But because we are also ignoring the excluded middle, this is not a problem. Actually, this becomes somewhat (!) clearer in formal logic.
Our premise is that the universe is not consistent: (Pv~P) { In English: There's some statement P such that both P and its negation, ~P, are true.}
From the logic on this page, we can arrive at arbitrary statement: Q
However, if we make statement Q more specific, we can say Q=~(P=P), thus "proving" the law of identity false.
We could also substitute Q=~(Pv~P)
Aha, you say, contradiction! And you'd be right ...except our premise is (Pv~P). Substituting, we get:
((Pv~P)v~(Pv~P))
Thus, we have proven that in an inconsistent universe, the excluded middle does and does not apply.
However, those of us who are not logicians will be glad to know that the concept of truth handed in its resignation some time ago. An inconsistent universe is in no way coherent, but could theoretically exist.
I understand that, but if the excluded middle does and does not apply, then it applies and doesn't not-apply. Then the initial "does and does not apply" is no longer the case instantly, and we have only applies.
(P & ¬P) & ¬(P & ¬P) --> ¬(P & ¬P) & ¬[(P & ¬P)]
Notice that ¬(P & ¬P) and ¬[(P & ¬P)] are equivalent.
So (P & ¬P) & ¬(P & ¬P) --> ¬(P & ¬P)
This is because ∀x([¬(P & ¬P)] ∈ x) ¬[(P & ¬P)]
For all worlds with ¬(P & ¬P), ¬[(P & ¬P)]
For all worlds with excluded middle, not-not-excluded middle. Since this supposed world has not-excluded middle and excluded middle, then not-not-excluded middle (because excluded middle). Therefore, excluded middle.
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