FOM: first order and second order logic: once more

[Roger Bishop Jones]

In response to Martin Davis Friday, September 01, 2000 8:00 PM

> At 09:49 AM 9/1/00 +0100, Roger Bishop Jones wrote:
> >Or, as I might put it: much better.
> >
> >For some purposes at least (e.g. saying what you mean).
> >
> >Roger Jones
>
> This is one of a series of comments in which Roger Jones extols the
virtues
> of his understanding of second order logic.

This is entirely false.
I have made no comments whatsoever on my understanding of second order
logic, which I am confident is limited by comparison with the average
contributor to the fom list.

> It's not a bad thing to find
> virtues where others find flaws, so I would invite Jones to provide a
> sample dialog in which these virtues become apparent, specifically in
which
> some clear propositions are being communicated using second order validity
> in some essential manner. .

As far as the virtues of second order logic is concerned there is absolutely
no need for me to provide examples, you yourself have provided copious
examples.

The difference between us is not in any technical matter about what can or
cannot be expressed in second order logic, it is about whether it is a good
thing or a bad thing to be able to express things, such as the propositions
of arithmetic, for which no complete (r.e.) proof system can ever be found.
Or at least, that seems to be what you are objecting to, but I could very
easly have misunderstood you.
In any case, it is not about the technical facts but about whether these are
"good" or "bad".

All that I have to say is that there is some *value* in second order,
because in second order logic propositions can be expressed which cannot be
expressed in first order logic.
So far as I am aware noone is disputing the facts, but some, for reasons
which still mystify me, seem to think that expressiveness is a bad thing and
are unwilling to accept that others with different purposes may reasonably
come to different value judgements.

Roger Jones