Tautologies
When a wff comes out true on every row of the truth table, that wff is a tautology.
So to determine whether a wff is a tautology, we do a truth table. Here's an
example. Let's see if (A
⊃ (B
A)) is a tautology:
| A | B | (B ⊃A) | (A ⊃(B ⊃A)) |
| T | T | T | T |
| T | F | T | T |
| F | T | F | T |
| F | F | T | T |
The two left-most columns are the base columns which show the possible assignments of truth and falsity to the two sentence letters. The third column computes the truth-value of the consequent of the conditional, and the final column computes the truth-value for the whole wff. We see that there are all "T"s in the final column. Hence (A ⊃ (B ⊃ A)) is a tautology.
Why does this wff come out always true? The consequent (B ⊃ A) is true on rows 1, 2 and 4. So that leaves row 3 as the only row where the whole conditional could be false. But the antecedent is also false on that row; hence the conditional is true on row 3 as well.
Let's look at another tautology.
| C | D | (C & ~C) | ((C & ~C) ⊃D) |
| T | T | F | T |
| T | F | F | T |
| F | T | F | T |
| F | F | F | T |
Notice that this the antecedent of this conditional comes out false on every row, and thus the conditional is true on every row, since a conditional is only false when its antecedent is true and its consequent false.
Certainly not all tautologies are conditionals. Consider:
| A | B | (~B v A) | (~A v (~B v A)) |
| T | T | T | T |
| T | F | T | T |
| F | T | F | T |
| F | F | T | T |
Notice that the truth-table for this wff bears a striking similarity to our first example. Can you explain why that's so?
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