1. With respect to a deductive system , a set of sentences such that for all formulae , if and only if , i.e., the members of are precisely its -consequences. An equivalent definition is that is an -theory if and only if is equal to its own deductive closure, i.e., .
2. In model theory for a deductive system , a set of sentences (the theory of the model ) whose members are precisely those sentences to which assigns the truth value of (i.e., are true in ) or some other designated value in the semantics for .