'When Are Relations Neither True Nor False? [Logic]

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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