Syntax Completeness and Its Properties on Formal Systems

Abstract

Syntax completeness is an important character of formal systems. In this paper, two questions are proposed that syntax completeness of a formal system S have relations with semantic completeness under what condition? What is relation between the syntax completeness of S and the syntax completeness of consistent extension of S? And that 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 first order predicate logic don't have syntax completeness.