I. Frege as a Realist

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.     

Gottlob Frege und Rudolf Eucken—Gesprächspartner in der Herausbildungsphase der modernen Logik

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.     

The truth and nothing but the truth,yet never the whole truth: Frege,Russell and the analysis of unities

It is widely assumed that Russell's problems with the unity of the proposition were recurring and insoluble within the framework of the logical theory of his Principles of Mathematics. By contrast, Frege's functional analysis of thoughts (grounded in a type-theoretic distinction between concepts and objects) is commonly assumed to provide a solution to the problem or, at least, a means of avoiding the difficulty altogether. The Fregean solution is unavailable to Russell because of his commitment to the thesis that there is only one ultimate ontological category. This, combined with Russell's reification of propositions, ensures that he must hold concepts and objects to be of the same logical and ontological type. In this paper I argue that, while Frege's treatment of the unity of the proposition has immediate advantages over Russell's, a deeper consideration of the philosophical underpinnings and metaphysical consequences of the two approaches reveals that Frege's supposed solution is, in fact, far from satisfactory. Russell's repudiation of the Fregean position in the Principles is, I contend, convincing and Russell's own position, despite its problems, conforms to a greater extent than Frege's with common sense and, furthermore, with certain ideas which are central to our understanding of the origins of the analytical tradition.     

Cantor on Frege's Foundations of Arithmetic: Cantor's 1885 Review of Frege's Die Grundlagen der Arithmetik

In 1885, Georg Cantor published his review of Gottlob Frege's Grundlagen der Arithmetik. In this essay, we provide its first English translation together with an introductory note. We also provide a translation of a note by Ernst Zermelo on Cantor's review, and a new translation of Frege's brief response to Cantor. In recent years, it has become philosophical folklore that Cantor's 1885 review of Frege's Grundlagen already contained a warning to Frege. This warning is said to concern the defectiveness of Frege's notion of extension. The exact scope of such speculations varies and sometimes extends as far as crediting Cantor with an early hunch of the paradoxical nature of Frege's notion of extension. William Tait goes even further and deems Frege ‘reckless’ for having missed Cantor's explicit warning regarding the notion of extension. As such, Cantor's purported inkling would have predated the discovery of the Russell–Zermelo paradox by almost two decades. In our introductory essay, we discuss this alleged implicit (or even explicit) warning, separating two issues: first, whether the most natural reading of Cantor's criticism provides an indication that the notion of extension is defective; second, whether there are other ways of understanding Cantor that support such an interpretation and can serve as a precisification of Cantor's presumed warning.     

The Role of Structural Reasoning in the Genesis of Graph Theory

The concept of quantity (Größe) plays a key role in Frege's theory of real numbers. Typically enough, he refers to this theory as ‘theory of quantity’ (‘Größenlehre’) in the second volume of his opus magnum Grundgesetze der Arithmetik (Frege 1903). In this essay, I deal, in a critical way, with Frege's treatment of the concept of quantity and his approach to analysis from the beginning of his academic career until Frege 1903. I begin with a few introductory remarks. In Section 2, I first analyze Frege's use of the term ‘source of knowledge’ (‘Erkenntnisquelle’) with particular emphasis on the logical source of knowledge. The analysis includes a brief comparison between Frege and Kant's conceptions of logic and the logical source of knowledge. In a second step, I examine Frege's theory of quantity in Rechnungsmethoden, die sich auf eine Erweiterung des Größenbegriffes gründen (Frege 1874). Section 3 contains a couple of critical observations on Frege's comments on Hankel's theory of real numbers in Die Grundlagen der Arithmetik (Frege 1884). In Section 4, I consider Frege's discussion of the concept of quantity in Frege 1903. Section 5 is devoted to Cantor's theory of irrational numbers and the critique deployed by Frege. In Section 6, I return to Frege's own constructive treatment of analysis in Frege 1903 and succinctly describe what I take to be the quintessence of his account.     

Propositions,Properties and Relations: Wittgenstein's “Notes on Logic” and the Tractatus

Frege famously argued that truth is not a property or relation. In the “Notes on Logic” Wittgenstein emphasised the bi‐polarity of propositions which he called their sense. He argued that “propositions by virtue of sense cannot have predicates or relations.” This led to his fundamental thought that the logical constants do not represent predicates or relations. The idea, however, has wider ramifications than that. It is not just that propositions cannot have relations to other propositions but also that they cannot have relations to anything at all. The paper explores the consequences of this insight for the way in which we should read the Tractatus. In the “Notes on Logic” the insight led to Wittgenstein's emphasis on “facts” in any attempt to understand the nature of symbolism. This emphasis is continued in the Tractatus. It is central to his view that propositions are facts which picture facts which prevent us from construing such picturing as a relation between what pictures and what is pictured. It illuminates the importance of context principle with regard to the distinction between showing and saying to which Wittgenstein attached so much importance and it underlies the non‐relational view of psychological propositions which he advocates. Finally, if propositions by virtue of sense cannot have predicates or relations the paradox at the end of a work which consist largely of propositions about propositions becomes intelligible.     

