Approximate Completeness of Hypersequent Calculus for First-Order Łukasiewicz Logic

Hypersequent calculus GŁ$\mathrm{\forall }$ for first-order Łukasiewicz logic was first introduced by Baaz and Metcalfe, along with a proof of its approximate completeness with respect to standard $[0,1]$-semantics. The completeness result was later pointed out by Gerasimov that it only applies to prenex formulas. In this paper, we will present our proof of approximate completeness of GŁ$\mathrm{\forall }$ for arbitrary first-order formulas by generalizing the original completeness proof to hypersequents.