P. T. Geach, notoriously, holds the Relative Identity Thesis, according to which a meaningful judgment of identity is always, implicitly or explicitly, relative to some general term. ‘The same’ is a fragmentary expression, and has no significance unless we say or mean ‘the same X’, where ‘X’ represents a general term (what Frege calls a Begriffswort or Begriffsausdruck). (P. T. Geach, Mental Acts (London: Routledge and Kegan Paul, 1957), p. 69. I maintain that it makes no sense to judge whether (...) things are ‘the same’, or remain ‘the same’, unless we add or understand some general term - ‘the same F’. (P. T. Geach, Reference and Generality, third Edition (Ithaca, N.Y.: Cornell University Press, 1980), pp. 63f. I am arguing for the thesis that identity is relative. When one says ‘x is identical with y’, this, I hold, is an incomplete expression; it is short for ‘x is the same A as y’, where ‘A’ represents some count noun understood from the context of utterance - or else, it is just a vague expression of a half-formed thought. (P. T. Geach, ‘Identity,’ Review of Metaphysics 21 (1967-8), p. 3.) One of the ways Geach seeks to support this is by tying it to the well nigh universally admired Fregean thesis about cardinality. (shrink)

A standard strategy for defending a claim of non-identity is one which invokes Leibniz’s Law. (1) Fa (2) ~Fb (3) (∀x)(∀y)(x=y ⊃ (∀P)(Px ⊃ Py)) (4) a=b ⊃ (Fa ⊃ Fb) (5) a≠b In Kalderon’s view, this basic strategy underlies both Moore’s Open Question Argument (OQA) as well as (a variant formulation of) Frege’s puzzle (FP). In the former case, the argument runs from the fact that some natural property—call it “F-ness”—has, but goodness lacks, the (2nd order) property of its (...) being an open question whether everything that instantiates it is good to the conclusion that goodness and F-ness are distinct. And in the latter case, the argument runs from the fact that that Hesperus has, but Phosphorus lacks, the property of being believed by the ancient astronomers to be visible in the evening sky to the conclusion that Hesperus and Phosphorus are distinct. Kalderon argues that both the OQA and FP fail because in neither case is there good reason to believe that both (1) and (2) are true. The reason we are tempted to believe that they are true is because we mistake de dicto claims for de re claims. In order for FP to go through, the truth of the following de re claims needs to be established: FP1) Hesperus was believed by the ancient astronomers to be visible in the evening sky. (shrink)

(No abstract is available for this citation).

In this paper, the authors discuss Frege''s theory of logical objects (extensions, numbers, truth-values) and the recent attempts to rehabilitate it. We show that the eta relation George Boolos deployed on Frege''s behalf is similar, if not identical, to the encoding mode of predication that underlies the theory of abstract objects. Whereas Boolos accepted unrestricted Comprehension for Properties and used the eta relation to assert the existence of logical objects under certain highly restricted conditions, the theory of abstract objects uses (...) unrestricted Comprehension for Logical Objects and banishes encoding (eta) formulas from Comprehension for Properties. The relative mathematical and philosophical strengths of the two theories are discussed. Along the way, new results in the theory of abstract objects are described, involving: (a) the theory of extensions, (b) the theory of directions and shapes, and (c) the theory of truth values. (shrink)

Due programmi diversi si intersecano nel lavoro di Frege sui fondamenti dell’aritmetica: • Logicismo: l’aritmetica `e riducibile alla logica; • Estensionalismo: l’aritmetica `e riducibile a una teoria delle estensioni. Sia nei Fondamenti che nei Principi, Frege articola l’idea che l’aritmetica sia riducibile a una teoria logica delle estensioni.

This paper presents a formalization of first-order arithmetic characterizing the natural numbers as abstracta of the equinumerosity relation. The formalization turns on the interaction of a nonstandard (but still first-order) cardinality quantifier with an abstraction operator assigning objects to predicates. The project draws its philosophical motivation from a nonreductionist conception of logicism, a deflationary view of abstraction, and an approach to formal arithmetic that emphasizes the cardinal properties of the natural numbers over the structural ones.

