Oran Magal

I argue that Quine’s early critique of Carnap’s conventionalism is in serious tension with the holism of "Two Dogmas of Empiricism", since his critique of convention- alism makes a compelling case for a privileged status either for logic, or for some other principle by means of which to derive consequences. Based on this, I call for a modification of Quinean holism, on grounds internal to Quine’s views. The result motivates a rehabilitation of Carnap’s notion of framework principles, and a rethinking of logic as a kind of relativised a priori.

The paper examines the interrelationship between mathematics and logic, arguing that a central characteristic of each has an essential role within the other. The first part is a reconstruction of and elaboration on Paul Bernays’ argument, that mathematics and logic are based on different directions of abstraction from content, and that mathematics, at its core it is a study of formal structures. The notion of a study of structure is clarified by the examples of Hilbert’s work on the axiomatization of geometry and Hilbert et al.’s formalist proof theory. It is further argued that the structural aspect of logic puts it under the purview of the mathematical, analogously to how the deductive nature of mathematics puts it under the purview of logic. This is then linked, in the second part, to certain aspects of Gödel’s critique of Carnap’s conventionalism, that ‘mere syntax’ cannot capture the full content of mathematics, which is revealed to be closely related to the characteristic of mathematics argued for by Bernays. Finally, this is connected with Gödel’s latter-day views about two kinds of formality, intensional and extensional, and the relationship between them.