Abstract

When faced with the question of whether to assert or deny a paradoxical sentence such as the liar, it seems that there are two plausible responses: neither asserting it nor denying it, or both asserting it and denying it. In this paper, I make this thought concrete by formulating bilateral proof systems (of both the natural deduction and sequent calculus variety) for the logics in the FDE family: K3, LP, and FDE. The different logics are simply the result of different choices of “coordination principles,” bilateral structural rules which coordinate the opposite speech acts of assertion and denial. I show that conceiving of these logics in bilateral terms has important philosophical consequences, most notably for the debate between “subclassical” and “substructural” approaches to paradox. In particular, I show how adopting Bilateral K3 enables one to endorse the “non-transitive” solution to paradox, as developed by Ripley, while maintaining that logical consequence (understood bilaterally) is transitive.