The Perils of Analysis

An argument mentioned by Bill Vallicella here neatly illustrates the danger of using modern predicate calculus to penetrate the logic of natural language. He cites an argument of Peter Van Inwagen, as follows.

Suppose that there exists nothing but my big parcel of land and such parts
as it may have. And suppose it has no proper parts but the six small parcels. .
. . Suppose that we have a bunch of sentences containing quantifiers, and that
we want to determine their truth-values: 'ExEyEz(y is a part of x & z is a
part of x & y is not the same size as z)'; that sort of thing. How many
items in our domain of quantification? Seven, right? That is, there are seven
objects, and not six objects or one object, that are possible values of our
variables, and that we must take account of when we are determining the
truth-value of our sentences. ("Composition as Identity," Philosophical
Perspectives 8 (1994), p. 213)

Van Inwagen's argument employs a method that is fundamental to all analytic philosophy. We have two ordinary language statements A and B below, and we want to decide whether B follows from A.

(A) There is a large parcel of land having two smaller parcels as proper parts.
(B) There are three things (the large parcel of land and the two smaller parcels)

If the inference is valid, then there is a third thing 'over and above' the two smaller things, and there is an 'ontological distinction' between the large parcel of land and its parts. Otherwise there are only two things, and the existence of the 'large parcel of land' simply reduces to the existence of the parts. It is not 'ontologically distinct'. This is an important philosophical conclusion, if we can establish it.

The procedure is to translate both statements into the language of predicate calculus, which has a determinate proof procedure, and see whether the inference holds. Thus

(A*) For some x, x is a large parcel of land having proper part x1 and proper part x2 and x1 /= x2.
(B*) For some x, for some y, for some z, x/=y and x/=z and y/=z

Clearly, given the additional premiss that no object x is identical to any of its proper parts (i.e. x /= x1 and x /= x2) we can establish that B* follows from A*. Thus the apparently simple translation from a natural language statement into the language of modern predicate calculus apparently leads to a philosophical conclusion. And a lot of modern analytic philosophy is like that. We are worried about whether the ordinary language statement A implies the ordinary language statement B. For ordinary language has no agreed and determinate proof procedure. So we translate A into a statement A* of the predicate calculus, which does have an agreed and determinate proof procedure, and we translate B into B*. Then we determine whether A* implies B*, which seems to solve the problem. But of course it doesn't, for the real question is whether the translation is correct. If we are unsure whether A implies B, how can we be sure that either of the translations (of A onto A* and B into B*) are correct? If the translation is obvious, how is it we were unsure of the implication in the first place?

The 'method of analysis' is not fundamentally unsound. If we are certain of the 'logical form' of an ordinary language statement - i.e. a form that makes inferences to other statements determinate and certain, then analysis is a useful technique. Otherwise it is not. What is the logical form of 'this big parcel consists of two small parcels'? If it is the same as 'This pair of shoes consists of 2 shoes', then we should proceed with caution.

Ontological dependence

My argument again. The following formally expressed statements are inconsistent.

1. (E x) number(x)
2. (x) number(x) implies not created(x)
3. (x) God created x

If (1) some x is a number, then (2) that x was not created. But (3) for all x, God created x, so God created that x. Contradiction.

Given this, it doesn't help to say that existentially quantified statements such as (1) don't really express or imply existence, or that (1) has no 'ontogical commitment'. This is irrelevant. Even if (1) is true of some x's to which we have no ontological commitment, it still logically follows that (3) is true of all x's, and so (presumably) is true of x's to which we are not ontologically committed. This is no way out.

Azzouni argues that our criterion for existence should be 'ontological independence'. If an object's properties depend wholly upon us (as in the case of fictional objects) then that entity does not exist. If our method for establishing the truth about an object is trivial (as in the case of mathematics) then it is ontologically dependent upon us. Accordingly, mathematical objects do not exist.

This does not help with the theological problem above. Even if (1) is true of ontologically dependent objects, there is still a contradiction, because there is nothing to prevent the universal quantifier ranging over such objects. If God created all things, then he made ontologically dependent things. But according to (2) some ontologically dependent things (numbers) are not created. Contradiction.

And Azzouni's suggestion creates another problem. I can meaningfully ask whether there are such things as 'ontologically dependent' objects. If yes, then why does Azzouni say that such objects do not exist? If not, in what sense is Azzouni offering any kind of solution at all? More on this later.

Azzouni on Ontological Commitment

Deflating Existential Consequence: A Case for Nominalism. Jody Azzouni, New York: Oxford University Press, 2004

I haven't got hold of a copy of this book yet, but the next best thing is a review. Three below.

Mark Colyvan
Thomas Hofweber
Joseph Melia