Plexus Summer School. Damián Szmuc: From Substructural to Parastructural Logics. Full Course, Part 1

In the past decade, attention has been put on systems like ST that, although non-standard in some sense, agree with classical logic in regards to their inferences. Some have argued that these systems are structural, because their logics alone do not present counterexamples to metainferences like Cut. Others have argued that they are not, because they can be extended to
systems presenting such counterexamples. In this course, we introduce this
debate as a motivation to discriminate between two different senses of nonstructurality in logic. The first conception, according to which frameworks like ST are structural, demands from logics to be non-Tarskian---and, in particular, non-transitive. The second conception, according to which
systems like ST are not structural, demands from logics to have onTarskian---and, in particular, non-transitive---extensions. Borrowing from the literature on logics invalidating the principle of Ex Falso Sequitur Quodlibet, we propose that this distinction should be set between substructural and parastructural logics. In a nutshell, we argue that the two senses of non-structurality discussed before mirror the two senses in which a logic can defy consistency---i.e. by having inconsistent theorems or by allowing for inconsistent extensions.

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