Path: utzoo!attcan!uunet!lll-winken!lll-tis!ames!pasteur!ucbvax!bloom-beacon!NRL-AIC.ARPA!hartley
From: hartley@NRL-AIC.ARPA (Ralph Hartley)
Newsgroups: comp.ai.digest
Subject: Re: Are all reasoning systems inconsistent?
Message-ID: <19880826032115.1.NICK@HOWARD-JOHNSONS.LCS.MIT.EDU>
Date: 26 Aug 88 03:21:00 GMT
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Approved: ailist@ai.ai.mit.edu

Date: Tue, 16 Aug 88 08:06 EDT
From: Ralph Hartley <hartley@nrl-aic.arpa>
Subject: Re: Are all reasoning systems inconsistent?
To: AILIST-REQUEST@LCS.MIT.EDU
Resent-Date: Thu, 18 Aug 88 14:12 EDT
Resent-From: Rob Austein <SRA@XX.LCS.MIT.EDU>
Resent-To: ailist-request@MC.LCS.MIT.EDU

Your problem lies in T2

>T2. Aa[P(s("~P(*)",a)) -> ~P(a)] ; If I can prove that I can't prove X,
>                                   then I can't prove X

This implies

Ea(~P(a))

i.e. that the system is consistent. Godel's 2nd (less well known) theorem
states that if it is possible to prove a system consistent within the system
then the system is NOT consistent. Therefore T2 cannot be a theorem in any
consistent system.

BTW - This is also a flaw in Hofstadter's reasoning about the prisoners dilema.
His argument goes as follows:
1. The other player uses the same reasoning as I do.
2. This reasoning produces a unique result (cooperate or defect but not both)
3. Therefor whatever I do he will do too.
4. So I should cooperate.

The problem, again, is that (1) and (2) imply that my logic is consistent -
therefore it is not.

			Ralph Hartley
			hartley@nrl-aic.ARPA