I would like to know if someone can explain in a somehow down to earth (almost logic free) way why is it true that in an axiom system where there is some statement $P$ such that $P$ and its negation $\lnot P$ are true, then every statement in the system is true?

I'm not sure if this can be done, but basically since I don't know any formal logic at all, I'm interested in seeing if at least the argument can be conveyed in an intuitive way, or if the idea can be explained without talking about first or second order logic and using symbols like $\top$, $\bot$, and $\vdash$.

This previous question is like the formal version (which I don't understand) so maybe my question can be thought of as a version for dummies of that question. 

Thanks a lot in advance.

I'll try to say this all in plain English:

Let's say we decide to accept the following two facts: (1) "I am a fish", and (2) "I am not a fish". Just keep those in mind.

Now let's pick any old statement, say: (3) "You can fly". Now let's prove that the statement is true!

Alright, we've already accepted that (1) "I am a fish". Of course, any time I have a true statement P, I can make a new true statement by making the statement "P or Q is true." Because to check if an 'or' statement is true, I only need to check that one of them is true. (If I tell you "My name is Dylan OR I can spit fire," you don't need to wait around with a fire extinguisher to tell if that statement is true. It's true because the first part of it is true).

So by this logic, the statement (4) "I am a fish or you can fly" must be true (since the first part is true.)

OK, but now let's say, in general, I have some 'or' statement "P or Q" and I know for a fact that the whole statement is true. If I also know that P is false then I can conclude that Q is true. Right? Because an 'or' statement is true if and only if at least one of the statements inside it is true, so if I rule out one of them the other one must be true. (So if I always tell the truth and I tell you that you have a billion dollars in your bank account OR I just ate a sandwich, you can check your bank account and quickly conclude that I just ate lunch... unless you're very wealthy.)

Alright, so far so good. We know the statement "I am a fish or you can fly" is definitely true. But wait, we also know that the statement "I am a fish" is false (remember, it's one of the things we assumed in the very beginning!). So that means, by what we just talked about, that the statement "You can fly" must be true.

So voilà! Using the magic of a contradictory system, we've proven you can fly!
