Re: Cantor's "diagonal argument". My Objection.

On Dec 11, 12:59 am, herbzet <herb...@xxxxxxxxx> wrote:
george wrote:

Interestingly, if we define P->Q in standard ways

That immediately gets into confusion between --> and ==>,
between implication and inference. I did not mention inference
rules at all. I just assumed that all sets of sound&complete
inference rules were equivalent, for this purpose.
If it is important to you to have only one, then resolution will do.
But I didn't think that that was important. What is important is
to prove that ONE finite set of sound&complete inference rules
EXISTS. Once that has been established, it HARDLY MATTERS WHICH
one you use.

What DOES matter is the BASE of axioms/validities/tautologies
that those inference rules are inferring FROM. That was why
I started where I did.

I chose ~ and either of ^ or v as primitive (because of duality,
in the presence of ~, either of ^ or v can be defined in terms of
the other), but you obviously could also choose --> and ~ .
More to the point, you could choose nand or nor as a primitive
connective AND THEN GET BY WITH ONLY ONE primitive
binary connective. To go, perhaps, with your one inference rule.
But that level of compression arguably doesn't gain anything
and arguably loses readibility in a MAJOR way.


Probably not so easy to have only these as an axiomatic basis.

Also, neither one looks very much like a rule of inference

Well, of course, but again, I didn't realize that the question was
about rules of inference. The question was about the validity
of a proof, which needed for the inference rules to be sound
and for them to be applied correctly, not for them to be few in
number.
.