On 07 Mar 2015, at 02:51, meekerdb wrote:
On 3/6/2015 7:24 AM, Bruno Marchal wrote:
That might depend on the context. Usually, in our computationalist
context it means true in the standard model of arithmetic, which is
"this reality" if you want.
In the modal context, it means true in this world (which in our
arithmetical context is NOT necessarily among the accessible world,
because we don't have []p -> p). With the logic of provability, we
cannot access the world we are in. p does not imply <>p
I wonder about such definitions of modal operators. WHY doesn't p
imply <>p? We could define <> so that it did. Is there some good
reason not to?
The modal logic are imposed by the fact that he box (and thus the
diamond) are the one describing the self-reference, by Solovay
theorem. The box is Gödel's beweisbar. It is an arithmetical
predicate. We really assume only Robinson (and Peano) arithmetic. We
don't have p -> <>p, because this would mean in particular t -> <>t,
and if that was a theorem of G, then <>t would be provable,
contradicting Gödel's incompleteness.
All modal logics are extracted from arithmetic. They are shortcuts
provided by Solovay's completeness theorem of G and G*, and the
Theaetetus' variants.
Bruno
Brent
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