0. P ∧ P’
1. P (simplification from 0)
2. P’ (simplification from 0)
3. P ∨ M (addition from 1)
4. M (disjunctive syllogism from 2 and 3) Q.E.D.
Or in words: P and not P are true. Therefore, P is true. Also, not P is true. Since P is true, P or some other statement M are true. But since not P is true, P or M must collapse into just M.
But we just made M up. It can be whatever we want. Let’s substitute in some actual sentences to see how this works:
P = It is raining
Not P = It is not raining
M = Earth is made out of cottage cheese
It is both raining and not raining. Therefore, it is raining. Also, it is not raining. If it’s raining, that means that we can say that it is true that it is raining or Earth is made out of cottage cheese. But it’s not raining, so in order for that to be true, Earth must be made out of cottage cheese.
The logic is valid (and fairly straightforward), but the conclusion is categorically wrong. Note that this relies on the logician’s use of inclusive as opposed to exclusive or. The latter is what we use in everyday language, but the former is what we mean by “or” in a deductive argument.
So if you’re ever examining your own beliefs and discover a contradiction, be warned!