Logical Consequence

We say that a set of formulas F1, F2, ..., Fn entails a formula G, written F1 &and F2 &and ... &and Fn ⊨ G, or that G is a logical consequence of the Fi, iff G is true in any interpretation where F1 &and F2 &and ... &and Fn is true. This is equivalent to saying that F1 &and F2 &and ... &and Fn &rarr G is valid.[The fact that &alpha ⊨ &beta iff &alpha &rarr &beta is valid is called the deduction theorem.]

F1 &and F2 &and ... &and Fn &rarr G is valid iff F1 &and F2 &and ... &and Fn &and ¬ G is inconsistent.

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 Fi by truth table, or by algebraically reducing one of the logical consequence formulas to true (false) as in the example above.