Example 6
We will prove:
| A |
| ~A |
| B |
This is wierd! We have A, ~A in the premises, and something completely unrelated to the premises in the conclusion. So if we can prove this, we can prove anything from the premises A, ~A together. But is this really so surprising? Remember that {A, ~A} is inconsistent., and any argument with inconsistent premises is valid. Since every valid argument in PL can be proved, (we'll state this again later, but not prove it) this argument has a proof.
How do you prove it? Just disjoin "A" with "B", and then use disjunctive syllogism. This shows that in PL, anything follows from an inconsistent set of premises. Is this a good thing?