Re: Is Validity Just a Hypothetical or Conditional Characteristic?

"galathaea" <galathaea@xxxxxxxxx> writes:

On Jan 29, 5:42 pm, herbzet <herb...@xxxxxxxxx> wrote:
Jan Burse wrote:
herbzet wrote:
herbzet wrote:

[...]

Yes. As a matter of fact, for any valid formula P -> Q it
is the case that (P & Q) -> Q is also valid, because it is
a tautology that

(i) (p -> q) <-> ((p & q) = p)

using "=" as a variant notation for "<->".

Badly put. (P & Q) -> Q is valid simpliciter. What I should
have said was

"As a matter of fact, for any valid formula P -> Q it
is the case that (P & Q) = P, because it is a tautology
that ..."

--
hz

Hm, yes A -> B :<=> A & B <-> A. But does this
lead us to somewhere? Your claim was:

> If you have a valid sentence P -> Q then there is a
> sentence P' = P of the form

> P1 & P2 & P3 ... & Pn (0 < n)

> and there is a sentence Q' = Q of the form

> P1 & P2 & P3 ... & Pm (0 < m <= n).

> That is, each conjunct of Q' is a conjunct of P'.

This relates trivially to P & Q <-> P. Namely put

P1 = Q
P2 = P

Then P1 & P2 <-> P, and P1 <-> Q.

Which means I will not be able to challenge you.

No, you will not be able to challenge my claim. The tautology

(i) (p -> q) <-> ((p & q) = p)

and all its substitution instances would allow me to conclude
from any valid formula P -> Q that the valid formula (P & Q) -> Q
trivially and un-illuminatingly satisfies my claim.

As a matter of fact, my claim can be proved non-trivially, with-
out assuming (i), and this is in fact how I first proved it to
myself. My proofs to you of the truth of that claim with regard
to (a & (a -> b)) -> b and the trivial formula (a & ~a) -> b
are themselves non-trivial and do not rely on (i) or some
variation thereof.

It was only later that I noticed the existence of (i). I cannot
make (i) go away. It is a fact that the conclusion of a valid
argument contains no more information than do the premises.

But there is more that can be said. The reason a valid argument
is valid is because the conclusion merely restates some or all
of what is stated by the premises, while adding nothing else.
To entertain the premises of a valid argument is already to have
entertained the conclusion.

This understanding of the nature of validity has been around at
least since the Peripatetics, but we now have an instrument, a
scientific language, in which this understanding can be rigorously
and concretely demonstrated.

What is particularly interesting is that this relation between
premise and conclusion is thus stated without any reference to truth
and falsehood whatever.

The relation of premise to conclusion is one of inclusion. Any
valid inference P -> Q can be non-trivially reduced to the form

(P1 & P2 & ... & Pn) -> (P1 & P2 & ... & Pm) (0 < m <= n)

with P = (P1 & ... & Pn) and Q = (P1 & ... & Pm)

(and no, the Pi need not be atomic). Obviously, a sentence of
this form cannot have a true premise and a false conclusion, but
any other combination of truth and falsehood is possible. The
validity of the argument in no way depends on the truth or falsity
of the Pi -- only on the relation of inclusion between premise and
conclusion.

My argument above assumes that there is a non-trivial demonstration
of my claim in FOL, which I have not yet found, as well as one
in the propositional calculus, which I have.

there is a class of formulas
in intuitionist predicate calculus
that are admissible but not derivable

There are some proof rules that are admissible, but not derivable;
I've not seen the terminology used for formulas.

(Also for classical logic, eg if there is a proof of P(x),
then there is a proof of all x P(x);
( whereas P(x)/all x P(x) is not a sound rule of inference ).

when you say that valid arguments
" To entertain the premises of a valid argument
is already to have entertained the conclusion. "
how does this entailment associate
in these nonstandard cases?

i mean
if derivation is the application of known truth preservation rules
what does only admissible validity _mean_?

-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar


--
Alan Smaill
.

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