Brandom’s inferentialist alternative to representationalism stresses material, meaning-oriented over formal, syntactic inference. Prior to the development of mathematical logic, philosophers typically used a mixture of reasoning about meanings with natural language analogues of simple formal reasoning. People in ordinary life still do this.
Where Brandom’s approach is distinctive is in its unprecedentedly thorough commitment to the reciprocal determination of meaning and inference. We don’t just do inference based on meanings grasped ready-to-hand as well as syntactic cues to argument structure, but simultaneously question and explicitate those very meanings, by bracketing what is ready-to-hand, and instead working out recursive material-inferential expansions of what would really be meant by application of the inferential proprieties in question.
For Brandom, the question of which logic to use in this explicitation does not really arise, because the astounding multiplication of logics — each with different expressive resources — is all in the formal domain. It is nonetheless important to note that formal logics vary profoundly in the degrees of support they offer for broad representationalist or inferentialist commitments.
Michael Dummet in The Logical Basis of Metaphysics argued strongly for the importance of constructive varieties of formal logic for philosophy. Constructive logics are inherently inference-centered, because construction basically just is a form of inference. (Dummet is concerned to reject varieties of realism that I would call naive, but seems to believe the taxonomy of realisms is exhausted at this point. This leads him to advocate a form of anti-realism. His book is part of a rather polarized debate in recent decades about realism and anti-realism. I see significant overlap between non-naive realisms and nonsubjective idealisms, so I would want to weaken his strong anti-realist conclusions, and I think Brandom helps us to do that.)
Without endorsing Dummet’s anti-realism in its strong form, I appreciate his argument for the philosophical preferability of constructive over classical logic. It seems to me that one cannot use modern “classical” formal logic without substantial representationalist assumptions, and a lot of assumed truth as well. If and when we do move into a formal domain, this becomes important.
As used in today’s computer science, constructive logic looks in some ways extremely different in its philosophical implications from Brouwer’s original presentation. Brouwer clouded the matter by mixing good mathematics with philosophical positions on intuition and subjectivity that were both questionable and not nearly as intrinsic to the mathematics as he seemed to believe. The formal parts of his argument now have a much wider audience and much greater interest than his philosophizing.
Constructive logic puts proof or evidence before truth, and eschews appeals to self-evidence. Expressive genealogy puts the material-inferential explicitation of meaning before truth, and eschews appeals to self-evidence. Both strongly emphasize justification, but one is concerned with proof, the other with well-founded interpretation. Each has its place, and they fit well together.