'When Are Relations Neither True Nor False?
in [Math]
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From: Jesse F. Hughes on 27 Mar 2010 09:27 Newberry <newberryxy(a)gmail.com> writes: > On Mar 26, 3:56 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> Newberry <newberr...(a)gmail.com> writes: >> > 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? >> >> Sure, they can, but "I close the switch" is not a proposition. "I am >> closing the switch" or "I have closed the switch" are propositions. > > I am not following. Well, it is not all that relevant. -- "If you are a mathematician, then you cannot dispute the result. If you dispute the result [...] then you are NOT a mathematician. Anyone who disputes this result [...] is not a mathematician. I am a mathematician, which is how I could find the result."--James S. Harris
From: Tim Golden BandTech.com on 27 Mar 2010 10:02 On Mar 26, 5:55 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Nam Nguyen wrote: > > Alan Smaill wrote: > >> Nam Nguyen <namducngu...(a)shaw.ca> writes: > >>>> Seriously, if you could demonstrate a truly absolute abstract truth > >>>> in mathematical reasoning, I'd leave the forum never coming back. > >>> If you can't (general "you") then I'm sorry: my duty to the Zen council, > >>> so to speak, is to see to it that "absolute" truths such as G(PA) is a > >>> thing of the past, if not of oblivion. > > >> one day you will realise that your duty to the Zen council > >> is to overcome your feeling of duty to what is purely subjective ... > > > I'm sure your belief in the "absolute" truth of G(PA) is subjective, which > > you'd need to overcome - someday. Each of us (including Godel) coming to > > mathematics and reasoning has our own subjective "baggage". > > > Is it FOL, or FOL=, that you've alluded to? For example. > > Note how much this physical reality has influenced and shaped our > mathematics and mathematical reasonings. Euclidean postulates had their > root in our once perception of space. From P(a) we infer Ex[P(x)] > wouldn't be an inference if the our physical reality didn't support > such at least in some way. And uncertainty in physics is a form > relativity. > > The point is relativity runs deep in reality and you're not fighting > with a lone person: you're fighting against your own limitation! > > Any rate, enough talk. Do you have even a single absolute truth you > could show me so that I'd realize I've been wrong all along? Let's > begin with the natural numbers: which formula in the language of > arithmetic could _you_ demonstrate as absolutely true? There is a fairly straightforward construction that can yield both boolean logic and continuous higher forms, and even a lower form that I will call universal. Constrain the real numbers to those values whose magnitude is unity. We see two options +1, -1 . Using polysign numbers extend this system to P3. ( http://bandtechnology.com/PolySigned ) One might initially consider there to be a three verticed logic here, but in the general form we see that the unity values now form a continuous circle. This is a nice exercise in the continuous/discrete paradigms of throught. In one dimension we see a discrete type, not unlike charge. In two dimensions we see that the same procedure yields a continuum of values, though there are arguably those three unique positions -1, +1, *1 . Inspecting the product logic back in P2 (the boolean or constrained real number case) - + = - + - = - + + = + - - = + and likewise in the three signed case (overlooking the above redundancy) - - = + - + = * - * = - + + = - + * = + * * = * Does a false false yield a true? The english language discourages the usage of double negatives, yet their use does exist within in it with such phrases as 'I am not an atheist.' Back in ordinary logic it is no problem to see that the math holds up in P2 so that Not(Not(A)) = A. The meaning of false and true cannot be reused in P3 and it is a nice human puzzle to consider that we and our dualistic thought patterns have artificially limited us. The P3 language is not sensible to the human mind, yet it may be entirely accurate. Treading on P1 is difficult for most, but there we see just one instance within this logical paradigm -1 . Thus the polysign allow for a universal but fairly inanimate form at the bottom of the hierarchy universality duality triality (not to be confused with Clifford form) ... By leaving the Euclidean and working the sphere these forms exist naturally. - Tim
From: Nam Nguyen on 27 Mar 2010 11:14 Daryl McCullough wrote: > Nam Nguyen says... >> Alan Smaill wrote: >> >>>> I'm sure your belief in the "absolute" truth of G(PA) is subjective, which >>>> you'd need to overcome - someday. Each of us (including Godel) coming to >>>> mathematics and reasoning has our own subjective "baggage". >>> Why on earth do you think I have some belief in the " "absolute" truth " >>> of G(PA) ? I don't even know what that *means* . >> OK. Then on the meta level, do you think it's correct to say that >> G(PA) can be arithmetically false? > > It is a *relative* truth. It's true in the standard interpretation > of the language of PA. So you've agreed "G(PA) can be arithmetically false"? What would be the relationship between "being arithmetically false" and "standard interpretation" do you think there is? The point being is "standard interpretation" is a relative notion.
