A 0-1 Law in Mathematical Fuzzy Logic

This article continues the theoretical study of weighted structures in mathematical fuzzy logic focusing on the finite model theory of fuzzy logics valued on arbitrary finite
$\mathrm{MTL}$
-chains. We show that for any first-order (or infinitary with finitely many variables) formula
$\varphi$
, there is a truth-value that
$\varphi$
takes almost surely in every finite many-valued model and such that every other truth-value is almost surely not taken. This generalizes a theorem in the fuzzy setting due to Robert Kosik and Christian G. Fermüller.