> On 4 Dec 2018, at 03:30, Brent Meeker <[email protected]> wrote: > > > > On 12/3/2018 7:31 AM, Bruno Marchal wrote: >>> On 2 Dec 2018, at 21:06, Brent Meeker <[email protected]> wrote: >>> >>> >>> >>> On 12/2/2018 4:52 AM, Bruno Marchal wrote: >>>> Language have no relation with truth a priori. Theories might have. >>>> Semantics are truth “by definition”, by relativising it to the notion of >>>> model/reality. >>>> >>> Then what is this "true" and "false" which you attribute to the >>> propositions of modal logic? >> In classical logic, truth is any object in a set of two objects, or it is >> the supremum in a Boolean algebra. In propositional logic a “world” is >> defined by any function from the set of atomic letters to {t, f}. > > Right. T and F are just formal markers in logic and the rules of inference > are supposed to preserve T. > >> >> Then if the theory is “rich enough”, truth can be meta-defined by “satisfied >> by the structure (N, 0, s, +, *). >> Of course, this presuppose the intuitive understanding of 2+2=4, etc. >> >> In our case, as all modal formula are arithmetical formula, it is the usual >> informal mathematical notion just above (arithmetical truth, satisfaction by >> the usual standard model). > > That's satisfaction relative to some particular axioms and rules of inference.
OK, but the modal logic just sum up purely arithmetical theorem. For example the fact that G proves <>t -> ~[]<>t really means that PA (or any Löbian entity) proves consistent(’t’) -> ~beweisbar(‘consistent(’t’)), And the fact that G proves []p -> [][]p means that for all arithmetical proposition p, PA proves beweisbar(‘p’) -> beweisbar(‘beweisbar(‘p’)’). The modal logic are imposed mathematically. G is the logic of (provable) self-reference, like G* \ G gives the true non provable proposition. The fact that G* \ G proves <>[]f means that the consistency of inconsistency is true, and non provable by PA. The entire theology, including physics, are constituted of true arithmetical formula. > >> >> That one can be define by V(‘p’) means the same as p. It is Tarski’s idea >> that ‘p’ is true when p, or when it is the case that p. Like wise, to say >> Provable-and-true(p) we use []p & p. > > That's the correspondence theory of truth, which is what ordinary discourse > and physics assume. Yes, except that with mechanism, the correspondence refers to the standard model of arithmetic (the non axiomatisable structure (N, 0, s, +, *). > So there are at least three kinds of "true”. They all derive from the standard model of arithmetic, that is, the elementary notions we learn in high school, although we don’t call it that way. > To which we might add the Trump theory of truth, "If it makes me look good > it's true.” Better to not add this one :) > >> >> I recommend the book by Torkel Franzen “Inexhaustibility” for a more >> detailed explanation of the concept of truth. > > I have the book but I haven't read it (so many books, so little time). I understand. Its other book on the misuse of Gödel’s incompleteness is also very good, and simpler to read. Bruno > > Brent > > >> >> We can come back, but I suggest to come back on this only when we need it, >> as this is an very rich and complex subject by itself. >> >> 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 https://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 https://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 https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
