I thought <>P meant P was possible? If so wouldn't P imply <>P? Or have I misremembered what <>P means?
On 7 March 2015 at 21:08, Bruno Marchal <marc...@ulb.ac.be> wrote: > > 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 > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to everything-list+unsubscr...@googlegroups.com. > To post to this group, send email to everything-list@googlegroups.com. > Visit this group at http://groups.google.com/group/everything-list. > For more options, visit https://groups.google.com/d/optout. > > > http://iridia.ulb.ac.be/~marchal/ > > > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to everything-list+unsubscr...@googlegroups.com. > To post to this group, send email to everything-list@googlegroups.com. > Visit this group at http://groups.google.com/group/everything-list. > For more options, visit https://groups.google.com/d/optout. > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
