Øøø Óò Blockinðù××óò × Ûððð Øøù׸óòðý Ôöóôö ×ùù××ø Óó Øøø ××ø Óó Äâ¹ôöóóó× Ò Óööøøøº

1 Of kits and mole ules Yiorgos Stavrinos∗ and Anastasia Veneti† MPLA, Department of Mathemati s University of Athens GR-15784 Zografou, Gree e 1 Introdu tion This arti le investigates the relation of the logi al systems \Interse tion Logi " and \Interse tion Syn hronous Logi ", abbreviated as IL and ISL, respe tively. They are both introdu ed (the former by Ron hi della Ro a and Roversi [RR01℄ and the latter by Pimentel, Ron hi and Roversi [PR05℄) to establish a logi al foundation for the interse tion types assignment system IT of Coppo and Dezani [CD80℄, a dedu tive system whi h assigns formulas, built from the intuitionisti impli ation and the interse tion, as types to terms of the untyped al ulus. The ne essity of their invention originates from the fa t that the impli ative and onjun tive fragment of Intuitionisti Logi , denoted LJ, fails as a logi al foundation for IT, i.e. fails to orrespond to IT through a de oration of its proofs with untyped -terms. A tually, LJ orresponds to the simple types assignment system → through a de oration whi h en odes the whole proofstru ture, i.e. the two systems relate through the Curry-Howard orresponden e. Even if we employ a de oration en oding the impli ative rules only, LJ still fails to be the logi behind IT. This is be ause the latter de oration works only in ase the premises of the rule introdu ing the onjun tion (depi ted below for all three systems) are en oded by the same -term, whi h then en odes the on lusion as well; thus, only a proper subset of the set of LJ-proofs an be de orated.