I got this from the thread https://philosophy.stackexchange.com/questions/34082/why-are-conditionals-with-false-antecedents-considered-true And this is what Stefan Perko answered.


>Suppose it is raining and it is not raining. Then it is raining. Hence, it is raining OR the moon is made of cheese (1). Since it is not raining, the moon is made of cheese (2).

>If you accept:

> 1. disjunction introduction, i.e. P -> (P or Q)
> 2. disjunctive syllogism, i.e. ((P or Q) and not P) -> Q
then you need to accept that contradictory assumptions imply everything.

>> Also there is a neat mathematical reason, why contradiction "should" imply everything. You can put a (pre)-order on the set of propositions by saying P is less then or equal to Q if and only, if P => Q. Then contradiction is the minimum in this ordered set, because it is less then every other proposition

The first part seems pretty straightforward except how anything can be true and false at the same time seems like a big no-no in logic.

The second quoted part about why contradiction "should" imply everything I can't really follow. Which is the preorder and on what propositions? Isn't it arbitrary to come up with the statement "P is less then or equal to Q? And why is contradiction minimum in this ordered set?