Review of Frege Making Sense, by Michael Beaney. London, U.K.: Duckworth, 1996. Pp. ix+358.
(Forthcoming in Mind, 1999)
Purporting to show how Frege's contributions to philosophy of language and philosophical logic were developed with the aim of furthering his logicist programme, the author construes him as more systematic than is often recognized. Centrally, the notion of sense as espoused in Frege's monumental articles of the Nineties had only an ostensible justification as an account of the informativeness of a posteriori identity statements. In fact its rationale was to help articulate the thesis that arithmetical truth is analytic, since, it is maintained, to sustain such a thesis the two sides of the identities at the heart of the logicist reconstruction must be shown to have the same sense. Yet the notion of sense required for the analyticity thesis was not, and could not have been, successfully deployed on behalf of Frege's logicism. For Frege also held that many arithmetical propositions, including, apparently, identities, are informative. But no proposition can be at once informative and analytic. Although systematic, Frege's work harbored a crucial internal tension.
Before the Nineties Frege would justify formalizations such as those that occurred in his concept script by invoking his notion of conceptual content (begrifflicher Inhalt), according to which two propositions have the same conceptual content iff they are provably logically equivalent. A symbolic representation of a sentence superficially of subject-predicate form in terms of function and argument is unexceptionable because our only aim in formalization is to preserve what is relevant to a sentence's inferential significance. (The practice would be objectionable if the sentence had been produced by a poet whose work we are translating into a different language.) Because P and Q can be provably logically equivalent without P iff Q being an analytic truth, however, conceptual content is too coarse-grained a notion to aid in formulation of the analyticity of arithmetic. For Frege a proposition is analytic if it is derivable from logical laws and definitions alone. Assuming that nothing is derivable from itself, Beaney takes this sufficient condition for analyticity to leave untouched those very logical laws and definitions. Taking it as well that these laws and definitions must be shown analytic if arithmetic is to be, Beaney sees the notion of sense as fashioned to settle the issue. According to the author's first and central gloss of the notion, two propositions have the same sense iff it is not possible to recognize the truth of one without recognizing the truth of the other (pp. 8-9; pp. 141-2; the generalization to the subsentential case is routine). This conception allows Frege to hold that P iff Q is analytic just in case P and Q have the same sense. But it collides with what Beaney terms Frege's Grundgedanke, which leads him to identify abstract objects with extensions of concepts, and thus for instance to maintain that
(*) the direction of a line is identical with the extension of the concept, line parallel to line a.
Beaney holds (p. 139) that one could have the concept of parallelism without that of direction, and infers that the displayed identification cannot involve two concepts having the same sense. At most they can have the same conceptual content. This is no cause for alarm, since Frege does not espouse the analyticity of geometry. However, the Grundgedanke also leads Frege to hold that
(**) The number 0 is the extension of the concept equinumerous to the concept #not identical with itself#.
Taking it that one could have the concept of number without having the concept of equinumerosity, Beaney infers that this identity is likewise not one between contents having the same sense. (**) cannot therefore be construed as analytic, and because it is unobvious identifications such as these that are crucial to the logicist enterprise Frege's aim in introducing the notion of sense cannot be achieved.
One is tempted to offer another criterion of sense identity on which P and Q express the same sense iff anyone who understands P and Q and who recognizes the truth of P can immediately recognize the truth of Q, and vice versa. Beaney terms this criterion SCE, and appears to hold that its generalization to the subsentential case counts (**) as analytic; yet it leaves room for the informativeness of arithmetic because for pairs of complex arithmetical propositions (e.g., 28 is an even perfect number; 28 is a Euclid number) a person could grasp both without being able immediately to recognize the truth of one on the basis of their recognition of the truth of the other. (It might take some calculation adverting to the definition's of perfect number and Euclid number.) SCE is rejected as useless, however, on the ground that if P and Q do express the same sense one cannot understand them without seeing that they do. Furthermore, the account must be rejected as circular, since we cannot but construe understanding P and Q as grasping their content. (p. 233)
The first five chapters of Beaney's book motivate the introduction of sense from the point of view of logicism and then explain how the notion cannot do the work for which it was fashioned. These chapters also contain an instructive comparison between Aristotelian and Fregean logic, an overview of some main themes in the history of mathematics in centuries previous to the Nineteenth, and an exposition of Frege's demand that all definitions be "fruitful." Chapter Six treats of some main problems in the theory of sense, focusing on empty names and application of the sense/reference distinction to sentences. Seven involves related topics in Frege's philosophical logic, particularly the quandary involving the concept horse, Russell's Paradox, and demonstratives and indexicality. (Four Appendices contain accounts of syllogistic theory, Leibniz's view of analysis, Frege's logical notation, and a chronology of Frege's life and works.)
