'When Are Relations Neither True Nor False?
in [Math]
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From: Jesse F. Hughes on 25 Mar 2010 13:39 Newberry <newberryxy(a)gmail.com> writes: >> Evidently, the empty set is not closed under the successor operation >> (according to you), since >> >> (Ay)( y in {} -> s(y) in {} ) >> >> is neither true nor false. So my question is this: since the empty >> set is not closed under the successor operation, would you say that >> >> (Ax)( (Ay)(y in x -> s(y) in x) -> (x is infinite) ) >> >> is a true statement? > > I think it is true. > >> (I hope not.) > > Why? Well, because that sort of reasoning will lead to trouble, I believe. Let me ask you a question. In general, is the statement (Ax)(Px -> Qx) -> (Ex)Px true? If so, we're going to have some problems. The following is a theorem of classical arithmetic and (I assume) also in your system. (An)((Ea,b)( a^2/b^2 = n ) -> (Ea,b)( a^2/b^2 = n and gcd(a,b) = 1 )). Note that this statement is not meaningless, since indeed (En)(Ea,b)(a^2/b^2 = 1) is true. From this, it follows that (Ea,b)( a^2/b^2 = 2 ) -> (Ea,b)( a^2/b^2 = 2 and gcd(a,b) = 1 ), right? Simple instantiation of 2 for n. Is that move legal? (I assume that at this point, we have no proof that 2 is irrational, since this is part of such a proof.) Hence, (Aa,b)( a^2/b^2 = 2 -> gcd(a,b) != 1 ) -> ~(Ea,b)( a^2/b^2 = 2 ). Now, if (Ax)(Px -> Qx) -> (Ex)Px is true in general, then we have both (Aa,b)( a^2/b^2 = 2 -> gcd(a,b) != 1 ) -> ~(Ea,b)( a^2/b^2 = 2 ) and (Aa,b)( a^2/b^2 = 2 -> gcd(a,b) != 1 ) -> (Ea,b)( a^2/b^2 = 2 ). Unfortunately, we also can prove (Aa,b)( a^2/b^2 = 2 -> gcd(a,b) != 1 ). Thus, we conclude both ~(Ea,b)( a^2/b^2 = 2 ) and (Ea,b)( a^2/b^2 = 2 ). Which is bad. Again, where did I go wrong? -- One these mornings gonna wake | Ain't nobody's doggone business how up crazy, | my baby treats me, Gonna grab my gun, kill my baby. | Nobody's business but mine. Nobody's business but mine. | -- Mississippi John Hurt
From: Jesse F. Hughes on 25 Mar 2010 13:40 Newberry <newberryxy(a)gmail.com> writes: > On Mar 24, 9:34 am, "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? -- 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: Daryl McCullough on 25 Mar 2010 14:00 Jesse F. Hughes says... > >Newberry <newberryxy(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. 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: MoeBlee on 25 Mar 2010 17:48 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. MoeBlee
From: Jesse F. Hughes on 25 Mar 2010 18:05
MoeBlee <jazzmobe(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.) 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" |