'When Are Relations Neither True Nor False?
in [Math]
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From: MoeBlee on 3 Mar 2010 12:05 On Mar 3, 10:52 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Mar 2, 10:43 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > Secondly, if you could formalize what he wrote into a full blown formal > > proof then in effect his proof was essentially a formal proof to begin > > with! > > Whatever you mean by "essentially", NO, the proof was given in German > and mathematical notation. The actual writeup Godel gave is NOT a > formal proof. That we can formalize it doesn't change that it ITSELF > is not a formal proof (a sequence of finite ....etc.) 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. It's just amazing to me that you RARELY seem to stop for a moment to think, "Hmm, this Aatu guy and some other people here do seem to know this material really well; maybe I should put aside my agenda for just a few moments to THINK and take in what they're saying." MoeBlee
From: Newberry on 3 Mar 2010 12:12 On Mar 3, 3:38 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Newberry <newberr...(a)gmail.com> writes: > > On Mar 2, 10:15 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > >> I just don't see any reason to give up a perfectly straightforward and > >> sensible argument (like that above) in order to fit your intuitions > >> that, if (3) is true, then (2) is neither true nor false. Perhaps > >> t-relevance logic is useful, but it does not seem to be satisfactory > >> for mathematical reasoning. In mathematics, the argument I outlined > >> above is a perfectly good argument for (3). > > >> Seems to me that you've got quite a job convincing mathematicians > >> otherwise. > > > I encourage you to look at the big picture and not just pick at one > > particular aspect. And do not worry about mathematicians. You can make > > your own judgement, can you not? > > I've made my judgment. There's absolutely nothing wrong or fishy with > the following argument. > > If there are counterexamples to FLT, then there are counterexamples > with prime exponent. > > There are not counterexamples with prime exponent. > > Therefore, there are no counterexamples to FLT. > > The idea that the second premise is not true unless the conclusion is > false is simply nonsense to me. > > By the way, it is not that I reject your aims just because of this one > example. For years now, I've told you that I haven't any clue why you > think that a vacuously true universal statement is not true. This > notion simply holds no intuitive plausibility for me, but it is the > single motivation for your attempts[1] at reforming logic. No. Please re-read my very first post in this thread. > Footnotes: > [1] Attempts that have, thus far, not introduced a single deductive > rule for predicate logic. > > -- > "[In the movie, Tom Green] delivers a child, severs the umbilicus with > his teeth and then swings the baby over his head before tenderly > handing it to the stunned, blood-spattered mother[...] This was, I > have to say, a bit much." -- New York Times movie reviewer A. O. Scott
From: Newberry on 3 Mar 2010 12:21 On Mar 3, 3:38 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Newberry <newberr...(a)gmail.com> writes: > > On Mar 2, 10:15 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > >> I just don't see any reason to give up a perfectly straightforward and > >> sensible argument (like that above) in order to fit your intuitions > >> that, if (3) is true, then (2) is neither true nor false. Perhaps > >> t-relevance logic is useful, but it does not seem to be satisfactory > >> for mathematical reasoning. In mathematics, the argument I outlined > >> above is a perfectly good argument for (3). > > >> Seems to me that you've got quite a job convincing mathematicians > >> otherwise. > > > I encourage you to look at the big picture and not just pick at one > > particular aspect. And do not worry about mathematicians. You can make > > your own judgement, can you not? > > I've made my judgment. There's absolutely nothing wrong or fishy with > the following argument. > > If there are counterexamples to FLT, then there are counterexamples > with prime exponent. > > There are not counterexamples with prime exponent. > > Therefore, there are no counterexamples to FLT. > > The idea that the second premise is not true unless the conclusion is > false is simply nonsense to me. > > By the way, it is not that I reject your aims just because of this one > example. For years now, I've told you that I haven't any clue why you > think that a vacuously true universal statement is not true. This > notion simply holds no intuitive plausibility for me, but it is the > single motivation for your attempts[1] at reforming logic. 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. > Footnotes: > [1] Attempts that have, thus far, not introduced a single deductive > rule for predicate logic. > > -- > "[In the movie, Tom Green] delivers a child, severs the umbilicus with > his teeth and then swings the baby over his head before tenderly > handing it to the stunned, blood-spattered mother[...] This was, I > have to say, a bit much." -- New York Times movie reviewer A. O. Scott
From: Jesse F. Hughes on 3 Mar 2010 12:27 Newberry <newberryxy(a)gmail.com> writes: > On Mar 3, 3:38 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > >> By the way, it is not that I reject your aims just because of this one >> example. For years now, I've told you that I haven't any clue why you >> think that a vacuously true universal statement is not true. This >> notion simply holds no intuitive plausibility for me, but it is the >> single motivation for your attempts[1] at reforming logic. > > No. Please re-read my very first post in this thread. Your first post is about the following: ,---- | I do claim that if for all a in the range of x | | (y)Aay (1) | | is neither true nor false then | | (x)(y)Aay (2) | | is neither true nor false. `---- Far as I can tell, this is an extension of your basic assumption, namely that (x)(Px -> Qx) presupposes (Ex)Px. If I'm mistaken on this, then you can clarify. Thus far, I'd say that your primary motivation is this claim. -- Jesse F. Hughes "I think the problem for some of you is that you think you are very smart. I AM very smart. I am smarter on a scale you cannot really comprehend and there is the problem." -- James S. Harris
From: Jesse F. Hughes on 3 Mar 2010 12:37
Newberry <newberryxy(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. The best way to do that is to deny that mathematical logic is the right logic. Instead, we will suppose that sentences which are vacuously true are in fact neither true nor false. Of course, your attempt is really *weird* at best. If 4.2 is true then 4.1 is neither true nor false, but 4.1 is used in the derivation of 4.2. So, if 4.2 is not true, then 4.1 may well be true after all! The situation is indeed very similar to my example. The fact is that I find my example not controversial in the least (nor your example, for that matter). If there are counterexamples to FLT, then there are counterexamples with prime exponent. There are not counterexamples with prime exponent. Therefore, there are no counterexamples to FLT. I sincerely doubt that you find this bit of reasoning dubious. I honestly cannot see anything the least bit controversial with it. Rather, it seems to me, you want to deny Goedel's theorem is true (for whatever reason) and so you want to deny that this bit of simple argument is correct. -- "Yup, you guessed it. If worse comes to worse, I *will* turn to the Army to help me with mathematicians. And then mathematicians don't think the NSA or CIA can save your asses, as generals LIKE me." -- James Harris's latest foray into mathematical logic. |