'When Are Relations Neither True Nor False?
in [Math]
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From: Nam Nguyen on 20 Mar 2010 19:07 Stefan Ram wrote: > Newberry <newberryxy(a)gmail.com> writes: >> When Are Relations Neither True Nor False? > > (I am late to this thread, so please excuse me if I > should repeat something that was already written.) > > Always. A relation is a set of pairs, and a set is > never true nor false. Therefore, every relation is > neither true nor false. > Technically speaking, what you said isn't quite true. A false relation is an empty set, while a true one isn't an empty set. A relation - a set - could be said to be neither true nor false if in defining it, it's impossible to know whether or not it's an empty set. For example, if you define a relation R such that the formula "There are infinitely many counter examples of GC" is true in that relation. Would that definition be _complete_ enough that you know the relation S be empty? or not?
From: Marshall on 20 Mar 2010 19:25 On Mar 20, 4:07 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Stefan Ram wrote: > > Newberry <newberr...(a)gmail.com> writes: > >> When Are Relations Neither True Nor False? > > > (I am late to this thread, so please excuse me if I > > should repeat something that was already written.) > > > Always. A relation is a set of pairs, and a set is > > never true nor false. Therefore, every relation is > > neither true nor false. > > Technically speaking, what you said isn't quite true. A false > relation is an empty set, while a true one isn't an empty set. > > A relation - a set - could be said to be neither true nor false > if in defining it, it's impossible to know whether or not it's > an empty set. Not quite. A relation could be said that it is unknown whether it is empty or not if, in defining it, it's impossible to know whether it's empty or not. However it's still one or the other. Marshall
From: Nam Nguyen on 20 Mar 2010 19:28 Nam Nguyen wrote: > Daryl McCullough wrote: >> Nam Nguyen says... >> >>> I don't go by belief in technical argument. >> >> There is no technical argument at play. > > There is a technical definition of truth in a model. I'm sure you > know that. > >> You reject standard >> mathematics for subjective reasons, > > I reject that because your "belief" in the natural number is just > that and doesn't conform with your own side's technical definition > of truth in model. > >> and I reject your mathematics >> for other subjective reasons: it doesn't sound like it produces >> any theorems of interest to anyone. For clarification, and for the record, I've never rejected the purely (finite) syntactical reasoning via formal systems and Hilbert-style rules of inference seen in Shoenfield's "Mathematical Logic" or other _textbooks_. It's your side dubious subjective belief that one would fully know the natural numbers to make it foundational that I'm rejecting. And so far your side has not been able to come up with technical reasons why the rejection is bad, other than the subjective and _dubious_ reason that is something like "because it doesn't fit with our subjective interest"! Pardon me, but is that a reasonable counter argument? > > Did you happen to heear hear me mentioning something like G2IT > (Godel-Goldbach Incompleteness Theorem of Knowledge) in other > thread sometimes ago. It "sounds" interesting to you isn't it? > No? It's hard to reason with you if you already have some _subjective_ > "bad" preconception and shut your mind out of objective analysis > of methods of reasoning. Iow, it could be interesting, Daryl, if > you further objectively investigate it. > > You're free of course to reject anything as you wish, including the > reasoning principles that I suspect you don't have valid reasons > to think of them as bad. > > All you do is staying unreasonable and attacking the messenger, instead > of technically confronting the 4 principles with technical analysis > and reasons.
From: J. Clarke on 20 Mar 2010 22:01 On 3/20/2010 7:07 PM, Nam Nguyen wrote: > Stefan Ram wrote: >> Newberry <newberryxy(a)gmail.com> writes: >>> When Are Relations Neither True Nor False? >> >> (I am late to this thread, so please excuse me if I >> should repeat something that was already written.) >> >> Always. A relation is a set of pairs, and a set is >> never true nor false. Therefore, every relation is >> neither true nor false. >> > > Technically speaking, what you said isn't quite true. A false > relation is an empty set, while a true one isn't an empty set. A relation is not a "set" at all, it establishes some rule by which elements of one set may be associated with elements of another set. If either or both sets are empty that does not mean that the relation is "true" or "false", it just means that it is a relation from some set into the empty set or vice versa. > A relation - a set - Which are you talking about, a relation or a set? > could be said to be neither true nor false > if in defining it, it's impossible to know whether or not it's > an empty set. For example, if you define a relation R such that > the formula "There are infinitely many counter examples of GC" > is true in that relation. Would that definition be _complete_ > enough that you know the relation S be empty? or not? You seem to be using the word "relation" in a very loose way that has little to do with the accepted mathematical definition.
From: Newberry on 20 Mar 2010 23:34
On Mar 20, 2:44 pm, r...(a)zedat.fu-berlin.de (Stefan Ram) wrote: > Newberry <newberr...(a)gmail.com> writes: > >When Are Relations Neither True Nor False? > > (I am late to this thread, so please excuse me if I > should repeat something that was already written.) > > Always. A relation is a set of pairs, and a set is > never true nor false. Therefore, every relation is > neither true nor false. Your point is not well taken. From the context we see that I mean "sentences with more than one variable." But since "when are sentences with more than one variable neither true nor false" sounds awkward I instead use the expression above. |