Abstract

Abstract. We unify the algebraic, relational and sequent methods used by various authors to investigate “dual intuitionistic logic”. We show that restricting sequents to “singletons on the left/right ” cannot capture “intuitionistic logic with dual operators”, the natural hybrid logic that arises from intuitionistic and dual-intuitionistic logic. We show that a previously reported generalised display framework does deliver the required cut-free display calculus. We also pinpoint precisely the structural rule necessary to turn this display calculus into one for classical logic. 1

Citations

122	Contraction-free sequent calculi for intuitionistic logic - Dyckhoff - 1992
98	Display logic - Belnap - 1982
50	Mathematical intuitionism, introduction to proof theory - Dragalin - 1988
36	Substructural logics on display - Goré - 1998
21	Cut-free display calculi for relation algebras - Goré - 1996
18	Semi-Boolean algebras and their applications to intuitionistic logic with dual operations - Rauszer - 1974
13	Subtractive logic - Crolard
13	Displaying Modal Logic - Wansing - 1998
11	Intuitionistic logic redisplayed - Gore - 1995
11	An algebraic and Kripke-style approach to a certain extension of intuitionistic logic - Rauszer - 1980
6	A remark on Gentzen’s calculus of sequents - Czermak - 1977
5	Dual-intuitionistic logic - Urbas - 1996
5	On logics with coimplication - Wolter - 1998
4	The logic of contradiction - Goodman - 1981
4	A uniform display system for intuitionistic and dual intuitionistic logic - Gore - 1995
3	Modal interpretation of Heyting-Brouwer logic - ̷Lukowski - 1996
1	The logic of contradiction (abstract - Goodman - 1978
1	C McKinsey and A Tarski. On closed elements in closure algebras - C - 1946