Re: The Law of the Excluded Middle again (long)

On Sun, 02 Dec 2007 16:44:23 +0000, I wrote:

This application of the Law of the Excluded Middle [...]
has a characteristic which I didn't notice at the time,
and which no-one else has pointed out either: it is applied only
to propositions containing only free variables.

I meant to write a bit more than that.

(Excuse: I was thinking about it while I was out doing some belated
Sunday afternoon shopping, and had forgotten some of it by the time
I got back home.)

Inside the scope of quantification, and applied only to propositions
not themselves containing any quantifiers, it seems impossible (i.e.
even more impossible!) to equate (or confuse) truth with provability.

There is no question of "proving" that x >= 1, when x is a free
variable not yet subject to quantification. So one cannot argue
against the validity of asserting "either x >= 1 or x < 1" on the
grounds that to assert this is to claim either to possess a proof
of x >= 1 or to possess a proof of x < 1. Indeed, any such claim
(about a statement containing free variables) would be nonsense.

The concept of truth has applicability even where the concept of
proof hasn't; so truth cannot reduce to provability (not even in
pure mathematics).

If LEM is to be rejected here, it has to be on other grounds than
those discussed in the "A quote (and question) about intuitionism"
thread. (Not that that discussion has finished yet, or that I have
even yet understood much of what was said in it.)

I imagine that the response to this argument will be along the lines
that I am still thinking within a classical framework, and my whole
concept of the significance of symbols like x, y and z in statements
like x^y + y^z + z^x > 1 will have to come into question.

And I will probably not understand this response! Nevertheless, the
argument will have made some slow progress, at least inside my skull.

Incidentally, it also seems quite impossible (at least to me, in my
pseudo-objective subjectivity ...) to conceive of the truth value
of the statement x >= 1 as being time-dependent or subject-dependent.
But of course this is not to claim that I know, with infallible and
God-like certainty, what the truth value of x >= 1 is! (I may be daft,
but I'm not that daft.)

Trying one more time to save myself some pain, by anticipating what
the response to this will be:

Even if I am required to think of x, y and z as necessarily denoting
some humanly-constructed entities (so that I must not imagine them
ranging freely over some objective continuum of ordered triples), I
cannot imagine even the most thoroughgoing constructivist believing
that x^y + y^z + z^x might be <= 1.

(No doubt here I'm failing to make some important distinction between
'w <= 1' and 'not(w > 1)'; but I cannot even imagine someone thinking
that x^y + y^z + z^x might not be > 1; and there's no point in me
trying to make a distinction that I don't understand.)

I'm once again coming up completely blank when I try to imagine how
a constructivist would actually feel about this "x^y + y^z + z^x > 1"
thing. Presumably they would be unhappy with the proof using LEM, but
I can't imagine the cause of that unhappiness; nor can I imagine being
motivated to search for some proof not using LEM. I suppose I'm just
going to have to wait until someone actually responds; and maybe they
will have understood a little more of the reasons for my bafflement,
and will therefore be better placed to enable me to understand their
thinking a little better.

(I can't put off the evil moment any longer, so I'll just post this.
I expect it's crap! But I just don't know any better, so I have to
start from here, otherwise I'd never get to anywhere else.) :-)
--
Angus Rodgers
(twirlip@ eats spam; reply to angusrod@)
Contains mild peril
.

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