'When Are Relations Neither True Nor False?
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From: Nam Nguyen on 21 Mar 2010 02:38 J. Clarke wrote: > 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, Then what _is_ a relation, to you? > 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? Either one. Since in this context a relation _is a set_ of certain n-tuples. [A relation-set has another name: an n-ary predicate.] > >> 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. Which standard textbook did you use that doesn't define a relation as a set?
From: Nam Nguyen on 21 Mar 2010 02:44 Marshall wrote: > 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. If it's _impossible_ to know whether or not it's empty then you haven't satisfied the set-hood definition for the "set", hence it's neither a true nor false set.
From: J. Clarke on 21 Mar 2010 08:55 On 3/21/2010 2:38 AM, Nam Nguyen wrote: > J. Clarke wrote: >> 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, > > Then what _is_ a relation, to you? > >> 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? > > Either one. Since in this context a relation _is a set_ of certain > n-tuples. [A relation-set has another name: an n-ary predicate.] >> >>> 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. > > Which standard textbook did you use that doesn't define a relation > as a set? What I get for posting in a fit of insomnia. However, if the definition of a "relation" is "a set of n-tuples", then by definition the empty set is not a "relation" and so your statement that "a false relation is an empty set" violates the definition. If you want to say that the empty set is not a relation that's fine, but if you want to say that it is a "false relation" please demonstrate that it is a relation at all.
From: Jesse F. Hughes on 21 Mar 2010 10:13 "J. Clarke" <jclarke.usenet(a)cox.net> writes: > However, if the definition of a "relation" is "a set of n-tuples", then > by definition the empty set is not a "relation" [...] Nonsense! The empty set is a set of n-tuples for *every* n. Every element of the empty set is an n-tuple (no matter the value of n). The empty set is a relation. -- Jesse F. Hughes "A year ago, my approval rating was in the 30s, my nominee for the Supreme Court had just withdrawn, and my vice president had shot someone. Ah, those were the good old days." G. W. Bush 3/28/07
From: J. Clarke on 21 Mar 2010 11:15
On 3/21/2010 10:13 AM, Jesse F. Hughes wrote: > "J. Clarke"<jclarke.usenet(a)cox.net> writes: > >> However, if the definition of a "relation" is "a set of n-tuples", then >> by definition the empty set is not a "relation" [...] > > Nonsense! The empty set is a set of n-tuples for *every* n. Every > element of the empty set is an n-tuple (no matter the value of n). > > The empty set is a relation. I thought I had you killfiled. Well, back you go. |