Finally, the completeness theorem of this α-resolution principle and the soundness theorem for the strong α-resolution are also proved. Then, an α-resolution principle, which can be used to judge if a first-order lattice-valued logical formula in LF(X) is false at a truth-valued level α (i.e., α-false), is established. Firstly, some concepts about lattice-valued resolution principle for LF(X) are introduced and the Herbrand theorem for LF(X) is proved. In the present paper, as a continuous work about α-resolution principle based on lattice-valued propositional logic LP(X) (Information Sciences 130 (2000) 1–29) whose algebra of truth-values is a relatively general lattice – lattice implication algebra (LIA), the lattice-valued resolution principle for the corresponding first-order lattice-valued logic system LF(X) is focused. The presented work provides a key theoretical support for automated reasoning approaches and algorithms in linguistic truth-valued logic, which can further support linguistic information processing for decision making, i.e., reasoning with words. Similarly, this conclusion also holds for linguistic truth-valued lattice-valued first-order $ () $į(X) is equivalently transformed into that for Firstly, the general form of α-resolution principle for lattice-valued propositional logic Concretely, the general form of α-resolution principle based on the above lattice-valued logic is equivalently transformed into another simpler lattice-valued logic system. This paper is focused on resolution-based automated reasoning theory in linguistic truth-valued lattice-valued logic based on linguistic truth-valued lattice implication algebra.
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