Consistency
We have introduced the notions of tautologyhood, contradiction, contingency and logical equivalence. The first three apply to individual wffs, and the last to pairs of wffs. Consistency and inconsistency are properties of sets of wffs. A set of wffs is consistent just in case there is at least one row of the truth table for the set where each wff comes out true. A set of wffs is inconsistent just in case there's no row of the truth table where all the wffs come out true.
Let's start with an obviously inconsistent set, the set {P, ~P}. This is a set of wffs, since both P and ~P are wffs. When P is true, ~P is false, and when P is false, ~P is true. So they can never be true "at the same time."
A less obvious example is the set {(P ≡ Q), ~((P ⊃ Q) & (Q ⊃ P))}. Let's do the truth table:
| P | Q | (P ≡ Q) | (P ⊃ Q) | (Q ⊃ P) | (P ⊃ Q) & (Q ⊃ P)) | ~((P ⊃ Q) & (Q ⊃ P)) |
| T | T | T | T | T | T | F |
| T | F | F | F | T | F | T |
| F | T | F | T | F | F | T |
| F | F | T | T | T | T | F |
If we compare the truth values for the two wffs in the set, indicated in the two
yellow columns, we see that there are no rows in which each member of the set gets the
truth value "T". So the set is inconsistent.
How about a consistent set? As an exercise, demonstrate that the following sets are consistent:
- {(P ⊃ Q), P}
- {(P v Q), ~P}
- {(P ⊃ Q), ~Q}
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