Abstract: In this talk, I examine interconnections between norms governing assertion, denial, questions and answers, and the common ground of a discourse. When we pay attention to the structure of norms governing polar (yes/no) questions, we can clarify the distinction between strong and weak denials, together with the parallel distinction between strong and weak assertion, and the way that these speech acts interact with the common ground.
With those connections established, I respond to two criticisms of the program sketched out in my 2005 paper “Multiple Conclusions”. First, that understanding the upshot of a valid sequent X ⊢ Y as enjoining us to not assert each member of X and deny each member of Y is altogether too weak to explain the inferential force of logical validity. Deriving X ⊢ A should tell us, after all, something about justifying A on the basis of X, rather than merely prohibiting A’s denial. Where is the force to actually conclude the conclusion of a proof? A second, related criticism is that the format of multiple conclusion sequents seems unsatisfactory, in that it has no place for distinguishing a single conclusion, and proofs, after all, seem to be proofs of individual claims.
I will argue that both of these concerns can be assuaged if we pay closer attention to the norms connecting assertions and denials along with justification requests — questions aiming at eliciting reasons for assertions or denials. Once we understand the connection between justification requests, definitionsand the common ground, we will see not only that the these two concerns can be met. A derivation of a sequent X ⊢ A,Y gives us an answer to a justification request “why A?” in any available context where each member of X has been ruled in and each member of Y has been ruled out, and a derivation of a sequent X,B ⊢ Y, similarly gives us an answer to the justification request “why not B?” in any such context. The picture that results utilises the full multiple premise, multiple conclusion sequent calculus of classical logic, and does due justice to the idea that a proof (or a refutation) proves (or refutes) one thing relative to background assumptions or premises. In addition, when we consider the connection between justification requests and the norms governing definitions, we can see more clearly what could be involved in taking the connective/quantifier rules of a logical system to define the concepts they introduce.
The talk is a Seminar at the Munich Centre for Mathematical Philosophy.