I’ve recently read both Dummett’s article “The Justification of Deduction” and Field’s article “Is Mathematical Knowledge Just Logical Knowledge?” and both seem to have a similar attitude towards the notion of logical consequence. In fact, the more I read of both Field and Dummett, the more similar they start to seem on a lot of issues (though they definitely have a very different way of talking about things). Basically, both of them argue that logical consequence is a notion that is prior to both the syntactic and semantic definitions that are usually given, and the reasons aren’t actually all that different. Field needs this to explain how we can do without the concepts that talk about mathematical objects (like models or formal proofs) and Dummett wants to show how it is that this practice is both justified and contentful and suggests that we can’t really be explaining either the syntactic or semantic notion in terms of the other. I think I should check out the Etchemendy book on this subject.
Logical Consequence11 03 2005