From: Marshall on 27 Mar 2010 11:20 On Mar 26, 8:57 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Marshall wrote: > > On Mar 26, 4:35 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> MoeBlee wrote: > >>> On Mar 26, 4:16 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >>>> MoeBlee wrote: > >>>>> On Mar 26, 3:11 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >>>>>> if you could demonstrate a truly absolute abstract truth in mathematical > >>>>>> reasoning, I'd leave the forum never coming back. > >>>>> Oh, sweet seduction, please don't tempt me so! > >>>> Go ahead and demonstrate one, or prove any of my 4 principles is wrong. > >>>> If you can. > >>> Oh, Nam, how you make my heart all aflutter! > >> Oh MoeBlee, how you fail time to time to technically show what I'd say > >> about the foundation issues of FOL reasoning be wrong. > > > That Moe has failed to do so in a way that you can understand > > is a failing, but it's not Moe's failing. > > Where did Moe successfully demonstrate, say, an absolute truth that > I failed to understand, or my 4 principles are wrong and I couldn't > understand his demonstration? You wouldn't be able to tell if he had. Your criteria for truth require that a sentence be true even if we redefine the terms in it. Under that criterion, of course anything can mean anything. In fact, I can completely agree with your "everything is relative" claim by redefining "everything" to mean "something" and "relative" to mean "absolute." > Or you're just full of babbling words with no technical substance, > as usual? [It seems like a habit of yours that when you couldn't > technically counter your opponent's argument then you just call him > a mad dog!] I have never called you a mad dog that I can recall. My term for you is "talentless bufoon." So much more apropos. Marshall
From: Nam Nguyen on 27 Mar 2010 11:25
Tim Golden BandTech.com wrote: > On Mar 26, 5:55 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Nam Nguyen wrote: >>> Alan Smaill wrote: >>>> Nam Nguyen <namducngu...(a)shaw.ca> writes: >>>>>> Seriously, if you could demonstrate a truly absolute abstract truth >>>>>> in mathematical reasoning, I'd leave the forum never coming back. >>>>> If you can't (general "you") then I'm sorry: my duty to the Zen council, >>>>> so to speak, is to see to it that "absolute" truths such as G(PA) is a >>>>> thing of the past, if not of oblivion. >>>> one day you will realise that your duty to the Zen council >>>> is to overcome your feeling of duty to what is purely subjective ... >>> I'm sure your belief in the "absolute" truth of G(PA) is subjective, which >>> you'd need to overcome - someday. Each of us (including Godel) coming to >>> mathematics and reasoning has our own subjective "baggage". >>> Is it FOL, or FOL=, that you've alluded to? For example. >> Note how much this physical reality has influenced and shaped our >> mathematics and mathematical reasonings. Euclidean postulates had their >> root in our once perception of space. From P(a) we infer Ex[P(x)] >> wouldn't be an inference if the our physical reality didn't support >> such at least in some way. And uncertainty in physics is a form >> relativity. >> >> The point is relativity runs deep in reality and you're not fighting >> with a lone person: you're fighting against your own limitation! >> >> Any rate, enough talk. Do you have even a single absolute truth you >> could show me so that I'd realize I've been wrong all along? Let's >> begin with the natural numbers: which formula in the language of >> arithmetic could _you_ demonstrate as absolutely true? > > There is a fairly straightforward construction that can yield both > boolean logic and continuous higher forms, and even a lower form that > I will call universal. > > Constrain the real numbers to those values whose magnitude is unity. > We see two options > +1, -1 . It's relative as to how many real numbers one could "constrain". So "constraint" is a relative notion, not an absolute one. In any rate, in all the below (including the URL) I still couldn't see an absolute truth. Could you state such truth here? > > Using polysign numbers extend this system to P3. > ( http://bandtechnology.com/PolySigned ) > One might initially consider there to be a three verticed logic here, > but in the general form we see that the unity values now form a > continuous circle. > This is a nice exercise in the continuous/discrete paradigms of > throught. In one dimension we see a discrete type, not unlike charge. > In two dimensions we see that the same procedure yields a continuum of > values, though there are arguably those three unique positions > -1, +1, *1 . > > Inspecting the product logic back in P2 (the boolean or constrained > real number case) > - + = - > + - = - > + + = + > - - = + > and likewise in the three signed case (overlooking the above > redundancy) > - - = + > - + = * > - * = - > + + = - > + * = + > * * = * > > Does a false false yield a true? The english language discourages the > usage of double negatives, yet their use does exist within in it with > such phrases as > 'I am not an atheist.' > Back in ordinary logic it is no problem to see that the math holds up > in P2 so that > Not(Not(A)) = A. > The meaning of false and true cannot be reused in P3 and it is a nice > human puzzle to consider that we and our dualistic thought patterns > have artificially limited us. The P3 language is not sensible to the > human mind, yet it may be entirely accurate. > > Treading on P1 is difficult for most, but there we see just one > instance within this logical paradigm > -1 . > Thus the polysign allow for a universal but fairly inanimate form at > the bottom of the hierarchy > universality > duality > triality (not to be confused with Clifford form) > ... > By leaving the Euclidean and working the sphere these forms exist > naturally. > > - Tim > |