Re: Is Validity Just a Hypothetical or Conditional Characteristic?



Jan Burse wrote:
Jan Burse wrote:
herbzet wrote:
Jan Burse wrote:

If you have a proof of the validity of
a sentence, you can look at the proof,
and will learn a lot of about the contribution
of the parts of the sentence to the validity
of the whole sentence.


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'.

Here is a counter example to your above claim.
The following sentence is valid, but I don't
see how you would arrive at your decomposition:

p & ~p -> q


Oops, one decomposition would be:

p & ~p & q -> q


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 "<->".

This is something of a "cook" for my main claim, in that (i)
can be used to "trivially" satisfy my claim in all cases, in the
manner you did. Annoying.

However, (i) does tell us that there is no more information in
the conclusion than there is in the premise, which is consonant
with, and indeed follows from, my claim.

Here is my other challenge:

a&(a->b) -> b (1)

Note: P=a&(a->b), Q=b.


a & (a -> b) = a & (~a v b) by definition

= (a & ~a) v (a & b) distributive law

= a & b law of expansion


and b is a conjunct of a & b.


(1) is one
of the simplest formulas I know that
need multiple applications of the
axiom schema A |- A. Here is a little
proof:

--------------
a, ~b |- a
----------- -------------- (Contrapositive)
a, b |- b a, ~a |- b
--------------------------------- (-> intro to the left)
a, (a->b) |- b
-------------------- (& intro to the left)
a&(a->b) |- b
-------------------- (-> intro to the right)
|- a&(a->b) -> b


What do you call this sort of tree demonstration? Is this
what they call a "sequent" proof?

In any case, I don't see the relevance of this proof to my
claim.

--
hz
.

Relevant Pages