## Dual intuitionistic logic revisited (2000)

Venue: | Automated Reasoning with Analytic Tableaux and Related Methods, St |

Citations: | 12 - 1 self |

### BibTeX

@INPROCEEDINGS{Goré00dualintuitionistic,

author = {Rajeev Goré},

title = {Dual intuitionistic logic revisited},

booktitle = {Automated Reasoning with Analytic Tableaux and Related Methods, St},

year = {2000},

pages = {252--267},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### 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

137 | Contraction-free Sequent Calculi for Intuitionistic Logic”, The - Dyckhoff - 1992 |

108 |
Display Logic
- Belnap
- 1982
(Show Context)
Citation Context ...to occurrences of structures in sequents as follows. In a sequent X ⊢ Y , an occurrence of Z is an antecedent part [succedent part] iff it occurs positively in X [Y ] or it occurs negatively in Y [X] =-=[Bel82]-=-. Two sequents X1 ⊢ Y1 and X2 ⊢ Y2 are structurally equivalent iff there is a derivation of the first sequent from the second (and vice-versa) using only display postulates from Figure 3. Due to the c... |

57 | Mathematical intuitionism: Introduction to proof theory, Transla- tions of mathematical monographs - DRAGALIN - 1988 |

39 | Substructural logics on display
- Goré
- 1997
(Show Context)
Citation Context ...at the “singletons on the left/right” restriction cannot provide the hybrid calculus we seek. We then give a cut-free display calculus δBiInt for BiInt, obtained from the general display framework of =-=[Gor98]-=-, and show that δBiInt does indeed capture Rauszer’s “intuitionistic logic with dual operators” by proving that δBiInt is sound with respect to Rauszer’s relational semantics and complete with respect... |

23 | Semi-boolean algebras and their applications to intuitionistic logic with dual operators - Rauszer - 1974 |

22 | Subtractive logic
- Crolard
- 2001
(Show Context)
Citation Context ... context”. Gentzen calculi for Int that permit multiples in the succedent do exist [Dra88, Dyc92]. A Gentzen calculus for DualInt with multiples in the antecedent has been explored by Tristan Crolard =-=[Cro0X]-=- who gives a calculus SLK 1 for BiInt, which he calls “subtractive logic”. The calculus SLK 1 puts a “singleton on the right” [“singleton on the left”] restriction on its rule for introducing implicat... |

21 | Cut-free display calculi for relation algebras
- Goré
- 1997
(Show Context)
Citation Context ...A & u �|= B) ⇒ v |= ˇ ∆]. ⊓⊔sTheorem6 (Completeness). A DualInt-valid ⇒ ⊢ A LDJ −< �⊃ -derivable. Proof. The Lindenbaum algebra formed from the equiprovable formulae of form a Brouwerian algebra. See =-=[Gor97]-=- for a similar proof. ⊓⊔ LDJ −< �⊃ Corollary7. A formula A is DualInt-valid iff ⊢ A is derivable in LDJ −< �⊃ . Corollary8. A sequent ⊢ A without ( −< ) is derivable in LDJ iff A[(B ⊃ C) := (∼ B ∨ C)]... |

15 |
An algebraic and Kripke-style approach to a certain extension of intuitionistic logic
- Rauszer
- 1980
(Show Context)
Citation Context ... formula built from PRP and ⊤ and ⊥ using only ∧, ∨, and →. A DualInt-formula is any formula built from PRP and ⊤ and ⊥ using only ∧, ∨ and −< . 2.1 Rauszer’s Relational Semantics for BiInt Rauszer’s =-=[Rau80]-=- semantics for BiInt extend the traditional Kripke semantics for Int, as Kripke semantics for tense logics extend those for modal logics. Rauszer actually considers the case of predicate-intuitionisti... |

15 |
Displaying Modal Logic
- Wansing
- 1998
(Show Context)
Citation Context ...tion we present a cut-free display calculus for BiInt that solves this problem. 4 A Display Calculus for Bi-Intuitionistic Logic For brevity we assume the reader is familiar with display calculi; see =-=[Wan98]-=-. The display calculus δBiInt we are about to define arises naturally from the generalised display calculus reported in [Gor98]. There, the connectives ⊗ and ⊕ are used for intensional conjunction and... |

