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

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From: Michael Stemper on 19 Feb 2010 13:39

In article <taofn.894$OJ6.99(a)newsfe22.iad>, Nam Nguyen <namducnguyen(a)shaw.ca> writes:

>You summarily dismissed Calvin's appropriately mentioning of CH in this
>thread topic, on the ground that it's philosophical, which I don't see
>as justified at all.
>
>Look, there's a false relation, e.g. the empty set {} for a 1-ary relation
>symbol, right?

I don't see any meaning to the word "false" here. {} is a relation,
period. It has no members, but it's a relation.

> There's also a true relation, a non-empty Universe U for
>a 1-ary relation symbol, right?

I'm going to try to decode what you're saying here. My guess is that
you mean "UxU" is a relation from U to U. Is my guess correct? If so,
it's a relation, just like {} is. These are the trivial cases, but
they're both valid relations.

The statement "(x,y) ∈ R" can be true or false. "R" is neither
true nor false. It's just a set (of ordered pairs).

--
Michael F. Stemper
#include <Standard_Disclaimer>
Time flies like an arrow.
Fruit flies like a banana.

From: Marshall on 19 Feb 2010 18:57

On Feb 19, 10:39 am, mstem...(a)walkabout.empros.com (Michael Stemper)
wrote:
> In article <taofn.894$OJ6...(a)newsfe22.iad>, Nam Nguyen <namducngu...(a)shaw..ca> writes:
> >You summarily dismissed Calvin's appropriately mentioning of CH in this
> >thread topic, on the ground that it's philosophical, which I don't see
> >as justified at all.
>
> >Look, there's a false relation, e.g. the empty set {} for a 1-ary relation
> >symbol, right?
>
> I don't see any meaning to the word "false" here. {} is a relation,
> period. It has no members, but it's a relation.

In some relational theories, relations of any arity >= 0 are allowed.
In such systems, sometime the 0-arity, empty relation is identified
with false and the 0-arity, non-empty relation is identified with
true. These relations are the identity and fixpoint values for
certain operators.

Marshall

From: Newberry on 19 Feb 2010 23:50

On Feb 19, 7:30 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
> Newberry <newberr...(a)gmail.com> writes:
> > You are asking me way too many questions: What gave me this peculiar
> > idea? Why it is an objection? What is the objection against? I do not
> > have a clue.
>
> Why then go on about such matters?

I do not have a clue.

> --
> Aatu Koskensilta (aatu.koskensi...(a)uta.fi)
>
> "Wovon man nicht sprechan kann, darüber muss man schweigen"
> - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

From: OP on 20 Feb 2010 00:15

Newberry wrote:
> On Feb 19, 7:30 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
>>
>> Why then go on about such matters?
>
> I do not have a clue.

Well it's settled then.

From: Nam Nguyen on 20 Feb 2010 01:59

Michael Stemper wrote:
> In article <taofn.894$OJ6.99(a)newsfe22.iad>, Nam Nguyen <namducnguyen(a)shaw.ca> writes:
>
>> You summarily dismissed Calvin's appropriately mentioning of CH in this
>> thread topic, on the ground that it's philosophical, which I don't see
>> as justified at all.
>>
>> Look, there's a false relation, e.g. the empty set {} for a 1-ary relation
>> symbol, right?
>
> I don't see any meaning to the word "false" here. {} is a relation,
> period. It has no members, but it's a relation.

Let's remember that in mathematical reasoning we're allowed to have
definitions, as long as they're technically correct. (If they reasonably
make sense then they'd be "sound" definitions, but they in the first
place must be technically correct).

In model-definition, a relation R is a set of n-tuples in which a n-ary
predicate P <-> P(x1, x2, ..., xn) [about individuals x1, x2, .., xn] will
be determined as true or false. And the way we determine the truth or
falsehood of P is by looking the set R: if the n-tuple (x1, x2, ..., xn)
is _in R_ then P is true, otherwise P is false.

But since {} contains *no* elements, if R = {} then *all* predicates
(formulas) of the form P(x1, x2, ..., xn) are _false_. On the other hand,
if R = the set of all n-tuples, then *all* predicates P(x1, x2, ..., xn)'s
are _true_. It just so happens that in the case of 1-ary relation,
the relation R is just a subset of U, the universe of individuals of the
model in question. Again, in this case, if R is an empty subset then it's
a false relation in the sense that all 1-ary predicate of the form P(x)
are _false_, while if R = U, then all predicates of the form P(x) are true,
hence R is a true relation in this sense.

We could generalize the definitions as:

- R is a true relation if the following formula is true in R:

Ax1...xn[P(x1, ..., xn)]

- R is a false relation if the following formula is true in R:

Ax1...xn[~P(x1, ..., xn)]

>> There's also a true relation, a non-empty Universe U for
>> a 1-ary relation symbol, right?
>
> I'm going to try to decode what you're saying here. My guess is that
> you mean "UxU" is a relation from U to U. Is my guess correct? If so,
> it's a relation, just like {} is. These are the trivial cases, but
> they're both valid relations.

Again I've explained the case of 1-ary where R would be just a subset
of U.

>
> The statement "(x,y) ∈ R" can be true or false. "R" is neither
> true nor false. It's just a set (of ordered pairs).

*****

Given what we've defined as true and false relations above, then the
answer to the (thread-title) question is extremely trivial: *in general*,
a relation is neither true or false, simply because in a relation there
might be true predicates as well as false ones in the relation, say, R.
For example P(x1,x2) might be true [i.e. (x1,x2) is in R], while P(x2,x1)
might be false [i.e. (x2,x1) isn't in R].

That's to say if we take the title-question "When Are Relations Neither
True Nor False?" rather _too literally_! If we take the liberty to give
the question a benefit of a doubt and grant it a slightly different
intention:

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

then that's a different question but it's still a technically valid
question the answer of which is either "never", or "yes, there are such
cases".

I already explained to AK the "yes" answer as the correct one to question
(1). Basically it's the cases when in defining a model/relation instance
we don't have complete one. But based on his terse/short 10-commandment-
like dismissing-assertions or questions throughout the thread, I don't
believe he has a desire to technically explore that answer. (He still
could prove me wrong in my belief here though).