Recent discussions on Fregean and neo-Fregean foundations for arithmetic and real analysis pay much attention to what is called either ‘Application Constraint’ ($AC$) or ‘Frege Constraint’ ($FC$), the requirement that a mathematical theory be so outlined that it immediately allows explaining for its applicability. We distinguish between two constraints, which we, respectively, denote by the latter of these two names, by showing how$AC$generalizes Frege’s views while$FC$comes closer to his original conceptions. Different authors diverge on the interpretation of$FC$and on (...) whether it applies to definitions of both natural and real numbers. Our aim is to trace the origins of$FC$and to explore how different understandings of it can be faithful to Frege’s views about such definitions and to his foundational program. After rehearsing the essential elements of the relevant debate (§1), we appropriately distinguish$AC$from$FC$(§2). We discuss six rationales which may motivate the adoption of different instances of$AC$and$FC$(§3). We turn to the possible interpretations of$FC$(§4), and advance a Semantic$FC$(§4.1), arguing that while it suits Frege’s definition of natural numbers (4.1.1), it cannot reasonably be imposed on definitions of real numbers (§4.1.2), for reasons only partly similar to those offered by Crispin Wright (§4.1.3). We then rehearse a recent exchange between Bob Hale and Vadim Batitzky to shed light on Frege’s conception of real numbers and magnitudes (§4.2). We argue that an Architectonic version of$FC$is indeed faithful to Frege’s definition of real numbers, and compatible with his views on natural ones. Finally, we consider how attributing different instances of$FC$to Frege and appreciating the role of the Architectonic$FC$can provide a more perspicuous understanding of his foundational program, by questioning common pictures of his logicism (§5). (shrink)

One of the more distinctive features of Bob Hale and Crispin Wright’s neologicism about arithmetic is their invocation of Frege’s Constraint – roughly, the requirement that the core empirical applications for a class of numbers be “built directly into” their formal characterization. In particular, they maintain that, if adopted, Frege’s Constraint adjudicates in favor of their preferred foundation – Hume’s Principle – and against alternatives, such as the Dedekind-Peano axioms. In what follows we establish two main claims. First, (...) we show that, if sound, Hale and Wright’s arguments for Frege’s Constraint at most establish a version on which the relevant application of the naturals is transitive counting – roughly, the counting procedure by which numerals are used to answer “how many”-questions. Second, we show that this version of Frege’s Constraint fails to adjudicate in favor of Hume’s Principle. If this is the version of Frege’s Constraint that a foundation for arithmetic must respect, then Hume’s Principle no more – and no less – meets the requirement than the Dedekind-Peano axioms do. (shrink)

Onofri [Onofri, A. 2018. ‘The Publicity of Thought.’ Philosophical Quarterly 68 (272): 521–541.] proposes an individuation criterion for thoughts that purports to satisfy both shareability (the notion that different thinkers, or a single thinker at different times, can and generally do think type-identical thoughts) and Frege's constraint (according to which two thoughts are different if it is possible for a rational subject to endorse one while rejecting the other). I argue that his proposal fails to satisfy Frege's (...) constraint. Then I propose a modification to Onofri's proposal to fix the problem. (shrink)

In his (Philosophical Perspectives 1: 455–480, 1987) and (Noûs, 40:361–368, 2006), Schiffer devised a puzzle about Salmón’s (in: Frege’s puzzle, MIT Press, 1986a) Millian-Russellian theory of belief reports, which Salmón resolved in his (Philosophical Perspectives 3:243–285, 1989) and (Noûs, 40:369–375, 2006). My paper has three objectives. First, I will argue that the strategy employed by Salmón (in: Noûs 40:369–375, 2006) to solve Schiffer’s puzzle and his argument for such a strategy are disputable. Second, I will raise a new puzzle, inspired (...) by ideas from Saul (in: Analysis 57:102–108, 1997) and Braun and Saul (in: Philos Stud 111:1–41, 2002), which achieves similar results to Schiffer’s puzzle but to which Salmón’s overall strategy for resolving the latter does not apply. Third, I will contend that the import of both puzzles is neither what Salmón maintains nor the alleged inadequacy of the Millian-Russellian semantics of belief reports as Schiffer suggests, but is the failure of Frege’s Constraint — a constraint to which several conceptions of modes of presentation, including Salmón’s (in: Frege’s puzzle, MIT Press, 1986a) in terms of guises and Schiffer’s (in: The things we mean, Oxford University Press, 2003) in terms of unstructured and fine-grained concepts/propositions, are committed. (shrink)

