The Bertrand Russell Society Quarterly - November 2005/February 2006

a copious harvest: frege and carnap in jena

James Connelly

Review of Erich H. Reck, Steve Awodey, Gottfried Gabriel, and Gottlob Frege, Frege’s Lectures on Logic: Carnap’s Student Notes, 1910-1914, Open Court Publishing, 2004.

Prior to his emergence as one of the most significant figures in analytic philosophy, Rudolf Carnap attended several courses offered between 1910-1914 by an aging Gottlob Frege at the University of Jena, where the latter had been a professor of logic and mathematics since 1874. The recent publication of Carnap’s notes from these courses, as Frege’s Lectures on Logic: Carnap’s Student Notes, 1910-1914, is a significant event in Frege scholarship in particular and the history of analytic philosophy in general. In addition to being of intrinsic interest as a documented philosophical interaction between these two seminal thinkers, the notes also provide extensive insight into the evolution of Frege’s logical system and the content of his teaching following Russell’s 1902 discovery and communication of his eponymous paradox to Frege.

As the editors of Carnap’s Notes point out, the volume sheds valuable light on those aspects of Frege’s thought that he felt could be retained despite the failure of his logicist project, thus indicating what he saw as the harvest of his life’s work (p. 4). The volume also illuminates an important source of Frege’s influence within the analytic tradition, namely, Carnap’s absorption and subsequent dispensation of Frege’s ideas as he encountered them in these lectures.

I In addition to the transcriptions of Carnap’s notes from three separate lecture courses, Begriffsschrift I, Winter 1910-11, Begriffsschrift II, Summer 1913, and Logic in Mathematics, Summer 1914, the volume contains two appendices meant to be part of either Begriffsschrift I or II (it is not clear in which lecture they belong). The transcriptions are accompanied by two introductory essays, which provide key historical, biographical, logical and philosophical background.

The first introductory essay is written by Gottfried Gabriel, the editor of the original German version of these lecture notes.¹ Gabriel compares the exposition found in the notes with those occurring in Frege’s Begriffsschrift (1879) and Grundgesetze der Arithmetik (The Basic Laws of Arithmetic), Vols. I (1893) and II (1903). He finds that while the exposition is by and large congruous with that offered in Grundgesetze – right down, for instance, to the employment of identical code numbers for the relevant laws and theorems – there are some important differences.

As in the Basic Laws, Frege employs additional rules of inference beyond the Begriffsschrift’s single modus ponens, and there is a corresponding reduction in the number of the Begriffsschrift’s basic laws. In fact, the number of basic laws is reduced even further in the notes than in the Grundgesetze; here basic laws IV, V and VI are all eliminated. While there is “no obvious reason” (p. 3) to dispense with basic law IV, Gabriel notes that the elimination of basic laws V and VI corresponds to Frege’s eliminating value ranges and the description operator, reflecting his retreat from the more constructionist ambitions of his logicism following Russell’s identification of the contradiction inherent in it.

Further evidence of Frege’s retreat from the logic of the Grundgesetze can be found in his analysis of the notion of ordering in a series, which dispenses with value ranges. Other such instances include his use of the term ‘content-stroke’, as he had in the Begriffsschrift, in lieu of his later phrase, ‘the horizontal’. Despite this change in terminology, however, “in substance … the conception of the Basic Laws dominates”, the content-stroke being characterized as “a special function of first-level, whose value for the argument ‘the true’ is the true and for all other arguments is the false” (p. 4).

Other highlights of Gabriel’s essay include a discussion of how the notes support but also belie Carnap’s later and somewhat controversial insistence that Frege defended the viability of logicism in these lectures. Gabriel notes that though Frege had “quietly drawn the consequences” of Russell’s paradox by eliminating value-ranges, he is nevertheless silent about the antinomy, a fact which may have “led Carnap to the premature conclusion that it presented no problem for him” (p. 7). His “casting doubt” at the outset of the third lecture course “on the representability of mathematical induction … in purely logical terms” (p. 6) confirms that though Frege treated the methods of proof in mathematics and geometry as logical, he did not in these lectures defend the stronger thesis that arithmetic (or geometry) is reducible to logic. Despite his evident abandonment of logicism, Frege nevertheless seems to have continued to conceive of numbers as non-logical objects and “attributions of number as statements about concepts” (p. 7).

