Re: Recursivity vs. Provability


Charlie-Boo wrote:
> > [ in particular, is there...]
> > > >Or a statement that is not provable and which is true?
>
> > H. Enderton wrote:
> > > True where?

I wrote:
> > I just had to quote that for emphasis.
> > This is one of the core terminological points
> > we sort of have to pound through the heads of
> > normal people: here in THIS weird room, there
> > is NO SUCH THING as "true", just PLAIN true.
> > You only get to be true (or false) UNDER AN
> > INTERPRETATION, or IN A MODEL.

C-B replied:
> But there are properties of truth that hold for
> all models, e.g. if P is true then ~P is false.

That's not the point. The point IS, as *I* said
(but you wrongly cut),

* ( ... if they were true in all models,
* we would be saying that they were provable,
* AS OPPOSED to saying that they were true).

The statements that are true under ALL interpretations
are SPECIAL. Sure they're true, but they're far MORE
than JUST true. Saying they're true is like saying
that Michael Jordan was a basketball player. The
IMPORTANT thing about these statements is NOT that
they're true in the standard model (which is what
just plain "true" would connote), even though they are.
It is that they are true in ALL models, and THEREFORE,
some OTHER much MORE HONORIFIC adjective (than just
plain "true") is appropriate for them.

In the case of a statement in 0th-order or "propositional"
logic like P v ~P, we say that it is a tautology or is
tautologous (NOT merely that it's true, though of course
it is). In the case of a first-order statement that is
true under all interpretations (like Ax[ P(x) v ~P(x) ]),
we say it is valid. In case we have some recursive set
S of axioms, and a sentence Th that is true WHENEVER these
axioms are, so that S -> Th is valid, so that Th is true in All
models OF S, we say that Th is a logical consequence of S,
or a theorem of S. This is a very different situation from
Th being true in only 1 model of S, or in only the standard
model, or in only non-standard models.

> These we can rely on.

Well, yes and no.
The nature of that "reliance" is philosophically
problematic. The kinds of statements that are
"always" true are actually not ALWAYS always true; whether
they are always true or not depends on what logic you are in.
If you don't know WHICH of P or ~P is true then there
are logics in which it is defensible to allege that neither
of them is true, and that therefore P v ~P isn't either.
The position that (at least) one of P & ~P *must* be either
true or false is called "the law of the excluded middle" and
while classical logic obeys it, some other logics that people
have thought up do not.

> I had considered also asking the question of what
> general properties of truth are there,

That's really backwards. Truth is a tiny little atomic
thing. It is something that is had BY OTHER bigger things.

> [I] didn't want to mix the 2 questions.
> However, now that I have, here are
> a few:
>
> 1. If P is true then ~P is false.
> 2. If P is false then ~P is true.
> 3. P has the same value as ~~P.
> 4. P is either true or false.
> 5. P is not both true and false.
>
> What other general properties of truth
> do we know

The above are NOT properties of truth.
They ARE properties of classical logic.
Other logics are possible. Just from the
fact that adjectives other than "true" occur
in the above, you should know that the above
also describes properties of falsity and of negation.
There is also more than one way to do negation.

The IMPORTANT thing if you are going to approach
things "from a logical point of view" is that NOTHING
has any INHERENT properties and EVERYthing has the
properties it has IN VIRTUE OF THE AXIOMS that we used
to DEFINE the concepts we are dealing with.
The 5 things above are not some inherent properties
of truth that we have to remember to adopt and conform
to if we want to do "truth" right. They are just
one way of doing it. It happens to be the standard
way, though.

.

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