'When Are Relations Neither True Nor False?
in [Math]
|
From: Alan Smaill on 5 Mar 2010 04:01 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Jesse F. Hughes wrote: > >> So, you want to deny that Goedel's theorem is true. > > The "crank" tends to assert Goedel's theorem is false. > The "standard theorist" would insist GIT is true. > > That leaves the "rebel" the only side who observes the > method in Godel's work is invalid. > > Except for the relativists, why should we care about invalid > truth or falsehood? Because the notions of "truth" and "validity" are not beyond dispute, and maybe we/you/I have got those wrong. -- Alan Smaill
From: Marshall on 5 Mar 2010 04:18 On Mar 5, 2:01 am, Alan Smaill <sma...(a)SPAMinf.ed.ac.uk> wrote: > Nam Nguyen <namducngu...(a)shaw.ca> writes: > > Jesse F. Hughes wrote: > > >> So, you want to deny that Goedel's theorem is true. > > > The "crank" tends to assert Goedel's theorem is false. > > The "standard theorist" would insist GIT is true. > > > That leaves the "rebel" the only side who observes the > > method in Godel's work is invalid. > > > Except for the relativists, why should we care about invalid > > truth or falsehood? > > Because the notions of "truth" and "validity" are not beyond dispute, and > maybe we/you/I have got those wrong. On usenet, nothing is beyond dispute. For example, try to get Nam to agree that the sky is blue. He has contested this in the past, sometimes with the argument that aliens might use the word "blue" to mean red. He also has contested the fact that formulas in the language of arithmetic are either true or false. Marshall
From: Jesse F. Hughes on 5 Mar 2010 06:15 Newberry <newberryxy(a)gmail.com> writes: > On Mar 4, 10:56 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> -----------------(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). "Probably"? I should give up a perfectly reasonable argument because you say I can *probably* prove the same claim some other way? I can't see any sense in that. > I still urge you to look at the big picture and not to pick at one > particular aspect. One aspect? I have presented two arguments that appear to be clearly valid from my perspective, but are likely invalid in your logic (though, since there is no deductive system, who can tell?). These are the two you've "addressed". You ignored the remainder of my comments in this post, I noticed. You're motivated by concerns over one aspect of logic: Goedel's theorem. I'm showing you various ways in which your proposed changes to classical logic result in seriously dubious judgments about validity. It seems to me I'm not the one missing the big picture. >> 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 - > -- Jesse F. Hughes "You are just flotsam dragged along with the current in something that is huger than anything civilization has ever faced before." -- James S. Harris on his own discoveries
From: Nam Nguyen on 5 Mar 2010 10:17 Marshall wrote: > On Mar 5, 2:01 am, Alan Smaill <sma...(a)SPAMinf.ed.ac.uk> wrote: >> Nam Nguyen <namducngu...(a)shaw.ca> writes: >>> Jesse F. Hughes wrote: >>>> So, you want to deny that Goedel's theorem is true. >>> The "crank" tends to assert Goedel's theorem is false. >>> The "standard theorist" would insist GIT is true. >>> That leaves the "rebel" the only side who observes the >>> method in Godel's work is invalid. >>> Except for the relativists, why should we care about invalid >>> truth or falsehood? >> Because the notions of "truth" and "validity" are not beyond dispute, and >> maybe we/you/I have got those wrong. > > On usenet, nothing is beyond dispute. So Godel's method(s) in his proof is disputable? > For example, try to get > Nam to agree that the sky is blue. Where in the past did I say Nam (I myself) doesn't "see" the sky as blue? > He has contested this in > the past, sometimes with the argument that aliens might use > the word "blue" to mean red. The aliens might. Who knows? Would you be able to "know" that, Marshall? > He also has contested the fact > that formulas in the language of arithmetic are either true or false. Some formulas, in certain contexts, yes. Are you able to refute what I said about those formulas in those contexts? > > > Marshall
From: Newberry on 5 Mar 2010 10:37
On Mar 4, 11:56 pm, David Bernier <david...(a)videotron.ca> wrote: > Newberry wrote: > > 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..... > > There was a debate of some kind about "denoting", etc. between > Bertrand Russell and Strawson, the author of the book you quote from. > > This is according to an article published by someone > under the name _Nearly_Anonymous_ here: > > http://bookstove.com/non-fiction/the-russell-strawson-debate-a-useful... > > Strawson writes "[...] if its subject class is empty" . > > He seems to be saying something about expressions such as, say, > "The current King of France" [there is no King of France today]. > > Either "The current King of France" is married, or (***) > "The current King of France" is not married. > > For Russell, (***) would have been a fine sentence and a true one. > But perhaps Strawson would have disagreed. > > David Bernier- Hide quoted text - As I mentioned there is entire literature on presuppositions. In fact you have to be speacialized in it if you want to know all of it. It is perhaps more popular among linguists than among logicians. Starwson indeed had disputes with Russell. Among other things he criticized his theory of descriptions. |