We now know of a number of ways of developing real analysis on a basis of abstraction principles and second-order logic. One, outlined by Shapiro in his contribution to this volume, mimics Dedekind in identifying the reals with cuts in the series of rationals under their natural order. The result is an essentially structuralist conception of the reals. An earlier approach, developed by Hale in his "Reals byion" program differs by placing additional emphasis upon what I here term Frege's (...) Constraint, that a satisfactory foundation for any branch of mathematics should somehow so explain its basic concepts that their applications are immediate. This paper is concerned with the meaning of and motivation for this constraint. Structuralism has to represent the application of a mathematical theory as always posterior to the understanding of it, turning upon the appreciation of structural affinities between the structure it concerns and a domain to which it is to be applied. There is, therefore, a case that Frege's Constraint has bite whenever there is a standing body of informal mathematical knowledge grounded in direct reflection upon sample, or schematic, applications of the concepts of the theory in question. It is argued that this condition is satisfied by simple arithmetic and geometry, but that in view of the gap between its basic concepts (of continuity and of the nature of the distinctions among the individual reals) and their empirical applications, it is doubtful that Frege's Constraint should be imposed on a neo-Fregean construction of analysis. (shrink)

In their book Relativism, Maria Baghramian and Annalisa Coliva (B&C; 2020) offer a panoramic view of various forms of relativism and their history. They mak.

Foundational projects disagree on whether pure and applied mathematics should be explained together. Proponents of unified accounts like neologicists defend Frege’s Constraint (FC), a principle demanding that an explanation of applicability be provided by mathematical definitions. I reconsider the philosophical import of FC, arguing that usual conceptions are biased by ontological assumptions. I explore more reasonable weaker variants — Moderate and Modest FC — arguing against common opinion that ante rem structuralism (and other) views can meet them. I dispel (...) doubts that such constraints are ‘toothless’, showing they both assuage Frege’s original concerns and accommodate neo-logicist intents by dismissing ‘arrogant’ definitions. (shrink)

Hilary Putnam and John McDowell have each argued against representational realist theories of perception and in favor of direct realist (or “common-sense realist”) alternatives. I claim that in both cases they beg the question against their representational realist opponents. Moreover, in neither case has any alternative been offered to the representational realist position where the solution to perceptual or demonstrative versions of Frege’s problem is concerned. In this paper I present a transcendental argument that some of our perceptions of external (...) objects must be direct in the sense that we perceive them and there is nothing else we perceive in virtue of which we do so. I also present a reply to standard objections to the claim that transcendental arguments can be used to support conclusions about the world and not simply about our own use of concepts. Finally, I present a theory in terms of which the relevant Frege problems can be solved without appeal to any of the sorts of representations in terms of which representational realism is defined. (shrink)

In this short monograph, John Horty explores the difficulties presented for Gottlob Frege's semantic theory, as well as its modern descendents, by the treatment of defined expressions. The book begins by focusing on the psychological constraints governing Frege's notion of sense, or meaning, and argues that, given these constraints, even the treatment of simple stipulative definitions led Frege to important difficulties. Horty is able to suggest ways out of these difficulties that are both philosophically and logically plausible and (...) Fregean in spirit. This discussion is then connected to a number of more familiar topics, such as indexicality and the discussion of concepts in recent theories of mind and language. In the latter part of the book, after introducing a simple semantic model of senses as procedures, Horty considers the problems that definitions present for Frege's idea that the sense of an expression should mirror its grammatical structure. The requirement can be satisfied, he argues, only if defined expressions--and incomplete expressions as well--are assigned senses of their own, rather than treated contextually. He then explores one way in which these senses might be reified within the procedural model, drawing on ideas from work in the semantics of computer programming languages. With its combination of technical semantics and history of philosophy, Horty's book tackles some of the hardest questions in the philosophy of language. It should interest philosophers, logicians, and linguists. (shrink)

