'When Are Relations Neither True Nor False?
in [Math]
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From: Frederick Williams on 4 Mar 2010 13:41 MoeBlee wrote: > > It's revealing that you choose to take your conversation with Aatu as > a "debate". With so many people involved we could call it a massdebate. Sorry.
From: MoeBlee on 4 Mar 2010 13:52 On Mar 4, 12:41 pm, Frederick Williams <frederick.willia...(a)tesco.net> wrote: > MoeBlee wrote: > > > It's revealing that you choose to take your conversation with Aatu as > > a "debate". > > With so many people involved we could call it a massdebate. Should be called 'therapy with an unwilling patient'. MoeBlee
From: Jesse F. Hughes on 4 Mar 2010 13:56 Newberry <newberryxy(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. 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
From: Nam Nguyen on 4 Mar 2010 23:09 Aatu Koskensilta wrote: > MoeBlee <jazzmobe(a)hotmail.com> writes: > >> P.S. And note that now you're arguing the OPPOSITE of your earlier >> challenge against the claim that Godel's proof can be formalized. > > My (very tentative) impression is that Nam didn't intend to argue this > opposite challenge. Which seems to tentatively mean MoeBlee didn't quite know what he was noting about here. [To me he didn't know what he was saying here]. > Rather, he is, rather bafflingly, arguing those who > assert that G�del's proof wasn't a formal proof but can be formalized > are contradicting themselves. This is a sound line of thought on the > bizarre and pointless definition of formal proof he gave earlier. I think you meant the passage I had earlier said: >> if there's a formal system where what he asserted is a theorem, then >> his proof is a formal proof. If not then his proof isn't. It's that's >> straight forward which doesn't require explanation on thing such as >> "ordinary mathematical proof", as you said. Why is that definition of a _formal proof_ "bizarre and pointless"? Or you just interjected into the debate baseless subjective comments whenever you feel pleased? Now, Shoenfield said "a formal system is the syntactical part of an axiom system" and then he said one of the only 3 components of this syntactical part is the rules of inference where and only where _formal system proofs_ can be had. This is what he said: "We need the third part of a formal system which will enable us to _conclude_ _theorems from the axioms_. This is provided by the rules of inference..." [The highlight are mine]. My definition of _formal proof_ is basically just a repeat of Shoenfield's definition of _formal system proof_. Are you saying Shoenfield's definition here is "bizarre and pointless"? The only reason I could think of why you had such very strange comment is perhaps you often flip-flop on the semantics of some technical terminologies, such as when you said "formalized" you actually meant "in-formalized" and in such case then yes what isn't a formal proof can be an "in-formalized" proof. But for crying out loud, why on Earth would one want flip the semantics of "formalized" into "in-formalized"?
From: Newberry on 4 Mar 2010 23:37
On Mar 4, 9:23 am, David Bernier <david...(a)videotron.ca> wrote: > Jesse F. Hughes 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. > > I read: > [ Newberry:] > " Gödel's sentence has the same form as (3.1):" > > I've been wondering if there's a typo. there and if it ought to > be numbered in the quote above (4.1) and not (3.1). It is out of context. he whole story is here http://www.scribd.com/doc/26833131/RelationsAndPresuppositions-2010-0.... > > > 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.) > > David Bernier- Hide quoted text - > > - Show quoted text - |