11 | Intuitionistic logic redisplayed
- Gore
- 1995
(Show Context)
Citation Context ...del-McKinsey-Tarski embedding of Int into S4 to give an embedding of BiInt into KtS4; see also [Luk96], which contains a typographical error in the translation. Consequently, we can use the method of =-=[Gor95b]-=- to give a cut-free display calculus for BiInt using the classical modal display logic framework of Wansing [Wan98]. Although Crolard does not prove cut-elimination for his SLK, it seems highly likely... |

7 |
A Remark on Gentzen’s Calculus of Sequents
- Czermak
- 1977
(Show Context)
Citation Context ...l methods to study “intuitionistic logic with dual operators” where the connective of “pseudo-difference” is the dual to intuitionistic implication. Rauszer does not consider Gentzen calculi. Czermak =-=[Cze77]-=- investigates “dual intuitionistic logic” by restricting Gentzen’s LK to “singletons on the left” which is the natural dual notion to Gentzen’s “singletons on the right” restriction for LJ. Goodman [G... |

7 | On logics with coimplication
- Wolter
(Show Context)
Citation Context ... A (reflexive and transitive) Kripke frame validates the formulae from Table1 iff it is symmetric. 6 Concluding Remarks My goal is to extend δBiInt to bi-intuitionistic modal and tense logics. Wolter =-=[Wol98]-=- studies intuitionistic modal logics obtained by extending BiInt withstense-logical modalities, and also extends the Gödel-McKinsey-Tarski embedding of Int into S4 to give an embedding of BiInt into K... |

5 |
Dual-intuitionistic logic
- Urbas
- 1996
(Show Context)
Citation Context ...ngly fails to give the crucial clause for satisfiability for his “pseudo-difference” connective. He also gives a “sequentcalculus” for his logic but does not investigate cut-elimination at all. Urbas =-=[Urb96]-=- highlights several deficiencies of Goodman’s analysis and defines several Gentzen calculi with the “singletons on the left” restriction, but ∗ Supported by a Queen Elizabeth II Fellowship from the Au... |

4 |
The logic of contradiction
- Goodman
- 1981
(Show Context)
Citation Context ...7] investigates “dual intuitionistic logic” by restricting Gentzen’s LK to “singletons on the left” which is the natural dual notion to Gentzen’s “singletons on the right” restriction for LJ. Goodman =-=[Goo81]-=- uses Brouwerian algebras to investigate the “logic of contradictions”. Goodman mentions that Kripke semantics also exist in which “any formula, once false, remains false”, but he annoyingly fails to ... |

4 | A uniform display system for intuitionistic and dual intuitionistic logic
- Goré
- 1995
(Show Context)
Citation Context ...uszer’s algebraic semantics. The display calculus δBiInt also captures both Int and DualInt separately. A first attempt to understand “dual intuitionistic logic” using display calculi can be found in =-=[Gor95a]-=-. But the framework proposed there was not general enough to cater for substructural logics. Such a framework can be found in [Gor98], and we dub this framework SLD. The current paper arose out of att... |

3 |
Modal interpretation of Heyting-Brouwer logic
- ̷Lukowski
- 1996
(Show Context)
Citation Context ...tionistic modal logics obtained by extending BiInt withstense-logical modalities, and also extends the Gödel-McKinsey-Tarski embedding of Int into S4 to give an embedding of BiInt into KtS4; see also =-=[Luk96]-=-, which contains a typographical error in the translation. Consequently, we can use the method of [Gor95b] to give a cut-free display calculus for BiInt using the classical modal display logic framewo... |

1 |
The logic of contradiction (abstract
- Goodman
- 1978
(Show Context)
Citation Context ...enko property; “Restricted” in that it does not extend to formulae containing implication since Czermak does not give “singletons on the left” rules for implication. Goodman, apparently independently =-=[Goo78]-=-, also attempts a “sequentcalculus” which we dub GDJ, whose core rules are the same as Czermak’s DJ and Urbas’ LDJ�⊃ , where this time, Goodman’s (¬) := (∼). Goodman’s GDJ also contains the following ... |

1 |
C McKinsey and A Tarski. On closed elements in closure algebras
- C
- 1946
(Show Context)
Citation Context ...been investigated to varying degrees of success using algebraic, relational, axiomatic and sequent perspectives. The ones I am aware of are listed below, there may well be others. McKinsey and Tarski =-=[MT46]-=- investigate algebraic properties of “closure” or Brouwerian algebras, the algebraic duals to Heyting algebras. Curry [Cur76] presents what he called “absolute implicational lattices” and “absolute su... |