Perplexed comments on Rationality Quotes: January 2011 - Less Wrong

2 Post author: wedrifid 03 January 2011 05:24AM

You are viewing a comment permalink. View the original post to see all comments and the full post content.

Article Navigation

Comments (268)
Tags: quotes

Comments (268)

You are viewing a single comment's thread. Show more comments above.

Comment author: HonoreDB 24 January 2011 11:13:44PM *  1 point [-]

So even if I throw (P->Q) -> (Q->P) as an additional axiom in I can't get a general explosion in first order logic.

Sure you can.

  1. R v ~R (axiom)
  2. R -> (R v ~R) (by 1)
  3. (R v ~R) -> R (by the new axiom)
  4. R (by 1 and 3)

Edit:

(P->Q) -> (Q->P) is not a fallacy. ∀P,Q: (P->Q) -> (Q->P) is a fallacy, and its negation is ∃P,Q: (P->Q)^~(Q->P) which is indeed a theorem in first order logic.

Comment author: Perplexed 25 January 2011 01:09:05AM 2 points [-]

Huh?? If you allow quantification over propositions, you are no longer using first order logic.

I think you were closer to being on track before your edit. The first thing to realize is that a fallacy is not a false statement. It is an invalid inference scheme or rule of inference.

So, with P and Q taken to be schematic variables (to be instantiated as propositions), the following is a fallacy (affirming the consequent):

P -> Q |- Q -> P

Or, you could have simply corrected the words "additional axiom" in the quoted claim to "additional axiom scheme".

Powered by Reddit Powered by Reddit