Corpus ID: 208527265

Negative Translations for Affine and Lukasiewicz Logic

  title={Negative Translations for Affine and Lukasiewicz Logic},
  author={Rob Arthan and Paulo Oliva},
We investigate four well-known negative translations of classical logic into intuitionistic logic within a substructural setting. We find that in affine logic the translation schemes due to Kolmogorov and Godel both satisfy Troelstra's criteria for a negative translation. On the other hand, the schemes of Glivenko and Gentzen both fail for affine logic, but for different reasons: one can extend affine logic to make Glivenko work and Gentzen fail and vice versa. By contrast, in the setting of… Expand
1 Citations
A Curry-Howard Correspondence for the Minimal Fragment of Łukasiewicz Logic
A term calculus is introduced which adds to the affine $\lambda$-calculus with pairing a new construct allowing for a restricted form of contraction and it is proved that ${\cal B}$ is strongly normalising and has the Church-Rosser property. Expand


On the Relation Between Various Negative Trans- lations
Several proof translations of classical mathematics into intuitionistic (or even minimal) mathematics have been proposed in the literature over the past century. These are normally referred to asExpand
Two Connections Between Linear Logic and Lukasiewicz Logics
It is proved that every MV-algebra can be embedded into a phase space, and every complete MV- algebra is isomorphic to some phase space. Expand
On Pocrims and Hoops
This work gives an algebraic framework for studying the semantics of double negation translations and uses a new indirect method to establish several important identities in the theory of hoops: in particular, it proves that the doubleNegation mapping in a hoop is a homormorphism. Expand
On Various Negative Translations
This paper defines a notion of a (modular) simplification starting from Kolmogorov translation, which leads to a partial order between different negative translations, in which Kuroda and Krivine are minimal elements. Expand
Computational Interpretations of Linear Logic
  • S. Abramsky
  • Mathematics, Computer Science
  • Theor. Comput. Sci.
  • 1993
Girard's linear logic is studied from the point of view of giving a concrete computational interpretation of the logic, based on the Curry—Howard isomorphism, which opens up a promising new approach to the parallel implementation of functional programming languages. Expand
The consequence relation in the logic of commutative GBL-algebras is PSPACE-complete
It is proved that both the equational theory and the quasiequational theory of commutative GBL-algebras are decidable (in contrast to the noncommutative case), but their complexity has not been studied yet. Expand
On Krivine's Realizability Interpretation of Classical Second-Order Arithmetic
Krivine's realizability interpretation of classical second-order arithmetic and its recent extension handling countable choice is investigated and a twostep interpretation is presented which first eliminates classical logic via a negative translation and then applies standard realizable interpretation. Expand
On the structure of generalized BL-algebras
Abstract.A generalized BL - algebra (or GBL-algebra for short) is a residuated lattice that satisfies the identities $$ x\Lambda y = ((x\Lambda y)/y)y = y(y\backslash (x\Lambda y)) $$ . It is shownExpand
Metamathematical investigation of intuitionistic arithmetic and analysis
Intuitionistic formal systems.- Models and computability.- Realizability and functional interpretations.- Normalization theorems for systems of natural deduction.- Applications of Kripke models.-Expand
Metamathematics of Fuzzy Logic
  • P. Hájek
  • Computer Science, Mathematics
  • Trends in Logic
  • 1998
This paper presents a meta-analysis of many-Valued Propositional Logic, focusing on the part of Lukasiewicz's Logic that deals with Complexity, Undecidability and Generalized Quantifiers and Modalities. Expand