Frege argued that though it is logically possible for an illogical community to exist, It is not possible that it should be right. Neither the assertion of false statements nor the acceptance of invalid arguments suffices to render a community illogical. The kinds of behavior which would suffice prove, On examination, To be very rare, But to justify frege's rather obscure remarks on illogicality and the universality of logical laws. The laws of logic are to be understood as constraints (...) on the kinds of rules governing linguistic exchanges needed if the language is to accommodate attempts to prove. (edited). (shrink)

Blackburn has outlined a formal account for moral expressivism, and we argued that the moral Frege-Geach problem can be solved formally by appending two rules for the boo-operator which are missing from his account. We then extended Blackburn’s formal account to generate a similar solution to the problem in modal context and showed that the validity of the modal argument can be preserved too in modal expressivism. However, the higher-order element endorsed by Blackburn does not seem necessary for solving the (...) Frege-Geach problem. Nor is his extension from moral expressivism to modal expressivism tenable, since the latter violates its own ontological constraint. A general moral is drawn on the basis on three observations made in evaluating Blackburn’s expressivism. (shrink)

ABSTRACT. Fodor characterizes concepts as consisting of two dimensions: one is content, which is purely denotational/broad, the other the Mentalese vehicle bearing that content, which Fodor calls the Mode of Presentation (MOP), understood "syntactically." I argue that, so understood, concepts are not interpersonally sharable; so Fodor's own account violates what he calls the Publicity Constraint in his (1998) book. Furthermore, I argue that Fodor's non-semantic, or "syntactic," solution to Frege cases succumbs to the problem of providing interpersonally applicable functional (...) roles for MOPs. This is a serious problem because Fodor himself has argued extensively that if Fregean senses or meanings are understood as functional/conceptual roles, then they can't be public, since, according to Fodor, there are no interpersonally applicable functional roles in the relevant senses. I elaborate on these relevant senses in the paper. (shrink)

A core commitment of Bob Hale and Crispin Wright’s neologicism is their invocation of Frege’s Constraint—roughly, the requirement that the core empirical applications for a class of numbers be “built directly into” their formal characterization. According to these neologicists, if legitimate, Frege’s Constraint adjudicates in favor of their preferred foundation—Hume’s Principle—and against alternatives, such as the Dedekind–Peano axioms. In this paper, we consider a recent argument for legitimating Frege’s Constraint due to Hale, according to which the primary (...) empirical application of the naturals is transitive counting, or answering ‘how many’-questions using numerals. We make two claims regarding Hale’s argument. First, it fails to legitimate Frege’s Constraint in virtue of resting on unsupported and highly contentious assumptions. Secondly, even if sound, Hale’s argument would vindicate a version of Frege’s Constraint which fails to adjudicate in favor of Hume’s Principle over alternative characterizations of the naturals. (shrink)

Neo-Russellians like Salmon and Braun hold that: the semantic contents of sentences are structured propositions whose basic components are objects and properties, names are directly referential terms, and a sentence of the form ‘n believes that S’ is true in a context c iff the referent of the name n in c believes the proposition expressed by S in c. This is sometimes referred to as ‘the Naive Russellian theory’. In this talk, I will discuss the Naive Russellian theory primarily (...) in connection with a problem known as Schiffer’s puzzle. Schiffer first presented the puzzle as an argument against the Naive Russellian theory. Schiffer’s argument proceeds in two steps. In step one, Schiffer argues that the Naive Russellian theory is committed to two principles regarding de re belief; the special-case consequence and Frege’s constraint. Then, in step two, Schiffer argues that the special-case consequence is not consistent with Frege’s constraint. Salmon and Braun reply to Schiffer’s argument that although the Naive Russellian theory is committed to Frege’s constraint, it is not committed to the special-case consequence. However, in this paper, I will argue with a new Schiffer-case that even if the Naive Russellian theory is not committed to the special-case consequence, it is still not consistent with Frege’s constraint. Concluding, I will discuss the possibility to reject Frege’s constraint within the Naive Russellian theory. (shrink)

The realist versions of mathematical structuralism are often characterized by what I call ‘the insubstantiality thesis’, according to which mathematical objects, being positions in structures, have no non-structural properties: they are purely structural objects. The thesis has been criticized for being inconsistent or descriptively inadequate. In this paper, by implementing the resources of a real-definitional account of essence in the context of Fregean abstraction principles, I offer a version of structuralism – essentialist structuralism – which validates a weaker version of (...) the insubstantiality thesis: mathematical objects have no non-structural essential properties. Next, I show how this rendition of structuralism alleviates a Fregean worry against insubstantiality, which is directed at the explanation of the applicability of mathematics from the structuralist perspective. (shrink)

