- Frege’s Conception of Logic by Patricia A. Blanchette
As is well known, Frege does not have a model-theoretic conception of logic and language; sentences of his logical language Begriffsschrift are always already fully meaningful. Frege also famously rejects Hilbert’s strategy for proving the independence and consistency of axioms. And he seems to have no particular interest in metatheory. Indeed, some have argued, Frege’s conception of logic precludes even the possibility of any metatheory. Given how well entrenched model theory, Hilbert-style proofs of independence and consistency, and metatheory are in mainstream logic today, it is worth asking why Frege’s conception of logic diverges, at least along these dimensions, so radically from our own. Blanchette’s thoughtful and interesting book seeks to answer the question by reflecting on the logicism Frege’s logic was to serve and in particular by close consideration of the nature and role of analysis in Frege’s logicist program.
On Kant’s account of it, the practice of mathematics has no need to analyze its concepts because, so he thought, they are always already clear. Developments in mathematical practice in the nineteenth century proved Kant wrong: the analysis of concepts is central to mathematics, at least as it was coming to be practiced. Instead of solving problems in the symbolic language of arithmetic and algebra using the sorts of constructive algebraic techniques that were first introduced by Descartes and then perfected over the course of the eighteenth century, this new form of mathematical practice was to proceed by deductive reasoning from concepts. And for this one needed to be much clearer than mathematicians had been hitherto about the contents of those concepts. Consider, for example, the notion of a rational number (which for the ancient Greeks is not even a number but instead a ratio of numbers). A rational number is understood in early modern mathematics to be a kind of expression (either a fraction or a repeating decimal). Such a conception is obviously of no use in inference, and Frege understands the notion of a rational number very differently: to be a (positive non-zero) rational number is to be such that there is at least one (positive non-zero) whole number that is a multiple of that number. And similarly for other arithmetical concepts, all are to be stripped of irrelevant (sensory) content, and ultimately shown to be purely logical. And truths depending on them similarly are to be shown to be grounded in logic alone. But again, as Blanchette emphasizes, this form of analysis does not thereby strip away all content and meaning. Frege’s logic is concerned with thoughts and their contentful concepts, not sentences syntactically characterized.
As the example of the concept of a rational number illustrates, analysis can seem quite radically to change our conception of that which is analyzed. A question naturally arises, [End Page 176] then, regarding the relationship between analysanda and analysantia, in particular, whether in Frege’s work “logical entailment … survives conceptual analysis” (4)—as it needs to for Frege’s logicism to succeed. Blanchette’s discussion of the question is thoughtful and quite rich. Blanchette provides as well a helpful discussion of the role the demand for sharp boundaries plays in Frege’s logic, an insightful and cogent account of Frege’s objections to Hilbert’s method of proving consistency and independence, and a sensible and nuanced discussion of the possibility and role of metatheory on Frege’s conception of logic.
But there are also limitations to Blanchette’s discussion insofar as she takes for granted many controversial claims about Frege’s conception of logic. She uncritically assumes, for example, that “Frege’s formal languages are ‘cleaned-up’ versions of natural languages” (49), essentially no different from our formal languages. That Frege’s language was modeled on the symbolic language of arithmetic and algebra, not on natural language, is ignored. She also uncritically assumes that Frege’s logicist project is reductive, although given the historical and mathematical context it is more plausible (as already indicated...