'When Are Relations Neither True Nor False?
in [Math]
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From: Newberry on 24 Feb 2010 00:24 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. It leads to paradoxes. And if you think that there is nothing paradoxical about Goedel then the Liar paradox certainly is. Anyway, whethere GIT suggests that two valued logic is impossible or not is not essential to my main point.
From: Newberry on 24 Feb 2010 00:28 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. 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. > > It is obvious where the gaps might be - > > in the so called "vacuously true" sentences. > > > So again, if > > > ~(Ex)[(x + n < 6) & (n = 8)] > > > is neither true nor false for any n (according to the logic of > > presuppositions.) Am I right that > > > ~(Ex)(Ey)[(x + y < 6) & (y = 8)] > > > is neither true nor false? > > I actually don't know what "The logic of presuppositions" is so I couldn't > comment on it. In FOL, depending on the exact axioms, you might have > some consistent theories where the formula would be true in any model, > and some others where it'd be be false. For example, in T with these > axioms: > > A1: Ax[Sx=0] > A2: Axy[x+y=0] > A3: (x < y) <-> x=y > > then ~((Ex)(Ey)[(x + y < 6) & (y = 8)]) would be _false_ in any model of T.
From: Nam Nguyen on 24 Feb 2010 00:43 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). > > 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: Nam Nguyen on 24 Feb 2010 00:52 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.
From: Jesse F. Hughes on 24 Feb 2010 06:22
Newberry <newberryxy(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. -- Jesse F. Hughes "They don't want a choice, they want something other than Windows." -- billwg explains that refusing Windows isn't a choice. |