Thread:Jacky720/@comment-32182236-20200429164606

If there's one objection to my logic series that hasn't already been mentioned that will, it's very likely that said objection will be the Liar's Paradox. (It has been said to mean logic isn't true...)

Simply put:This statement is false.

Is it false? Or is it true? Or.. Is it something entirely?

Well, first, we have to put this into our formal logic.

P:¬P

This statement is identical to the one I just gave, as it asserts itself to be false.

So, how would we determine the truth value of this statement?

Well, let's go through the first few steps.


 * P1:¬P1
 * C1:P1->¬P1
 * C2:¬P1→P1
 * C3:¬P1→¬¬P1

That's actually all we need for right now. Phase 1 has already been complete. Here's why...


 * P2:P1∨¬P1
 * C4:¬P1 (P2, C1, [what is this rule actually called? If A implies B, and A or B is true, then B is true.. But I forget the name of that rule!])
 * C5:¬¬P1 (C4, C1, modus ponnens)
 * Cω+1:¬…¬P1 (Cω, C1, modus ponnens)
 * Cω+1:¬…¬P1 (Cω, C1, modus ponnens)

So, we're left with an infinite number of negation symbols. The system "freezes" at this point, because 1+ω=ω. So this infinite set is the "final" answer. This is the answer to the Liar's paradox. So, let's apply the negation symbols to P1, one by one...

True, not true, true, not true, true, not true...

1, 0, 1, 0, 1, 0...

1-1+1-1+1-1+1-1...

Wait.. That last one seems pretty familiar...

In fact, it has been proven that this series goes to 1/2.

Now, 1/2 is not a superposition of 1 and 0.. It's something completely different.

Therefore, it is neither true or false, but something else. It appeared to be a contradiction, but after the infinite steps, we have discovered what it really is.

So, does logic break? ...Doesn't look like it. (Though we should probably say that it's not a proposition, as propositions are either true, false, or a superposition of the two, also known as indeterminate. That's what happens when we bring the Law of Excluded Middle and turn it from classical to quantum.)

Also, the similar paradox of "The statement below is false", "The statement above is true" is similar.


 * P:¬Q
 * Q:P

We can substitute Q for P, and we get that statement P is ¬P. It's the same thing.

..Actually, though.. If there's a set of all propositions, then it's possible to prove that the Liar's Paradox cannot be a member of that set.. But that's given an assumption that such a set even exists. (After all, we can already prove that the number of propositions is infinite. P, ¬P, ¬¬P, ¬...¬P)

But it does tell us the problem isn't with logic itself. If we apply everything correctly, we learn that it's not true, and it's not false. It's not both. It's neither. LNC still holds.

So, are there other ways to get a truth value of 1/2? Has logic become useless?

Nope. All propositions refer to a certain object holding a certain property. A proposition is true if the object holds the property in question, and false if it doesn't.

The only way to get "1/2" is with an infinite regress, like the Liar's Paradox.

Undertale gives us true premises to start with, meaning foundationalism holds, meaning infinitism doesn't.

Therefore no infinite statements exist, except those that couldn't be proven or disproven anyway.

Thus, logic is not broken! (At least, not in Undertale.)

Edit:Wait.. That means P2 isn't true..

Which means the proof doesn't hold..

Guess I'll have to see if I can prove there's a set of all statements... 