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

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From: Nam Nguyen on 20 Feb 2010 13:00

Newberry wrote:
> On Feb 20, 9:42 am, Frederick Williams <frederick.willia...(a)tesco.net>
> wrote:
>> Nam Nguyen wrote:
>>
>>> Frederick Williams wrote:
>>>> Aatu Koskensilta wrote:
>>>>> As usually understood it makes no sense to say of a relation that it is
>>>>> or is not true.
>>>> It seems ok to me to take "such and such a relation is false" to mean
>>>> that no objects in the domain of discourse have the relation to one
>>>> another. For example "x is the mother of y" could be called false in
>>>> the domain {Aatu, Fred}. Ok, you might say "not satisfiable" but so
>>>> what?
>>> For example, given a language L(P1,P2) where P1, P2 are 1-ary symbols,
>>> let's consider the following T:
>>> A1: P1(x) <-> x=x
>>> A2: P2(x) <-> ~P1(x)
>>> It's obvious in any model of T, the relation in which A1 is true is
>>> a true relation, and the one in which A2 is true is a false relation.
>>> Apparently, "satisfiable" isn't even an issue here.
>> This is one of those threads that causes me to think "would that the
>> contributors could find something more interesting to discuss."
>>
> I also thought that we should have rather discussed the substance. But
> given all this and concurring that the terminology is less important,
> is there perhaps a better way to express my intent rather than "when
> are relations neither true nor false" to avoid any potential confusion?

First I've already corrected this particular post of mine with:

> My mistake, let's break that into 2 T's: T1 (with A1) and T2 (with A2).

As for your "original" intention, I also already made a guess and suggested:

> (1) When is a relation R such that a particular formula F would
> be neither true nor false in it?
>

I think there's a huge _substance_ we could have for furthering the discussion.
That's to say, as you've alluded to, if we really care for the substance, instead
of the "peripherals".

From: Nam Nguyen on 20 Feb 2010 13:15

Nam Nguyen wrote:
> Newberry wrote:
>> On Feb 20, 9:42 am, Frederick Williams <frederick.willia...(a)tesco.net>
>> wrote:
>>> Nam Nguyen wrote:
>>>
>>>> Frederick Williams wrote:
>>>>> Aatu Koskensilta wrote:
>>>>>> As usually understood it makes no sense to say of a relation that
>>>>>> it is
>>>>>> or is not true.
>>>>> It seems ok to me to take "such and such a relation is false" to mean
>>>>> that no objects in the domain of discourse have the relation to one
>>>>> another. For example "x is the mother of y" could be called false in
>>>>> the domain {Aatu, Fred}. Ok, you might say "not satisfiable" but so
>>>>> what?
>>>> For example, given a language L(P1,P2) where P1, P2 are 1-ary symbols,
>>>> let's consider the following T:
>>>> A1: P1(x) <-> x=x
>>>> A2: P2(x) <-> ~P1(x)
>>>> It's obvious in any model of T, the relation in which A1 is true is
>>>> a true relation, and the one in which A2 is true is a false relation.
>>>> Apparently, "satisfiable" isn't even an issue here.
>>> This is one of those threads that causes me to think "would that the
>>> contributors could find something more interesting to discuss."
>>>
>> I also thought that we should have rather discussed the substance. But
>> given all this and concurring that the terminology is less important,
>> is there perhaps a better way to express my intent rather than "when
>> are relations neither true nor false" to avoid any potential confusion?
>
> First I've already corrected this particular post of mine with:
>
> > My mistake, let's break that into 2 T's: T1 (with A1) and T2 (with A2).
>
> As for your "original" intention, I also already made a guess and
> suggested:
>
> > (1) When is a relation R such that a particular formula F would
> > be neither true nor false in it?
> >
>
> I think there's a huge _substance_ we could have for furthering the
> discussion.
> That's to say, as you've alluded to, if we really care for the
> substance, instead of the "peripherals".

One of the substances we could have from the question (1) is the possibility
of formally classifying formulas of a particular groups: arithmetically
truth-unassigned-able formulas.

Anyone cares to constructively contribute to the classification. Even if just
to say that's impossible and explain why it is so.

From: Frederick Williams on 20 Feb 2010 13:18

Nam Nguyen wrote:
>
> Newberry wrote:
> > On Feb 20, 9:42 am, Frederick Williams <frederick.willia...(a)tesco.net>
> > wrote:
> >> Nam Nguyen wrote:
> >>
> >>> Frederick Williams wrote:
> >>>> Aatu Koskensilta wrote:
> >>>>> As usually understood it makes no sense to say of a relation that it is
> >>>>> or is not true.
> >>>> It seems ok to me to take "such and such a relation is false" to mean
> >>>> that no objects in the domain of discourse have the relation to one
> >>>> another. For example "x is the mother of y" could be called false in
> >>>> the domain {Aatu, Fred}. Ok, you might say "not satisfiable" but so
> >>>> what?
> >>> For example, given a language L(P1,P2) where P1, P2 are 1-ary symbols,
> >>> let's consider the following T:
> >>> A1: P1(x) <-> x=x
> >>> A2: P2(x) <-> ~P1(x)
> >>> It's obvious in any model of T, the relation in which A1 is true is
> >>> a true relation, and the one in which A2 is true is a false relation.
> >>> Apparently, "satisfiable" isn't even an issue here.
> >> This is one of those threads that causes me to think "would that the
> >> contributors could find something more interesting to discuss."
> >>
> > I also thought that we should have rather discussed the substance. But
> > given all this and concurring that the terminology is less important,
> > is there perhaps a better way to express my intent rather than "when
> > are relations neither true nor false" to avoid any potential confusion?
>
> First I've already corrected this particular post of mine with:
>
> > My mistake, let's break that into 2 T's: T1 (with A1) and T2 (with A2).

But you now need to correct your:

It's obvious in any model of T, the relation in which A1 is true is
a true relation, and the one in which A2 is true is a false relation.

--
..... A lamprophyre containing small phenocrysts of olivine and
augite, and usually also biotite or an amphibole, in a glassy
groundmass containing analcime.

From: Nam Nguyen on 20 Feb 2010 13:24

Frederick Williams wrote:
> Nam Nguyen wrote:

>> > My mistake, let's break that into 2 T's: T1 (with A1) and T2 (with A2).
>
> But you now need to correct your:
>
> It's obvious in any model of T, the relation in which A1 is true is
> a true relation, and the one in which A2 is true is a false relation.
>

Can you elaborate as to what is there to correct, after breaking T into
T1 and T2?

From: Frederick Williams on 20 Feb 2010 13:27

Nam Nguyen wrote:
>
> Frederick Williams wrote:
> > Nam Nguyen wrote:
>
> >> > My mistake, let's break that into 2 T's: T1 (with A1) and T2 (with A2).
> >
> > But you now need to correct your:
> >
> > It's obvious in any model of T, the relation in which A1 is true is
> > a true relation, and the one in which A2 is true is a false relation.
> >
>
> Can you elaborate as to what is there to correct, after breaking T into
> T1 and T2?

The thing that I quoted mentions T but you longer have T, you have T1
and T2.

--
..... A lamprophyre containing small phenocrysts of olivine and
augite, and usually also biotite or an amphibole, in a glassy
groundmass containing analcime.