"By looking at Frege's lectures on logic through the eyes of the young Carnap, this book casts new light on the history of logic and analytic philosophy. As two introductory essays by Gottfried Gabriel and by Erich H. Reck and Steve Awodey explain, Carnap's notes allow us to better understand Frege's deep influence on Carnap and analytic philosophy, as well as the broader philosophical matrix from which both continental and analytic styles of thought emerged in the 20th century."--BOOK (...) JACKET. (shrink)

This essay provided a summary of my book, _Belief and Meaning, showing how I thought that it was possible to combine a view of intentional content that was externalist as well as one which keeps faith with Frege's constraint raised by his puzzle about identity and Kripke's puzzle about belief. The account criticizes current externalist accounts owing to Putnam, Burge, McDowell and Davidson. A great deal turns on meaning and intentional content being contextual in a way that the (...) book elaborates. (shrink)

This note discusses Onofri's recent argument that no theory of concepts can jointly satisfy the publicity constraint and Frege's constraint, because these constraints are inconsistent. I show that this argument relies on the publicity constraint having an implication that it does not have.

Being in a state of confusion is, unfortunately, an ordinary part of human experience. What is going on in our heads exactly, when we are confused about something? Surely, there are many different ways of being confused, but in his engaging new book Talking About, Elmar Unnsteinsson takes on the special case of what he calls identity confusion. He suggests that understanding what happens when we are in a state of identity confusion – and what kinds of things this confusion (...) allows us to, or inhibits us from doing – will dissolve Frege's puzzle. And more importantly, it will pave the road to a new theory of reference which relies on what he calls the Edenic Constraint – a constraint that makes acts of reference unavailable to victims of identity confusion. I argue that this account is convincing if the Edenic Constraint is restricted to certain cases of combinatory confusion – namely those that involve conflicting intentions – but that it is less plausible that the reference mechanism fails in cases of combinatory confusion that don't involve conflicting intentions, or in any cases of separatory confusion. (shrink)

In broadest terms, this thesis is concerned to answer the question of whether the view that arithmetic is analytic can be maintained consistently. Lest there be much suspense, I will conclude that it can. Those who disagree claim that accounts which defend the analyticity of arithmetic are either unable to give a satisfactory account of the foundations of mathematics due to the incompleteness theorems, or, if steps are taken to mitigate incompleteness, then the view loses the ability to account for (...) the applicability of mathematics in the sciences. I will show that this criticism is not successful against every view whereby arithmetic is analytic by showing that the brand of "conventionalism" about mathematics that Rudolf Carnap advocated in the 1930s, especially in Logical Syntax of Language, does not suffer from these difficulties. There, Carnap develops an account of logic and mathematics that ensures the analyticity of both. It is based on his famous "Principle of Tolerance", and so the major focus of this thesis will to defend this principle from certain criticisms that have arisen in the 80 years since the book was published. I claim that these criticisms all share certain misunderstandings of the principle, and, because my diagnosis of the critiques is that they misunderstand Carnap, the defense I will give is of a primarily historical and exegetical nature. Again speaking broadly, the defense will be split into two parts: one primarily historical and the other argumentative. The historical section concerns the development of Carnap's views on logic and mathematics, from their beginnings in Frege's lectures up through the publication of Logical Syntax. Though this material is well-trod ground, it is necessary background for the second part. In part two we shift gears, and leave aside the historical development of Carnap's views to examine a certain family of critiques of it. We focus on the version due to Kurt Gödel, but also explore four others found in the literature. In the final chapter, I develop a reading of Carnap's Principle - the `wide' reading. It is one whereby there are no antecedent constraints on the construction of linguistic frameworks. I argue that this reading of the principle resolves the purported problems. Though this thesis is not a vindication of Carnap's view of logic and mathematics tout court, it does show that the view has more plausibility than is commonly thought. (shrink)

This volume contains all of Frege's extant unpublished writings on philosophy and logic other than his correspondence, written at various stages of his career.

Translation of Frege's 'Über Begriff und Gegenstand' (1892). Translation by Peter Geach, revised by Max Black.

