A Focused Approach to Combining Logics

Chuck Liang (1) , Dale Miller (2, 3)

1 Computer Science Department [Hempstead]
2 PARSIFAL - Proof search and reasoning with logic specifications
3 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]

We present a compact sequent calculus LKU for classical logic organized around the concept of polarization. Focused sequent calculi for classical, intuitionistic, and multiplicative-additive linear logics are derived as fragments of the host system by varying the sensitivity of specialized structural rules to polarity information. We identify a general set of criteria under which cut elimination holds in such fragments. From cut elimination we derive a unified proof of the completeness of focusing. Furthermore, each sublogic can interact with other fragments through cut. We examine certain circumstances, for example, in which a classical lemma can be used in an intuitionistic proof while preserving intuitionistic provability. We also examine the possibility of defining classical-linear hybrid logics.

Domaines: Logique en informatique [cs.LO]

Chuck Liang, Dale Miller. A Focused Approach to Combining Logics. Annals of Pure and Applied Logic, 2011, ⟨10.1016/j.apal.2011.01.012⟩. ⟨hal-00772736⟩