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Carinthium comments on Skepticism about Probability - Less Wrong Discussion
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The logical sceptic could argue that they are showing that logic is self-defeating- that when logic is taken to its ultimate conclusion it is shown to be false, therefore logically logic should be rejected. This is precisely what I would argue.
As for the matter of reality- if it exists, then of course it doesn't go away when we stop believing it. But how do we know that?
If logic is pennies, and blue is a sock, then why are cats punctuation marks? Yes, because your mask is Maryland, and WiFi smokes bananas.
(This is what happens when you live in a world without logic, and is the only response you should have to someone who is a logic skeptic).
See my below response gedymin.
Logically, "self-defeating" is not equal to "not self-defeating". If the skeptic rejects logic, then he should accept that "self-defeating" is equal to "not self-defeating". Therefore, if logic is self-defeating, then logic is also not self-defeating.
As for the second point - the epistemic perspective is more important than the ontological one. Seriously, read the conclusion of the "Simple truth".
This debate is getting silly, I'm out of here.
On the first point, if you get to a conclusion within logic which marks it as "self-defeating", then from a logical perspective logic doesn't work. Non-logic doesn't matter for those who aren't logical, but for a logical person logic matters.
On the second point, once you start postulating actually viable alternatives to the world not existing, and considering the Evil Demon Argument, there is nothing in there which is actually dealt with.