Abstract
This paper is an attempt to develop the many-valued first-order fuzzy logic. The set of its truth, values is supposed to be either a finite chain or the interval 〈0, 1〉 of reals. These are special cases of a residuated lattice 〈L, ∨, ∧, ⊗, →, 1, 0〉. It has been previously proved that the fuzzy propositional logic based on the same sets of truth values is semantically complete. In this paper the syntax and semantics of the first-order fuzzy logic is developed. Except for the basic connectives and quantifiers, its language may contain also additional n-ary connectives and quantifiers. Many propositions analogous to those in the classical logic are proved. The notion of the fuzzy theory in the first-order fuzzy logic is introduced and its canonical model is constructed. Finally, the extensions of Godel's completeness theorems are proved which confirm that the first-order fuzzy logic is also semantically complete.
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