[FOM] n-th order ZFC
Robert Black
mongre at gmx.de
Mon Jul 11 12:58:05 EDT 2011
Am 11.07.11 01:35, schrieb meskew at math.uci.edu:
> Why does second-order logic help the realist position? One could just say
> that the powerset operation is fully determinate, even though it cannot be
> categorically described in first-order logic. In second order logic you
> have a logic with very restricted semantics, but one can also restrict
> semantics in first-order logic by picking out a class of distinguished
> models of a given theory, perhaps even a one-element class. So a prior
> commitment to realism about powersets seems to obviate any new
> philsophical lessons of second-order logic. Conversely, believing that
> standard 2nd-order semantics make sense seems to require prior commitment
> to realism about powersets.
>
I agree entirely: the crucial point is the determinacy of the power-set
operation (or some equivalent in terms of Fregean concepts or plural
quantification). But once you believe that, second-order logic is an
obviously neat way of expressing categorical theories. It's worth noting
that second-order logic is a very natural system - it's not a
coincidence that when Frege created modern logic in the
_Begriffssschrift_ he went straight to second-order logic. Of course he
needed to do that to get the ancestral construction, but also
(disregarding what we now know about incompleteness) once you've got
individual constants and variables and predicate constants it's really
rather odd not to allow predicate variables.
Robert
--
Robert Black
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