Quantified propositional Gödel logics

Baaz, Matthias, Agata Ciabattoni, and Richard Zach. 2000. “Quantified Propositional Gödel Logic.” In Logic for Programming and Automated Reasoning. 7th International Conference, LPAR 2000, edited by Andrei Voronkov and Michel Parigot, 240–56. LNCS 1955. Berlin: Springer. DOI: 10.1007/3-540-44404-1_16

It is shown that $$\mathbf{G}^\mathrm{qp}_\uparrow$$, the quantified propositional Gödel logic based on the truth-values set $$V_\uparrow = {1 – 1/n : n = 1, 2, . . .} \cup {1}$$, is decidable. This result is obtained by reduction to Büchi’s theory S1S. An alternative proof based on elimination of quantifiers is also given, which yields both an axiomatization and a characterization of $$\mathbf{G}^\mathrm{qp}_\uparrow$$ as the intersection of all finite-valued quantified propositional Gödel logics.