Classical polarizations yield double-negation translations

Zakaria Chihani (1) , Danko Ilik (1) , Dale Miller (1)

1 PARSIFAL - Proof search and reasoning with logic specifications

Double-negation translations map formulas to formulas in such a way that if a formula is a classical theorem then its translation is an intuitionistic theorem. We shall go beyond just examining provability by looking at correspondences between inference rules in classical proofs and in intuitionistic proofs of translated formulas. In order to make this comparison interesting and precise, we will examine focused versions of proofs in classical and intuitionistic logics using the LKF and LJF proof systems. We shall show that for a number of known double-negation translations, one can get essentially identical (focused) intuitionistic proofs as (focused) classical proofs. Thus the choice of a common double-negation translation is really the same choice as a polarization of classical logic (of which there are many).

Domaines: Logique en informatique [cs.LO]; Logique [math.LO]

Zakaria Chihani, Danko Ilik, Dale Miller. Classical polarizations yield double-negation translations. 2016. ⟨hal-01354298⟩