First-order Gödel logics

Abstract

First-order Gödel logics are a family of finite-or infinite-valued logics where the sets of truth values V are closed subsets of [0, 1] containing both 0 and 1. Different such sets V in general determine different Gödel logics G V (sets of those formulas which evaluate to 1 in every interpretation into V ). It is shown that G V is axiomatizable iff V is finite, V is uncountable with 0 isolated in V , or every neighborhood of 0 in V is uncountable. Complete axiomatizations for each of these cases are given. The r.e. prenex, negation-free, and existential fragments of all first-order Gödel logics are also characterized.

Monadic Fragments of Gödel Logics: Decidability and Undecidability Results

“…In other words, arbitrary formulas are not equivalent to prenex formulas, in general. However, we have the following (stated in [4] without proof): Proposition 1. The following quantifier shift laws, where x is not free in B, and where Q denotes either ∃ or ∀ (uniformly over a formula) are valid in all Gödel logics:…”

“…Note that the schemes that are dual to (4) and (5) are not valid in general (but are valid in G ↑ and G n , n ≥ 2; see Proposition 3). Counterexamples are readily obtained for standard Gödel logic G [0,1] .…”

“…Therefore (9) is valid in G ↓ , in G ↑ , and in all G n for n ≥ 2, but not, e.g., in G [0,1] . In fact there are infinitely many different infinite-valued first-order Gödel logics, according to [4]. The conjecture that there are just countable many different Gödel logics has recently been settled in [5].…”

“…In contrast to the propositional case, where there is only one infinite-valued Gödel logic with respect to validity, different sets V of truth values determine different first-order Gödel logics G V , in general. As shown in [4], G V is recursively axiomatizable only when V is either finite or is order isomorphic to [0,1] or to {0} ∪ [ 1 2 , 1].…”

“…This bound implies a considerable gain in efficiency for the corresponding fragments of finite-valued Gödel logics (including classical logic). An elegant axiomatization for the untangled fragment of G ↑ can also be extracted from the decision procedure, contrasting the fact that G ↑ is not recursively axiomatizable [3,4].…”