How Open Can Radically Open Reason Relations Be?

Reason (or material) relations of implication and incompatibility hold in virtue of the meanings of linguistic expressions in a given language. As opposed to formal or logical consequence relations, reason relations are often structurally open, i.e., unclosed under principles such as monotonicity or transitivity. In their recent book, Hlobil and Brandom go even further, allowing for such relations to be radically open, i.e., unclosed under any structural principle. In particular, they reject the restrictive forms of monotonicity and transitivity known as cautious monotonicity (CM) and cumulative transitivity (CT). In the present paper, I argue that such radical openness is objectionable on the grounds that it does not allow us to construct for every set even a weak version of Tarski’s notion of a theory, thereby undermining the very notion of consequence. I then lay out minimal principles under which such a construction is possible, and suggest that reason relations should be closed under those principles.