It seems clear to me that second-order logic is topic neutral.
Second-order quantifiers do not need to be understood as ranging over
sets; it is sufficient to understand them as ranging over properties.

For example, to be a natural number is to be an object which has all
the properties which belong to 0 and belong to the successor of any
object having them (once we decide what "0" and "successor" mean).
The specifically mathematical content of this definition is all in the
definitions of "0" and "successor".

Talk of properties does not belong to any particular subject; in any
subject whatever, we will use language about properties.  If we adopt
the position that there "aren't any properties" (whatever this means),
the problem of explaining away talk about properties is a general
philosophical problem -- it belongs to philosophy, not to mathematics;
it may even be taken to belong to logic (see below)!  If we adopt the
position that there "are" properties and that we can quantify over
them, this is again a philosophical position which admits application
in every area of knowledge.

Historically, it is my understanding that Frege originally presented
his logic (the ancestor of modern first-order logic) in a way which
admitted quantification over predicates.  Moreover, I believe that the
first-order theories of arithmetic and analysis are adaptations of
theories originally stated in second-order form before set theory was
ever formally proposed (or at about the same time).

Second-order logic is not set theory in disguise.  It may be the
theory of universals, but this is not a mathematical issue -- the
issue of existence or nonexistence of universals is of universal
interest (topic-neutral).

I agree that second-order logic is not a formal system of deduction.
If formal systems of deduction exhaust logic, then "second-order
logic" is not "logic".

I think that second-order logic is _obviously_ logic, and so (by
reductio) formal systems don't exhaust logic.  Logic includes the
study of the form of sentences; the maneuver involved in second-order
logic (quantifying over predicates) is certainly of logical interest
and calls for logical analysis.

I'm looking at a dictionary definition of the word "logic".  The 2nd
definition of the word given is "a system of formal principles of
deduction or inference".  I have already granted (as I must) that
second-order logic is not that.  (second-order logic is not a logic(2)).

The first definition given is "a science that deals with the canons
and criteria of validity in thought and demonstration and that
traditionally comprises the principles of definition and
classification and correct use of terms and {\em the principles of
correct predication\/} (my italics) and the principles of reasoning
and demonstration".  It appears obvious that the question of the
meaningfulness of second-order quantifiers falls under this
definition; second-order logic is part of the subject matter of logic
(though one might belong to a school of logic which regards
second-order logic as inadmissible: "thou shalt not quantify over
predicates").  Second-order logic falls within the scope of logic(1);
the question of the admissibility of second-order logic as a
fundamental system of reasoning is part of the analysis of
predication, which is a logical(1) issue.

If one's "principles of correct predication" allow one to quantify
over predicates (part of one's logic(1)) then a logic(2) (formal
system of inference) which one would accept would be the system in
which the rules of first-order logic are extended to cover
second-order quantifiers and comprehension axioms are provided to
assert the existence of particular properties.  This is of course
formally also a first-order two-sorted system.  (Further logical(1)
reasoning would lead one to accept stronger and stronger logics(2) in
a way which no meta-logic(2) can capture formally, e.g. by considering
G\"odel sentences of successive theories).

The "principles of definition" clause of the definition of logic(1) is
also of interest.  The natural numbers and the reals are definable up
to isomorphism in second-order logic; this is not the case in any
first-order theoryd and the deliberately obtuse might | holmes@math.idbsu.edu
not glimpse the wonders therein. |