Chris P, Chris M, and Pat, (01)
I just wanted to add a few words to support Pat's comment: (02)
CP> I find large tracts of the reference data are irredeemably
> higherorder.... (03)
PH> Irredeemably?? I am quite unconvinced that universals really
> need true semantic higherorder predication. That would be a
> very strong claim to defend. Are you sure that the cardinality
> of the set of universals is really uncountable? (04)
The fact that you can quantify over predicates without going beyond
FOL can be shown by using a construction defined by Quine (1953): (05)
Quine, Williard Van Orman (1953) "Reduction to a dyadic predicate,"
reprinted in Quine (1966), pp. 224226. (06)
Quine, Willard Van Orman (1966) Selected Logic Papers, Enlarged
Edition, Harvard University Press, Cambridge, MA, 1995. (07)
What Quine does is to show that any theory that has any set of
predicates that take any number of arguments can be replaced by
an equivalent theory in which there is only one dyadic predicate F. (08)
Instead of using Quine's notation, I'll assume the "apply" operator
of LISP, which is slightly more perspicuous. Let apply be a dyadic
operator, whose first argument is any Nadic predicate P and whose
second argument is any list of N items, (09)
L = (a1, a2, ..., aN). (010)
Then define (011)
P(a1, a2, ..., aN) = apply(P,L). (012)
This apply operator applies P to the list of arguments in L.
If you adapt this approach to predicate calculus, you get a
formalism with apply as the only predicate, which takes one
of the predicates of the original theory as its first argument
and a list of N items as its second argument. (013)
Now quantifiers can freely range over P or the elements of L without
making the theory higher order. This is exactly what CL does, but
it eliminates the pretense of using the "apply" operator. (014)
This construction illustrates a crucial point: what creates problems
in traditional higerorder logic is *not* the syntax, but the semantics.
With Quine's construction, the apply operator, or the CL syntax, the
crucial feature that keeps the logic within the realm of FOL is the
semantics  i.e., that there is a single domain D over which all the
quantifiers range. (015)
What makes HOL complicated is the assumption of a hierarchy of domains: (016)
1. Ordinary variables range over a set D0 of individuals. (017)
2. Variables that represent firstorder predicates range over D1,
whose cardinality is 2^card(D0). (018)
3. Variables that represent secondorder predicates range over D2,
whose cardinality is 2^card(D1). (019)
4...... (020)
Notice that the cardinality of these domains grows exponentially
(and if the starting domain is countably infinite, the Ds grow
very fast indeed). (021)
If you just assume a single domain D, in which everything of interest
is a member, you never go beyond FOL. (022)
John (023)
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