. The problem which occupies Frege in GGI §10 is the general problem according to which we ‘cannot yet decide’, for any arbitrary function, what value ‘’ has if ‘ℵ’ is a canonical value-range name. This (...) is a problem with the ‘reference’ of value-range names, but only in the weak sense that, if we do not exercise care, value-range terms might become ‘bedeutungslos’ for purely formal reasons. Frege addresses the general problem only for the primitive function- and object-names he has already introduced into his concept-script. I argue that this methodology was perfectly intentional: his intention for GG in general, on display in GGI §10, is to check, for each primitive function- and object-name, as it is introduced into concept-script, whether it interacts with the other primitive names which have already been introduced in such a way that these atomic combinations of primitive names do not become bedeutungslos. If there is a risk of producing a bedeutungslos combination, Frege will make an arbitrary stipulation to ensure that logical hygiene is maintained. I argue that this interpretation does not violate some of the other principal commitments of GG. (shrink)