pluralism and proofs

Beall and Restall’s Logical Pluralism [2] characterises pluralism about logical consequence in terms of the different ways cases can be selected in the analysis of logical consequence as preservation of truth over a class of cases. This is not the only way to understand or to motivate pluralism about logical consequence. Here, I will examine pluralism about logical consequence in terms of different standards of proof. We will focus on sequent derivations for classical logic, imposing two different restrictions on classical derivations to produce derivations for intuitionistic logic and for dual intuitionistic logic. The result is another way to understand the manner in which we can have different consequence relations in the one language. Furthermore, the proof-theoretic perspective gives us a different explanation of how the one concept of negation can have three different truth conditions, those in classical, intuitionistic and dual-intuitionistic models. motivation for pluralism about logical consequence » I am a pluralist about logical consequence [1, 2]. From the beginning of my interest in pluralism about logical consequence, my motivation has remained unchanged: I take it that there are particular, concrete arguments, for which the question ‘is this argument valid?’ has more than one correct and informative answer. My first example was arguments of the form of disjunctive syllogism (from p ∨ q and ¬p to q) and explosion (from p and ¬p to q). In both of these cases, I became convinced that the most appropriate answer was twofold: ∗See http://consequently.org/writing/pluralism_and_proofs/ for the latest version of the paper, to post comments and to read comments left by others. ¶ Thanks to audiences at the “Logical Pluralism” workshop in Tartu, the Logic Seminar at the University of Melbourne, and the Logic or Logics Workshop of the Foundations of Logical Consequence Project at Arché at the University of St Andrews for comments on this paper. ¶ I was supported by the ARC Discovery Grants dp0556827 and dp1094962, and Tonio K’s Life in the Foodchain.