In this article, a sound and complete tableau system for Rational Pavelka Logic~RPL! is introduced. Extended formulas are used as the counterpart of the graded formulas. In this calculus, if we want to show that the graded formula~x, r! is tableau provable~in the finite fuzzy theory F, respectively!, we develop a tableau for the extended formula @r, x#~for the set of extended formulas $@r, x# , @x 1 , a 1 # , . . . , @x n , a n # %, respectively!. If this tableau closes we claim that~x, r! is tableau provable~in the fuzzy theory F, respectively!. We claim also that x is valid at the degree equal to the l.u.b. that allows the closure of the tableaux. Our tableaux are a first step toward efficient procedures of automated deduction in narrow fuzzy logic with truth constants.
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