= 27d567bc2d48d9f40e9ccc20ea370b15)
DanielFilan comments on Stupid Questions December 2014 - Less Wrong
You are viewing a comment permalink. View the original post to see all comments and the full post content.
= 783df68a0f980790206b9ea87794c5b6)
Loading…= b715d02e85f7b3b4ae65ca3d3e450868)
Subscribe to RSS Feed
Powered by Reddit
= f037147d6e6c911a85753b9abdedda8d)
You are viewing a comment permalink. View the original post to see all comments and the full post content.
Comments (341)
The whole point of this discussion is that I don't think that your proof structure is valid. To be honest, I'm not sure where your confusion lies here. Do you think that all statements that are true in PA are provable in PA? If not, how are you deriving provable(if x then q) from (if x then q)?
In regards to your above comment, just because you have provable(x or not(x)) doesn't mean you have provable(not(x)), which is what you need to deduce provable(if x then q).
To answer the below: I'm not saying that provable(X or notX) implies provable (not X). I'm saying...I'll just put it in lemma form(P(x) means provable(x):
If P( if x then Q) AND P(if not x then Q)
Then P(not x or Q) and P(x or Q): by rules of if then
Then P( (X and not X) or Q): by rules of distribution
Then P(Q): Rules of or statements
So my proof structure is as follows: Prove that both Provable(P) and not Provable(P) imply provable(P). Then, by the above lemma, Provable(P). I don't need to prove Provable(not(Provable(P))), that's not required by the lemma. All I need to prove is that the logical operations that lead from Not(provable(P))) to Provable(P)) are truth and provability preserving
Breaking my no-comment commitment because I think I might know what you were thinking that I didn't realise that you were thinking (won't comment after this though): if you start with (provable(provable(P)) or provable(not(provable(P)))), then you can get your desired result, and indeed, provable(provable(P) or not(provable(P))). However, provable(Q or not(Q)) does not imply provable(Q) or provable(not(Q)), since there are undecideable questions in PA.
Ohhh, thanks. That explains it. I feel like there should exist things for which provable(not(p)), but I can't think of any offhand, so that'll do for now.