FOM: Re: Book recommendation

[Roger Bishop Jones]

Steve Simpson says:

> Of course I am delighted by
> Manzano's point about first-order logic subsuming the other systems.
> But the basic point seems obvious to me.  I guess I am a believer in
> what Martin Davis (Mon, 22 Feb 1999 14:36:21 -0800) called "Hilbert's
> Thesis", in the "second-order logic is a myth" thread last year (FOM,
> February-March 1999).

Whether first order logic is universal among "logics" depends upon what one
accepts to be a reasonable way of reducing one logic to another.

If one took (for example) "one-reducibility" as acceptable, not only
first-order logic, but any formal system whose theorems form a creative set,
is universal (and there are some examples much simpler than first order
logic).

The usual interpretation of Godel's incompleteness results gives us a more
exacting standard of reducibility, which is particularly relevant to that
special class of logical statements which are claims about the consistency
of formal systems.
The measure of strength which comes out of this is "proof theoretic
strength", and first order logic is (without "non-logical" axioms) too
"weak" even to get on the scale.

Historical uses of the term "Logic" are so diverse that we must make our own
minds up how the term should be used, or simply accept that it will continue
to have diverse usage.

For my money however, true claims about logical consistency are logical
truths, and it follows from Godel's incompleteness results that there is no
universal (formal, r.e.) logic.

Roger Jones