Re: Recursivity vs. Provability

george wrote:
> 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).

I didn't cut that from the post I replied to. My response to that
is: I'm talking about properties of truth and falsity as they were
originally used, not someone's more recent generalization or
variation on these notions. I'm also not talking about statements
that are true under all interpretations, but rather properties of truth
and falsity that hold for true and false as they were originally used.

> 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.

Not that it's relevant, but interpretations don't really apply to
statements. They apply to an expression that needs an interpretation
to form a statement. 1+1=2 is true or false because 1, 2, + and = are
all constants that have definitions. No interpretation is needed. P^Q
is not per se true or false because it contains variables and thus is
not a statement. It is an expression that, given values for P and Q
that are statements, forms a statement.

I think it is a mistake to mix specific ("constant") statements
like 1+1=2 with general expressions like P&Q. It gives the impression
that there is more than one truth. Interpretations are really a way to
create statements, but each statement is true or false independent of
how it was created.

> > 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.

You're talking about re-definitions of Logic that people have come up
with over the years. I am talking about the original definition of
Logic, a term which I personally don't think should be used outside
of its original definition.

> > 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.

That is the redefining of terms that I think is a mistake. Sure, it
gives some authors the opportunity to say that they have generalized
the notion of logic or truth, but that is not really so. They are only
talking about ways to create statements, which themselves really have
nothing to do with the interpretation.

> 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 properties exist already, and the axioms should reflect those
existing, inherent properties. This is the same principle as above:
Axioms are a way to formalize mathematical objects, but the objects
themselves have their properties independent of the axioms. This is
very similar to the mistake of using interpretations to create
statements, then saying that a specific statement is true or false
depending on the interpretation. It is simply referring to variables
and values substituted for them. That is how it should be seen, IMHO.

C-B

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