# On Lukasiewicz logic with truth-constants

Canonical completeness results for L$\displaystyle (\mathcal{C})$ , the expansion of Lukasiewicz logic with a countable set of truth-constants $\displaystyle \mathcal{C}$ , have been recently proved for the case when the algebra of truth constants $\displaystyle \mathcal{C}$ is a subalgebra of the rational interval $\displaystyle [0, 1] \cap \mathbb{Q}$ . The case when $\displaystyle C \not \subseteq [0, 1] \cap \mathbb{Q}$ was left as an open problem. In this paper we solve positively this open problem by showing that L$\displaystyle (\mathcal{C})$ is strongly canonical complete for finite theories for any countable subalgebra $\displaystyle \mathcal{C}$ of the standard Lukasiewicz chain $\displaystyle [0,1]_{L}$ .