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 <[email protected]> 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 [email protected]. > To post to this group, send email to [email protected]. > 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 [email protected]. > To post to this group, send email to [email protected]. > 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 [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
