It is well known that Frege was an extensionalist, in the following sense: he held that the truth-value of a sentence is always a function only of the references of its parts. One consequence of this view is that expressions occurring in certain linguistic contexts – for example, the that-clauses of propositional attitude ascriptions – do not have their usual references, but refer instead to what are usually their senses. But although a number of philosophers have objected to this result, (...) no one has yet attempted to see what happens to Frege's views – and, in particular, to his theory of sense and reference – if his extensionalism is abandoned while his other views are maintained. This paper thus does two things. First, it clears the way for such an attempt, by arguing that recent defences of Frege's extensionalism – by Tyler Burge, Saul Kripke, Terence Parsons and Christopher Peacocke – fall short. Second, it sketches a positive proposal for a non-extensionalist application of Frege's theory of sense and reference to sentences of indirect discourse and ascriptions of propositional attitude. (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)

This paper argues against the claim that Frege is committed to an infinite hierarchy of senses. Carnap and Kripke, along with many others, argue the contrary; I expose where all such arguments go astray. Invariably these arguments assume (without citation) that Frege holds that sense and reference are always distinct. This is the fulcrum upon which the hierarchy is hoisted. The counter to this assumption is based on two important but neglected passages. The locution ‘indirect sense’ has no (...) ontological significance for Frege. It was just a ‘short expression’ (his term) to indicate that the customary relation between sign, sense, and reference is suspended in what he called the exception case of indirect context. Once the hierarchy is abandoned, the alleged problem of just what indirect sense is, as well as questions as to why Frege said so little about it and why he did not examine iterative indirect contexts are resolved. Of course, with no hierarchy, Davidson’s “unlearnability argument” turns out to be a non-starter. Some objections to my interpretation are considered and answered. Additionally, comparisons to Dummett’s one-level view of sense, which is similar to mine are made, as well as to some two-level views (Parsons and Skiba). (shrink)

It is often claimed that the theory of function levels proposed by Frege in Grundgesetze der Arithmetik anticipates the hierarchy of types that underlies Church’s simple theory of types. This claim roughly states that Frege presupposes a type of functions in the sense of simple type theory in the expository language of Grundgesetze. However, this view makes it hard to accommodate function names of two arguments and view functions as incomplete entities. I propose and defend an alternative interpretation of (...) first-level function names in Grundgesetze into simple type-theoretic open terms rather than into closed terms of a function type. This interpretation offers a still unhistorical but more faithful type-theoretic approximation of Frege’s theory of levels and can be naturally extended to accommodate second-level functions. It is made possible by two key observations that Frege’s Roman markers behave essentially like open terms and that Frege lacks a clear criterion for distinguishing between Roman markers and function names. (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)

The main objective of this paper is to examine how theories of truth and reference that are in a broad sense Fregean in character are threatened by antinomies; in particular by the Epimenides paradox and versions of the so-called Russell-Myhill antinomy, an intensional analogue of Russell’s more well-known paradox for extensions. Frege’s ontology of propositions and senses has recently received renewed interest in connection with minimalist theories that take propositions (thoughts) and senses (concepts) as the primary bearers of truth and (...) reference. In this paper, I will present a rigorous version of Frege’s theory of sense and denotation and show that it leads to antinomies. I am also going to discuss ways of modifying Frege’s semantical and ontological framework in order to avoid the paradoxes. In this connection, I explore the possibility of giving up the Fregean assumption of a universal domain of absolutely all objects, containing in addition to extensional objects also abstract intensional ones like propositions and singular concepts. I outline a cumulative hierarchy of Fregean propositions and senses, in analogy with Gödel’s hierarchy of constructible sets. In this hierarchy, there is no domain of all objects. Instead, every domain of objects is extendible with new objects that, on pain of contradiction, cannot belong to the given domain. According to this approach, there is no domain containing absolutely all propositions or absolutely all senses. (shrink)

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 ...

