'When Are Relations Neither True Nor False?
in [Math]
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From: Nam Nguyen on 22 Mar 2010 00:03 Nam Nguyen wrote: > Jesse F. Hughes wrote: >> "J. Clarke" <jclarke.usenet(a)cox.net> writes: >> >>> On 3/21/2010 11:31 AM, Nam Nguyen wrote: >>>> J. Clarke wrote: >>>> >>>>> 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. >>>> As long as you understand the empty set is a set, then you'd understand >>>> a demonstration. Here is one. A predicate formula P(x1,x2,...,xn) is >>>> defined >>>> to be true in the relation set R iff the n-tuple (x1,x2,...,xn) is >>>> in R, >>>> and >>>> *_defined_ to be _false_ iff the n-tuple (x1,x2,...,xn) _is not_ in R*. >>>> >>>> But since the empty set is defined so that it has no element, >>>> (x1,x2,...,xn) >>>> is not in R is true, no matter what n-tuple one would happen to have. >>> Oh, I see, you're making up a definition of "truth" and playing games >>> with it. OK. >>> >> >> Nam's more or less right here. >> >> A binary relation R over A and B is a subset of A x B. Let a in A and >> b in B. We say that the formula R(a,b) is true if <a,b> in R and >> false otherwise. >> >> The empty set is a subset of A x B (for every pair of sets A and >> B). Let's denote this relation F. For every a in A and b in B, we >> see that F(a,b) is false. Thus, the empty set as a relation is >> sometimes referred to as the relation "false". >> >> The set A x B is also a subset of A x B and hence also a relation over >> A and B. Let's denote it T. For every a in A and b in B, clearly >> T(a,b) is true and hence T is sometimes referred to as the relation >> "true". >> >> Obviously, this generalizes to n-ary relations. >> >> All of this is perfectly standard. Nam's presentation is a bit odd to >> me (as when he writes "A predicate formula P is defined to be true in >> the relation set R...." -- this terminology doesn't sound familiar to >> me), but the gist is correct. >> > > I don't think it'd sound odd at all if one remembers that in this context > "predicate" and "relation" are equivalent/interchangeable. This is what > Shoenfield wrote: > > A subset of the set of n-tuples in A is called an _n-ary predicate_ in > A. If P represents such a predicate, then P(a1,...,an) means that the > n-tuple (a1,...,an) is in P. > > In a another technical note, section 2.5 pg.19, he defined a particular > formula A to be true in a structure M as: > > If a is pa1...an where p is not =, we let It should have been: "If A is pa1...an where p is not =, we let" > > M(A) = T iff pm(M(a1),...,M(an)) > > (i.e, iff the n-tuple (M(a1),...,M(an)) belongs to the predicate pm) > > > [He actually used a graphical symbol (in both upper case and lower case) > for 'M' and 'm' which is a structure; I just couldn't type that symbol > out of course]. > > My usage is actually correct and I sympathize with it sounding a bit odd. > The culprit is that "predicate" and "relation" are used interchangeably > (and I already gave a caveat) but "predicate" is a more formal system term > used in advanced textbook, while "relation" is more of an abstract algebra > term used in undergraduate textbook. But they both are set in this context > of models and structures. > > I doubt that J. Clarke would have believed that relation in this context is > a set, had I not used a more formal definition such as "predicate" from a > textbook about formal system. > > Hope this has clarified what I said. > >
From: J. Clarke on 22 Mar 2010 08:17 On 3/22/2010 12:03 AM, Nam Nguyen wrote: > Nam Nguyen wrote: >> Jesse F. Hughes wrote: >>> "J. Clarke" <jclarke.usenet(a)cox.net> writes: >>> >>>> On 3/21/2010 11:31 AM, Nam Nguyen wrote: >>>>> J. Clarke wrote: >>>>> >>>>>> 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. >>>>> As long as you understand the empty set is a set, then you'd >>>>> understand >>>>> a demonstration. Here is one. A predicate formula P(x1,x2,...,xn) is >>>>> defined >>>>> to be true in the relation set R iff the n-tuple (x1,x2,...,xn) is >>>>> in R, >>>>> and >>>>> *_defined_ to be _false_ iff the n-tuple (x1,x2,...,xn) _is not_ in >>>>> R*. >>>>> >>>>> But since the empty set is defined so that it has no element, >>>>> (x1,x2,...,xn) >>>>> is not in R is true, no matter what n-tuple one would happen to have. >>>> Oh, I see, you're making up a definition of "truth" and playing >>>> games with it. OK. >>>> >>> >>> Nam's more or less right here. >>> >>> A binary relation R over A and B is a subset of A x B. Let a in A and >>> b in B. We say that the formula R(a,b) is true if <a,b> in R and >>> false otherwise. >>> >>> The empty set is a subset of A x B (for every pair of sets A and >>> B). Let's denote this relation F. For every a in A and b in B, we >>> see that F(a,b) is false. Thus, the empty set as a relation is >>> sometimes referred to as the relation "false". >>> >>> The set A x B is also a subset of A x B and hence also a relation over >>> A and B. Let's denote it T. For every a in A and b in B, clearly >>> T(a,b) is true and hence T is sometimes referred to as the relation >>> "true". >>> >>> Obviously, this generalizes to n-ary relations. >>> >>> All of this is perfectly standard. Nam's presentation is a bit odd to >>> me (as when he writes "A predicate formula P is defined to be true in >>> the relation set R...." -- this terminology doesn't sound familiar to >>> me), but the gist is correct. >>> >> >> I don't think it'd sound odd at all if one remembers that in this context >> "predicate" and "relation" are equivalent/interchangeable. This is what >> Shoenfield wrote: >> >> A subset of the set of n-tuples in A is called an _n-ary predicate_ in >> A. If P represents such a predicate, then P(a1,...,an) means that the >> n-tuple (a1,...,an) is in P. >> >> In a another technical note, section 2.5 pg.19, he defined a particular >> formula A to be true in a structure M as: >> >> If a is pa1...an where p is not =, we let > > It should have been: "If A is pa1...an where p is not =, we let" > >> >> M(A) = T iff pm(M(a1),...,M(an)) >> >> (i.e, iff the n-tuple (M(a1),...,M(an)) belongs to the predicate pm) >> >> >> [He actually used a graphical symbol (in both upper case and lower case) >> for 'M' and 'm' which is a structure; I just couldn't type that symbol >> out of course]. >> >> My usage is actually correct and I sympathize with it sounding a bit odd. >> The culprit is that "predicate" and "relation" are used interchangeably >> (and I already gave a caveat) but "predicate" is a more formal system >> term >> used in advanced textbook, while "relation" is more of an abstract >> algebra >> term used in undergraduate textbook. But they both are set in this >> context >> of models and structures. >> >> I doubt that J. Clarke would have believed that relation in this >> context is >> a set, had I not used a more formal definition such as "predicate" from a >> textbook about formal system. >> >> Hope this has clarified what I said. Definition of "truth". Where does it come from?
