Intersection Logic in sequent calculus style

The intersection type assignment system has been designed directly as deductive system for assigning formulae of the implicative and conjunctive fragment of the intuitionistic logic to terms of lambda-calculus. But its relation with the logic is not standard. Between all the logics that have been proposed as its foundation, we consider ISL, which gives a logical interpretation of the intersection by splitting the intuitionistic conjunction into two connectives, with a local and global behaviour respectively, being the intersection the local one. We think ISL is a logic interesting by itself, and in order to support this claim we give a sequent calculus formulation of it, and we prove that it enjoys the cut elimination property.Comment: In Proceedings ITRS 2010, arXiv:1101.410

Linear Aspects of Intersection

Linear logic deals with two kinds of conjunction, multiplicative ⊗ and additive &. For these connectives both sequents A⊗B ⇒ A &B and A&B ⇒ A⊗B are not derivable. Another system with two kinds of conjunction (and disjunction) is Intersection and Union Logic IUL [3,6,5] which aims to give a logical foundation for intersection and union types [1]. In IUL, conjunction ∧ is an asynchronous connective and has a multiplicative definition whereas intersection ∩ being synchronous is necessarily additive [3]. For these connectives, A ∧B ⇒ A ∩B is not derivable whereas A ∩B ⇒ A ∧B is and thus intersection ∩ behaves as a special synchronous conjunction. To investigate further the nature of these connectives we define a translation ( ) ◦ of IUL in linear logic and prove a full embedding. In the translation of ∩ we take into account the additive aspect of ∩ and therefore (A ∩B) ◦ = A ◦ & B ◦, whereas for ∧ and ⊗ we consider general elimination rules [4,2] and thus we translate (A ∧ B) ◦ =!A ◦ ⊗!B ◦. Since!A ⊗!B ⇒ A&B is derivable (and not conversely), the embedding with & is in a sense tighter than the embedding with ⊗, which is dual to the relation of ∩ to ∧