On 22 Oct 2013, at 19:07, meekerdb wrote:

On 10/22/2013 6:19 AM, Bruno Marchal wrote:

"[]p -> p" is correctness. It is trivially true for the machine I consider, because they are correct by definition/choice.

Consistency is correctness on the f: []f -> f. It is a very particular case of correctness. There are machines which are not correct, yet consistent. For example Peano-Arithmetic + the axiom beweisbar('f').

Believing '0=1', does not make you inconsistent. Only non correct.

?? But in ordinary logic a false proposition allows you to prove anything. So if I prove '0=1' then I can prove any proposition - which is the definition of inconsistency.

OK. I should have written "Believing '0=1'", does not make you inconsistent. Only non correct.

Or if your prefer: believing "believing '0=1' does not make you inconsistent. Only non correct.

If you add the axiom "0=1" to PA, it get inconsistent, as, like you say, ordinary logic will imply that you can prove all proposition.

But here I was not adding "0=1" as an axiom to PA, I was adding "believe "0=1"" as an axiom, and from this you cannot prove all propositions. In particular you cannot prove "0=1". You can only prove that you can prove all propositions. That might make you stupid, and certainly unsound, but not inconsistent.

Just keep in mind that Bf -> f is not provable by PA. And keep well the difference between PA asserts (proves, believes) p, and PA asserts Bp.

Bruno






Brent



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