FOM: second-order logic is a myth
Robert.Black at nottingham.ac.uk
Thu Mar 4 06:18:06 EST 1999
I don't think Steve (Wed, 3 Mar 1999 16:11:02) has quite got the intended
thrust of my argument.
Firstly, let me say that I agree that set theory is normally studied in
first-order logic, and rightly so. (How else could one study the universe
of sets except by examining consequences of axioms while using an
underlying logic in which a correct proof is effectively recognizable as
I also agree that if you choose to demarcate the bounds of 'logic' via
recursive enumerability of its theorems then (1) that's a defensible
demarcation marking something important and (2) it rules out second-order
logic from being 'logic'.
But my post was directed against a different claim I took Steve to have
made, namely that if you take topic-neutrality as your criterion of
demarcation of logic, *that* ruled out second-order logic.
My argument goes in the form of a dilemma. Take a non-first-orderizable
statement about some arbitrary subject matter (I took Boolos's 'Napoleon is
not one of my ancestors'). Then:
EITHER You claim with Boolos that this second-order statement is to be
understood in terms of plural quantification (and I'm here expecting
agreement that Boolos has indeed shown that monadic second-order logic
amounts to plural quantification - or if you prefer, plural quantification
amounts to monadic second-order logic), and is therefore not about sets.
Then clearly the topic-neutrality criterion admits second-order logic as
part of logic (though you can still exclude set theory if you want).
OR You insist that plural quantification is just a sneaky way of talking
about sets. (Despite the joky way in which I presented it, this is
actually my own preferred option, and it appears to be Steve's too. From
what Steve said in reply to Sazonov on 25 February, it seems that he's only
a platonist at weekends, so let's take him on a Saturday when he can
entertain the thought that Napoleon isn't one of his ancestors together
with its commitment to sets. God knows what he thinks about his line of
descent on a Tuesday!) But then it seems that you will have to concede
that set theoretical reasoning pervades our thought concerning just about
any topic, and by the topic-neutrality criterion set theory will be part of
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