[FOM] First and a half order logic
Dean Buckner
Dean.Buckner at btopenworld.com
Sat May 17 12:17:30 EDT 2003
Harvey:
>There appears to be a minority of philosophers of mathematics and
>philosophical logicians who properly understand the issues and
>nonissues regarding first order/second order axiomatizations for
>f.o.m.
I'm with the majority. But isn't the difficulty that there is something in
between, and we have failed to recognise this? I have argued that we can
quantify over plurals, and that plurals are not sets. Here is the argument
again. We assume
(1) a exists and b exists
and introduce the compound name "a & b" as follows
(2) a exists and b exists iff (def) a & b exist
then introduce the simple name "S":
(3) S = a&b
where we are just replacing the compound name "a&b" with the simple name
"S". Finally, allow quantification over simple names, as
(4) (exists X) X = S = a & b
Plural logic is simply first order logic that allows substitution of
compound names in the argument space. Isn't this set theory by the back
door? No, because no existence assumption is required to get from (1) to
(4). By contrast, we cannot get to
(4*) (exists S) S = {a,b}
without a further assumption or axiom (specifically pair set axiom). {a,b}
is another object, separate from a, b or a and b together, whose existence
does not directly follow from the mere existence of a and b.
I think Harvey's idea is that we can get from plural logic to set
theoretical logic by means of a translation or interpretation from one
symbolic language to another. Let me repeat that is impossible. Despite
the similarity between (4) and (4*), we cannot get from the one to the other
without extra assumptions which are themselves interpretable in plural
logic.
Finally, we define the relation "one of".
x is one of a&b iff (def) x=a or x=b
which, once again, resembles but is not interpretable as the set-membership
sign.
What's the deal? - something stronger than simple first order logic (i.e.
without set-theoretical assumptions). Sentences like
(S)(E T)(x) [ x oneof S iff x oneof T ]
cannot be interpreted in simple first order. But they can be interpreted
using plural names and quantification. And they do not involve assumptions
about "sets".
While plural quantification is stronger than first-order, it is insufficient
to generate Cantor's Theorem. That is because there is no singleton set, no
null set, and most importantly no infinite set. You can only derive a
weaker version of the Theorem, that the elements of a finite set cannot
count all combinations of elements (which is obvious).
Can we get real numbers? I don't see why not, but then I don't see exactly
how we can, either. Plural logic has to be stronger than PA, since it
supports predicates and allows us to quantify over any finite set
imaginable. Surely that is strong enough for decent mathematics?
Dean
Dean Buckner
London
ENGLAND
Work 020 7676 1750
Home 020 8788 4273
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