"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)

The first complete English translation of a groundbreaking work. An ambitious account of the relation of mathematics to logic. Includes a foreword by Crispin Wright, translators' Introduction, and an appendix on Frege's logic by Roy T. Cook. The German philosopher and mathematician Gottlob Frege (1848-1925) was the father of analytic philosophy and to all intents and purposes the inventor of modern logic. Basic Laws of Arithmetic, originally published in German in two volumes (1893, 1903), is Freges magnum opus. It (...) was to be the pinnacle of Freges lifes work. It represents the final stage of his logicist project the idea that arithmetic and analysis are reducible to logic and contains his mature philosophy of mathematics and logic. The aim of Basic Laws of Arithmetic is to demonstrate the logical nature of mathematical theorems by providing gapless proofs in Frege's formal system using only basic laws of logic, logical inference, and explicit definitions. The work contains a philosophical foreword, an introduction to Frege's logic, a derivation of arithmetic from this logic, a critique of contemporary approaches to the real numbers, and the beginnings of a logicist treatment of real analysis. As is well-known, a letter received from Bertrand Russell shortly before the publication of the second volume made Frege realise that his basic law V, governing the identity of value-ranges, leads into inconsistency. Frege discusses a revision to basic law V written in response to Russells letter in an afterword to volume II. The continuing importance of Basic Laws of Arithmetic lies not only in its bearing on issues in the foundations of mathematics and logic but in its model of philosophical inquiry. Frege's ability to locate the essential questions, his integration of logical and philosophical analysis, and his rigorous approach to criticism and argument in general are vividly in evidence in this, his most ambitious work. Philip Ebert and Marcus Rossberg present the first full English translation of both volumes of Freges major work preserving the original formalism and pagination. The edition contains a foreword by Crispin Wright and an extensive appendix providing an introduction to Frege's formal system by Roy T. Cook. Readership: Scholars and advanced students in philosophy of logic, philosophy of mathematics, and early analytic philosophy. (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.

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

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.

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

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

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

In this paper, we characterize the strength of the predicative Frege hierarchy, , introduced by John Burgess in his book [J. Burgess, Fixing frege, in: Princeton Monographs in Philosophy, Princeton University Press, Princeton, 2005]. We show that and are mutually interpretable. It follows that is mutually interpretable with Q. This fact was proved earlier by Mihai Ganea in [M. Ganea, Burgess’ PV is Robinson’s Q, The Journal of Symbolic Logic 72 619–624] using a different proof. Another consequence of the (...) our main result is that is mutually interpretable with Kalmar Arithmetic . The fact that interprets EA was proved earlier by Burgess. We provide a different proof. Each of the theories is finitely axiomatizable. Our main result implies that the whole hierarchy taken together, , is not finitely axiomatizable. What is more: no theory that is mutually locally interpretable with is finitely axiomatizable. (shrink)

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

The paper analyses Frege's approach to the identity conditions for the entity labelled by him as Sinn. It starts with a brief characterization of the main principles of Frege's semantics and lists his remarks on the identity conditions for Sinn. They are subject to a detailed scrutiny, and it is shown that, with the exception of the criterion of intersubstitutability in oratio obliqua, all other criteria have to be discarded. Finally, by comparing Frege's views on Sinn with (...) Carnap's method of extension and intension and the method of intensional isomorphism, it is proved that these methods do not provide a criterion for the identity of Frege's Sinn, even for extensional contexts, that the concept of intension does not coincide, as stated by Carnap, in these contexts, with Frege's concept of Sinn, and that Carnap's claim that in oratio obliqua Frege's semantics leads to an infinite hierarchy of Sinn entities can be questioned at least hypothetically in the light of certain new historical facts. (shrink)

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 'Über den Zweck der Begriffsschrift' (1882).

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

Frege's essay "Der Gedanke.Eine logische Untersuchung" was first published in the Beitrage zur Philosophie des Deutschen Idealismus for 1918-1919 and is one of three related logical studies published as a complete work by Gunther Patzig entitled Logische Untersuchungen in Gottingen, 1966 .

