The semantics and proof theory of linear logic

Linear logic is a new logic which was recently developed by Girard in order to provide a logical basis for the study of parallelism. It is described and investigated in [9]. Girard's presentation of his logic is not so standard. In this paper we shall provide more standard proof systems and semantics. We shall also extend part of Girard's results by investigating the consequence relations associated with Linear Logic and by proving corresponding Strong completeness theorems. Finally, we shall investigate the relation between Linear Logic and previously known systems, especially Relevance Logics.