Two Connections Between Linear Logic and Lukasiewicz Logics

In this work we establish some syntactical and semantical links between Łukasiewicz Logics and Linear Logic. First we introduce a new sequent calculus of infinite-valued Łukasiewicz Logic by adding a new rule of inference to those of Affine Linear Logic. The only axioms of this calculus have the form A ⊢ A. Then we compare the (provability) semantics of both logics, respectively given by MV-algebras and phase spaces. We prove that every MV-algebra can be embedded into a phase space, and every complete MV-algebra is isomorphic to some phase space. In fact, completeness is necessary and sufficient for the existence of the isomorphism. Our proof is constructive.