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

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From: Nam Nguyen on 20 Feb 2010 12:36

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.

From: Frederick Williams on 20 Feb 2010 12:42

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."

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

From: Jesse F. Hughes on 20 Feb 2010 12:43

Nam Nguyen <namducnguyen(a)shaw.ca> writes:

> 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.

What does "in any model of T, the relation in which A1 is true is a
true relation" *mean*? (Let's ignore, for now, the fact that there
are no models of T.)

--
Quincy (age 5): Baba, play some [computer games].
Mama: Quincy, if you want [Baba] to live, don't make those
suggestions.
Quincy: Make those suggestions. Got it.

From: Nam Nguyen on 20 Feb 2010 12:48

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.

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

From: Newberry on 20 Feb 2010 12:52

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 relatons neither true nor false" to avoid any potential confusion?