I try to reconstruct how Frege thought to reconcile the cognitive value of arithmetic with its analytical nature. There is evidence in Frege's texts that the epistemological formulation of the context principle plays a decisive role; it provides a way of obtaining concepts which are truly fruitful and whose contents cannot be grasped beforehand. Taking the definitions presented in the Begriffsschrift,I shall illustrate how this schema is intended to work.

Frege's and Russell's views are obviously different, but because of certain superficial similarities in how they handle certain famous puzzles about proper names, they are often assimilated. Where proper names are concerned, both Frege and Russell are often described together as "descriptivists." But their views are fundamentally different. To see that, let's look at the puzzle of names without bearers, as it arises in the context of Mill's purely referential theory of proper names, aka the 'Fido'-Fido theory.

Chapter: Sense and Intentionality A: Reference and Sense — Preliminary Remarks Few people during Frege's lifetime paid due attention to his work and its ...

This is the first single-volume edition and translation of Frege's philosophical writings to include his seminal papers as well as substantial selections from ...

Examines Frege's theory of judgement, according to which a judgement is, paradigmatically, the assertion that a particular object falls under a given concept. Throughout the book the aim is to both state Frege's views clearly and concisely, and to defend, modify or reject these where appropriate.

Call a family F of subsets of a set E inductive if ∅ ∈ F and F is closed under unions with disjoint singletons, that is, if ∀X∈F ∀x∈E–X(X ∪ {x} ∈ F]. A Frege structure is a pair (E.

This volume bears witness to the continuing importance and influence of that agenda.

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. Many thanks for helpful comments to Wilfrid Hodges, to Bob Hale, and to an anonymous referee. (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)

Abstract Though it has been claimed that Frege's commitment to expressions in indirect contexts not having their customary senses commits him to an infinite number of semantic primitives, Terrence Parsons has argued that Frege's explicit commitments are compatible with a two-level theory of senses. In this paper, we argue Frege is committed to some principles Parsons has overlooked, and, from these and other principles to which Frege is committed, give a proof that he is indeed committed to an infinite number (...) of semantic primitives?an intolerable result. (shrink)

Tyler Burge presents a collection of his seminal essays on Gottlob Frege (1848-1925), who has a strong claim to be seen as the founder of modern analytic philosophy, and whose work remains at the centre of philosophical debate today. Truth, Thought, Reason gathers some of Burge's most influential work from the last twenty-five years, and also features important new material, including a substantial introduction and postscripts to four of the ten papers. It will be an essential resource for any historian (...) of modern philosophy, and for anyone working on philosophy of language, epistemology, or philosophical logic. (shrink)

The paper scrutinizes Frege's Euclideanism - his view of arithmetic and geometry as resting on a small number of self-evident axioms from which non-self-evident theorems can be proved. Frege's notions of self-evidence and axiom are discussed in some detail. Elements in Frege's position that are in apparent tension with his Euclideanism are considered - his introduction of axioms in The Basic Laws of Arithmetic through argument, his fallibilism about mathematical understanding, and his view that understanding is closely associated with inferential (...) abilities. The resolution of the tensions indicates that Frege maintained a sophisticated and challenging form of rationalism, one relevant to current epistemology and parts of the philosophy of mathematics. (shrink)

This book surveys the assortment of methods put forth for fixing Frege's system, in an attempt to determine just how much of mathematics can be reconstructed in ...

Hintikka and Sandu have recently claimed that Frege's notion of function was substantially narrower than that prevailing in real analysis today. In the present note, their textual evidence for this claim is examined in the light of relevant historical and biographical background and judged insufficient.

