FOM: second-order logic is a myth

[Pat Hayes]

Stephen Simpson writes:

>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.
>

But surely (I speak in some trepidation, being a newcomer to these august
electronic halls), our very idea of logical truth now comes from Tarski,
whose account of the meaning of 'forall' seems to use the notion of set at
its very center: that of the universe of quantification. 'forall x P(x)' is
true (in an interpretation I) just when I(P) holds of every thing in the
universe of I, which is a set. True, this requires only a modest dip into
set *theory*, but it does seem to involve sets centrally in the very
business of logic.

Or is your point that the subjectmatter of set theory appears when we first
talk of a power set? Maybe that is the crucial point at which we pass from
what can nowadays be thought of as simple everyday logic into a genuine
mathematical *topic*. This does seem to have been the sticking point for a
number of people in history (Brouwer and Lesniewski, for example) and
certainly the axiom of choice seems to be the place where reasonable
mathematicians might legitimately disagree about something, and therefore
find alternative foundations more or less congenial.

Pat Hayes


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