Frege's Conception of Logic

Frege's Conception of Logic

Frege's Conception of Logic

Frege's Conception of Logic

Synopsis

In Frege's Conception of Logic Patricia A. Blanchette explores the relationship between Gottlob Frege's understanding of conceptual analysis and his understanding of logic. She argues that the fruitfulness of Frege's conception of logic, and the illuminating differences between that conception and those more modern views that have largely supplanted it, are best understood against the backdrop of a clear account of the role of conceptual analysis in logical investigation.

The first part of the book locates the role of conceptual analysis in Frege's logicist project. Blanchette argues that despite a number of difficulties, Frege's use of analysis in the service of logicism is a powerful and coherent tool. As a result of coming to grips with his use of that tool, we can see that there is, despite appearances, no conflict between Frege's intention to demonstrate the grounds of ordinary arithmetic and the fact that the numerals of his derived sentences fail to co-refer with ordinary numerals.

In the second part of the book, Blanchette explores the resulting conception of logic itself, and some of the straightforward ways in which Frege's conception differs from its now-familiar descendants. In particular, Blanchette argues that consistency, as Frege understands it, differs significantly from the kind of consistency demonstrable via the construction of models. To appreciate this difference is to appreciate the extent to which Frege was right in his debate with Hilbert over consistency- and independence-proofs in geometry. For similar reasons, modern results such as the completeness of formal systems and the categoricity of theories do not have for Frege the same importance they are commonly taken to have by his post-Tarskian descendants. These differences, together with the coherence of Frege's position, provide reason for caution with respect to the appeal to formal systems and their properties in the treatment of fundamental logical properties and relations.

Excerpt

The purpose of this book is to provide a clear account of Frege’s conception of logic, with a particular emphasis on those aspects of his conception that diverge from mainstream views of logic today. the reason to pursue the differences that separate modern views from Frege’s is in part simply scholarly: we can’t get Frege right if we take into account only those of his views that have survived intact. But more importantly, the goal is to take Frege’s views seriously on their own terms in an effort to determine to what extent he was right.

Frege’s view of logic is closely connected with his conception of, and his commitment to, a certain sort of theoretical reduction. His guiding intellectual project was the attempt to establish his logicist thesis, the thesis that arithmetic is reducible to logic. the reduction Frege pursued was of a simple kind, at least in outline: the thesis is that each arithmetical truth is provable from purely logical premises. This immediately puts logic at center stage in two ways: one needs an identifiable collection of logical truths to stand as premises, and a reliable means of employing logic in the course of proof.

There is little that is controversial about the general idea that to prove the claims of one theory using just the resources of another is to effect an important reduction, one that shows that the reduced science is grounded, in an explanatorily useful way, in the reducing one. As Frege puts it, the effect of such a reduction of arithmetic to logic would be to show that arithmetic is “simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one.” But considerably less straightforward is Frege’s idea that his way of proceeding would in fact have constituted a demonstration that arithmetical truths are provable from logical ones. Leaving aside the infamous difficulty about his purportedly logical truths, i.e. that some of them are now known not to be logical and arguably not even to be truths, a further difficulty surrounds the purportedly arithmetical ones. the truths Frege actually proves are not easily-recognizable claims about e.g. the commutativity of addition or about the number 7. They are instead truths about complex and apparently-unfamiliar objects, functions, and relations. the immediate question to press on Frege is that of how the provability . . .

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