'When Are Relations Neither True Nor False?
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From: Newberry on 4 Mar 2010 23:49 On Mar 4, 9:30 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Newberry <newberr...(a)gmail.com> writes: > > Let us clarify this once and for all. Here is Goedel's original > > theorem. > > Quoting Gödel's original statement is nothing but pointless > obscurantism. I thought it was pretty obvious that the reason I quoted the original was to show that there was nothibg about truth. I contrasted it with another formulation that explicitly included truth. But in any case why is quoting Gödel's original statement pointless obscurantism? > > Also if you want to claim that there are TRUE but unprovable formulae > > I do not know if you can do it by purely syntactic means as some > > people are claiming in this thread. > > Who are these people? If we state a theorem using the notion of > arithmetical truth obviously we must introduce and make use of this > concept also in the proof. > > But let's consider the following variant of the incompleteness theorem > in which we certainly don't invoke in any essential way the notion of > arithmetical truth: > > For any formal theory in which statements of the form "the Diophantine > equation D(x1, ..., xn) = 0 has no solutions" can be expressed, either > the theory proves "the Diophantine equation D(x1, ..., xn) = 0 has no > solutions" for some Diophantine equation D(x1, ..., xn) = 0 that has > solutions, or there are infinitely many Diophantine equations D(x1, ...., > xn) = 0 which have no solutions but for which "the Diophantine equation > D(x1, ..., xn) = 0 has no solutions" is not provable. > > How would you apply your logical ideas, about statements neither true > nor false, about meaninglessness, about presuppositions, here? > > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, darüber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Nam Nguyen on 4 Mar 2010 23:52 Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Apparently, debating to you means you simply ignore what people ask and >> then say whatever you feel pleased. > > Well, what relevance do you take your somewhat odd reasoning about "0=0" > to have to the observation that G�del's proof doesn't involve any model > theoretic considerations? > Well, don't you suspect for a moment that somewhere in his proof he might have _implicitly_ utilize the truth of the natural numbers, as the purported standard model of say PA? For example, in his definition of Prim(x): Prim(x) <-> ~(Ez)[ z <= x & z =/= 1 & z =/= x & x/z] & x >1 don't you think he'd mean something like Prim(2) is _true_ somewhere in his proof, again _implicitly_?
From: Newberry on 5 Mar 2010 00:03 On Mar 4, 10:56 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Newberry <newberr...(a)gmail.com> writes: > > On Mar 4, 6:24 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > >> Newberry <newberr...(a)gmail.com> writes: > >> > On Mar 3, 9:37 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > >> >> Newberry <newberr...(a)gmail.com> writes: > >> >> > This is my main motivation: > > >> >> > Gödel's sentence has the same form as (3.1): > > >> >> > ~(Ex)(Ey)(Pxy & Qy) (4.1) > > >> >> > Pxy means that x is the proof of y, where x, y are Gödel numbers of > >> >> > wffs or sequences of wffs. Q has been constructed such that only one y > >> >> > = m satisfies it, and m is the Gödel number of (4.1). > >> >> > Assume that Gödel's sentence (4.1) is not derivable, i.e. that > > >> >> > ~(Ex)Pxm (4.2) > > >> >> > is true. Then (4.1) is ~(T v F). Thus if Gödel's sentence is not > >> >> > derivable it is neither true nor false. > > >> >> So, you want to deny that Goedel's theorem is true. > > >> > We better get this straigh first. No. I do not want to deny that > >> > Goedel's theorem is true. > > >> Well, I'm sorry if I misrepresented your opinion, but you *just* > >> suggested that if (4.2) is true, then (4.1) is neither true nor false > >> and hence is not true. > > >> Maybe what you mean is this: Goedel's theorem is a theorem of PA in > >> classical logic, but you are interested in a different logic. In this > >> different logic, the analog to Goedel's theorem is neither true nor > >> false. Is that a correct assessment? (Of course, there is no real > >> alternative logic yet, but let us ignore that fact for now.) > > > Let us clarify this once and for all. [...] > > I apologize for mis-representing your views on Goedel's theorem. > > Will you be responding to my other comments any time soon? > > -----------------(My other comments)---------------------------------- > > A similar bit of reasoning that seems utterly uncontroversial to me is > this. Suppose I have a proof of > > ~(Ex)(Px & Qx) (1) > > and also a proof of > > ~(Ex)(Px & ~Qx). (2) > > Then I may conclude > > ~(Ex)Px. (3) > > This seems perfectly sensible