'When Are Relations Neither True Nor False?
in [Math]
|
From: Newberry on 27 Mar 2010 00:03 On Mar 26, 3:49 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > 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 mean that there is a plausible explanation why there is no > >inconsistency. I do not like inconsistencies. > > The Liar sentence is not *expressible* in any standard mathematical > theory (PA or ZFC). So you don't have to do anything to keep the Liar > from spoiling the consistency of those languages. Why you think you have to tell me that I do not know. If you lool a few lines above you will see that I was talking about the Liar paraox in the natural language.
From: Newberry on 27 Mar 2010 00:09 On Mar 26, 3:53 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Newberry <newberr...(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. You can picture 2 + 2 = 4. For example the union of a set of two red apples with a set of two green apples is a set of four apples. The number 2 is the set of all sets of cardinality 2. The number 4 is the set of all sets of cardinality 4. From the natural numbers you construct rational numbers and real numbers and Cartesian producst of real numbers ... > > > > > > >> 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.- Hide quoted text - > > - Show quoted text -- Hide quoted text - > > - Show quoted text -
From: Jesse F. Hughes on 27 Mar 2010 00:17 Newberry <newberryxy(a)gmail.com> writes: > On Mar 26, 3:49 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) > wrote: >> 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 mean that there is a plausible explanation why there is no >> >inconsistency. I do not like inconsistencies. >> >> The Liar sentence is not *expressible* in any standard mathematical >> theory (PA or ZFC). So you don't have to do anything to keep the Liar >> from spoiling the consistency of those languages. > > Why you think you have to tell me that I do not know. If you lool a > few lines above you will see that I was talking about the Liar paraox > in the natural language. > But what you say is a non-sequitur. Perhaps, if the liar paradox is neither true nor false in English, then it is not a paradox. But this observation has nothing at all to do with your primary aim: that is, to deny that vacuously true universal statements are true. -- Jesse F. Hughes "You people are the diminishment of a world." -- James S. Harris, to mathematicians.
From: Jesse F. Hughes on 27 Mar 2010 00:22 Newberry <newberryxy(a)gmail.com> writes: > On Mar 26, 3:53 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> Newberry <newberr...(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. > > You can picture 2 + 2 = 4. For example the union of a set of two red > apples with a set of two green apples is a set of four apples. The > number 2 is the set of all sets of cardinality 2. The number 4 is the > set of all sets of cardinality 4. From the natural numbers you > construct rational numbers and real numbers and Cartesian producst of > real numbers ... Well, yes and no. I can construct them in a certain sense, but I surely can't picture them in any reasonable sense. I'm a simple housewife, and I find myself utterly incapable, for instance, of picturing the difference between a regular polygon with 999 sides and a regular polygon with 1000 sides. I'm just that limited. And yet, you want me to picture infinite dimensional spaces before I can assent that theorems regarding those spaces are true. Some unanswered points are left below. >> > 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. -- Jesse F. Hughes "Conviction of fraud can mean jail time. It can mean social censor. It can mean big headlines where mathematicians take "perp walks" before a jeering public." -- JSH on the "censor" that awaits mathematicians.
From: Newberry on 27 Mar 2010 00:34
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. > > -- > "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)- Hide quoted text - > > - Show quoted text - |