maximal consistent set in formal logic, any set of sentences S that is consistent – i.e., no contradiction is provable from S – and maximally so – i.e., if T is consistent and S 0 T, then S % T. It can be shown that if S is maximally consistent and s is a sentence in the same language, then either s or – s (the negation of s) is in S. Thus, a maximally consistent set is complete: it settles every question that can be raised in the language. See also COMPLETENESS , SET THEOR. P.Mad.