We say that a set of formulas *F _{1}, F_{2}, ..., F_{n}*

*F _{1} &and F_{2} &and ... &and F_{n} &rarr G*
is valid iff

Proof: *¬ (F &rarr G) = ¬ (¬ F &or G) = (F &and ¬ G)*.

Note that * anything* is a logical consequence of *□* (or an
inconsistent set of clauses).

Proof: *□ &rarr X = ¬ □ &or X = True &or X = True*.

Thus, if the result that is derived from a set of clauses is to be meaningful, the set of clauses must be consistent.

We could prove that a formula *G* follows from formulas *F _{i}* by truth
table, or by algebraically reducing one of the logical consequence formulas
to true (false) as in the example above.