FOM: second-order logic is a myth
Stephen G Simpson
simpson at math.psu.edu
Wed Mar 3 16:11:02 EST 1999
In 28 Feb 1999 21:29:05 I argued against the idea of second-order
logic, by arguing that set-theoretic realism should not be considered
part of logic. This is because set theory is subject with a specific
subject matter, sets, while logic is a common method or background
used by all scientific subjects.
Robert Black 2 Mar 1999 18:33:15 writes:
> this is ... 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.
and uses a geneological example to argue that set theory is part of
our everyday vocabulary, hence (according to Black) part of logic.
My reply to Black's overall point is that, pace Frege, most set
theorists nowadays correctly regard set theory as a first-order
theory, or perhaps more accurately a collection of first-order
theories. When the set theorists write things like ZF, ZFC, ZFC+GCH,
ZF+V=L, ZF+MA, ZF+AD+DC, etc., those acronyms are referring to
specific first-order theories. Like all first-order theories, these
first-order theories consist of a logical part (the predicate
calculus) and a non-logical part. The non-logical part includes all
of the specifically set-theoretic axioms. In this way the set
theorists themselves acknowledge that set theory is not part of the
underlying logic. They pay due respect to the sharp distinction,
formalized in the predicate calculus, between logical axioms and
subject-matter-specific, in this case set-theoretic, axioms. This
current attitude is the result of a long historical development of the
predicate calculus and axiomatic set theory. Some of the key people
were Frege, Russell, Cantor, Hilbert, Zermelo, Fraenkel, and von
Neumann.
My reply to the geneological example is that, even in this example,
the sets should be viewed not as part of the logic but rather as part
of the subject matter. Black himself could have made this clearer by
saying `there exists a set of people such that ...' instead of the
less precise `there are some people such that ...'.
To summarize, the standard modern view of the matter is that notions
such as `and', `or', `not', `there exists', `for all' belong to logic,
while notions such as `for all subsets of a given set' belong to set
theory.
-- Steve
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