Intuitionistic phase semantics is almost classical

We study the relationship between classical phase semantics for classical linear logic (LL) and intuitionistic phase semantics for intuitionistic linear logic (ILL). We prove that (i) every intuitionistic phase space is a subspace of a classical phase space, and (ii) every intuitionistic phase space is phase isomorphic to an “almost classical” phase space. Here, by an “almost classical” phase space we mean a phase space having a double-negation-like closure operator. Based on these semantic considerations, we give a syntactic embedding of propositional ILL into LL.