A family of Gödel hybrid logics

Didier Galmiche, Yakoub Salhi

Abstract: In this paper, we define a family of fuzzy hybrid logics that are based on Gödel logic. It is composed of two infinite-valued versions called GH∞<math><msub is="true"><mi mathvariant="sans-serif" is="true">GH</mi><mo is="true">∞</mo></msub></math> and WGH∞<math><msub is="true"><mi mathvariant="sans-serif" is="true">WGH</mi><mo is="true">∞</mo></msub></math>, and a sequence of finitary valued versions (GHn)0<n<∞<math><msub is="true"><mrow is="true"><mo stretchy="false" is="true">(</mo><msub is="true"><mi mathvariant="sans-serif" is="true">GH</mi><mi is="true">n</mi></msub><mo stretchy="false" is="true">)</mo></mrow><mrow is="true"><mn is="true">0</mn><mo is="true">&lt;</mo><mi is="true">n</mi><mo is="true">&lt;</mo><mo is="true">∞</mo></mrow></msub></math>. We define decision procedures for both WGH∞<math><msub is="true"><mi mathvariant="sans-serif" is="true">WGH</mi><mo is="true">∞</mo></msub></math> and (GHn)0<n<∞<math><msub is="true"><mrow is="true"><mo stretchy="false" is="true">(</mo><msub is="true"><mi mathvariant="sans-serif" is="true">GH</mi><mi is="true">n</mi></msub><mo stretchy="false" is="true">)</mo></mrow><mrow is="true"><mn is="true">0</mn><mo is="true">&lt;</mo><mi is="true">n</mi><mo is="true">&lt;</mo><mo is="true">∞</mo></mrow></msub></math> that are based on particular sequents and on a set of proof rules dealing with such sequents. As these rules are strongly invertible the procedures naturally allow one to generate countermodels. Therefore we prove the decidability and the finite model property for these logics. Finally, from the decision procedure of WGH∞<math><msub is="true"><mi mathvariant="sans-serif" is="true">WGH</mi><mo is="true">∞</mo></msub></math>, we design a sound and complete sequent calculus for this logic.