The deductive closure of a set S of propositions with respect to a set R of deductive rules is the set C of propositions deducible from propositions in S by means of rules in R. (Note that the union of S and C is the same as C itself.)
In the case where S is the null set and R is the set
of deductive rules used in first-order logic, the
deductive closure C is the set of tautologies
of first-order logic.
Let us consider another example. Suppose S = { p } and let R contain the deductive rule r:
X --> ~~X (where X is a placeholder, -->
symbolizes material implication and ~
symbolizes negation). C contains at least the proposition p; by rule r, it follows that ~~p. So C contains at least the elements p and ~~p. And from these, the propositions ~~p (which we already know is in C) and ~~~~p follow. Continuing this indefinitely, we see that the deductive closure C with respect to R of S is { p, ~~p, ~~~~p, ~~~~~~p, ... }. In other words, C contains p with either zero or some positive even number of ~s.