If you draw implications on a false asumption then the result is useful only to show that an assumption is false.
So if PA → 1=2 then PA → 1<>2. How is that useful?
If PA → 6 is prime then PA also → 6 is not prime.
Once you assume that PA implies something that PA actually implies is false, you get a logical contradiction. Either PA is inconsistent or PA does not imply the false thing.
How can it be useful to reason about what we could prove from false premises? What good is it to pretend that PA is inconsistent?
Larry, one of them is counterfactual.
If you draw implications on a false asumption then the result is useful only to show that an assumption is false.
So if PA → 1=2 then PA → 1<>2. How is that useful?
If PA → 6 is prime then PA also → 6 is not prime.
Once you assume that PA implies something that PA actually implies is false, you get a logical contradiction. Either PA is inconsistent or PA does not imply the false thing.
How can it be useful to reason about what we could prove from false premises? What good is it to pretend that PA is inconsistent?