On Affine Logic and Łukasiewicz Logic

The multi-valued logic of Lukasiewicz is a substructural logic that has been widely studied and has many interesting properties. It is classical, in the sense that it admits the axiom schema of double negation elimination, [DNE]. However, our understanding of Lukasiewicz logic can be improved by separating its classical and intuitionistic aspects. The intuitionistic aspect of Lukasiewicz logic is captured in an axiom schema, [CWC], which asserts the commutativity of a weak form of conjunction. We explain how this axiom is equivalent to a restricted form of contraction. We then show how Lukasiewicz Logic can be viewed both as an extension of classical affine logic with [CWC], or as an extension of what we call intuitionistic Lukasiewicz logic with [DNE], intuitionistic Lukasiewicz logic being the extension of intuitionistic affine logic by the schema [CWC]. At first glance, intuitionistic Lukasiewicz logic seems to be a very weak fragment of intuitionistic logic. We show that [CWC] is a surprisingly powerful (restricted) form of contraction, implying for instance all the intuitionistically valid De Morgan’s laws. However the proofs can be very intricate. These results are presented using derived connectives to clarify and motivate the proofs. Applications include: a simpler proof of the Ferreirim-Veroff-Spinks theorem, proof that idempotent elements of a hoop form a sub-hoop, proof that double negation is a hoop homomorphism, and proofs for the above mentioned De Morgan dualities. We conclude by showing that our homomorphism result on the doublenegation mapping in particular implies that all negative translations of classical into intuitionistic Lukasiewicz coincide (as they do in full intuitionistic logic). This is in contrast with affine logic for which we show, by appeal to results on semantics proved in a companion paper, that both the Gentzen and the Glivenko translations fail.