Clarence Lewis Protin

Several different proof translations exist between classical and intuitionistic logic (negative translations), and intuitionistic and linear logic (Girard translations). Our aims in this paper are: (1) to consider extensions of intuitionistic linear logic corresponding to each of these systems, and (2) using this common logical basis, to develop a uniform approach to devising and simplifying proof translations. Through this process of “simplification” we recover most of the well-known translations in the literature.

We propose a new modal logic endowed with a simple deductive system to interpret Aristotle's theory of the modal syllogism. While being inspired by standard propositional modal logic, it is also a logic of terms that admits a (sound) extensional semantics involving possible states-of-affairs in a given world. Applied to the analysis of Aristotle's modal syllogistic as found in the Prior Analytics A8-22, it sheds light on various fine-grained distinctions which when made allow us to clarify some ambiguities and obtain a completely consistent system and prove all of the modal syllogisms considered valid by Aristotle. This logic allows us also to make a connection with the axioms of modern propositional modal logic and to perceive to what extent these are implicit in Aristotle's reasoning. Further work will involve addressing the question of the completeness of this logic (or variants thereof) together with an extension of the logic to include a calculus of relations (for instance the relational syllogistic treated in Galen's Introduction to Logic) which Slomkowsky has argued is already found in the Topics.