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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. G ottlob Frege's work in logic and the foundations of mathematics centers on claims of logical entailment; most important among these is the claim that arithmetical truths are entailed by purely logical principles. Occupying a less central but nonetheless important role in Frege's work are claims about failures of entailment. Here, the clearest examples are his theses that the truths of geometry are not entailed by the truths of logic or of arithmetic, and that some of them are not entailed by each other. As he, and we, would put it: the truths of Eluclidean geometry are independent of the truths of logic, and some of them are independent of one another.'Frege's talk of independence and related notions sounds familiar to a modern ear: a proposition is independent of a collection of propositions just in case it is not a consequence of that collection, and a proposition or collection of propositions is consistent just in case no contradiction is a consequence of it. But some of Frege's views and procedures are decidedly tinmodern. Despite developing an extremely sophisticated apparattus for demonstrating that one claim is a consequience of others, Frege offers not a single demonstration that one claim is not a conseqtuence of others. Thus, in partictular, he gives no proofs of independence or of consistency. This is no accident. Despite his firm commitment to the independence and * Versions of this paper were deliveied at the University of Notre Dame, the Univeisity of Califoinia/River-side, and the 1995 meeting of the Society for Exact PhiJosophy. Thanks to memiiber-s of eachi atidience for lhelpful commillents, and especially to Mic Detlefsen and Aldo Antonelli for comimients and criticism. Thanks also to the Center for the Study of Langtiage and Infor-mation at Stanfor-d University for its hiospitality. See, for example, Die Grundlagen der Aritlmetik, tr anslated as Tuae Foundations of 318 THE JOURNAL OF PHILOSOPHY consistency claims just mentioned, Frege holds that independence and consistency cannot systematically be demonstrated.2Frege's view here is particularly striking in light of the fact that his contemporaries had a fruitful and systematic method for proving consistency and independence, a method which was well known to him. One of the clearest applications of this method in Frege's day came in David Hilbert's 1899 Foundations of Geometry,3 in which he establishes via essentially our own modern method the consistency and independence of various axioms and axiom systems for Euclidean geometry. Frege's reaction to Hilbert's work was that it was simply a failure: that its central methods were incapable of demonstrating consistency and independence, and that its usefulness i...

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In 1906, Frege proposes a method, one which rests on the "formal" nature of logical laws, for proving mathematical independence claims. There are many curious features of the 1906 proposal, including the fact that Frege seems subsequently to have found it unacceptable. This essay explores Frege's proposal, and rejection, of the 1906 independence-test, with the goal of clarifying Frege's understanding of the nature of logical entailment and of the "formal" nature of logical laws.