Why in an inconsistent axiom system every statement is true? (For Dummies) 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.

In an inconsistent system you can prove a false proposition. Suppose $F$ is a false proposition. Let $P$ be a proposition. We will show that $P$ is true.
Since $F$ is false, $\lnot F$ is true. It follows that $\lnot F \vee P$ is true. Why? Because for every true propostion $X$, $X\vee Y$ is also true for any proposition $Y$.
What can we deduce from $\lnot F \vee P$? We assumed that $F$ is true, from which it follows that $\lnot F$ is  false. But if $\lnot F \vee P$ is true, which we showed, at least one of $\lnot F$ or $P$ must be true. Since $\lnot F$ is false, $P$ must be true.
Alternative proof:
$X$ implies $Y$ is equivalent to $\lnot\ X \vee Y$. Let's say $X$ is false. It follows that $\lnot X$ is true, which implies $\lnot X \vee Y$, which is equivalent to $X$ implies $Y$. Since we assumed $X$ to be true, we can conclude that $Y$ is true.
