The curious and amusing thing is that in FU, Smullyan call that error the "beginners error" (page 46). It consists in believing that the formula (a-> -b) & (a -> b) is a contradiction, where actually the formula is true in case a is false. A simpler example is (p-> -p). This is true for p false. What is curious and amusing is that Smullyan made that very error page 42 (of the first edition, and it is wrongly corrected in the second edition). Can you see it. Morality: consistent machines *can* be inconsistent, as any Loebian machine believes (-Bf -> -B- Bf). Well; that is neither a justification nor a consolation ...
At 12:26 23/09/04 +0200, Bruno Marchal wrote:
In the second paragraph of the "physics and sensations" section of my paper "the origin of physical laws and sensations" I made a rather stupid error (what a shame!).
Indeed I say "Note that neither G nor G* does prove it [where it is for Bp -> -B-p]". This is ridiculous, because G* proves Bp->p, for any p, and thus G* proves B-p -> -p, and thus (by contraposition) G* proves p->-B-p, and by propositional calculus Bp->-B-p. Worst, my justification was that Bf->B-f (where f = false). This is correct, but I infer from that that Bf->-B-f is not provable by G*. But G* proves both Bf->B-f and Bf->-B-f, that is Bf->f.
The error has no consequences for the rest of the paper, but still, why did I wrote that???
Please, don't hesitate to ask ANY questions, so that perhaps other errors will be single out. You can also propose more general critics. Don't be shy. (You can also ask questions about FU).