🚀 go-pugleaf

In article <1992Dec30.230419.4341@guinness.idbsu.edu> holmes@opal.idbsu.edu (Randall Holmes) writes:

>
>Restricted quantifiers, in ZFC or NF, are nothing more than formulae
>with one free variable; (for all x in range Q)(P) means nothing more
>than (for _all_ x)(if "Q holds of x" then P).  There is no other way
>of introducing restricted quantifiers in either theory.
>
I'm not at all interested in the points being disputed in this thread, but 
as an advocate of restricted quantification, I have to point out that this 
is dead wrong, unless I am confused about the nature of ZFC and NF.

The point of restricted quantification, as I see it, is to avoid the
nutty consequences that arise from using "if".  One such nutty consequence
is giving clear truth conditions in cases where Q holds of nothing. With
restricted quantification, the semantics returns "tilt" when Q is true
of nothing:(All: Qx)(Px) is not evaluable when Qx does not hold of any x, 
because no x get plugged into P.  With unrestricted quantification, 
the proposition is true.  These kinds of cases, which might be summed up
as "results which do not seem consistent with natural language," multiply
on further investigation.  Even if I'm wrong about this, it is still wrong 
say that the two are equivalent.  Unrestricted quantification forces you
to split up the domain during evaluation, using your system.  Restricted 
quantification allows you to stipulate a subpart of the domain, in which
to evaluate a given proposition.  Further, restricted quantification 
makes formulae more closely mirror natural language

If this has been discussed early in the thread, just forgive me, don't 
bother flaming.  I'm aware that I don't really know what's going on in
the thread.

Bob

r-knippen@uchicago.edu