Wednesday, September 08, 2010

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.


David Brightly said...

Hi O,

(1) This seems very sensible. But deciding on a 'logical form' would be choosing one of two extremes---mere plurality or composite individual---of what seems to be a spectrum. Consider

an asteroid belt, a boulder field, a scree, a pile of stones, a dry-stone wall, a Pyramid, a house.

All of these can be seen as pluralities and as wholes. Our thought seems readily to slide between the two poles.

(2) I was struck by one of your comments at BV's:

Some writers have developed plural or collective versions of predicate calculus, to capture statements of the form

These people = Peter and Paul


Some X's = A & B

I'm wondering if these calculi are powerful enough to express

These people = Peter and Paul
Those people = Paul and Mary
Some of these people = some of those people

If so, then we seem precious close to an embedding of naive set theory in predicate calculus. The referring terms 'these people' and 'those people' behave analogously with sets. Isn't this what Frege, Russell, and Whitehead were aiming for?

Edward Ockham said...

Hi David - the only person whose work I am familiar with in any detail is Tom McKay.

On your point about the similarity with naive set theory - well McKay has an axiom of unrestricted comprehension that he claims is not vulnerable to the paradoxes but it's some time since I have read him. (Sorry I can't help further).

David Brightly said...

OK, thanks, that's interesting. I ought to get McKay.