From: Jesse F. Hughes on 22 Mar 2010 09:02 "J. Clarke" <jclarke.usenet(a)cox.net> writes: > On 3/22/2010 12:03 AM, Nam Nguyen wrote: >> Nam Nguyen wrote: >>> Jesse F. Hughes wrote: >>>> >>>> A binary relation R over A and B is a subset of A x B. Let a in A and >>>> b in B. We say that the formula R(a,b) is true if <a,b> in R and >>>> false otherwise. >>>> >>>> The empty set is a subset of A x B (for every pair of sets A and >>>> B). Let's denote this relation F. For every a in A and b in B, we >>>> see that F(a,b) is false. Thus, the empty set as a relation is >>>> sometimes referred to as the relation "false". >>>> >>>> The set A x B is also a subset of A x B and hence also a relation over >>>> A and B. Let's denote it T. For every a in A and b in B, clearly >>>> T(a,b) is true and hence T is sometimes referred to as the relation >>>> "true". >>>> >>>> Obviously, this generalizes to n-ary relations. >>>> >>>> All of this is perfectly standard. Nam's presentation is a bit odd to >>>> me (as when he writes "A predicate formula P is defined to be true in >>>> the relation set R...." -- this terminology doesn't sound familiar to >>>> me), but the gist is correct. >>>> >>> >>> I don't think it'd sound odd at all if one remembers that in this context >>> "predicate" and "relation" are equivalent/interchangeable. This is what >>> Shoenfield wrote: >>> >>> A subset of the set of n-tuples in A is called an _n-ary predicate_ in >>> A. If P represents such a predicate, then P(a1,...,an) means that the >>> n-tuple (a1,...,an) is in P. >>> >>> In a another technical note, section 2.5 pg.19, he defined a particular >>> formula A to be true in a structure M as: >>> >>> If a is pa1...an where p is not =, we let >> >> It should have been: "If A is pa1...an where p is not =, we let" >> >>> >>> M(A) = T iff pm(M(a1),...,M(an)) >>> >>> (i.e, iff the n-tuple (M(a1),...,M(an)) belongs to the predicate pm) >>> >>> >>> [He actually used a graphical symbol (in both upper case and lower case) >>> for 'M' and 'm' which is a structure; I just couldn't type that symbol >>> out of course]. >>> >>> My usage is actually correct and I sympathize with it sounding a bit odd. >>> The culprit is that "predicate" and "relation" are used interchangeably >>> (and I already gave a caveat) but "predicate" is a more formal system >>> term >>> used in advanced textbook, while "relation" is more of an abstract >>> algebra >>> term used in undergraduate textbook. But they both are set in this >>> context >>> of models and structures. >>> >>> I doubt that J. Clarke would have believed that relation in this >>> context is >>> a set, had I not used a more formal definition such as "predicate" from a >>> textbook about formal system. >>> >>> Hope this has clarified what I said. > > Definition of "truth". Where does it come from? It's a perfectly standard definition of truth for relations in set theory. It is essentially the same as model theory uses, except that in model theory, there is a more explicit difference between the syntax and semantics (a relation is not literally a set, but rather its interpretation is a set). Any introductory text on predicate logic or set theory should include this material. -- Damn John Jay. Damn everyone who won't damn John Jay. Damn everyone who won't put lights in his windows and sit up all night damning John Jay. -- Political graffiti from late 18th c. Boston
From: Aatu Koskensilta on 22 Mar 2010 10:41 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > If that's case, and you have to say if it is or isn't, then an example > of a "nontrivial" metamathematical theorem would be the following modified > GIT: > > (1) _If_ PA is consistent, then for any consistent formal system T > sufficient strong to carry out basic notions of arithmetic, > there's a formula G(T) which is syntactically undecidable in T > but of which a certain encoded formula, say, encoded(G(T)) is > provable in PA. > > Would (1) be an interesting theorem? It's hard to say, since your statement of this theorem is rather opaque. For instance, what is encoded(G(T))? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 22 Mar 2010 10:50
Marshall <marshall.spight(a)gmail.com> writes: > And the best opinions in the world do not, IMHO, rise to the level > of the worst evidence. I'm calling round 1 for TP here. Bah! Even barely educated guesses rise far above the level of the worst evidence. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |