Partial connexivity and the logic CG ʹ3

In its initial formulation, CG₃^ is a three-valued paraconsistent calculus derived from specific modifications to the matrix of Gödel logic G₃. We show that enriching CG₃^ with the so-called Aristotle’s theses, or their equivalent variants, leads to a trivial logic. To resolve this issue, we define certain proper subsystems of CG₃^ in a manner that prevents trivialization. However, unlike prevailing perspectives in connexive logic, we do not assume that certain classically valid formulas must be rejected due to their counter-intuitive nature. By extending some subsystems of CG₃^ with Aristotle’s theses, our objective is to develop calculi that validate as many CG₃^-valid formulas as possible while remaining non-trivial.