Semantics of weakening and contraction

Abstract

The shriek modality \\s! of linear logic performs two tasks: it restores in annotated from both weakening and contraction. We separate these tasks by introducing two modalities: ! w for weakening and ! c for contraction. These give rise to two logics which are “inbetween” linear and intuitionistic logic: in affine (or weakening) logic one always has a weakening and a ! w for contraction and in relevant (or contraction) logic one always has a contraction and a ! w weakening. The semantics of these logics is obtained from special kinds of monads, introduced by Anders Kock in the early seventies. As subtle point is how to retrieve the \\s! of linear logic from ! w and ! c . Technically this will be achieved in terms of distributive laws—introduced by Jon Beck. We find models where one has \\s! = ! w ! c and also models with \\s! = ! c ! w . It will be shown that on the category of complete lattices one has comonads ! w and ! c with ! w ! c = \\s!= ! c ! w .