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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., Comment: Research supported by FWF grant P 36571

This paper develops a general methodology to connect propositional and first-order interpolation. In fact, the existence of suitable skolemizations and of Herbrand expansions together with a propositional interpolant suffice to construct a first-order interpolant. This methodology is realized for lattice-based finitely-valued logics, the top element representing true. It is shown that interpolation is decidable for these logics.

Propositional logics in general, considered as a set of sentences, can be undecidable even if they have "nice" representations, e.g., are given by a calculus. Even decidable propositional logics can be computationally complex (e.g., already intuitionistic logic is PSPACE-complete). On the other hand, finite-valued logics are computationally relatively simple - at worst NP. Moreover, finite-valued semantics are simple, and general methods for theorem proving exist. This raises the question to what extent and under what circumstances propositional logics represented in various ways can be approximated by finite-valued logics. It is shown that the minimal $m$-valued logic for which a given calculus is strongly sound can be calculated. It is also investigated under which conditions propositional logics can be characterized as the intersection of (effectively given) sequences of finite-valued logics., Comment: arXiv admin note: substantial text overlap with arXiv:math/0203204

Any intermediate propositional logic (i.e., a logic including intuitionistic logic and contained in classical logic) can be extended to a calculus with epsilon- and tau-operators and critical formulas. For classical logic, this results in Hilbert's $\varepsilon$-calculus. The first and second $\varepsilon$-theorems for classical logic establish conservativity of the $\varepsilon$-calculus over its classical base logic. It is well known that the second $\varepsilon$-theorem fails for the intuitionistic $\varepsilon$-calculus, as prenexation is impossible. The paper investigates the effect of adding critical $\varepsilon$- and $\tau$-formulas and using the translation of quantifiers into $\varepsilon$- and $\tau$-terms to intermediate logics. It is shown that conservativity over the propositional base logic also holds for such intermediate $\varepsilon\tau$-calculi. The "extended" first $\varepsilon$-theorem holds if the base logic is finite-valued G\"odel-Dummett logic, fails otherwise, but holds for certain provable formulas in infinite-valued G\"odel logic. The second $\varepsilon$-theorem also holds for finite-valued first-order G\"odel logics. The methods used to prove the extended first $\varepsilon$-theorem for infinite-valued G\"odel logic suggest applications to theories of arithmetic.

A logic satisfies the interpolation property provided that whenever a formula {\Delta} is a consequence of another formula {\Gamma}, then this is witnessed by a formula {\Theta} which only refers to the language common to {\Gamma} and {\Delta}. That is, the relational (and functional) symbols occurring in {\Theta} occur in both {\Gamma} and {\Delta}, {\Gamma} has {\Theta} as a consequence, and {\Theta} has {\Delta} as a consequence. Both classical and intuitionistic predicate logic have the interpolation property, but it is a long open problem which intermediate predicate logics enjoy it. In 2013 Mints, Olkhovikov, and Urquhart showed that constant domain intuitionistic logic does not have the interpolation property, while leaving open whether predicate G\"odel logic does. In this short note, we show that their counterexample for constant domain intuitionistic logic does admit an interpolant in predicate G\"odel logic. While this has no impact on settling the question for predicate G\"odel logic, it lends some credence to a common belief that it does satisfy interpolation. Also, our method is based on an analysis of the semantic tools of Olkhovikov and it is our hope that this might eventually be useful in settling this question.

We give examples of calculi that extend Gentzen's sequent calculus LK by unsound quantifier inferences in such a way that (i) derivations lead only to true sequents, and (ii) proofs therein are non-elementarily shorter than LK-proofs., Comment: 21 pages. July 2017 preprint

G\"odel logic with the projection operator Delta (G_Delta) is an important many-valued as well as intermediate logic. In contrast to classical logic, the validity and the satisfiability problems of G_Delta are not directly dual to each other. We nevertheless provide a uniform, computational treatment of both problems for prenex formulas by describing appropriate translations into sets of order clauses that can be subjected to chaining resolution. For validity a version of Herbrand's Theorem allows us to show the soundness of standard Skolemization. For satisfiability the translation involves a novel, extended Skolemization method., Comment: 23 pages, accepted for LMCS (Logical Methods in Computer Science)

First-order Goedel logics are a family of 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 Goedel 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 Goedel logics are also characterized., Comment: 37 pages

The prenex fragments of first-order infinite-valued Goedel logics are classified. It is shown that the prenex Goedel logics characterized by finite and by uncountable subsets of [0, 1] are axiomatizable, and that the prenex fragments of all countably infinite Goedel logics are not axiomatizable., Comment: 6 pages; forthcoming in ISMVL 2003

Bernays introduced a method for proving underivability results in propositional calculi by truth tables. In general, this motivates an investigations of how to find, given a propositional logic, a finite-valued logic which has as few tautologies as possible, but which has all the valid formulas of the given logic as tautologies. It is investigated how far this method can be carried using (1) one or (2) an infinite sequence of finite-valued logics. It is shown that the best candidate matrices for (1) can be computed from a calculus, and how sequences for (2) can be found for certain classes of logics (including, in particular, logics characterized by Kripke semantics)., Comment: 20 pages

It is shown that G-up, the quantified propositional Goedel-Dummett logic based on the truth-values set V-up = {1 - 1/n : n >= 1} u {1}, is decidable. This result is obtained by reduction to Buechi's theory S1S. An alternative proof based on elimination of quantifiers is also given, which yields both an axiomatization and a characterization of G-up as the intersection of all finite-valued quantified propositional Goedel logics., Comment: v.2: 17 pages, revised published version (v.1: 15 pages)

Takeuti and Titani have introduced and investigated a logic they called intuitionistic fuzzy logic. This logic is characterized as the first-order Goedel logic based on the truth value set [0,1]. The logic is known to be axiomatizable, but no deduction system amenable to proof-theoretic, and hence, computational treatment, has been known. Such a system is presented here, based on previous work on hypersequent calculi for propositional Goedel logics by Avron. It is shown that the system is sound and complete, and allows cut-elimination. A question by Takano regarding the eliminability of the Takeuti-Titani density rule is answered affirmatively., Comment: v.2: 15 pages. Final version. (v.1: 15 pages. To appear in Computer Science Logic 2000 Proceedings.)