Dieser Band enthält die vier Arbeiten Freges: Begriffsschrift, eine der arithmetischen nachgebildeten Formelsprache, 1879; Anwendungen der Begriffsschrift, 1879; Über den Briefwechsel Leibnizens und Huggens mit Papin, 1881; Über den Zweck der Begriffsschrift, 1883; Über die wissenschaftliche Berechtigung einer Begriffsschrift, 1882. Frege's research work in the field of mathematical logic is of great importance for the present-day analytic philosophy. We actually owe to Frege a great amount of basical insight and exemplary research, which set up a new standard also in (...) other fields of knowledge. As the founder of mathematical logic he severely examindes the syllogisms on which arithmetic is built up. In doing so, Frege recognized that our colloquial language is inadequate to define logic structures. His notional language corresponded to the artaivicial logical language demandes by Leibniz. Frege's achievement in the field of logic were so important, that they radiated into the domain of philosophy and influenced the development of mathematical logic decisively. (shrink)

It is a consensus to locate the origin of the reflexive foundations of modern semantics in Frege's work. Since Frege's distinction between two components of meaning (sense and reference), however, semantics has been forced to lead a double life. Among its first receptions, in Russell's famous article (1905), the first unresolved criticism of this solution was that: It is not possible to split semantics into a theory about two classes of objects without their yielding one and the same (...) thing under lower and higher conditions of instantiation (depending on the function used to identify it). But even Russell could not avoid a crisis. It is not possible to reconcile semantic coordination for a set of non-classical extension of instantiation and encoding (possible instances, counterfactual truth values, etc.) while preserving the classical properties of signification. This article covers these moments with a rough diagnosis: modern semantics has a reflexive ceiling. It is unable to model the contingent features of an "object" without oversizing itself to deal with various constraints on that object adapted to various strategies of intensional and modal specification. In order to model idealized conditions of assertability (Putnam), one must filter the sentences that pass Tarskian test using non-sematic parameters – like the parameter of coherence of a scientific paradigm. It cannot keep that model without stopping being semantic. We conclude with a response to attempts to give semantic status to complex scientific reasoning, and a suggestion as to how to locate the philosophical origin of this claim. (shrink)

[Translation of Frege's 'Über das Trägheitsgesetz].

[Translation of Frege's 'Gedankengefüge' (1923)].

This is the 1991 (2nd) edition of the 1986 book (MIT Press), considered to be the classic defense of Millianism. The nature of the information content of declarative sentences is a central topic in the philosophy of language. The natural view that a sentence like "John loves Mary" contains information in which two individuals occur as constituents is termed the naive theory, and is one that has been abandoned by most contemporary scholars. This theory was refuted originally by philosopher Gottlob (...) Frege. His argument that the naive theory did not work is termed Frege's puzzle, and his rival account of information content is termed the orthodox theory. In this detailed study, Nathan Salmon defends a version of the naive theory and presents a proposal for its extension that provides a better picture of information content than the orthodox theory gives. He argues that a great deal of what has generally been taken for granted in the philosophy of language over the past few decades is either mistaken or unsupported, and consequently, much current research is focused on the wrong set of questions. Salmon dissolves Frege's puzzle as it is usually formulated and demonstrates how it can be reconstructed and strengthened to yield a more powerful objection to the naive theory. He then defends the naive theory against the new Frege puzzle by presenting an idea that yields both a surprisingly rich and powerful extension of the naive theory and a better picture of information content than that of the original orthodox theory. Nathan Salmon is Professor of Philosophy, University of California at Santa Barbara. (shrink)

This volume contains English translations of Frege's early writings in logic and philosophy and of relevant reviews by other leading logicians. Professor Bynum has contributed a biographical essay, introduction, and extensive bibliography.

The idea that mathematics is reducible to logic has a long history, but it was Frege who gave logicism an articulation and defense that transformed it into a distinctive philosophical thesis with a profound influence on the development of philosophy in the twentieth century. This volume of classic, revised and newly written essays by William Demopoulos examines logicism's principal legacy for philosophy: its elaboration of notions of analysis and reconstruction. The essays reflect on the deployment of these ideas by the (...) principal figures in the history of the subject - Frege, Russell, Ramsey and Carnap - and in doing so illuminate current concerns about the nature of mathematical and theoretical knowledge. Issues addressed include the nature of arithmetical knowledge in the light of Frege's theorem; the status of realism about the theoretical entities of physics; and the proper interpretation of empirical theories that postulate abstract structural constraints. (shrink)

