Goedel logics: Prenex fragments
Matthias Baaz, Mariami Gamsakhurdia
TL;DR
The paper solves when first-order Gödel logics admit logically equivalent prenex normal forms, revealing that only finite-valued logics and $G_\uparrow$ have this property, with $G_\uparrow$ equaling the intersection of all finite Gödel logics. It introduces the Gluing Lemma to analyze accumulation-point structures and uses Skolemization and the $\Delta$ operator to map prenex forms to the corresponding logics, yielding a complete classification across finite, countable, and uncountable truth-value sets. It further analyzes effective equivalence between validity and prenex validity, showing recursive enumerability criteria: logics with finite $V$, uncountable $V$ with $0$ isolated, or with $0$ having uncountable neighbourhoods are r.e.; otherwise, no uniform effective translation exists, though the prenex fragment remains r.e. In the presence of $\Delta$, finite-valued cases retain prenex forms, while $G_\uparrow^\Delta$ loses the intersection-property and lacks a general equivalent prenex form. The results have implications for decision procedures and illuminate open questions for countable Gödel logics, suggesting a conjecture that effective validity-equivalence holds precisely for arithmetical logics.
Abstract
In this paper, we provide a complete classification for the first-order Goedel logics concerning the property that the formulas admit logically equivalent prenex normal forms. We show that the only first-order Goedel logics that admit such prenex forms are those with finite truth value sets since they allow all quantifier-shift rules and the logic $G_\uparrow$ with only one accumulation point at 1 in the infinite truth value set. In all the other cases, there are generally no logically equivalent prenex normal forms. We will also see that $G_\uparrow$ is the intersection of all finite first-order Goedel logics. The second part of this paper investigates the existence of effective equivalence between the validity of a formula and the validity of some prenex normal form. The existence of such a normal form is obvious for finite valued Goedel logic and $G_\uparrow$. Goedel logics with an uncountable truth value set admit the prenex normal forms if and only if every surrounding of 0 is uncountable or 0 is an isolated point. Otherwise, uncountable Goedel logics are not recursively enumerable, however, the prenex fragment is always recursively enumerable. Therefore, there is no effective translation between the valid formula and the valid prenex normal form. However, the existence of effectively constructible validity equivalent prenex forms for the countable case is still up for debate.