'When Are Relations Neither True Nor False?
in [Math]
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From: Jesse F. Hughes on 26 Mar 2010 06:53 Newberry <newberryxy(a)gmail.com> writes: > 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. Well, have at it! >> 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. So? You said that I have to be able to picture it. >> 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. No, I'm not saying what the world would look like if x = x + 1. I'm merely pointing out a single consequence of that equation. Indeed, this consequence is *true* in those structures in which x = x + 1. (As Nam pointed out, such structures do exist, you know.) >> >> >> [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. -- "You are beneath contempt because you betray mathematics itself, and spit upon the truth, spit upon decency, and spit upon the intelligence of the world. You betrayed the world, and now it's time for the world to notice." -- James S. Harris awaits Justice for crimes against Math.
From: Jesse F. Hughes on 26 Mar 2010 06:56 Newberry <newberryxy(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. -- "Maya Nahib is not a Checotah Indian! [...] Maya Nahib is an Englishman!" "Are you telling us that a civilized white man could kill and ravish and destroy with all the brutality of a savage?" -- Adventures by Morse radio program (1944)
From: Daryl McCullough on 26 Mar 2010 07:24 Newberry says... > >On Mar 25, 3:38=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> Newberry says... >> >> >If you take the position that there are truth value gaps then the Liar >> >papradox is solvable in English. >> >> What does it mean to be "solvable" and why do you want it to be solvable? > >It means that there is a plausible explanation why there is no >inconsistency. I do not like inconsistencies. I don't see how truth gaps help in the Liar paradox. Suppose you have a truth predicate T(x) and you have a sentence L (with Godel number #L) of the form forall x, A(x) -> ~T(x) and you have a theorem forall x, A(x) <-> x=#L Then L cannot be true, and cannot be false. So L falls into a "truth gap". But then what about the sentence "L is not true" which is formalized by ~T(#L) Is that true, or does that have a truth gap, as well? We just agreed that L was not true, so if T(x) is a truth predicate, that should be formalized by ~T(#L). From that, surely it follows that "forall x, if x=#L, then ~T(x)" Since x=#L <-> A(x), then surely it follows that "forall x, A(x) -> ~T(x)" So L follows from the claim that L falls in the truth gap. Truth gaps *don't* help with the Liar paradox. -- Daryl McCullough Ithaca, NY
From: Jesse F. Hughes on 26 Mar 2010 10:24 Transfer Principle <lwalke3(a)lausd.net> writes: > On Mar 22, 5:18 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> Transfer Principle <lwal...(a)lausd.net> writes: >> > Notice that J. Clarke here is expressing similar ideas regarding the >> > empty set and vacuous truth as the OP of this thread, Newberry. >> No, he's not. He's claiming that the empty set is not a relation. He >> is not claiming that vacuously true statements are neither true nor >> false. >> Why you think this is even remotely related to Newberry's pet project >> is beyond me. > > There must be some reason that Clarke claims that the empty set is > not a relation, in contradiction to ZFC and FOL. First, let's note that *even if* Clarke claims the empty set is not a relation, that does not mean Clarke agrees with Newberry. Again, even if I thought that Clarke's claim was a philosophical position, I see no reason you think that it's similar to Newberry's ideas. > Let's go back to a post made by Rotwang: > > "And if a physicist accepts all > the axioms of ZFC, as well as the usual rules of FOL, then he must > accept all of the resulting theorems whether he realises it or not." > > Applying contapositive to Rotwang's statement, we see that if a > physicist _doesn't_ accept all the theorems of ZFC/FOL, then he > doesn't accept at least one of the axioms of ZFC/FOL. > > Of course, we don't know whether Clarke or Newberry are even > physicists, since after all: > >> [1] I'm not saying that Newberry or Clarke is untrained in >> mathematics [or physics, for that matter]. I have no idea of their >> backgrounds, of course. > > But still, I have yet to see why the contrapositive to Rotwang's > statement can't be generalized to any poster. So any poster who > doesn't accept all of the theorems of ZFC/FOL must reject at least > one of the axioms of ZFC/FOL. And since Clarke rejects "the empty > set is a relation," which is a theorem of ZFC/FOL, then we conclude > that he rejects at least one of the axioms. And I'd like to know which > axiom that is (to allow for the possibility of a theory with takes the > rejected axiom and replace it by, say, its negation -- a theory which > hopefully would be more acceptable to Clarke). You honestly seem incapable of realizing that sometimes people make mistakes. One can make a mistake for any of the following reasons: (1) One can simply be ignorant of set theory and its conventions, as Clarke seems to be in this thread. (2) One can be so devoted to a particular conclusion that his reasoning regarding that conclusion is repeatedly invalid. WM's reasoning is an example of this. He is not just proposing an alternative to ZFC, but offers fallacious proof after fallacious proof that ZFC is invalid. (3) One can both be ignorant and prone to bad reasoning. AP seems a good example of this. Sometimes, people are just wrong in their opinions, but you seem to neglect this possibility. Rather, you think the proper response is to find a theory that for each mistaken statement about set theory so that the statement is correct in that theory. It's an odd hobby, but have at it, I suppose. >> Why you think this is even remotely related to Newberry's pet project >> is beyond me. > > So Hughes doubts my claim -- and we already know that the most > common criticism of me by far is misinterpretation and incorrect > guessing of what others are saying. > > So how can I reduce this bad habit of mine, and avoid misinterpreting > what's in Clarke's post. We know that Clarke objects to _some_ axiom > of ZFC/FOL, but how can I find out which one? We know no such thing. Every indication is that Clarke was ignorant of the definitions regarding relations and truth in set theory. > I could try asking Clarke directly. But I must do so in a way so as > not to appear to be a standard theorist or a "bully." I'm not trying > to _interrogate_ Clarke in order to criticize him, but to find out > which axiom he rejects in order to come up with an alternative > theory, without the standard theorists accusing me of > misinterpreting posts. Yes, by all means, ask him. That's obviously the single best approach. You think he rejects some axiom of ZFC. Just ask him which. -- Jesse F. Hughes "You're terrified of your daughters dreaming about me." -- James S. Harris, on why mathematicians fear him
From: Alan Smaill on 26 Mar 2010 10:37
Nam Nguyen <namducnguyen(a)shaw.ca> writes: > 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. > > That's a grossly erroneous over-generalization. It's many of those who > defend standard mathematics who erroneously object to the counter-intuitive > _but real nature_ of mathematics: relativity of truth and provability. > > To date they can't cite any absolutely true formula, without the formula > being false in another similar context, and yet they'd believe such > "absolute" truth is intuitive. > >> The ultimate logic >> would be one in which it is impossible to prove any result that >> you couldn't already guess was true. > > The ultimate logic is one which is relativistic. Is that an absolute truth, then? I know, it's an old ploy, but your position just begs the question. -- Alan Smaill email: A.Smaill at ed.ac.uk School of Informatics tel: 44-131-650-2710 University of Edinburgh |