Summary: Lattice-valued logic with truth values in a lattice is a kind of an important nonclassical logic. This paper is a continuation of our work which aims at providing a logical foundation for uncertain information processing theory, especially for fuzziness, incomparability in uncertain information, as well as transformation of uncertain information in reasoning. Based on the gradational \(L\)-type lattice-valued propositional logic \(L_{\nu pl}\), established previously by the authors [Inf. Sci. 114, 205-235 (1999; Zbl 0936.03023)], with truth-values in a lattice implication algebra, and following the discussion [the authors, Int. J. Gen. Syst. 29, 53-79 (2000; Zbl 0953.03028)] about the semantic systems of gradational \(L\)-type lattice-valued first-order logic \(L_{\nu fl}\) with truth values in a lattice implication algebra, the corresponding syntax systems of \(L_{\nu fl}\), are investigated in this paper, where the \(L\)-type \(\alpha\)-Soundness Theorem, \((\alpha,\beta)\)-Completeness Theorem, \((\alpha,\beta)\)-Consistency Theorem and \((\alpha,\beta, \theta)\)-Deduction Theorem are proved, and the compactness of \(L_{\nu fl}\) is also discussed.