Adding truth-constants to logics of continuous t-norms: Axiomatization and completeness results

Francesc Esteva, Joan Gispert, Lluís Godo, Carles Noguera

Abstract: In this paper we study generic expansions of logics of continuous t-norms with truth-constants, taking advantage of previous results for Łukasiewicz logic and more recent results for Gödel and Product logics. Indeed, we consider algebraic semantics for expansions of logics of continuous t-norms with a set of truth-constants {r¯|r∈C}<math><mo stretchy="false" is="true">{</mo><mover accent="true" is="true"><mrow is="true"><mi is="true">r</mi></mrow><mrow is="true"><mo stretchy="true" is="true">¯</mo></mrow></mover><mspace width="0.16em" is="true"></mspace><mo is="true">|</mo><mspace width="0.16em" is="true"></mspace><mi is="true">r</mi><mo is="true">∈</mo><mi is="true">C</mi><mo stretchy="false" is="true">}</mo></math>, for a suitable countable C⊆[0,1]<math><mi is="true">C</mi><mo is="true">⊆</mo><mo stretchy="false" is="true">[</mo><mn is="true">0</mn><mo is="true">,</mo><mn is="true">1</mn><mo stretchy="false" is="true">]</mo></math>, and provide a full description of completeness results when (i) the t-norm is a finite ordinal sum of Łukasiewicz, Gödel and Product components, (ii) the set of truth-constants covers all the unit interval in the sense that each component of the t-norm contains at least one value of C different from the bounds of the component, and (iii) the truth-constants in Łukasiewicz components behave as rational numbers.