Gottlob Frege has exerted an enormous influence on the evolution of twentieth-century philosophy, yet the real significance of that influence is still very much a matter of debate. This book provides a completely new and systematic account of Frege's philosophy by focusing on its cornerstone: the theory of sense and reference. Two features distinguish this study from other books on Frege. First, sense and reference are placed absolutely at the core of Frege's work; the author shows that no adequate account (...) of the theory can avoid analysing the notion of thought that underpins it, or explaining how it has clarified our concept of judgement. Second, the theory is situated within the development of Frege's thought; the author reveals how the theory caused Frege to alter many of his fundamental views. In doing so the author presents a clearer picture of the problems the theory was intended to solve, and delineates more sharply the characteristic features of Frege's philosophy. (shrink)

The objects of credence are the entities to which credences are assigned for the purposes of a successful theory of credence. I use cases akin to Frege's puzzle to argue against referentialism about credence: the view that objects of credence are determined by the objects and properties at which one's credence is directed. I go on to develop a non-referential account of the objects of credence in terms of sets of epistemically possible scenarios.

"One of Frege's main semantic principles, is however, missing in Dummett's book, [Frege: philosophy of language] and it is has been ignored by most Frege scholars. That principle is the thesis concerning the ambiguity of the word 'is'. Angelelli come close to attending to it when he makes some remarks on identity and predication, and Matthias Schirn puts special emphasis on the role of the thesis in Frege's work. However, the great majority of Frege scholars have neglected the ambiguity doctrine, (...) even when they have commented on each of the allegedly different meanings of 'is' separately. This is strange in view of the fact that it was Frege and Russell who proposed the thesis and established it as one of the basic ingredients of modern logic. They have in fact been followed by most philosophers. For instance, in the Tractatus Ludwig Wittgenstein emphasizes the ambiguity of the verb 'to be' and stresses the.. (shrink)

In the opening sentence of his Methods of Logic, W. V. O. Quine writes, “Logic is an old subject, and since 1879 it has been a great one.”1 Quine is referring to the year in which Gottlob Frege presented his Begriffschrift, or “concept-script,” one of the first published accounts of a logical system or calculus with quantification and a function-argument analysis of propositions. There can be no doubt as to the importance of these introductions, and, indeed, Frege’s orientation and advances, (...) if not his particular system, have proven to be highly significant for much of mathematical logic and research pertaining to the foundations of mathematics. Nevertheless, Quine’s assessment, often repeated, obscures other .. (shrink)

Two claims the present author has made about Frege's philosophy are defended against Michael Dummett's criticisms (The Interpretation of Frege's Philosophy and ?Objectivity and Reality in Lotze and Frege?, this journal, 1982). The claim that Frege was concerned primarily with epistemological problems rather than with the theory of meaning, and the claim (this journal, 1978) that the ascription of Wirklichkeit to Thoughts is evidence of Frege's realism, are clarified and defended. Dummett's own characterization of Frege's realism is considered and rejected.

An explanation of Frege's change from objective idealism to platonism is offered. Frege had originally thought that numbers are transparent to reason, but the character of his Axiom of Courses of Values undermined this view, and led him to think that numbers exist independently of reason. I then use these results to suggest a view of Frege's mathematical epistemology.

In this note the claim is defended that Frege was a realist in the sense that he attributed causal efficacy to certain abstract objects. The arguments of Dummett and Sluga (cf. Inquiry, Vols. 18, 19, and 20 [1975?77]) to the contrary are criticized.

