A Study on Actions for Atomic Logics

Nowadays there is a large number of non-classical logics, each one best suited for reasoning about some issues in abstract fields, such as linguistics or epistemology, among others. Proving interesting properties for each one of them supposes a big workload for logicians and computer scientists. We want an approach into this problematic that is modular. To adress this issue, the report shows new insights in the construction of Atomic Logics introduced by Guillaume Aucher. Atomic Logics let us represent very general left and right introduction rules and they come along a new kind of rules based on display logics and residuation. A new approach is taken into the definition of Atomic Logics, which is now built on a class of actions for which we prove cut-elimination. We show that some of them are equivalent to Aucher's Atomic Logics and we prove cut-elimination and Craig Interpolation for a class of them. The introduced theory is applied to the non-associative Lambek Calculus throughout the report. It is accompanied by a computer-checked formalisation of the original syntax in the proof assistant Coq.