Abstract:

Canonical completeness results for {\L}$(\mathcal{C})$, the expansion of {\L}ukasiewicz logic {\L} with a countable set of truth-constants $\mathcal{C}$, have been recently proved in \cite{eggn06} for the case when the algebra of truth constants $\mathcal{C}$ is a subalgebra of the rational interval $[0, 1] \cap \mathbb{Q}$. The case when $C \not \subseteq [0, 1] \cap \mathbb{Q}$ was left as an open problem. In this paper we solve positively this open problem by showing that {\L}$(\mathcal{C})$ is strongly canonical complete for finite theories for {\em any} countable subalgebra $\mathcal{C}$ of the standard {\L}ukasiewicz chain $[0,1]_{\L}$.

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