'When Are Relations Neither True Nor False?
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From: Newberry on 25 Feb 2010 00:54 On Feb 24, 12:31 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > > > > > > > >On Feb 24, 7:14=A0am, 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 ~=3D 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. > > I would say that there is a quantifier ordering ambiguity at > work here: > > Godel's theorem says that for every theory T of the right type, > there is a formula Phi such that neither Phi nor ~Phi is provable > in T. This is correct. But it would be incorrect to claim that every such Phi in every such theory must be interpreted as true. It doesn't say: There is a formula Phi such that for any > theory T of the right type, neither Phi nor ~Phi is provable. > > In other words, the fact that neither Phi nor ~Phi is provable > in a *particular* theory T does not suggest that Phi is "neither > true nor false", because there will always be some other theory T' in > which it *is* provable. > > No formula is *absolutely* unprovable (that is, unprovable by > any sound theory). If the theory is sound then what it can or cannot prove will depend on the interpretation. If there are sentences that are neither true nor false then it will not be able to (dis)prove them.
From: Nam Nguyen on 25 Feb 2010 00:54 Newberry wrote: > 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. To be frank I'm not one of those who'd be easily throwing buzzwords around like "classical logic", "constructive logic", etc... Model truth or falsehood in this context do have a precise definition, immaterial to whether or not one would call that "classical", "constructively valid" or what have we. Everything will boil down to either one conforms or not conforms to the definition. And nothing else would matter. So, if you don't use model to define truth/falsehood for "WFFs that are neither true nor false" then what would be your definition of a formula being true or false in the first place? And what does your definition have to do with GIT, it being dependent on the naturals as a model of some formal system?
From: Newberry on 25 Feb 2010 01:00 On Feb 24, 9:17 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Newberry <newberr...(a)gmail.com> writes: > > Anyway, whethere GIT suggests that two valued logic is impossible or > > not is not essential to my main point. > > Well, I don't really have any interest in your main point. I was just > wondering why the incompleteness theorem suggests to you that "two > valued logic is impossible". Apparently it's impossible to say. I do not know why you have any interest in knowing why GIT suggests that bi-valent logic is impossible, considering that you have already made up your mind.
From: Frederick Williams on 25 Feb 2010 06:50 Newberry wrote: > > On Feb 24, 2:13 pm, Frederick Williams <frederick.willia...(a)tesco.net> > wrote: > > Daryl McCullough wrote: > > > No formula is *absolutely* unprovable (that is, unprovable by > > > any sound theory). > > > > Is that what you meant? > > Me? No. No, Daryl. Clearly x =/= x is unprovable by any sound theory. So I fancy he meant that there is no phi such that both phi and not-phi are unprovable by any sound theory.
From: Frederick Williams on 25 Feb 2010 06:55
Newberry wrote: > If the theory is sound then what it can or cannot prove will depend on > the interpretation. What do you mean by interpretation? If you mean a model plus a function that assigns entities in the model to linguistic entities in the theory, then that's not so: proof is a syntactic matter and does not depend on interpretation (in the sense I use the word). > If there are sentences that are neither true nor > false then it will not be able to (dis)prove them. |