Abstractionism, which is a development of Frege's original Logicism, is a recent and much debated position in the philosophy of mathematics. This volume contains 16 original papers by leading scholars on the philosophical and mathematical aspects of Abstractionism. After an extensive editors' introduction to the topic of abstractionism, the volume is split into 4 sections. The contributions within these sections explore the semantics and meta-ontology of Abstractionism, abstractionist epistemology, the mathematics of Abstractionis, and finally, Frege's application constraint (...) within an abstractionist setting. (shrink)

Translation of Frege's 'Über die Begriffsschrift des Herrn Peano und meine eigene' (1896). Translated by V.H. Dudman.

The question of the applicability of mathematics is an epistemological issue that was explicitly raised by Kant, and which has played different roles in the works of neo-Kantian philosophers, before becoming an essential issue in early analytic philosophy. This paper will first distinguish three main issues that are related to the application of mathematics: indispensability arguments that are aimed at justifying mathematics itself; philosophical justifications of the successful application of mathematics to scientific theories; and discussions on the application of real (...) numbers to the measurement of physical magnitudes. A refinement of this tripartition is suggested and supported by a historical investigation of the differences between Kant’s position on the problem, several neo-Kantian perspectives, early analytic philosophy, and late 19th century mathematicians. Finally, the debate on the cogency of an application constraint in the definition of real numbers is discussed in relation to a contemporary debate in neo-logicism, in order to suggest a comparison not only with Frege’s original positions, but also with the ideas of several neo-Kantian scholars, including Hölder, Cassirer, and Helmholtz. (shrink)

I argue that a certain initially appealing Fregean conception of our shared semantic competence in our shared language cannot be made good. In particular, I show that we must reject two fundamental Fregean principles‐what I call Frege's Adequacy Condition and what I call Frege's Cognitive Constraint on Reference Determination. Frege's adequacy condition says that in an adequate semantic theory, sentence meanings must have the same fineness of grain as attitude contents. The Cognitive Constraint on Reference (...) Determination says that in an adequate semantic theory mechanisms of reference determination will fix routes of epistemic access to referents and will mediate one‐sided recognition judgments about referents. I argue against Frege's adequacy condition that if meanings are sliced as finely as it requires there will be too many meanings to go around. I argue against the cognitive constraint on reference determination that even in an adequate theory of reference, mechanisms of reference determination must be sharply distinguished from routes of epistemic access. If so, we have little reason to expect an adequate theory of reference determination to directly illuminate Frege's cognitive puzzles or be sufficient to explain the basis of our one‐sided recognition judgments. (shrink)

(Translation of Frege's Grundgesetze II, §§ 86-137).

A perspective in the philosophy of mathematics is developed from a consideration of the strengths and limitations of both logicism and platonism, with an early focus on Frege’s work. Importantly, although many set-theoretic structures may be developed each of which offers limited isomorphism with the system of natural numbers, no one of them may be identified with it. Furthermore, the timeless, ever present nature of mathematical concepts and results itself offers direct access, in the face of a platonist account which (...) generates a supposed problem of access. Crucially too, pure mathematics has its own distinctive method of confirming or validating results - mathematical proof - which supplies a higher level of confidence and objectivity than that available elsewhere. The dichotomy of invention and discovery is too jejune a framework for analysing creative mathematical activity. The Gödelian platonist perspective is evaluated and queried through scrutiny of the part played by mathematical resources and constraints in relation to human activity. It appears that there can be non-causal mathematical explanations and mathematical constraint on purely natural processes. Valuable implications of Quine’s naturalism are explored, but one must be cautious of his thesis of confirmational holism. The distinction between algebraic and non-algebraic mathematical theories usefully contributes to our understanding of the internally differentiated nature of the subject. (shrink)

Number words seemingly function both as adjectives attributing cardinality properties to collections, as in Frege’s ‘Jupiter has four moons’, and as names referring to numbers, as in Frege’s ‘The number of Jupiter’s moons is four’. This leads to what Thomas Hofweber calls Frege’s Other Puzzle: How can number words function as modifiers and as singular terms if neither adjectives nor names can serve multiple semantic functions? Whereas most philosophers deny that one of these uses is genuine, we instead argue that (...) number words, like many related expressions, are polymorphic, having multiple uses whose meanings are systematically related via type shifting. (shrink)