The final chapter returns to what the author describes as the main theme of the book, namely that in developing logical systems or in providing philosophical analyses Frege is reformulating senses rather than excogitating contents that are there to be grasped. Logical reconstruction, when successful, issues in the refinement of the concepts we bring to the table, enabling us to develop those concepts (such as those of function, series, some, or extension) into conceptions. The definitions and axioms central to the logicist enterprise issue in theorems that can be informative, something they could not be if all such theorems were ones whose proof depended only on general logical laws and definitions that preserved sense. Instead those definitions need only preserve conceptual content so long as it is recognized that, rather than excogitating a person- or community-independent set of arithmetic contents, the logicist programme crystallizes the senses of arithmetical propositions. Such a crystallization must preserve all the inferential prowess of any proposition of which it treats, but in not being beholden to the precise content of those propositions may effect conceptual innovation. (pp. 149-50)
The author's aim to discern organic unity in Frege is laudable, and he has succeeded in providing a wide and illuminating perspective on this philosopher. This reviewer is, however, inclined to suspect the contention that the notion of sense must be impaled either on the horn of arithmetic's analyticity or of its informativeness, and this for two reasons. First, the suggested reformulation of the criterion for sense identity as SCE has surely been rejected too swiftly. Understanding need not be construed as having a proposition limpidly before the mind; if the issue is the utility of a notion of sense identity employing that of understanding we may discern evidence for an agent's grasp of a proposition in her ability to use a sentence expressing it as a premise in reasoning. One's competence with two distinct sentences need not imply awareness that they express the same proposition if they do, so it can in a given case be open whether one who understands two distinct sentences knows that they express ththought. Beaney's failure to address such a reply to his objection to SCE is particularly surprising given the familiarity of this conception of sense identity from, for instance, Evans' treatment of the Intuitive Criterion of Difference.
Second, the issue of the analyticity of the logical laws and definitions used in the logicist reconstruction simply need not arise, or may be settled by the stipulation that every proposition is derivable from itself. (For the case of definitions the derivation would consist in the removal of a vertical line in the movement from -A to |-A.) In either case, the fact that identifications such as (**) might not appear to be analytic truths in the traditional sense need not vitiate the logicism. For they can be informative while being logically unexceptionable, and in the latter case they may be used to show that arithmetic rests only upon logic and definitions. Although, as Beaney recognizes, Frege inherited much of Kant's intellectual repertoire, he departed from the Kantian conception of analyticity in a more dramatic way than perhaps has been appreciated. (For further discussion see J. Horty, "Frege on the Psychological Significance of Definitions," Philosophical Studies 72 (1993): 223-63.) Frege's own account of analyticity need not be seen as an inadequate attempt to reconstruct his predecessor's notion. Rather it is a co-opting for his own purposes of established terminology with little attempt to respect its traditional meaning. So Frege's conception of the analyticity of arithmetic may be taken quite literally, and in spite of the counterintuitive identifications flowing from the Grundgedanke, had the logicism worked out he would have had no difficulty concluding that arithmetical facts are reason's closest kin.
Frege's notion of sense is problematic no matter how construed, but if this reviewer's last objection is correct then it need not be further burdened with the task of supporting the analyticity of the identifications central to the logicist enterprise. (If the former objection is correct then even if it is so invoked then the results are not obviously disastrous.) These qualms leave untouched the suggestion that the notion of sense was a handmaiden to logicism, although the notion must now be understood as helping to answer the question, How are informative arithmetical identity statements possible? Such a view of the notion of sense may lead to further developments in its interpretation, and as a result Beaney's is a welcome contribution to Frege scholarship. It is also accessible to advanced undergraduates and beginning graduate students, and particularly because of its comprehensive treatment would make a good secondary source for courses on Frege or on the development of analytic philosophy.