Frege and Eucken were colleagues in the faculty of philosophy at Jena University for more than 40 years. At times they had close scientific contacts. Eucken promoted Frege's career at the university. A comparison of Eucken's writings between 1878 and 1880 with Frege's writings shows Eucken to have had an important philosophical influence on Frege's philosophical development between 1879 and 1885. In particular the classification of the Begriffsschrift in the tradition of Leibniz is influenced by Eucken. Eucken also influenced Frege's (...) choice of philosophical and logical terms. Finally, there are analogous positions concerning relations between concepts and their expressions in natural language, Frege was probably also influenced by Eucken's use of the term ?tone?. Eucken used Frege's arguments in his own fight against psychologism and empiricism. (shrink)

Several scholars have argued that Wittgenstein held the view that the notion of number is presupposed by the notion of one-one correlation, and that therefore Hume's principle is not a sound basis for a definition of number. I offer a new interpretation of the relevant fragments on philosophy of mathematics from Wittgenstein's Nachlass, showing that if different uses of ‘presupposition’ are understood in terms of de re and de dicto knowledge, Wittgenstein's argument against the Frege-Russell definition of number turns out (...) to be valid on its own terms, even though it depends on two epistemological principles logicist philosophers of mathematics may find too ‘constructivist’. (shrink)

This paper concentrates on some aspects of the history of the analytic-synthetic distinction from Kant to Bolzano and Frege. This history evinces considerable continuity but also some important discontinuities. The analytic-synthetic distinction has to be seen in the first place in relation to a science, i.e. an ordered system of cognition. Looking especially to the place and role of logic it will be argued that Kant, Bolzano and Frege each developed the analytic-synthetic distinction within the same conception of scientific rationality, (...) that is, within the Classical Model of Science: scientific knowledge as cognitio ex principiis . But as we will see, the way the distinction between analytic and synthetic judgments or propositions functions within this model turns out to differ considerably between them. (shrink)

This paper continues Michael Dummett's and my discussion of Frege in The Philosophy of Michael Dummett [2007]. Most of it is about Dummett’s change in view on Frege’s senses and objects. The issues include: the cognitive order versus the ontological order for the forward road; the nature and identity of senses and the different senses of "intension;" the nature of saturation; whether special quantifiers are now needed for senses; and Frege’s earlier and later permutation arguments. I discuss the implications of (...) the forward road for intuitionism as well. (shrink)

In this paper, I want to contribute to understanding and improving on Keenan'sintriguing equivalence result about reducible type quantifiers (Keenan, 1992).I give an alternative proof of his result which generalizes to type quantifiers, andI show how the reduction of a reducible type quantifier to (the composition of) ntype quantifiers can be effected.

This paper attempts to confine the preconceptions that prevented Frege from appreciating Hilbert?s Grundlagen der Geometrie to two: (i) Frege?s reliance on what, following Wilfrid Hodges, I call a Frege?Peano language, and (ii) Frege?s view that the sense of an expression wholly determines its reference.I argue that these two preconceptions prevented Frege from achieving the conceptual structure of model theory, whereas Hilbert, at least in his practice, was quite close to the model?theoretic point of view.Moreover, the issues that divided Frege (...) and Hilbert did not revolve around whether one or the other allowed metalogical notions.Frege, e.g., succeeded in formulating the notion of logical consequence, at least to the extent that Bolzano did; the point is rather that even though Frege had certain semantic concepts, he did not articulate them model?theoretically, whereas, in some limited sense, Hilbert did. (shrink)

This paper has three goals: (i) to show that the foundational program begun in theBegriffsschrift, and carried forward in theGrundlagen, represented Frege's attempt to establish the autonomy of arithmetic from geometry and kinematics; the cogency and coherence ofintuitive reasoning were not in question. (ii) To place Frege's logicism in the context of the nineteenth century tradition in mathematical analysis, and, in particular, to show how the modern concept of a function made it possible for Frege to pursue the goal of (...) autonomy within the framework of the system of second-order logic of theBegriffsschrift. (iii) To address certain criticisms of Frege by Parsons and Boolos, and thereby to clarify what was and was not achieved by the development, in Part III of theBegriffsschrift, of a fragment of the theory of relations. (shrink)