Frege's theory of indirect contexts and the shift of sense and reference in these contexts has puzzled many. What can the hierarchy of indirect senses, doubly indirect senses, and so on, be? Donald Davidson gave a well-known 'unlearnability' argument against Frege's theory. The present paper argues that the key to Frege's theory lies in the fact that whenever a reference is specified (even though many senses determine a single reference), it is specified in a particular way, so (...) that giving a reference implies giving a sense; and that one must be 'acquainted' with the sense. It is argued that an indirect sense must be 'immediately revelatory' of its reference. General principles for Frege's doctrine of sense and reference are sated, for both direct and indirect quotation, to be understood iteratively. I also discuss Frege's doctrine of tensed and first person statements in the light of my analysis. The views of various other authors are examined. The conclusion is to ascribe to Frege an implicit doctrine of acquaintance similar to that of Russell. (shrink)

Translation of Frege's 'Über den Zweck der Begriffsschrift' (1882).

Dieser Band enthält die vier Arbeiten Freges: Begriffsschrift, eine der arithmetischen nachgebildeten Formelsprache, 1879; Anwendungen der Begriffsschrift, 1879; Über den Briefwechsel Leibnizens und Huggens mit Papin, 1881; Über den Zweck der Begriffsschrift, 1883; Über die wissenschaftliche Berechtigung einer Begriffsschrift, 1882. Frege's research work in the field of mathematical logic is of great importance for the present-day analytic philosophy. We actually owe to Frege a great amount of basical insight and exemplary research, which set up a new standard also in (...) other fields of knowledge. As the founder of mathematical logic he severely examindes the syllogisms on which arithmetic is built up. In doing so, Frege recognized that our colloquial language is inadequate to define logic structures. His notional language corresponded to the artaivicial logical language demandes by Leibniz. Frege's achievement in the field of logic were so important, that they radiated into the domain of philosophy and influenced the development of mathematical logic decisively. (shrink)

(Translation of Frege's "Le nombre entier", 1895).

In lieu of an abstract, here is a brief excerpt of the content:MEANING IN A '.REALIST PERSPECTivE STEPHEN THERON National University of Lesotho Lesotho I DISCUSSION OF meaning and ref,erring in the terms laid down in a classic article of Frege's has generated a stereotyped attitude to the question in the minds of many. It is simply assumed that meaning is, as it were, the contrary of reference. In logic this is 11eflected by the assumed pamdigm of there being (...) formal systems which al'e.called purely formal to the extent that they are ' uninterpreted ', i.e. ·divol'ced from all questions of reference.1 If as a matter of history we accept that modern logic has grown basically out of Kantian soil, then it should not surprise that it is generally linked with a form of apriorism. The violence of the movement against ' psychologism ' shows how strong the danger is of linking the necessities of logic with those supposed epistemic necessities the non-Kantian realist can only regard as spurious. Anyhow this paper will be less·directly about logic than about those supposed necessities. I mean such constraints as Jonathan Bennett calls in general the " veil of perception " 2, ·as if perceiving, which should be an un- :veiling or bringing to lighrt, at the same time obscures what is seen or thought. The same idea recurs in the recent theory of 1 Of. Henry Veateh, Intentional Logic (Yale, 1952): In trying to subsume ordinary intentional relations like that of subject. predicate under the more abstract forms of a completely uninterpreted relational pattern... treated simply as a one-term relation on a par with all the... n-term real relations •. • mathematical logicians have in effect changed the character of such intentional relations completely.... (pp. 7778 ) :i See J. L. Mackie's Problems from Locke. ~9 30 STEPHEN THERON 'internal realism', just a more confusing name for transcendental idealism so far as I can see. There is of course no sense to the idea of checking what we claim to know against what we do not yet know. Yet he:re we should not be discussing precisely truth but the quite separate ontological question whether " THE WORLD is independent of any particular representrution we have of it" (Putnam's notion of the 'metaphysical realism' he rejects), whether "reality is altogether independent of experience," as Michael Devitt argues.3 This is a question of the being of the world, which existed before I d1d. Even as concerns the present moment, the wor1d is such that I or you might not exist, and this contingency of cognitive beings, severally or generally, to the actual world, as a possibility inherent in it, may not without more ado be identified with the quite different possibility of another world, like to this e.x:cept for the absence of me or you. For the former is a real if passive possibility in a thing, viz., the world, the latter is merely an alternative conception or idea of a different and merely possible thing. The concept of my dog with a broken leg is a different concept from that of my dog as healthy, but it is of the same dog, and that is why I don't want him to break his 1eg. The world without me was and would be the same world, though differently conceived. This is the basic requirement of realism, which philosophers have often violated, confusing the metaphysical with the epistemological, the real with the certain. For example Descartes reasoned fallaciously from his being certain, as he thought, of the soul's but not of the body's existence to the ontological separateness of these two. Kant, again, seemed to reason from duty as a pure concept or epistemic entity to the real separwteness, as a positivie value, of the motive of duty from any motive of pleasure or aspiration to happiness. Yet it is possible, indeed probable, that in reality soul and body, a M. Devitt," Realism and the Renegade Putnam," Nous 1983, p. 297. MEANING IN A REALIST PERSPECTIVE 31 though distinguishable, do not have separate acts of existence,,and, in any human existence, that the motive of duty both cannot and ought not to be separated from the distinguishable... (shrink)

