Going back at least to Making It Explicit (1994), Brandom has championed a nonstandard approach to semantics, in accordance with his thesis that inference is prior to representation in the order of explanation.
More conventional approaches to semantics standardly invoke model theory, which employs universal algebra and mathematical relations to study mathematical structures that can be regarded as interpretations of the formal theories studied by proof theory. This is already interesting, insofar as it invokes relations, structures, universal algebra, proof theory, and a (formal) kind of interpretation; but to begin with, this approach is natively designed to work with formal theories, and there are nontrivial issues with its application to natural language. Then also, its principal concern is with formal models as representations.
Brandom is more interested in the ethical and epistemic uses of natural language than in formal theories. Haskell Curry already spoke in the 1950s of natural language as the “use” language implicitly presupposed as a metalanguage in all formal developments. (Natural language is said to contain an indefinite regress of metalanguages.)
Brandom adopts Wittgenstein’s dictum that meaning is use. He understands this as referring principally not to instances of use but to norms governing proprieties of use, and the norms in question not as descriptions of empirical fact about what is said to be right, but about what actually ought to be, understood as a genuine question that ought to have an objective answer, even if the objectivity of any particular putative answer can never be assumed, but has to be shown through reasoning.
The intuition behind inferential semantics is that meaning in real life is primarily constituted through processes of valid implicit informal reasoning in natural language that can be made explicit in terms of material inference and material incompatibility. Unlike the model-theoretic approach, this means it’s semantics all the way down. We have to find our way to what is honestly good usage in terms of immanent criteria in an emergent process, without the convenience of a postulated syntax as an artificially tidy starting point. After all, real life is much messier than that. (See also Conceptual, Representational; Aristotelian Dialectic; Syntax, Semantics, Ethics.)
Any given explicitation of meaning through material inference and material incompatibility can always be converted to an appropriately defined static structure, which can then be nominalized for convenient reference. In contradistinction to the structure involved in attempts to apply model-theoretic semantics to natural language, however, this sort is implicitly constructed in the process of developing our explicitation, not purportedly found in empirical data.
Constructive formal logics seem to me to also naturally push semantics in an inferential direction, but this is not the path Brandom takes. Apart from some discussion of paraconsistency related to belief revision, he prefers to remain noncommittal on preferred formalisms. (See also Categorical Evil.)
Brandom actually puts normative pragmatics before any semantics, and sees the two as very closely related. He sees representational truth as derivative from rational justification.
In passing, I would note that contemporary structural operational semantics used in programming language research also seems to have a broadly inferential flavor, but is concerned with formal rather than material inference.
(I have previously used the term “semantics” in a very general way, barely distinguishable from the way I have used “hermeneutics”. When I speak, e.g., of Aristotelian semantics, all I mean is a serious and extended inquiry into meaning. When I speak of “hermeneutics”, I just mean some kind of methodologically self-aware approach to interpretation.)