Re: Inconsistent = all sentences provable?



george wrote:

[...]

There are inference rules and
conventions FROM WHICH IT FOLLOWS that this
follows from that. THERE IS NOT some final factual
TRUTH of the matter,

I disagree, although I will grant that the matter
strongly depends on what we agree to mean by "fact",
"truth", and "follows from".

(That might merit a "DUH!" from you, but let's
face it, what we mean by "fact" and "truth" is
problematic in this context.)

My position, briefly, is that as soon as we reach
agreement on what our utterances are to mean,
then the meaning of some utterances include what
is meant by other utterances, and it is in this
that "following from" consists.

[...]

Peter_Smith sez:

RAA tells you that if P entails a contradiction
you can infer not-P. But that isn't inferring
anything from a contradiction,

Classically, there is no difference: EFQ is already
tautologous, so you could simply perform the relevant
instance of it within the context of the reductio;
YOU HAVE an F to work with, from that point.

it is inferring something from the fact
that P entails a contradiction.

Access to that kind of meta-fact ISN'T EVEN AVAILABLE
in the relevant context. RAA is just noticing the fact that
(P -> (Q&~Q)) -> ~P is tautologous.
(Q&~Q) -> ~P BY ITSELF is equally tautologous.
There is NOT a distinction of level here (AS there is,
say, with universal generalization in FOL, where you
infer something from the existence of an inference).
The fact that -> is one of the connectives in the underlying
language makes THAT distinction between ->and =>
irrelevant.

The meaning of this last sentence is a little murky here,
(at least to me), but I agree that our classical propositional
logic does not make a distinction "of level" between tautologies;
they're all (extensionally) equivalent to P -> P. I see this as
something that needs to be fixed, although it would, of course,
be wearisome to recast all formalized logic/mathematics into
a new mold, and why bother?

I've got an idea about how to do this, but it might turn
out to be nothing. I need to think about it a lot more.

I just wanted to get my two cents in.

--
hz
.

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