Reasoning About Knowledge In Linear Logic: Modalities and Complexity

In a recent paper, Jean-Yves Girard commented that ”it has been a long time since philosophy has stopped intereacting with logic”[17]. Actually, it has not been such a ”long” time since, e.g., Dag Prawitz and Michael Dummett developed philosophical arguments within the paradigm of Gentzen-style systems in favour of the adoption of intuitionistic logic. But recent developments within logic have left philosophers far behind. Prawitz’s timely book Natural Deduction [?], along with a key result obtained at around the same time, the Curry-Howard isomorphism [21], initiated deep changes within logic; to wit, the development of the substructural logics [?, ?]. Dummett developed within proof-theoretical semantics a mature version of his anti-realist challenge, closely allied to Prawitz’s own philosophical considerations. But within the debate generated since by Dummett’s anti-realism, philosophers have for the most part not paid much attention to the developments within logic since the 1970s. The almost complete absence of any discussion of linear logic within this context should convince any one that there is something seriously amiss here. The present paper is related to an attempt by Jacques Dubucs to provide a new impetus to the anti-realism debate by providing a radical anti-realist line of argument that also ties the debate more closely to issues of concern within substructural logics [?, 9, 10]. We shall only make brief remarks about his argument in section 1. Our intention is to push it a few steps further and explore the possibilities for a radically anti-realist epistemic logic. In a nutshell, it has been suggested that a radical form of anti-realism should force one to look at structural rules within Gentzen-style systems that are responsible for the idealizations of the full structural logic. By ’idealization’ we merely mean features of structural logic which allow for infinities to creep in, so to speak, and which should not go without notice within the interpretation of proofs as actions that has dominated much of the thinking about prooftheoretical semantics since the days of Prawitz and Dummett (e.g., in the work