(and some consequence)

In this short paper, I compare and contrast the kind of symmetric treatment of negation favoured in different ways by Huw Price (in “Why ‘Not’?”) and by me (in “Multiple Conclusions”) with Robert Brandom’s analysis of scorekeeping in terms of commitment, entitlement and incompatibility. Both kinds of account are what Brandom calls a normative pragmatics. They are both semantic anti-realist accounts of meaning in the significance of vocabulary is explained in terms of our rule-governed (normative) practice (pragmatics). These accounts differ from intuitionist semantic anti-realism by providing a way to distinguish the inferential significance of “A” and “A is warranted.” Although proof plays a central role, in neither accont is verification the primary bearer of meaning. Our accounts make these distinctions in terms of a subtle analysis of our practices. On the one hand according to Price and me, we assert as well as deny; on the other, Brandom distingushes downstream commitments from upstream entitlements and the notion of incompatibility definable in terms of these. In this paper I will examine a number connections between these different approaches, and end with a discussion of the kind of account of proof that might emerge from these considerations. # This paper consists of a series of vignettes exploring the issue of the structure of inferential relationships between premises and conclusions. The sections are connected in two ways. (1) Each takes the dual role of assertion and denial to be central in characterising logical consequence. (2) As a whole, they form a plea for philosophers and logicians to take a liberal view of the kinds of structure of proof. This paper is a part of a larger project of using insights Critical comments and discussion posted at http://consequently.org/writing/adcei are most welcome. { This research is supported by the Australian Research Council, through grant DP0343388. { Thanks to Karen Green, Mark Lance, Allen Hazen, Huw Price, Gøran Sudnholm and Barry Taylor for helpful discussions, and to an anonymous referee of Logical Studies for feedback helping me clarify the presentation.