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

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From: Daryl McCullough on 21 Mar 2010 16:28

J. Clarke says...
>
>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.

Why in the world would you have done that? Jesse is one of
the most sensible posters in this group.

--
Daryl McCullough
Ithaca, NY

From: J. Clarke on 21 Mar 2010 22:04

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.

From: Nam Nguyen on 21 Mar 2010 22:43

J. Clarke wrote:
> 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.

No. You're wrong and your telling people about "making up a definition" here
is idiotic: it's in, e.g., Shoenfield's "Mathematical Logic", sections 2.1
("Functions and Predicates") and 2.5 ("Structures").

From: Jesse F. Hughes on 21 Mar 2010 22:55

"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.

--
"When you go to class today, if your professor talks about algebraic
number theory, or misuses Galois Theory[,] I want you to carefully
notice how you feel. Hold on to that feeling so that you never forget
it." --James S. Harris, on channeling rage via Galois theory.

From: Nam Nguyen on 21 Mar 2010 23:59

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

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.