The question whether Frege’s theory of indirect reference enforces an infinite hierarchy of senses has been hotly debated in the secondary literature. Perhaps the most influential treatment of the issue is that of Burge (1979), who offers an argument for the hierarchy from rather minimal Fregean assumptions. I argue that this argument, endorsed by many, does not itself enforce an infinite hierarchy of senses. I conclude that whether or not the theory of indirect reference can avail itself (...) of only finitely many senses is pending further theoretical development. (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)

THERE IS GREAT MOTIVATION WITHIN FREGE'S THEORY TO\nCONSTRUE THE CARDINAL NUMBERS AS QUANTIFIERS, WHICH ARE\nHIGHER LEVEL CONCEPTS. BUT FREGE ARGUED THAT THE CARDINAL\nNUMBERS ARE OBJECTS, NOT CONCEPTS, AND DEFINED THEM\nACCORDINGLY. MOREOVER, FREGE'S HIERARCHY OF CONCEPTS\nPREVENTED HIM FROM CONSTRUING THE NUMBERS AS CONCEPTS. MY\nPURPOSE IS TO BRING OUT THE QUANTIFICATIONAL NATURE OF THE\nNUMBERS IN THE FACE OF THESE OBSTACLES. THE PAPER PRESSES\nTHE QUANTIFICATIONAL VIEW ONTO FREGE'S CONCEPT OF NUMBER AS\nIT TRACES ITS DEVELOPMENT FROM THE "BEGRIFFSSCHRIFT",\nTHROUGH THE 1880S, (...) INTO ITS FORMALIZATION IN THE\n"GRUNDGESETZE". THE THEORY YIELDS SURPRISINGLY EASILY UNDER\nPRESSURE FROM THE QUANTIFICATIONAL VIEW, AND ONE\nSYMPATHETIC WITH THAT VIEW MAY TAKE ENCOURAGEMENT FROM THE\nFACT THAT IT WAS FREGE'S HIERARCHY OF CONCEPTS, NOT HIS\nACTUAL ARGUMENTS, THAT PREVENTED HIS CONSTRUING NUMBERS AS\nQUANTIFIERS. (shrink)

I investigate why Frege rejected the theory of types, as Russell presented it to him in their correspondence. Frege claims that it commits one to violations of the law of excluded middle, but this complaint seems to rest on a dogmatic refusal to take Russell’s proposal seriously on its own terms. What is at stake is not so much the truth of a law of logic, but the structure of the hierarchy of the logical categories, something Frege seems to (...) neglect. To come to a better understanding of Frege’s response, I proceed to investigate his conception of the nature of the logical categories, and how it differs from Russell’s. I argue that, for Frege, our grasp of the logical categories cannot be severed from our grasp of the Begriffsschrift notation itself. Russell, on the other hand, attaches no such importance to notation. From Frege’s point of view, Russell has not succeeded in presenting an alternative conception of the logical hierarchy, since such a conception must go in tandem with the development of a notation. Moreover, Frege has good reasons to think that Russell’s proposal does not admit of a suitable notation. (shrink)

The paper aims at studying, in full generality, logics defined by imposing a variable inclusion condition on a given logic \. We prove that the description of the algebraic counterpart of the left variable inclusion companion of a given logic \ is related to the construction of Płonka sums of the matrix models of \. This observation allows to obtain a Hilbert-style axiomatization of the logics of left variable inclusion, to describe the structure of their reduced models, and to locate (...) them in the Leibniz hierarchy. (shrink)

A British coffee chain’s fundraising practices constitute a background for this study to examine ideological discourses behind British charitable giving. The charity executes projects in coffee growing communities by providing education for children in disadvantaged neighbourhoods. The study takes a critical stance from a discursive paradigmatic perspective to analyse visual contents used by the charity. The applied visual critical discourse analysis was inspired by Barthes’ semiotic theory. Findings suggest that the adverts’ interpretative repertoires can serve ideologies that sustain the donors’ (...) social-cultural dominance over the recipients by justifying group-based social hierarchies. The chains of signifiers do not question the spectators’ moral position but rationalise status quo and maintain undisturbed consumption. Findings suggest the possibility that the discourses of prosocial behaviour incited under consuming conditions do not challenge but may even justify global inequality. This finding presents challenges facing charitable organisations balancing their work within wider commercial and cultural contexts. (shrink)

