FOM: second-order logic is a myth

Cristian Cocos ccocos at
Wed Mar 17 01:18:15 EST 1999

Stephen G Simpson wrote:

> John Mayberry 11 Mar 1999 16:28:35 writes:
>  > first order logic doesn't provide a model of reasoning either:
>  ...
>  > it works for some such definitions (e.g. the definition of "group")
>  > but not for others (e.g. the definition of "complete ordered field"
>  > or "topological space").
> I disagree.  All of this reasoning, including the definitions of
> complete ordered field and topological space, is captured in
> first-order logic.  More generally, almost all of mathematics is
> formalizable in ZFC, and ZFC is a first-order system.  These
> mathematical facts are well known, but you (and other advocates of
> second-order logic, such as Shapiro) don't seem to fully accept them.

What about Boolos' (...and others'...) nonfirstorderizable sentences?
You're probably going to say that you're going to involve set theoretic
notions to account for them too. On the other hand, intuitively speaking
they don't appear to be sentences of mathematics.

Anyway, I think however that the real threat against a first-order
thinking comes from a *semantic* perspective and not from the point of
view of the syntax. The proponent of a first-order discourse is doomed
to live with non-cathegoricity -- something which, to put it mildly,
extremely unappealing in the eyes of the of the struggling scientist
(...definitely *other* than 'pure' mathematicians).

>  > logic - formal, mathematical logic - is part of set theory.
> This is historically incorrect.  Logic existed long before set theory.
> Logic goes back to Aristotle; set theory goes back to Cantor.
> It's also scientifically incorrect.  The correct view of the matter is
> that logic is a framework or common background for all scientific
> theories.  Set theory is one of those theories.  All of the set
> theories commonly considered in set theory textbooks (ZF, ZFC, ZF+V=L,
> ZF+DC+AD, etc etc) are in this framework, the underlying logic being
> first-order logic.

I must confess that this is a very appealing cliché: set
theory/mathematics as a theory like any other (perhaps even competing
with some others) while logic embraces them all. It sounds like you
develop, say, thermodynamics or any other (serious) scientific theory,
however modest, totally independent(ly) of any specific mathematical
assumptions. I myself am very curious how such a (non-trivial)
scientific theory would look like.

Cristian Cocos

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