Goedel logics: Prenex fragments

TL;DR

This paper classifies first-order Goedel logics based on their ability to admit equivalent prenex normal forms, revealing that only finite and certain infinite cases allow such forms, with implications for their logical properties and computability.

Contribution

It provides a complete classification of first-order Goedel logics regarding prenex normal forms and investigates the effective equivalence of validity between formulas and their prenex forms.

Findings

Finite and $G_\uparrow$ logics admit prenex forms.

Uncountable logics generally do not admit effective translation.

Prenex fragment is always recursively enumerable.

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.