Computability theory originated with the seminal work of Gödel, Church, Turing, Kleene and Post in the 1930s. This theory includes a wide spectrum of topics, such as the theory of reducibilities and their degree structures, computably enumerable sets and their automorphisms, and subrecursive hierarchy classifications. Recent work in computability theory has focused on Turing definability and promises to have far-reaching mathematical, scientific, and philosophical consequences. Written by a leading researcher, Computability Theory provides a concise, comprehensive, and authoritative introduction to (...) contemporary computability theory, techniques, and results. The basic concepts and techniques of computability theory are placed in their historical, philosophical and logical context. This presentation is characterized by an unusual breadth of coverage and the inclusion of advanced topics not to be found elsewhere in the literature at this level. The book includes both the standard material for a first course in computability and more advanced looks at degree structures, forcing, priority methods, and determinacy. The final chapter explores a variety of computability applications to mathematics and science. Computability Theory is an invaluable text, reference, and guide to the direction of current research in the field. Nowhere else will you find the techniques and results of this beautiful and basic subject brought alive in such an approachable and lively way. (shrink)

Ce recueil a pour but de montrer que Heidegger n'est pas seulement, et peut-etre pas d'abord, ce penseur solitaire de Foret-Noire pretant l'oreille a la voix de l'Etre, comme on s'est souvent plu a le presenter, et comme il a sans doute lui-meme contribue a le faire accroire: c'est un penseur extraordinairement attentif aux developpements philosophiques de son temps, dont la pensee s'est elaboree en grande partie en dialogue non seulement avec les Grecs, la pensee chretienne, l'existentialisme de Kierkegaard ou (...) encore la phenomenologie husserlienne, mais egalement avec ses predecesseurs immediats (Dilthey, Brentano, Bergson, Braig, York von Wartenburg), les ecoles neokantiennes de Bade (Rickert, Lask) et de Marbourg (Cohen, Natorp, Cassirer), ainsi qu'avec certains contemporains, tels Jaspers ou Scheler. Des ses debuts, sa pensee s'avere ouverte aussi bien aux avancees de la logique mathematique (Frege et Russell) que, plus tard, aux developpements de l'anthropologie et de la biologie de son epoque (von Uexkull). Cette mise en perspective, qui nous parait plus que jamais necessaire, permet ainsi de mettre en lumiere des aspects negliges de l'oeuvre, de mieux comprendre les sources et, peut-etre, certaines limites de celui qui demeure, malgre l'ombre qui pese sa pensee, l'un des plus grands philosophes du XXe siecle. (shrink)

Computability theory concerns information with a causal–typically algorithmic–structure. As such, it provides a schematic analysis of many naturally occurring situations. Emil Post was the first to focus on the close relationship between information, coded as real numbers, and its algorithmic infrastructure. Having characterised the close connection between the quantifier type of a real and the Turing jump operation, he looked for more subtle ways in which information entails a particular causal context. Specifically, he wanted to find simple relations on reals (...) which produced richness of local computability-theoretic structure. To this extent, he was not just interested in causal structure as an abstraction, but in the way in which this structure emerges in natural contexts. Post’s programme was the genesis of a more far reaching research project. In this article we will firstly review the history of Post’s programme, and look at two interesting developments of Post’s approach. The first of these developments concerns the extension of the core programme, initially restricted to the Turing structure of the computably enumerable sets of natural numbers, to the Ershov hierarchy of sets. The second looks at how new types of information coming from the recent growth of research into randomness, and the revealing of unexpected new computability-theoretic infrastructure. We will conclude by viewing Post’s programme from a more general perspective. We will look at how algorithmic structure does not just emerge mathematically from information, but how that emergent structure can model the emergence of very basic aspects of the real world. (shrink)