What identifies Classical Logic as such? What is a Paraconsistent Logic? As easy and straightforward as answering these questions seem, in this talk, I devote to discussing some overlooked subtleties that, I argue, need to be taken into account in providing a cogent answer to them. The main contribution of this work consists, therefore, in offering a novel approach to how Classical and Paraconsistent Logics should be defined, which can account for these overlooked details in a systematic manner. So, my focus will be on finding an appropriate identity criterion for logical systems. My proposal is motivated by reflecting upon different aspects connected with the abstract notion of logical consequence. Firstly, I analyze what could be called a logic of metainferences —i.e., of inferences between inferences—which are valid in the non-transitive logic ST (Strict-Tolerant). As is well known, ST and CL have the same set of validities. Nevertheless. ST invalidates the Cut rule. Consequently, as pointed out e.g. by Barrio, Rosenblatt & Tajer in [1], some classically valid metainferences—such as meta Modus Ponens, meta Explosion, and others, on which more below—fail to be valid in it. In fact, the logic of the metainferences of ST is the paraconsistent logic LP. So, this talk presents an impossibility result concerning the extensional characterization of Classical Logic, i.e. the identification of Classical Logic with its set of valid inferences of some inferential level. Secondly, I consider cases of failures of the rule of metaexplosion that could be described as substructural paraconsistency. Finally, I show how it is possible to obtain a sequence of logics which can be progressively ordered regarding their degree of classicality. To argue for this I present a collection of ST-related logics which coincide with Classical Logic in more and more inferential levels as we move forward in the sequence—although, for all of them, there is some inferential level at which these systems start to behave non-classically.

Reference

[1] E. Barrio, L. Rosenblatt, and D. Tajer. "The Logics of Strict-Tolerant Logic". Journal of Philosophical Logic, 44(5):551–571, 2015.