FOM: Book recommendation

[Stephen G Simpson]

Roger Bishop Jones writes:

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

The context of this discussion thread is Manzano's book "Extensions of
First Order Logic" (Cambridge University Press, 1996).  In that book,
Manzano shows how to reduce many of the best known so-called
alternative logics (second- and higher- order logic, dynamic logic,
temporal logic, modal logic, etc) to many-sorted first-order logic, in
precise ways, which Manzano spells out in great detail.

I regard Manzano's reductions of alternative logics to first-order
logic as being very reasonable.  Don't you agree?

 > 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).

This is definitely not the kind of reducibility that Manzano is
talking about.  It is much weaker.

 > it follows from Godel's incompleteness results that there is no
 > universal (formal, r.e.) logic.

This depends on what you mean by "universal".  Manzano makes a
detailed case that first-order logic is in fact universal among
alternative (formal, r.e.) logics, in a reasonable sense of
"universal".

Note also that G"odel's incompleteness results are usually stated and
proved in the context of first-order logic, not alternative logics.

-- Steve