Contradictions
A contradiction is a wff which is always false. To test a wff to see whether
it is a contradiction, we construct the truth-table for the wff and look at the final
column. If the final column has all "F"s, then the wff is a contradiction.
Let's test ((A v B) & (C
⊃ ~A)) to see
whether it is a contradiction:
| A | B | C | (A v B) | ~A | (C ⊃ ~A) | ((A v B) & (C ⊃ ~A)) |
| T | T | T | T | F | F | F |
| T | T | F | T | F | T | T |
| T | F | T | T | F | F | F |
| T | F | F | T | F | T | T |
| F | T | T | T | T | T | T |
| F | T | F | T | T | T | T |
| F | F | T | F | T | T | F |
| F | F | F | F | T | T | F |
((A v B) & (C ⊃ ~A)) is not a contradiction, because it is true on at least one row of the truth table. In contrast, consider (A & (B & ~A)):
| A | B | ~A | (B &~A) | (A & (B & ~A)) |
| T | T | F | F | F |
| T | F | F | F | F |
| F | T | T | T | F |
| F | F | T | F | F |
This wff is false on all rows (in the final column), and so is a contradiction.
You should begin to think about the relationship between contradictions and tautologies. Tautologies are always true. But what happens if we put a tilde in front of a tautology? What happens if we put a tilde in front of a contradiction? These relationships are explored further in the exercise: Logical Status.
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