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.

So-called 'Frege cases' pose a challenge for anyone who would hope to treat the contents of beliefs (and similar mental states) as Russellian propositions: It is then impossible to explain people's behavior in Frege cases without invoking non-intentional features of their mental states, and doing that seems to undermine the intentionality of psychological explanation. In the present paper, I develop this sort of objection in what seems to me to be its strongest form, but then offer a response to it. (...) I grant that psychological explanation must invoke non-intentional features of mental states, but it is of crucial importance which such features must be referenced. -/- It emerges from a careful reading of Frege's own view that we need only invoke what I call 'formal' relations between mental states. I then claim that referencing such 'formal' relations within psychological explanation does not undermine its intentionality in the way that invoking, say, neurological features would. The central worry about this view is that either (a) 'formal' relations bring narrow content in through back door or (b) 'formal' relations end up doing all the explanatory work. Various forms of each worry are discussed. The crucial point, ultimately, is that the present strategy for responding to Frege cases is not available either to the 'psycho-Fregean', who would identify the content of a belief with its truth-value, nor even to someone who would identify the content of a belief with a set of possible worlds. It requires the sort of rich semantic structure that is distinctive of Russellian propositions. There is therefore no reason to suppose that the invocation of 'formal' relations threatens to deprive content of any work to do. (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.

Frege's logicist program requires that arithmetic be reduced to logic. Such a program has recently been revamped by the "neologicist" approach of Hale and Wright. Less attention has been given to Frege's extensionalist program, according to which arithmetic is to be reconstructed in terms of a theory of extensions of concepts. This paper deals just with such a theory. We present a system of second-order logic augmented with a predicate representing the fact that an object x is the (...) extension of a concept C, together with extra-logical axioms governing such a predicate, and show that arithmetic can be obtained in such a framework. As a philosophical payoff, we investigate the status of the so-called Hume's Principle and its connections to the root of the contradiction in Frege's system. (shrink)

It is Frege's third contribution that makes the point of departure for the present paper. Not merely did Frege show how to manipulate symbols more exactly; he also gave a searching account of what these symbols mean. Consider a philosophical problem that arises out of the simplest arithmetic. When we say that 5 = 2 + 3, what do we mean? Do we mean that 5 is identical with 2 + 3? But in some ways 5 and 2 + (...) 3 are obviously different. Or do we mean that 5 and 2 + 3 are equal but not identical, equality being a relation that falls short of complete identity? But in that case, some ordinary ways of speaking in mathematics must be false. Suppose a pupil is asked for the positive square root of 25. The phrasing of the question implies that there is one and only one. No doubt he may answer, '5'. But then it follows that the answer '2 + 3' is not allowable, since ex hypothesi 5 and 2 + 3, though equal, are not identical. Yet ordinarily the answer '2 + 3' would be regarded as strange but not as wrong. (shrink)

Gottlob Frege's Grundgesetze der Arithmetik, or Basic Laws of Arithmetic, was intended to be his magnum opus, the book in which he would finally establish his logicist philosophy of arithmetic. But because of the disaster of Russell's Paradox, which undermined Frege's proofs, the more mathematical parts of the book have rarely been read. Richard G.

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.