'When Are Relations Neither True Nor False?
in [Math]
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From: Newberry on 24 Feb 2010 09:52 On Feb 23, 9:52 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Newberry wrote: > > On Feb 23, 7:52 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> Newberry wrote: > >>> On Feb 22, 11:00 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >>>> Frederick Williams wrote: > >>>>> This is one of those threads that causes me to think "would that the > >>>>> contributors could find something more interesting to discuss." > >>>> This thread to me is an interesting one in that (at least) its title > >>>> suggests a way to generalize GIT into a more comprehensive statement > >>>> about incompleteness in mathematical reasoning, through FOL. > >>> I do not know if I would call it generalization but it is something of > >>> that sort. > >> I'm not sure I get what you said here. If it's something of that sort > >> of "generalization", why you wouldn't call it "generalization"? [But I > >> don't think we have to argue about it so if you don't feel like answering > >> the question that's fine with me.] > > >>> Goedel's incompeteness theorem suggests that two valued > >>> logic is impossible. > >> In what way though? > > >>> Furthermore Goedel's second theorem does not > >>> apply to theories with gaps. > >> This is the 2nd time (iirc) you mentioned "theories with gaps". What > >> would you mean by "gaps"? > > > WFFs that are neither true nor false. > > WFFs are just formulas. So what's your _precise technical definition_ > of a formula that's neither true nor false? > > > > > I have heard a lot about models. No matter what model you have you > > still have to decide if a vacuous sentence is true or not in that > > model or any model for that matter. > > If you do actually have a model, there's no vacuous formula: every > formula is either true or false. So there's no such a thing as a "gap" > formula in a genuine model. You are assuming classical logic. I am not.
From: Newberry on 24 Feb 2010 09:59 On Feb 23, 9:43 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Newberry wrote: > > On Feb 23, 8:10 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > >> Newberry <newberr...(a)gmail.com> writes: > >>> Goedel's incompeteness theorem suggests that two valued logic is > >>> impossible. > >> Anything may suggest anything to someone. If we are to discuss this you > >> need to spell out in some detail this suggestion, and what you mean by > >> saying that "two valued logic is impossible". > > > It is impossible to prove all true sentences. > > That's a little vague. An inconsistent formal systems would prove > all true sentences, as well as false ones, as well as ones that > are neither true or false,.... So it's not impossible! (Iotw, we can't > talk about "prove" without mentioning a formal system). Let me rephrase my answer. I was taken a little aback by the question. Like Chesterton said if somebody asked why you preferred civilization to savagery you would not immediately know what to answer even though it seems so obvious. Goedel's theorem states that there is a sentence G such that neither it nor ~G are provable (in a rather large class of formal systems.) This to me suggests non-bivalence. After all if there are sentences ~(T v F) then it is not surprising that neither them nor their negations are not provable. > > It leads to paradoxes. And if you think that there is nothing > > paradoxical about Goedel then the Liar paradox certainly is. > > What specific mathematical language is the Liar paradox written in? > > > > > Anyway, whethere GIT suggests that two valued logic is impossible or > > not is not essential to my main point. > > So what's your main point? And what road map would you use to defend it?
From: Daryl McCullough on 24 Feb 2010 10:14 Newberry says... >Goedel's theorem states that there is a sentence G such that neither >it nor ~G are provable (in a rather large class of formal systems.) >This to me suggests non-bivalence. *Why* does it suggest non-bivalence? If you have a theory that is incapable of proving 0 ~= 1 (and also incapable of proving its negation), does that suggest non-bivalence? Not to me; it only suggests that the theories axioms are not strong enough to prove some interesting true statements. At what point would you ever be justified in saying, for any formula Phi, "It's not that our axioms are inadequate to prove Phi or ~Phi, it's that Phi is neither true nor false"? -- Daryl McCullough Ithaca, NY
From: Newberry on 24 Feb 2010 10:24 On Feb 24, 3:22 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Newberry <newberr...(a)gmail.com> writes: > > On Feb 23, 7:53 am, Frederick Williams <frederick.willia...(a)tesco.net> > > wrote: > >> Newberry wrote: > >> > > Frederick Williams wrote: > >> > > > This is one of those threads that causes me to think "would that the > >> > > > contributors could find something more interesting to discuss." > > >> > [...] Goedel's incompeteness theorem suggests that two valued > >> > logic is impossible. > > >> Wow! If that were so I'd withdraw my remark. > > > What remark? > > There are two quotes attributed to Frederick in the post you replied > to. The second quote was about withdrawing a remark and was probably > about a prior remark, don't you think? > > So, look above and see if you can find a remark made by Frederick. > Look carefully! Once you see that remark, chances are good that it's > the remark he meant. > Your argument is sound and to the point. I find it refreshing. You do not see it very often on this board.
From: Newberry on 24 Feb 2010 10:27
On Feb 24, 7:14 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > >Goedel's theorem states that there is a sentence G such that neither > >it nor ~G are provable (in a rather large class of formal systems.) > >This to me suggests non-bivalence. > > *Why* does it suggest non-bivalence? If you have a theory that is > incapable of proving 0 ~= 1 (and also incapable of proving its > negation), does that suggest non-bivalence? Not to me; it only > suggests that the theories axioms are not strong enough to prove > some interesting true statements. At what point would you ever > be justified in saying, for any formula Phi, "It's not that our > axioms are inadequate to prove Phi or ~Phi, it's that Phi is > neither true nor false"? a) Goedel shows that the axioms CANNOT be made strong enough. b) By defining a semantics according to which a formula is ~(T v F). |