FOM: second-order logic is a myth
Robert.Black at nottingham.ac.uk
Tue Mar 2 13:33:15 EST 1999
There are various ways in which one can try to delimit the scope of logic,
and thereby distinguish it from, or allow it to embrace some or all of,
mathematics. And some of these ways will exclude second-order logic from
being really logic. Second-order logic won't be logic if we insist that
the logical truths be recursively enumerable, for example. Another idea
for possible demarcation with a tradition running from Kant to Hartry Field
is based on the thought that pure logic shouldn't entail the existence of
anything, and EXAx not-Xx is a second-order validity which is naturally
interpreted as asserting the existence of the empty set.
However, neither of these is the line Steve is running in his posting of 28
Feb. Rather, his argument is:
>the correct view of the matter (going back to Aristotle) is
>that logic is a method or common background shared by all scientific
>subjects, not only mathematics. This key scientific/philosophical
>distinction is reflected in the usual predicate calculus distinction
>between logical and `non-logical' axioms. The logical axioms are
>common to all subjects (i.e. theories), while the `non-logical' ones
>are subject-matter specific. A major defect of second-order logic is
>that it blurs this key scientific/philosophical distinction by
>pretending that axioms regarding sets (a specific subject matter) are
>part of the underlying logic.
Now this is also a long tradition of demarcation - Frege for example
identified the scope of logic via generality/topic-neutrality. But of
course Frege didn't think that ruled out second-order logic, and it's not
at all natural to think it does so.
To take a Frege-Boolos example, suppose I tell someone that there are some
people such that I am one of them and any parent of one of them is one of
them but Napoleon is not one of them. This non-first-orderizable statement
seems to have as its subject matter genealogy rather than set theory.
Similarly the well-known Geach-Kaplan example 'Some critics admire only one
another' seems to belong to the special subject matter of sociology.
Maybe that's just appearance. Maybe anyone who believes that Napoleon is
not one of their ancestors has to be a set-theoretic platonist. But if so,
that shows how deeply embedded commitment to sets is in our talk about
*all* subject matters. And that will in turn put set theory into logic by
Steve's criterion. After all, there can be sets of anything, sets of
cabbages, sets of kings and the (empty - I agree with Martin Davis here)
set of postmodernist marxists under Steve's bed.
Could I recommend that Steve look at chaps 3-5 of Boolos's _Logic, Logic
and Logic_, particularly the discussion of the topic neutrality of logic on
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