scholarly journals On Various Negative Translations

2011 ◽  
Vol 47 ◽  
pp. 21-33 ◽  
Author(s):  
Gilda Ferreira ◽  
Paulo Oliva
Keyword(s):  
Negative Translations

Glivenko theorems and negative translations in substructural predicate logics

2012 ◽  
Vol 51 (7-8) ◽  
pp. 695-707 ◽  
Author(s):  
Hadi Farahani ◽  
Hiroakira Ono
Keyword(s):  
Negative Translations

scholarly journals An Isbell duality theorem for type refinement systems

2017 ◽  
Vol 28 (6) ◽  
pp. 736-774
Author(s):  
PAUL-ANDRÉ MELLIÈS ◽  
NOAM ZEILBERGER
Keyword(s):  
Proof Theory ◽  
Sequent Calculus ◽  
Context Effects ◽  
Linear Logic ◽  
Embedding Theorem ◽  
Hoare Logic ◽  
Covariant Representation ◽  
Refinement Types ◽  
Positive Representation ◽  
Negative Translations

Any refinement system (= functor) has a fully faithful representation in the refinement system of presheaves, by interpreting types as relative slice categories, and refinement types as presheaves over those categories. Motivated by an analogy between side effects in programming andcontext effectsin linear logic, we study logical aspects of this ‘positive’ (covariant) representation, as well as of an associated ‘negative’ (contravariant) representation. We establish several preservation properties for these representations, including a generalization of Day's embedding theorem for monoidal closed categories. Then, we establish that the positive and negative representations satisfy an Isbell-style duality. As corollaries, we derive two different formulas for the positive representation of a pushforward (inspired by the classical negative translations of proof theory), which express it either as the dual of a pullback of a dual or as the double dual of a pushforward. Besides explaining how these constructions on refinement systems generalize familiar category-theoretic ones (by viewing categories as special refinement systems), our main running examples involve representations of Hoare logic and linear sequent calculus.

Sign in / Sign up

Export Citation Format

Share Document