The second introductory essay, by Eric H. Reck and Steve Awodey, explicates key ideas and notation prominent in the logical system developed in the notes and provides sketches of Frege as a person and lecturer by people who came into contact with him while Carnap studied at Jena. Carnap’s own reflections are included as well as Wittgenstein’s. Frege appears as a somewhat frail and unapproachable older gentleman, possessed of an unquestionable charisma, perhaps as a result of the keen intellect and immense passion for logical and scientific work he continued to display despite his advancing years.

II The first lecture course, Begriffsschrift I, resembles the sort of introduction to Frege’s key logical and semantic contributions one might get in any contemporary North American philosophy department. Frege begins by explaining such rudimentary elements of his notation as the content-, judgment-, conditional- and negation-strokes, showing how these operate as functions from the truth values of the component sentences they take as arguments to the truth values of the compound statements formed from them, and how other truth-functions, such as conjunction and disjunction, may be built up in turn out of these more primitive ones.

Frege presents several key rules of inference, the most basic of which is transportation (or contraposition (p. 160)), in which an upper term negated takes the place of a lower term, and the lower term negated takes the place of the upper term. Other more intricate forms of inference, like ‘cut’ and ‘negation’, are also introduced (pp. 33, 60-63). Frege then analyzes rules of inference involving generality, culminating in the classical square of opposition presented in his own function-theoretic and quantificational notation.

Frege notes that the propositions displayed in the square of opposition are identified only for the purposes of showing the connection between his own system and that of traditional logic and that the distinction between subject and predicate, characteristic of the traditional Aristotelian analysis of these forms of judgment, “does violence to the nature of things” (p. 71). The course concludes with a discussion of such semantic distinctions as that between meaning and sense as it applies in the cases of proper names, sentences, concepts, and indirect discourse, as well as such logical distinctions as that between first and second order functions. Interestingly, Frege insists on treating concept words as names of certain sorts of quasi-objects, i.e., concepts (p. 74), despite continuing to adhere to a rigorous distinction between concepts and objects and despite eliminating concept-extensions.

Following Begriffsschrift I are two appendices. In the first, Frege analyzes the ontological proof of the existence of God, noting that existence is a ‘feature’ (Beschaffenheit) rather than a ‘characteristic’ (Merkmal) of a concept; in the second he analyzes statements of number as statements about concepts. The appendices are followed by Begriffsschrift II. It begins by recapitulating some of the basic logical and semantic notions covered in Begriffsschrift I, building on these notions to present a more systemic and advanced treatment of formal deduction.

Frege first shows how his notation can be used to define two key mathematical notions, namely, the continuity of an analytic function at a particular point and the limit of a function for positive arguments increasing towards infinity (pp. 88, 91). Following a four page gap in the notes, which the editors conjecture is where Frege introduced Axioms I and II, he then introduces Axiom III, using it to derive such properties of identity as Leibniz’s law, reflexivity, and symmetry (pp. 37, 93-97). This is followed by two proofs, the first that two numbers are equal if each is greater than the other when increased by an arbitrarily small amount and the second that limits are unique. These examples are provided, Frege says, for the purpose of showing “how one can conduct proofs with our notation” (p. 98).

Frege rounds out Begriffsschrift II by stressing the importance of rigour in mathematical proof, along with relevant distinctions between the psychological and the logical, functions and their values, real and apparent variables, as well as signs and what those signs signify. He considers several examples from differential and integral calculus, employing them to show that failure to maintain the requisite philosophical distinctions leads to the result that “one contradicts oneself continually” (p. 133). He then concludes by recommending the various questions considered to the student “for further reflection” (ibid.).

