Field addresses many of the points raised in my previous post in his essay by this title (reprinted in “Realism, Mathematics, and Modality”) and I realize that I should have had an idea of what he would do anyway, having read the introduction to that book. He believes in a primitive modal notion of logical consequence that may in fact be quite powerful, to which both the syntactic and semantic consequence relations described by proof theorists and model theorists are approximations. The fact that one is an upper bound and the other is a lower bound, and that in some languages they coincide, is evidence that these approximations are exact for those languages. However, these results (and even the theories themselves) only really make sense from a platonist perspective, because they discuss abstract entities like deductions and models.
On Conservativeness and Incompleteness8 03 2005