It is not possible to model semantics of natural language unless one can clearly establish the structure of contextual information and how "context" guides the construction of semantic identities of linguistic units. We claim that this cannot be done unless one can overcome the representational locality imposed by extensional formalisms. In this paper we present a formalism aiming at setting up a non-extensional framework allowing the expression of a notion of context. The very notion of identity is revisited and some forms of non-extensional identity are proposed. In the last section we suggest a representation of metaphor understanding that implements a form of interaction between global contextual information and local semantic identities.