Starting from an exact correspondence between linear approximations and non-idempotent intersection types, we develop a general framework for building systems of intersection types characterizing normalization properties. We show how this construction, which uses in a fundamental way Melliès and Zeilberger's łtype systems as functorsž viewpoint, allows us to recover equivalent versions of every well known intersection type system (including Coppo and Dezani's original system, as well as its non-idempotent variants independently introduced by Gardner and de Carvalho). We also show how new systems of intersection types may be built almost automatically in this way.

We introduce a new graphical representation for multiplicative and exponential linear logic proof-structures, based only on standard labelled oriented graphs and standard notions of graph theory. The inductive structure of boxes is handled by means of a box-tree. Our proof-structures are canonical and allows for an elegant definition of their Taylor expansion by means of pullbacks.

This paper is at the same time a first step towards an "implementation" of the inferentialist view of meaning and a first proposal for a logical structure which describes an argumentation. According to inferentialism the meaning of a statement lies in its argumentative use, its justifications, its refutations and more generally its deductive relation to other statements. In this first step we design a simple notion of argumentative dialogue. Such dialogues can be either carried in purely logical terms or in natural language. Indeed, a sentence can be mapped to logical formulas representing the possible meanings of the sentence, as implemented with some categorial parsers. We then present our version of dialogical logic, which we recently proved complete for first order classical logic. Next we explain, through examples, how argumentative dialogues can be modeled within our version of dialogical logic.Finally, we discuss how this framework can be extended to real argumentative dialogues, in particular with a proper treatment of axioms.