A Natural Deduction System for Intuitionistic Fuzzy Logic Matthias Baaz 1⋆ , Agata Ciabattoni 1⋆⋆ , and Christian G. Ferm¨ uller 2 1 Institut f¨ ur Algebra und Computermathematik E118.2, Technische Universit¨at Wien, A–1040 Vienna, Austria, 2 Institut f¨ ur Computersprachen E185, Technische Universit¨at Wien, A–1040 Vienna, Austria {baaz, agata, chrisf}@logic.at Abstract. Intuitionistic fuzzy logic IF was introduced by Takeuti and Titani. This logic coincides with the first-order G¨odel logic based on the real unit interval [0, 1] as set of truth-values. We present a natural de- duction system NIF for IF . NIF is defined by suitably translating a first-order extension of Avron’s hypersequent calculus for G¨odel logic. Soundness, completeness and normal form theorems for NIF are pro- vided. 1 Introduction Intuitionistic fuzzy logic IF was defined by Takeuti and Titani [18] as the logic of the complete Heyting algebra over the real unit interval [0, 1]. IF turned out to coincide with the first-order G¨ odel logic based on the truth-value set [0, 1]. The finite-valued propositional versions of this logic were introduced by G¨odel in 1933 to show that intuitionistic logic does not admit a characteristic finite ma- trix [11]. Dummett later generalized these to the infinite set of truth-values [0, 1], and showed that the set of its tautologies LC is axiomatized as intuitionistic logic extended by the linearity axiom (A ⊃ B) ∨ (B ⊃ A). Together with  Lukasiewicz and product logic, LC is also one of the most important formalizations of fuzzy logic [12]. Cut-free sequent calculi for LC have been defined in [16, 1, 8]. However these calculi make use of rules with an arbitrary number of premises. A cut-free calcu- lus for LC which does not have this drawback has been introduced in [2]. This calculus, called GLC, is based on hypersequents — a simple and natural gener- alization of Gentzen’s sequents to multisets of sequents (see [3] for an overview). Contrary to the other calculi for LC , in GLC the rules introducing the connec- tives are exactly those of Gentzen’s sequent calculus LJ for intuitionistic logic. In [7] the hypersequent calculus GLC ∀∃ was defined by extending GLC with rules for quantifiers. It was shown that GLC ∀∃ is sound and complete for IF . ⋆ Research supported by the Austrian Science Fund under grant P–12652 MAT ⋆⋆ Research supported by EC Marie Curie fellowship HPMF–CT–1999–00301