Logic in Mathematics, the third lecture course, picks up where Begriffsschrift II leaves off, that is, in a more philosophical vein than the earlier material, which consists, by and large, of a technical, if rudimentary, exposition of Frege’s logical system. Frege opens the course asking: “Are the inferences in mathematics purely logical? Or are there specifically mathematical inferences that are not governed by general laws of logic?” (p. 135) He then examines a proof of the proposition ‘(a + b) + n = a + (b + n)’ via mathematical induction, which he identifies as an inference of the later, specifically mathematical sort (ibid.). After a discussion of this proof, Frege concludes that “every mathematical inference is analyzed into a general mathematical theorem or axiom and a purely logical inference” (p. 134), thus rendering questionable Carnap’s claim that at the time of these lectures Frege adhered to the logicist program.

Frege goes on to detail the role played by logical inference within the sort foundational project which he does intend to endorse, which involves supplementing purely logical laws with “axioms, postulates, and perhaps definitions” (p. 138). These, he maintains, should be limited to as few a number as possible in the interest of discovering “that kernel out of which all of mathematics can be developed” (p. 137).

Following cursory remarks on postulates and axioms, Frege shifts to a detailed discussion of definitions, which he characterizes as “stipulations that a group of signs can be replaced by simple signs” (p.139), and which he argues are “logically superfluous, but psychologically valuable” (p.140). The discussion leads him to consider some contemporary views of definition and to a critical discussion of various putative definitions of the concept of number reminiscent of that undertaken in the Foundations of Arithmetic. In particular, Weirstrass’s definition that “a number is a group of similar things ... (and) a numerical magnitude results from the repeated positing of similar elements,” comes up for consideration, leading Frege to remark that “(a)ccording to Weierstrass a railroad train would be a number…(which) now comes racing along from Berlin” (pp. 139-144).

In the remainder of the course, Frege develops some familiar themes in an extended discussion of distinctions between the psychological and the logical, the sense and meaning of proper names and sentences, concepts and objects, as well as first and second order functions. Frege also repeats ideas developed in Begriffsschrift II on the importance of distinguishing between a function and its value, particularly when one seeks to identify a complex function as comprised of two simpler component functions, e.g., ‘(1 + 2x)²’ from ‘(1+ 2x)’ and ‘ζ²’ (p. 154).

Some important ideas introduced here include Frege’s insistence on the importance of clear and sharp boundaries for concepts, and on the philosophically essential role played by elucidations: “what a function is cannot be defined, it cannot be reduced logically to something more simple; one can only hint at it, elucidate it” (p. 152). The course concludes with reflections on the distinction between direct and indirect proof, with Frege giving examples from geometry which show that false propositions can be employed in constructing sound proofs, provided those propositions are never asserted but are rather explicitly taken throughout the proof as antecedents of conditional statements.

III I have tried to give a sense of the quality and content of the volume by tracing a path through it, highlighting some of the elements which seem to me most interesting and relevant. Specialists in Frege’s logical and mathematical work are likely to discover much of value in the volume which has not been touched on here at all, or else very briefly – for example, Frege’s discussion of indirect proof vis-à-vis non-Euclidean geometry in the latter portions of the course on Logic and Mathematics. By contrast, specialists in philosophy of language are likely to be intrigued by the various discussions of key themes in Fregean semantics developed throughout the volume, which are worth examining both in connection with their reception by Carnap and in light of developments in Frege’s system after the failure of his logicist program. For example, Carnap’s claim in Meaning and Necessity that Frege held concept-extensions to be meanings of concepts is contradicted in the notes and examined by the editors. Even a non-specialist will benefit from the editors’ and Frege’s own presentation of key logical and semantic innovations and from the wealth of historical and biographical information concerning both Carnap and Frege. The volume is a first rate piece of scholarship which I recommend to anyone working on or interested in Frege in particular or the history of analytic philosophy in general regardless of their specific level of expertise.

NOTES