FOM: sterility?

[Randall Holmes]

I understand by "second-order logic" first-order logic with the
addition of quantifiers over predicates (there's more to it, of
course).  In this language, the issue of identity conditions for
second-order "objects" does not arise for purely grammatical reasons.
One could introduce P = Q (where P and Q are second-order variables)
as an abbreviation for (forall x.Px iff Qx) if one favored an
extensional approach to equality; there doesn't seem to be a way to
introduce any other notion of equality.

I think that some formal treatments of second-order logic do allow
equations between second-order variables; in a such a treatment, I
suppose one could adopt extensionality or not.

I'm partial myself to the view that equations between properties are
either senseless (this would go along with their exclusion by the
grammar of one's formal language) or to be understood extensionally,
but this is a substantial philosophical question which requires
careful consideration.  I don't think that it affects the mathematical
applications of second-order logic considered here; I would be
interested to hear from anyone who thinks it does bear on the
mathematics.

In third-order logic, equality for second order objects becomes
formally definable! (x^2 = y^2 =def for all P^3, P^3(x^2) iff
P^3(y^2)).  Philosophers might or might not like this definition?  But
I prefer to eliminate third order logic in favor of second-order logic
with two first-order sorts; I don't think that nth order logic for n>2
adds anything essential to the logical expressiveness of 2nd order
logic (nth order logic is stronger than 2nd order logic for n>2, but
is equivalent to 2nd order systems with additional "nonlogical"
hypotheses).

And God posted an angel with a flaming sword at | Sincerely, M. Randall Holmes
the gates of Cantor's paradise, that the       | Boise State U. (disavows all) 
slow-witted and the deliberately obtuse might | holmes at math.idbsu.edu
not glimpse the wonders therein. | http://math.idbsu.edu/~holmes