This paper focuses on investigating the inference rules with generalized quantifiers in a lattice-valued first-order logic system lF(X) with truth-values in a linguistic truth-valued lattice implication algebra (L-LIA). Since the qualifier constraint may become more important when considering the semantic of natural language, so in using multi-valued logic as a tool to model approximate reasoning, how to control the truth-value transfer during the inference process for some rule with qualifiers is very important, which is not the case in classical logic in which the logic deduction system is symbolic reasoning with strict syntactical proof, the semantics are too simple to be considered. This work put more effort on semantic interpretation and truth-valued transfer, especially present and investigate some reasoning rules with generalized quantifiers (rather than universal quantifiers and existential quantifier only) for lattice-valued first-order logic system lF(X). We prove the satisfiability and validity of these inference rules, where inference rules are interpreted by the semantic truth value transfer, which shows the control of truth-value level during the inference process.
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