Kant,wittgenstein and the limits of logic

This paper has two purposes. (1) To justify the claim that there is an important distinction underlying the saying/showing distinction of the Tractatus; the distinction which Kant characterises as that between historical and rational knowledge. (2) To argue that it is because the Tractatus accepts Frege/Russell logic as a complete representation of all thought according to laws, that what is shown cannot be recognised as knowledge. This is done by interpolating Frege's logical innovations between the views of Kant and Wittgenstein on logic and mathematics.     

Conceptual Analysis and Analytical Definitions in Frege

Logical (or conceptual) analysis is in Frege primarily not an analysis of a concept but of its sense. Five Fregean philosophical principles are presented as constituting a framework for a theory of logical or conceptual analysis, which I call analytical explication. These principles, scattered and sometime latent in his writings are operative in Frege's critique of other views and in his constructive development of his own view. The proposed conception of analytical explication is partially rooted in Frege's notion of analytical definition. It may also be the basis of what is required of a reduction of one domain to another, if it is to have the philosophical significance many reductions allegedly have.     

Zur Miete bei Frege – Rudolf Hirzel und die Rezeption der stoischen Logik und Semantik in Jena

It has been noted before in the history of logic that some of Frege's logical and semantic views were anticipated in Stoicism. In particular, there seems to be a parallel between Frege's Gedanke (thought) and Stoic lekton; and the distinction between complete and incomplete lekta has an equivalent in Frege's logic. However, nobody has so far claimed that Frege was actually influenced by Stoic logic; and there has until now been no indication of such a causal connection. In this essay, we attempt, for the first time, to provide detailed evidence for the existence of this connection. In the course of our argumentation, further analogies between the positions of Frege and the Stoics will be revealed. The classical philologist Rudolf Hirzel will be brought into play as the one who links Frege with Stoicism. The renowned expert on Stoic philosophy was Frege's tenant and lived in the same house as the logician for many years.

In der Geschichte der Logik ist häufig bemerkt worden, dass einige der logischen und semantischen Auffassungen Freges in der Stoa antizipiert worden sind. Genannt wurden insbesondere die Parallelen zwischen dem Fregeschen Gedanken und dem stoischen Lekton sowie die Unterscheidung zwischen vollständigen und unvollständigen Lekta, die bei Frege ihre Entsprechung hat. Ein Wirkungszusammenhang ist allerdings nicht behauptet worden. Dazu gab es bislang auch keinen Anlass. Der vorliegende Beitrag versucht erstmalig, einen detaillierten Indizienbeweis für das Bestehen eines solchen Zusammenhangs vorzulegen. Dabei werden weitere charakteristische Übereinstimmungen zwischen Frege und der Stoa aufgewiesen. Als Mittelsmann wird der Altphilologe Rudolf Hirzel vorgestellt. Er wohnte lange Jahre als Mieter zusammen mit Frege im selben Haus und war ein anerkannter Experte der stoischen Philosophie.     

On Russell's vulnerability to Russell's paradox

Influenced by G. E. Moore, Russell broke with Idealism towards the end of 1898; but in later years he characterized his meeting Peano in August 1900 as ‘the most important event’ in ‘the most important year in my intellectual life’. While Russell discovered his paradox during his post-Peano period, the question arises whether he was already committed, during his pre-Peano Moorean period, to assumptions from which his paradox may be derived. Peter Hylton has argued that the pre-Peano Russell was thus vulnerable to (at least one version of) Russell's paradox and hence that the paradox exposes a pre-existing difficulty in Russell's Moorean philosophy. Contrary to Hylton, I argue that the Moorean Russell adhered to views which insulated him against the paradox. Further, I argue that Russell became vulnerable to his paradox as a result of changes in his Moorean position occasioned, first, by his acceptance of Cantor's theory of the transfinite, and, second, by his correspondence with Frege. I conclude with some general comments regarding Russell's acceptance of naïve set theory.     

I. Frege's ‘Kernsätze zur Logik’

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.     

Frege: two theses,two senses

One particular topic in the literature on Frege's conception of sense relates to two apparently contradictory theses held by Frege: the isomorphism of thought and language on one hand and the expressibility of a thought by different sentences on the other. I will divide the paper into five sections. In (1) I introduce the problem of the tension in Frege's thought. In (2) I discuss the main attempts to resolve the conflict between Frege's two contradictory claims, showing what is wrong with some of them. In (3), I analyse where, in Frege'ps writings and discussions on sense identity, one can find grounds for two different conceptions of sense. In (4) I show how the two contradictory theses held by Frege are connected with different concerns, compelling Frege to a constant oscillation in terminology. In (5) I summarize two further reasons that prevented Frege from making the distinction between two conceptions of sense clear: (i) the antipsychologism problem and (ii) the overlap of traditions in German literature contemporary to Frege about the concept of value. I conclude with a hint for a reconstruction of the Fregean notion of ‘thought’ which resolves the contradiction between his two theses.     

