Other Gappy Truth Table Tests

In the last section we showed how the  Gappy Truth Table Method streamlines our test for the validity of arguments.  That same method can be used to test for all the semantic properties. The strategy is the same: We only compute rows which are relevant for testing the semantic property in question.  Here we'll discuss how to test tautologies, contradictions, and contingencies.  We'll leave testing for consistency and logical equivalence as an exercise.

Tautologies:  A tautology is true on every row of the truth table. To test for tautologies using the Gappy Method we attempt to show that a wff is not a tautology.  To do that we just have to find one row of the truth table where the wff comes out false. If we discover such a row we can stop. The wff in question is not a tautology. If the wff comes out true on a row, we go on to another row. If we wind up testing all rows, and the wff is true on all of them, then it is a tautology.

Contradictions: A contradiction is false on every row of the truth table. To test for contradictions using the Gappy Method we attempt to show that a wff is not a contradiction.  To do that we just have to find one row of the truth table where the wff comes out true. If we discover such a row we can stop. The wff in question is not a contradiction. If the wff comes out false on a row, we go on to another row. If we wind up testing all rows, and the wff is false on all of them, then it is a contradiction.

Contingencies: A contingent wff comes out true on at least one row and false on at least one row. So we can stop computing the truth values once we've found one row where the wff is true and one where it is false.

 We now turn to a much more elegant semantic method, the Truth Tree Method.