to me. It seems sensible because you are used to it from classical logic. If (3) then you can probably prove it directly and you do not have to do it in roundabout manner through (1) and (2). I still urge you to look at the big picture and not to pick at one particular aspect. > Unfortunately, according to your > statements on presuppositions, if I were to conclude (3), then (1) and > (2) are not true (since they are neither true nor false) and hence, I > assume, cannot be used in a proof of (3). Oops! > > Again, I can't give you any examples of this form of reasoning from > ordinary mathematics. Perhaps someone else can. > > One last example: suppose I have two counties, B and C. In county B, > there are no Republicans. In C, no one voted for Obama. Then > > In counties B and C, there are no Republicans who voted for Obama. (a) > > is true. Unfortunately, the two consequences > > In B, there are no Republicans who voted for Obama. (b) > > and > > In C, there are no Republicans who voted for Obama. (c) > > are neither true nor false (according to you), even though (in > classical logic, at least) the statement (a) is equivalent to the > conjunction (b) & (c). I just don't see why I should think that (a) > is true, but (b) is neither true nor false. > > Finally, it seems that your notions are very sensitive to what we take > to be atomic predicates. In a language in which "is large and round" > is a predicate R and "is square" a predicate S, > > ~(Ex)(Rx & Sx) > > is true. In a language in which "is large" is a predicate L and "is a > round square" is a predicate T, the equivalent statement > > ~(Ex)(Lx & Tx) > > is neither true nor false, even though it means the same thing. This > situation strikes me as a problem. > > -- > Jesse F. Hughes > "The future is a fascinating thing, and so is history. And you people > are a fascinating part of history, for those in the future." > -- James S. Harris is fascinating, too- Hide quoted text - > > - Show quoted text -
From: Nam Nguyen on 5 Mar 2010 00:05 Nam Nguyen wrote: > Aatu Koskensilta wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> Apparently, debating to you means you simply ignore what people ask and >>> then say whatever you feel pleased. >> >> Well, what relevance do you take your somewhat odd reasoning about "0=0" >> to have to the observation that G�del's proof doesn't involve any model >> theoretic considerations? >> > > Well, don't you suspect for a moment that somewhere in his proof he might > have _implicitly_ utilize the truth of the natural numbers, as the > purported > standard model of say PA? For example, in his definition of Prim(x): > > Prim(x) <-> ~(Ez)[ z <= x & z =/= 1 & z =/= x & x/z] & x >1 > > don't you think he'd mean something like Prim(2) is _true_ somewhere in > his proof, again _implicitly_? Note also B. R. Braithwaite's introduction of Godel's paper has this on the Godel's numbering (or "arithmetization"): "For example, the metamathematical statement that the series s of formulas is a 'proof' of the formula f is true if and only if a certain arithmetical relation holds between Godel's numbers of s and of f ..." Tell us, Aatu, which of the following are you're *protesting*: (1) "a certain arithmetical relation holds between [...] numbers" means *_true_ in the naturals*. (2) The natural numbers isn't a model of say PA ?
From: Nam Nguyen on 5 Mar 2010 00:17
Nam Nguyen wrote: > Nam Nguyen wrote: >> Aatu Koskensilta wrote: >>> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >>> >>>> Apparently, debating to you means you simply ignore what people ask and >>>> then say whatever you feel pleased. >>> >>> Well, what relevance do you take your somewhat odd reasoning about "0=0" >>> to have to the observation that G�del's proof doesn't involve any model >>> theoretic considerations? >>> >> >> Well, don't you suspect for a moment that somewhere in his proof he might >> have _implicitly_ utilize the truth of the natural numbers, as the >> purported >> standard model of say PA? For example, in his definition of Prim(x): >> >> Prim(x) <-> ~(Ez)[ z <= x & z =/= 1 & z =/= x & x/z] & x >1 >> >> don't you think he'd mean something like Prim(2) is _true_ somewhere in >> his proof, again _implicitly_? > > Note also B. R. Braithwaite's introduction of Godel's paper has this on the > Godel's numbering (or "arithmetization"): > > "For example, the metamathematical statement that the series s of > formulas > is a 'proof' of the formula f is true if and only if a certain > arithmetical > relation holds between Godel's numbers of s and of f ..." > > Tell us, Aatu, which of the following are you're *protesting*: > > (1) "a certain arithmetical relation holds between [...] numbers" means > *_true_ in the naturals*. > > (2) The natural numbers isn't a model of say PA I meant to ask if you're protesting: (2) The natural numbers is a model of say PA > ? |