§§3-4 of the Begriffsschrift present Frege’s objections to a dominant if murky nineteenth-century semantic picture. I sketch a minimalist variant of the pre-Fregean picture which escapes Frege’s criticisms by positing a thin notion of semantic content which then interacts with a multiplicity of kinds of truth to account for phenomena such as modality. After exploring several ways in which we can understand the existence of multiple truth properties, I discuss the roles of pointwise and setwise truth properties in modal logic. (...) I argue that thinking of supertruth and determinate truth as setwise truth properties allows an understanding of supervaluationist approaches to vagueness which escapes both Williamson’s objections to and a needless metalinguistic orientation of traditional supervaluationism. (shrink)

In this essay, I discuss some observations by Peirce which suggest he had some idea of the substantive metalogical differences between logics which permit both quantifiers and relations, and those which do not. Peirce thus seems to have had arguments?which even De Morgan and Frege lacked?that show the superior expressiveness of relational logics.

Annotation This volume collects classic essays by Dummett on Frege and related subjects.

In his Grundlagen , Frege held that geometrical truths.are synthetic a priori , and that they rest on intuition. From this it has been concluded that he thought, like Kant, that space and time are a priori intuitions and that physical objects are mere appearances. It is plausible that Frege always believed geometrical truths to be synthetic a priori ; the virtual disappearance of the word 'intuition' from his writings from after 1885 until 1924 suggests, on the other hand, that (...) he became dissatisfied with the notion of intuition as he had employed it in Grundlagen . The belief that a priori intuition is a source of knowledge does not in itself entail idealism: that is a question about what it is that makes true the propositions known in this way. In Grundlagen , Frege expressly states that geometrical truths are objective in the sense of being independent of our intuition. This shows that, even at that period, Frege did not draw the idealist conclusion drawn by Kant. (shrink)

Frege held that logical objects are objective but not wirklich, and that psychologism follows from the mistake of believing whatever is not wirklich to be subjective. It has been suggested that Frege's use of the terms ?objective? and ?wirklich? is in line with that found in Lotze's Logic; from this it has been inferred that Frege's doctrines have been misinterpreted as being ontological in character, but that they really belong to epistemology. In fact, Lotze held that something may be the (...) same for all thinkers, and yet may exist only in thought, not independently of it. For Frege, by contrast, there is nothing intermediate between the content of a single consciousness and what exists independently of being thought at all. This crucial disagreement underlies the divergence between Frege's realism and Lotze's idealism. (shrink)

The short fragment of Frege's Nachlass which bears the above title, given to it by the editors, is in fact a sequence of connected comments by him on the Introduction to Lotze's Logik, or, more exactly, a response by him to that Introduction. It is thus very probably the earliest piece of writing from Frege's pen on the philosophy of logic surviving to us, and, when it is read in this light, the motivation for its author's puzzling selection of remarks (...) and the turns of phrase he employs become intelligible. We see here an early attempt by Frege to attain clarity about a topic that was to preoccupy him throughout his entire philosophical career, the nature of thoughts. (shrink)

H. Sluga (Inquiry, Vol. 18 [1975], No. 4) has criticized me for representing Frege as a realist. He holds that, for Frege, abstract objects were not real: this rests on a mistranslation and a neglect of Frege's contextual principle. The latter has two aspects: as a thesis about sense, and as one about reference. It is only under the latter aspect that there is any tension between it and realism: Frege's later silence about the principle is due, not to his (...) realism, but to his assimilating sentences to proper names. Contrary to what Sluga thinks, the conception of the Bedeutung of a name as its bearer is an indispensable ingredient of Frege's notion of Bedeutung, as also is the fact that it is in the stronger of two possible senses that Frege held that Sinn determines Bedeutung. The contextual principle is not to be understood as meaning that thoughts are not, in general, complex; Frege's idea that the sense of a sentence is compounded out of the senses of its component words is an essential component of his theory of sense. Frege's realism was not the most important ingredient in his philosophy: but the attempt to interpret him otherwise than as a realist leads only to misunderstanding and confusion. (shrink)

This highly acclaimed book is a major contribution to the philosophy of language as well as a systematic interpretation of Frege, indisputably the father of ...