FOM: Re: ``Reduction'' of Second-Order Logic to First-Order Logic

[Roger Bishop Jones]

Thanks to those who pointed out my elementary blunder in guessing that
satisfiable sentences are r.e.

I only wish that equally decisive and concise opinions had been offered on
the main point of dispute.

In response to Matt Insall's last, I propose a closure.

So far as I can tell Matt has now not only conceded the point I sought to
make, but has called it trivial.
So far as I can see, his other objections are wholly concerned with claims
which I have never made.

What I intended to do when I unwittingly started this thread, was to resist
(what I took to be) the use by Steven Simpson of reductions of various
logics to first order logic published by Manzano to bolster the attack on
second-order logic in the thread "the myth of second-order logic".
My intention was simply to observe that since it is possible in second order
logic (with standard semantics) to express claims which cannot be expressed
in first order logic, there can be no reduction to first order logic (with
the standard semantics) which will be wholly satisfactory to those who wish
to use second order logic because of its greater expressive power.
To be more precise about this negative claim I mentioned grounds for denying
that there can be an effective truth preserving reduction of standard second
order logic to first order logic.

Believing that this point is no longer under dispute, I am now content.

Roger Jones