Re: SUO: Logic and Ontology
On Tue, Mar 05, 2002 at 04:10:08PM +0100, West, Matthew R SITI-ITPSIE wrote:
> I was recently sent a copy of Axiomathes, and noticed a paper
> with the above title.
>
> Cocchiarella, Nino B.; "Logic and Ontology", Axiomathes 12: 117-150, 2001
>
> The part of the paper that interested me was the different
> interpretations that could be made of predication, depending on
> whether your basic ontology is based on nominalism, conceptualism, or
> realism (or some graduation between).
>
> One of these, attributed to Quine, sems to coincide with my own
> interpretation of predication, and I repeat it here.
>
> "Quine's understanding of his ontology as platonistic and of sets
> as universals is based on a rather involuted argument,
To say the least.
> the essentials of which are as follows:
>
> if we were to adopt platonism as a theory of universals as represented
> by higher order logic in which predicate as well as individual
> variables can be bound, then
>
> 1. predicate quantifiers can be given a referential ontological
> interpretation only if predicates are (mis)construed as singular terms
> (i.e. terms that can occupy the argument or subject positions of
> predicates); and
There is a false assumption here, namely, that because the predicates of
natural language cannot be grammatical subjects, it is therefore somehow
wrong to allow the predicates of a *logical* language to occur as
logical subjects (i.e., to occur in subject position in atomic
formulas). But this is a confusion. True enough, natural language
predicates like "is a philosopher" cannot occupy the subject positions
of natural language sentences; we can't say, e.g., "Is a philosopher is
a property". Rather, you have to choose some nominalized counterpart,
e.g., the gerund "being a philosopher" if you want to have a legitimate
noun phrase. But it is a confusion to argue that it is therefore
illegitimate in a *logical* language to allow predicates to occur in
subject position, e.g., to allow both
(1) (philosopher Quine)
and
(2) (property philosopher).
But why? A nominalized predicate denotes exactly what the predicate
itself expresses. "Being a philosopher" denotes exactly the property we
attribute to Quine when we say "Quine is a philosopher". Why should we
therefore not allow a single logical expression to play the roles of
both natural language expressions? We can tell those roles apart simply
by observing where the logical expression occurs: "philosopher" in (1),
occurring as it does in predicate position, corresponds to the NL
predicate "is a philosopher"; "philosopher" in (2), occurring as it does
in subject position, corresponds in this instance to the gerundive form
of the predicate, viz., "being a philosopher". Simple.
If you MUST have something corresponding to nominalization, introduce a
term forming operator ^ that applies to predicates; then instead of (2)
you can write
(2') (property ^philosopher).
But note that ^ plays NO semantic role whatsoever and hence is only
superfluous syntactic sugar, and in fact a potential source of
confusion that is best avoided.
> 2. assuming extensionality,
If you want, though there are tons of troublesome counterexamples
(discussed here at great length a long time back). I'll simply note,
with no intention of defending it, that you can't do justice to modal
intuitions on this assumption, unless you are willing to countenance
merely possible objects.
> 3. predicates, as singular terms, can only denote sets, ...
Assuming the preceding highly problematic assumption.
> 4. [which] must then also be the universals that are the values of the
> predicate variables in predicate positions; and therefore
>
> 5. predication must be the same as membership, in which case
>
> 6. we might as well replace predicate variables by individual
> variables (thereby accepting nominalism's exclusion of bound predicate
> variables) and take sets as values of the individual variables,
> arriving thereby at
>
> 7. a first order theory of membership (set theory) which
Yes, you could perhaps do that, but now I *do* think you lose something
important, as all predication on this approach is reduced to a single
exemplification (or, if you will, membership) predicate. Seems to me
that this *does* miss something important about meaning that is revealed
in natural language. When we say "Quine is a philosopher", we mean
simply that Quine is a philosopher; we are predicating that property
directly of Quine. We do *not* mean that Quine and being a philosopher
stand in the exemplification (or, if you will, membership) relation,
which, if I'm understanding correctly, is what we would have to be
expressing on the Quinean view. Of course, it is *true* that Quine and
being a philospoher stand in the exemplification (membership) relation
if, and only if, Quine is a philosopher; my claim is only that that is
not what we *mean* when we say "Quine is a philosopher"; it is rather
what we mean when we say "Quine exemplifies being a philosopher" and
hence refer to the exemplification relation explicitly. Hence, the
reduction to a single exemplification (or membership) relation seems to
miss an important fact about meaning.
> 8. is platonist because it has abstract entities as values of its one
> type of variable.
>
> Thus beginning with higher order logic with bound predicate variables
> as a version of platonism, we arrive at the nominalist position to
> recognise only quantification with respect to individual variables (or
> the subject positions of predicates) but with individual variables
> that can have abstract sets as their values, which are therefore
> really universals (i.e. entities that have a predicable nature)"
A good rendition of the argument, I think; but serious problems with
steps 1 and 2, as noted.
> Well I'm sure there are some bits of that that have gone over my head,
> but the bottom line is that predication = membership is a possible
> interpretation of predication,
Yes, though, as noted, this follows from highly dubious premises.
> The bottom line here as far as I can see is that without stating a
> particular meaning for predication, KIF is ambiguous.
No more than any formal language that requires interpretation.
> Or perhaps I should say that when you use KIF, you need to state the
> meaning of predication you are using.
Well, yes; it's called semantics. KIF has a well-defined semantics. In
KIF 3.0, predication was in fact taken to be membership. On the latest
semantics (for the above reasons, among others) it is not, though it is
*consistent* with the semantics to understand predication as membership;
that is, there are formal models of KIF in which properties are sets,
and hence in which predication is membership. But this is not an
assumption of the semantics itself.
> It strikes me that using different interpretations of predication adds
> to the difficulties of integrating ontologies with different
> ontologies of logic.
Well, then don't use different interpretations. Not sure what your
point is here.
Regards,
-chris
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