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- Matthias Baaz, Richard Zach
- CSL
- 2000

Takeuti and Titani have introduced and investigated a logic they called intuitionistic fuzzy logic. This logic is characterized as the first-order Gödel logic based on the truth value set [0, 1]. The logic is known to be axiomatizable, but no deduction system amenable to prooftheoretic, and hence, computational treatment, has been known. Such a system is… (More)

- Matthias Baaz, Norbert Preining, Richard Zach
- Ann. Pure Appl. Logic
- 2007

First-order Gödel 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 Gödel logics GV (sets of those formulas which evaluate to 1 in every interpretation into V ). It is shown that GV is axiomatizable iff V is finite, V is… (More)

- Matthias Baaz, Richard Zach
- ISMVL
- 1998

Entailment in propositional Gödel logics can be defined in a natural way. While all infinite sets of truth values yield the same sets of tautologies, the entailment relations differ. It is shown that there is a rich structure of infinite-valued Gödel logics, only one of which is compact. It is also shown that the compact infinite-valued Gödel logic is the… (More)

- Matthias Baaz, Christian G. Fermüller, Richard Zach
- Bulletin of the EATCS
- 1993

MUltlog is a system which takes as input the specification of a finitely-valued first-order logic and produces a sequent calculus, a natural deduction system, and a calculus for transforming a many-valued formula to clauses suitable for many-valued resolution. All generated rules are optimized regarding their branching degree. The output is in the form of a… (More)

- Matthias Baaz, Norbert Preining, Richard Zach
- ISMVL
- 2003

The prenex fragments of first-order infinite-valued Gödel logics are classified. It is shown that the prenex Gödel logics characterized by finite and by uncountable subsets of [0,1] are axiomatizable, and that the prenex fragments of all countably infinite Gödel logics are not axiomatizable.

- Matthias Baaz, Alexander Leitsch, Richard Zach
- CSL
- 1995

It is shown that the infinite-valued first-order Gödel logic G based on the set of truth values {1/k : k ∈ ω \ {0}} ∪ {0} is not r.e. The logic G is the same as that obtained from the Kripke semantics for first-order intuitionistic logic with constant domains and where the order structure of the model is linear. From this, the unaxiomatizability of Kröger’s… (More)

- Richard Zach
- Synthese
- 2003

After a brief flirtation with logicism in 1917–1920, David Hilbert proposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays and Wilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the development of axiomatic systems for ever stronger and more… (More)

We exhibit a construction principle for natural deduction systems for arbitrary finitely-many-valued first order logics. These systems are systematically obtained from sequent calculi, which in turn can be extracted from the truth tables of the logics under consideration. Soundness and cut-free completeness of these sequent calculi translate into soundness,… (More)