From qualitative differences to a continuum of development: commentary on Leslie Smith's epistemological principles for developmental psychology

In his article Epistemological principles for developmental psychology Leslie Smith helps to re-open some of the key issues Piaget explored through his genetic epistemology. Smith shows the important parallels between logician Gottlieb Frege's understanding of rational thought, and the way in which Piaget developed such notions in his own theory. But while Frege's theory helps set the parameters for whether thought can be judged as rational, or if it even should be judged as rational, it also shows the logicians' disdain for exploring anything resembling development of rationality. Thus Frege might have an important, but necessarily mediated impact on the field of human development. Piaget's carefully crafted theory of epistemological development potentially serves as such a mediating device. It can be argued that Piaget crafted together arguments of logicians such as Frege, and epistemologists such as Lévy-Bruhl, to develop his extraordinary achievement of a genetic epistemology that leads to an understanding of the human condition. One of Piaget's accomplishments was to develop a continuum out of the logicians' dichotomy between non-logical and logical in which the non-rational flows into the rational.     

ANALYSIS,ABSTRACTION PRINCIPLES,AND SLINGSHOT ARGUMENTS

Frege's views regarding analysis and synomymy have long been the subject of critical discussion. Some commentators, led by Dummett, have argued that Frege was committed to the view that each thought admits of a unique ultimate analysis. However, this interpretation is in apparent conflict with Frege's criterion of synonymy, according to which two sentence express the same thought if one cannot understand them without regarding them as having the same truth–value. In a recent article in this journal, Drai attempts to reconcile Frege's criterion of synonymy with unique ultimate analysis by holding that, for Frege, if two sentences satisfy the criterion without being intensionally isomorphic, at most one of them is a privileged representation of the thought expressed. I argue that this proposal fails, because it conflicts not only with Frege's views of abstraction principles but also with slingshot arguments (including one presented by Drai herself) that accurately reflect Frege's commitment to the view that sentences alike in truth–value have the same Bedeutung. While Drai helpfully connects Frege's views of abstraction principles with such slingshot arguments, this connection cannot become fully clear until we recognise that Frege rejects unique ultimate analysis.     

Sense data and logical relations: Karin Costelloe-Stephen and Russell’s critique of Bergson

ABSTRACT

Though scholarship has explored Karin Costelloe-Stephen’s contributions to the history of psychoanalysis, as well as her relations to the Bloomsbury Group, her philosophical work has been almost completely ignored. This paper will examine her debate with Bertrand Russell over his criticism of Bergson. Costelloe-Stephen had employed the terminology of early analytic philosophy in presenting a number of arguments in defence of Bergson’s views. Costelloe-Stephen would object, among other things, to Russell’s use of an experiment which, as she points out, was first conducted by Carl Stumpf. Russell appeals to Stumpf's experiment in his attempt to prove that sense data are terms in logical relations, a thesis presupposed by the project of logical analysis outlined in Our Knowledge of the External World. A reformulated version of Costelloe-Stephen's argument put forth by this paper shows that Russell's argument fails to provide adequate proof for his thesis. Further modifications of the argument can also address a reconstruction (based on contemporary reports) of Russell's reply to Costelloe-Stephen. In his reply, Russell would use, already in 1914, the term ‘analytic philosophy’ in contrasting his and Moore’s approach to a continental one, exemplified by Bergson and Costelloe-Stephen.     

Frege's Cardinals Do Not Always Obey Hume's Principle

Hume's Principle, dear to neo-Logicists, maintains that equinumerosity is both necessary and sufficient for sameness of cardinal number. All the same, Whitehead demonstrated in Principia Mathematica's logic of relations (where non-homogeneous relations are allowed) that Cantor's power-class theorem entails that Hume's Principle admits of exceptions. Of course, Hume's Principle concerns cardinals and in Principia's ‘no-classes’ theory cardinals are not objects in Frege's sense. But this paper shows that the result applies as well to the theory of cardinal numbers as objects set out in Frege's Grundgesetze. Though Frege did not realize it, Cantor's power-theorem entails that Frege's cardinals as objects do not always obey Hume's Principle.     

Frege's Begriffsschrift is Indeed First-Order Complete

It is widely taken that the first-order part of Frege's Begriffsschrift is complete. However, there does not seem to have been a formal verification of this received claim. The general concern is that Frege's system is one axiom short in the first-order predicate calculus comparing to, by now, standard first-order theory. Yet Frege has one extra inference rule in his system. Then the question is whether Frege's first-order calculus is still deductively sufficient as far as the first-order completeness is concerned. In this short note we confirm that the missing axiom is derivable from his stated axioms and inference rules, and hence the logic system in the Begriffsschrift is indeed first-order complete.     

Frege's ‘On the Concept of Number’ (1891) – an unnoticed publication

A short piece by Frege, heretofore overlooked, containing a précis of his views on the concept of number, is presented, after some very brief questions about Frege's possible involvement in the wider intellectual milieu.