II. Frege as Idealist and then Realist

Michael Dummett argued that Frege was a realist while Hans Sluga countered that he was an objective idealist in the rationalist tradition of Kant and Lotze. Sluga ties Frege's idealism to the context principle which he argues Frege never gave up. It is argued that Sluga has correctly interpreted the pre‐1891 Frege while Dummett is correct concerning the later period. It is also claimed that the context principle was dropped prior to 1891 to be replaced by the doctrine of unsaturated entities.     

Beltrami's model and the independence of the parallel postulate

In their correspondence in 1902 and 1903, after discussing the Russell paradox, Russell and Frege discussed the paradox of propositions considered informally in Appendix B of Russell's Principles of Mathematics. It seems that the proposition, p, stating the logical product of the class w, namely, the class of all propositions stating the logical product of a class they are not in, is in w if and only if it is not. Frege believed that this paradox was avoided within his philosophy due to his distinction between sense (Sinn) and reference (Bedeutung). However, I show that while the paradox as Russell formulates it is ill-formed with Frege's extant logical system, if Frege's system is expanded to contain the commitments of his philosophy of language, an analogue of this paradox is formulable. This and other concerns in Fregean intensional logic are discussed, and it is discovered that Frege's logical system, even without its naive class theory embodied in its infamous Basic Law V, leads to inconsistencies when the theory of sense and reference is axiomatized therin. therein.     

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.     

II. Frege's realism

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'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,August Bebel and the Return of Alsace-Lorraine: The dating of the distinction between Sinn and Bedeutung

A detailed chronology is offered for the writing of Frege's central philosophical essays from the early 1890s. Particular attention is given to (the distinction between) Sinn and Bedeutung. Suggestions are made as to the origin of the examples concerning the Morning Star/Evening Star and August Bebel's views on the return of Alsace-Lorraine. Likely sources are offered for Frege's use of the terms Bestimmungsweise, Art des Gegebenseins and Sinn und Bedeutung.     

Dummett and Frege on sense and Selbständigkeit

As part of his attack on Frege’s ‘myth’ that senses reside in the third realm, Dummett alleges that Frege’s view that all objects are selbständig (‘self-subsistent’, ‘independent’) is an underlying mistake, since some objects depend upon others. Whatever the merits of Dummett’s other arguments against Frege’s conception of sense, this objection fails. First, Frege’s view that senses are third-realm entities is not traceable to his view that all objects are selbständig. Second, while Frege recognizes that there are objects that are dependent upon other objects, he does not take this to compromise the Selbständigkeit of any objects. Thus, Frege’s doctrine that objects are selbständig does not make the claim of absolute independence that Dummett appears to have taken it to make. Nevertheless, in order to make a good case against Frege based on the dependency of senses, Dummett need only establish his claim that senses depend upon expressions: appeal to an absolute conception of independence is unnecessary. However, Dummett’s arguments for the dependency of senses upon expressions are unsuccessful and they show that Dummett’s conception of what it is to be an expression also differs significantly from Frege’s.     

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.     

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.     

Thought's Social Nature

Abstract: Wittgenstein, throughout his career, was deeply Fregean. Frege thought of thought as essentially social, in this sense: whatever I can think is what others could think, deny, debate, investigate. Such, for him, was one central part of judgement's objectivity. Another was that truths are discovered, not invented: what is true is so, whether recognised as such or not. (Later) Wittgenstein developed Frege's idea of thought as social compatibly with that second part. In this he exploits some further Fregean ideas: of a certain generality intrinsic to a thought; of lack of that generality in that which a thought represents as instancing some such generality. (I refer to this below as the ‘conceptual‐nonconceptual’ distinction.) Seeing Wittgenstein as thus building on Frege helps clarify (inter alia) his worries, in the Blue Book, and the Investigations, about meaning, intending, and understanding, and the point of the rule following discussion.     

Frege on Syntax,Ontology, and Truth's Pride of Place

Frege's strict alignment between his syntactic and ontological categories is not, as is commonly assumed, some kind of a philosophical thesis. There is no thesis that proper names refer only to objects, say, or that what refers to an object is a proper name. Rather, the alignment of categories is internal to Frege's conception of what syntax and ontology are. To understand this, we need to recognise the pride of place Frege assigns within his theorising to the notion of truth. For both language and the world, the Fregean categories are logical categories, categories, that is, of truth. The elaboration of this point makes clear the incoherence of supposing that they might not align.     

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.     

Grundlagen,Section 64: Frege's Discussion of Definitions by Abstraction in Historical Context

I offer in this paper a contextual analysis of Frege's Grundlagen, section 64. It is surprising that with so much ink spilled on that section, the sources of Frege's discussion of definitions by abstraction have remained elusive. I hope to have filled this gap by providing textual evidence coming from, among other sources, Grassmann, Schlömilch, and the tradition of textbooks in geometry for secondary schools (including a textbook Frege had used when teaching in a Privatschule in Jena in 1882–1884). In addition, I put Frege's considerations in the context of a widespread debate in Germany on ‘directions’ as a central notion in the theory of parallels.     

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.     

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.     

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.     

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.