'When Are Relations Neither True Nor False?
in [Math]
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From: Newberry on 26 Mar 2010 00:59 On Mar 25, 10:26 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Newberry <newberr...(a)gmail.com> writes: > > On Mar 24, 3:32 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > >> Newberry <newberr...(a)gmail.com> writes: > >> > Plus > > >> > (x)((x = x + 1) -> (x = x + 2)) > > >> > does not look particularly meaningful to me. > > >> I don't believe you. > > > Trust me. > > >> You know what it means. It's perfectly clear > >> what it means. It means that whenever x = x + 1, then x = x + 2..[1] > > > The sentence "if it rains then some roads are wet" describes a > > possible state of affairs. I can picture to myself what it means. I > > can even picture "if it rains then no roads are wet." It is still > > conceivable although very unlikely. "If it rains and does not rain > > then the roads are wet" does not describe any possible state of > > affairs. I cannot picture to myself what it expresses. > > Is the statement "Honesty is a virtue" meaningful? What do you > picture when you think about that statement? It can certainly be analyzed into something imaginable. > As usual, your claim that meaning involves picturing various states of > affairs is silliness. I can understand various theorems about, say, > infinite dimensional spaces. I daresay that I know those theorems are > meaningful, even though I cannot picture a space with more than three > dimensions. This argument is indeed silly. These theorems are about Certesian products R x R x R x R ... If you understand numbers, real numbers and cartesian products then you of course understand statements about sets of n-tuples of real numbers. If the product has less than 4 dimensions then it can also be understood as staments about the physical space. > > Of course, as Daryl points out, it is very easy to "picture" what the > above sentence means. It means the same thing as > > (Ax)( ~(x = x + 1) or (x = x + 2) ). > > I see no problem understanding that sentence at all. > > > The analytic sentences are rather odd. But even then given "all > > bachelors are unmarried" if you examine every bachelor you will find > > that he is umarried. Given "all married bachelors are unmarried > > bachelors" is just like "when it rains and does not rain ..." I cannot > > picture anything. > > > Similaly I cannot picture (x)(x = x+1) -> (x = x+2) any better than I > > can picture anything being attributing to married bachelors. > > As I said previously, I understand the meaning of that sentence and > can even immediately see that it is true, through the following > perfectly simple reasoning. You are saying what the world would look like if x = x + 1. No such word is possible so it is not possible to say or even to imagine what such a world would look like. > > >> [1] In fact, this statement seems obviously true! Suppose > >> x = x + 1. Then we may substitute x + 1 for x in the right hand side > >> of the equation x = x + 1, thus: > > >> x = x + 1 > >> = (x + 1) + 1 > >> = x + 2. > > >> I see nothing the least bit fishy about this reasoning. > > -- > Jesse F. Hughes > > "As you can see, I am unanimous in my opinion." > -- Anthony A. Aiya-Oba (Poeter/Philosopher)- Hide quoted text - > > - Show quoted text -
From: Newberry on 26 Mar 2010 01:05 On Mar 25, 11:00 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Jesse F. Hughes says... > > > > >Newberry <newberr...(a)gmail.com> writes: > >> On Mar 24, 9:34=A0am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > > >>> Paradoxical in what sense? > > >> Does not everybody know what the paradox of material implication is? > > >I'm just a simple housewife (with somewhat hairy legs). Why not tell > >me what you mean by "the paradox of material implication" and why it's > >a paradox? > > Wikipedia has a list of theorems of classical logic that it calls > "paradoxes of material implication": > > http://en.wikipedia.org/wiki/Paradoxes_of_material_implication > > There's nothing paradoxical about any of them, other than the > fact that they may be counter-intuitive to someone who is a > complete newbie to formal logic. Are you saying in a roundabout manner that in classical logic (P & ~P) -> Q is a tautology? Well we know that. But what does it have to do with anything? > > Ultimately, the people on this newsgroup who object to standard > mathematics are really objecting to the idea that there can be > such a thing as a counter-intuitive result. The ultimate logic > would be one in which it is impossible to prove any result that > you couldn't already guess was true. > > -- > Daryl McCullough > Ithaca, NY
From: Newberry on 26 Mar 2010 01:10 On Mar 25, 3:05 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > MoeBlee <jazzm...(a)hotmail.com> writes: > > On Mar 25, 1:00 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) > > wrote: > > >> Wikipedia has a list of theorems of classical logic that it calls > >> "paradoxes of material implication": > > >>http://en.wikipedia.org/wiki/Paradoxes_of_material_implication > > >> There's nothing paradoxical about any of them > > > The discussion there about (P&Q) -> R |- (P -> R) v (Q -> R) > > > is at least somewhat interesting. > > Yes, but their example ("If I close switch A and switch B, the light > will go on. Therefore, it is either true that if I close switch A the > light will go on, or that if I close switch B the light will go on.") > is poorly chosen, since P, Q and R stand for propositions, while "I > close switch A (or B)" is an action. (I'm not sure what type of > sentence "The light will go on," is -- it's not an action, in the > sense of dynamic logic, but rather it describes a change in the > world.) Do you think that propositions cannot be about actions? > > Their example is better understood in dynamic logic rather than > propositional logic. > > No matter, the fix is easy -- indeed, I'll go fix it now. > > -- > Meaningless movies > on the screen behind the band that's blowing Waterboys, > throwing shapes "My Love is My Rock > Half of the music is on tape in the Weary Land"
