@article { 3315, title = {Expanding the propositional logic of a t-norm with truth-constants: completeness results for rational semantics}, journal = {Soft Computing - A Fusion of Foundations, Methodologies and Applications}, volume = {14}, year = {2010}, pages = {273-284}, publisher = {Springer}, abstract = {In this paper we consider the expansions of logics of a left-continuous t-norm with truth-constants from a subalgebra of the rational unit interval. From known results on standard semantics, we study completeness for these propositional logics with respect to chains de?ned over the rational unit interval with a special attention to the completeness with respect to the canonical chain, i.e. the algebra over $[0, 1] \cap Q$ where each truth-constant is interpreted in its corresponding rational truth-value. Finally, we study rational completeness results when we restrict ourselves to deductions between the so-called evaluated formulae.}, author = {Francesc Esteva and Llu\'{\i}s Godo and Carles Noguera} }