Relation between Semantic Completeness and Syntax Completeness on General Formal Systems

Semantic completeness and syntax completeness are important characters of the general formal systems, two the completeness have different meaning for a formal system. This paper analyze the relation between semantic completeness of a formal system S and syntax completeness under certain condition and the relation between the syntax completeness of S and the syntax completeness of consistent extension of S, and following results are proved. (1) If S has syntax completeness then S must has semantic completeness when S has soundness. (2) If S doesn´t has syntax completeness then any consistent extensions of S don´t have syntax completeness either, the converse not holds. (3) Any extensions of the first order predicate logic don´t have syntax completeness.