From: Transfer Principle on 26 Mar 2010 01:20 On Mar 23, 8:06 am, James Burns <burns...(a)osu.edu> wrote: > MoeBlee wrote: > > A more interesting question would be to look for > > a theorem concerning only (in some sense of > > "concerning") real numbers that physicists apply > > but that is not a theorem of Z-"regularity". > If 0.999... and 1 are supposed to be the result > of some measurement (not too unreasonable an > assumption, I hope, given that we are talking > about physicists), then the physicists' view would > be that they must have error bars larger than zero. > Clearly, 0.999... and 1 do not differ significantly, > no matter what non-zero error bounds one names. > However, differing /significantly/ is part of what > is implied by "0.999... < 1", in the physicist's > view. Or, at least, this is how it seems to me. Interesting argument. In trying to figure out why so many participants in the poll (however informal the poll may be) disagree with ZFC on what 0.999... is, it was mentioned in this thread that some of the posters were confused as to what 0.999... even is. For example, one participant in a 0.999... poll once claimed that his _physics_ teacher told him that 0.999...<1 -- because the teacher told him that a mark of 90% in the class was the minimum to earn an "A," and adamantly stated that there would be no rounding, so 89.999% is a "B." Of course, the teacher really meant "89+999/1000%," not "89.999...%" at all. Another poster made a similar conclusion after receiving a GPA of 2.999 (i.e., 2+999/1000 -- which was actually based on a bad grading/rounding policy on the part of the _university_). I wonder how many posters in any poll (whether formal or informal) would vote for 0.999...=1 once they realize that 0.999... has _infinitely_ many nines, and so must differ by 1 by less than any measurable (as Burns points out here) amount. This discussion concerns the theory Z-Regularity and what it proves (as opposed to ZF-Regularity). Yet the proof of 0.999...=1 to which I linked explicitly lists the Replacement Schema as one of the axioms that is used in the proof. In trying to determine exactly where an instance of the Replacement Schema is used in the proof, it appears that the schema is used to construct the real numbers. It turns out that the proof that 1 is the multiplicative identity on N is the first proof that requires Replacement: http://us.metamath.org/mpegif/mulidpi.html yet the definition of multiplication on N in terms of the multiplication of finite ordinals requires only Extensionality, Separation Schema, Powerset, and Pairing. Of course, someone is likely to point out that this is due to my overreliance on free sites such as Metamath, and if I could afford an actual _textbook_, I'd learn exactly how to construct the real numbers without Replacement Schema.
From: Transfer Principle on 26 Mar 2010 01:42
On Mar 25, 9:25 pm, Marshall <marshall.spi...(a)gmail.com> wrote: > On Mar 25, 11:00 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) > wrote: > > Ultimately, the people on this newsgroup who object to standard > > mathematics are really objecting to the idea that there can be > > such a thing as a counter-intuitive result. The ultimate logic > > would be one in which it is impossible to prove any result that > > you couldn't already guess was true. > QFT I don't mind counterintutive results per se -- but sometimes I wonder why some counterintuitive results are preferred over other (possibly counterintuitive as well) results. For example, many so-called "cranks" regularly oppose (Dedekind) infinite sets, because the idea that a set may have the same cardinality as some of its proper subsets is counterintuitive. But if we were to come up with a new theory which avoids this "problem," then we'd introduce another problem -- namely that the existence of a bijection between two sets is no longer sufficient for the sets to have the same cardinality -- which also can be said to contradict the intuitive notion that we can pair up elements, one from each set, in a bijection to determine whether they have the same size. Thus, the "cranks" who oppose bijections to determine set size due to their being counterintuitive can only offer alternatives that are also counterintuitive. My own goal isn't to avoid all counterintuitive results, but to allow at least for theories in which the "cranks'" intutions are spared even at the expense of the standard theorists', yet have such theories still be "useful" by the standard theorists' criteria (which, since I'm replying to Spight here, include being as powerful and easy to use as ZFC). I like to think of there being some sort of "symmetry" or "dualism" here (which apparently isn't quite the same as the "symmetry" as mentioned by Nguyen): ZFC is intuitive because set size is based on bijections. ZFC is counterintutive because proper subsets can have the same size. (for some alternate theory T, yet to be discovered): T is intuitive because proper sets don't have the same size. T is counterintuitive because set size might not be based on bijections. and have ZFC and T be equally easy to use and equally powerful (so ZF-Infinity doesn't cut it, unless one can show that ZF-Infinity is just as powerful as ZFC, which has yet to be done). |