June 21, 2010

Unrestricted Quantifiers and Fundamental Quantifiers

According to Ted Sider, ontology is concerned with determining what objects are in the scope of the 'unrestricted' universal quantifier. Sider argues that ontological questions thus have genuine objective answers, for there can be no vagueness in the meaning of the unrestricted quantifier. Suppose, says Sider, that there are two precisifications, ∀1 and ∀2 of the universal quantifier ∀. Then, he says,

there must be some thing, x, that is in the extension of one, but not the other, of ∀1 and ∀2. But in that case, whichever of ∀1 and ∀2 lacks x in its extension will fail to be an acceptable precisification of the unrestricted quantifier. It is quite clearly a restricted quantifier since there is something - x - that fails to be in its extension. (Four-Dimensionalism, pp. 128-129)

Now, I agree with Sider that ontological disagreements are genuine disagreements, but I think Sider's reasoning here is mistaken, and I think the mistake effects much of the rest of his metaphysics. For, according to this view, we can pick out the unrestricted quantifier - the quantifier we care about in ontology - because it has the biggest extension possible. Now, presumably there is some sort of restriction about the statements expressed being true. So it seems that Sider's assumption is that if we can make true statements using a quantifier that includes a thing, x, in its domain, then x is in the domain of the unrestricted quantifier, and so belongs in our ontology.

From this, much of the picture given in Four-Dimensionalism will follow. For instance, consider the following three controversial principles:

  1. Mereological Universalism (MU): For any objects, the ys, there exists an x which is composed of the ys

  2. Doctrine of Arbitrary Undetached Parts (DAUP): Suppose there is a region R of space exactly filled by an object x. Then for any subregion R* of R, there is an object y that exactly fills R*.

  3. Temporal Doctrine of Arbitrary Undetached Parts (TDAUP): Suppose an object x persists through a period P of time. Then for any sub-period P* of P, there is an object y such that, at each moment in P*, y occupies the same region of space as x, and y exists all and only during P*.

These principles are, I say, controversial. They are also at odds with common sense. For instance, we would not typically say that there is an entity composed of my left sock, my computer, and the Eiffel tower. We would also not say that there is some entity consisting of all of my parts except my hand. Finally, we would not say that there is an entity sitting in my chair typing at my computer which did not exist five minutes ago. However, if we think that the important quantifier is the one with the biggest domain, then all three theses are trivialized, since we can introduce talk of all of these entities and use that way of speaking to say true things. For instance, we can say, "one part of the entity composed of my left sock, my computer, and the Eiffel tower is located in Paris," and this will be true. Sider elsewhere denies that he means to make such a weak claim: he remarks on p. 62 that not even the opponents of four-dimensionalism deny that it is possible to introduce a linguistic framework in which four-dimensionalism is true. (I note, however, that when philosophers like David Lewis say that mereological universalism does not violate the principle of parsimony - as if Lewis cared about parsimony! - because wholes are nothing over and above their parts, the trivial reading is strongly suggested.) But if the fact that we can introduce a linguistic framework in which we make true claims about temporal parts (etc.) is not sufficient to guarantee the truth of four-dimensionalism, then why is the fact that a quantifier leaves out an entity included by another quantifier a guarantee that it is not an acceptable precisification of the unrestricted quantifier?

Consider a more radical example: "there are more characters in Shakespeare's Much Ado About Nothing than in Sartre's No Exit." This sentence is no doubt true, and it quantifies over fictional characters. So any quantifier which does not have fictional characters in its domain would seem not to be Sider's unrestricted quantifier.

This all seems mistaken to me. We don't want the biggest quantifier domain when we are doing ontology. We want the smallest domain that leaves nothing out. This is, after all, much like what is going on with Quine's ontological commitment criterion: if we can paraphrase an entity away, then we don't want it in the domain of the quantifier we use in fundamental ontology. Call this the fundamental quantifier.

Now, once we have introduced this notion of a fundamental quantifier, MU, DAUP, and TDAUP all begin to look quite suspect. For instance, suppose we list all the intrinsic properties of my left sock, all the intrinsic properties of my computer, all the intrinsic properties of the Eiffel tower, and all the relations which obtain between these three entities, all the while supposing there is nothing they jointly compose. Would anything be added if we went on to list the intrinsic properties of the (stipulated) entity composed of my left sock, my computer, and the Eiffel Tower? I suspect not.

If this is right, then we should introduce an entity composed of a collection of parts only if there are some facts about that entity which are not constituted by facts about the parts. Now, there are some definitions of 'supervene' such that the whole might still supervene on the parts. For instance, there might be physical, psycho-physical, or metaphysical laws in our world which connect the properties of the whole to the properties of the parts, so that there can be no change in the whole without corresponding changes in the parts, but there might be a property F such that the proposition that the whole is F is not identical to any proposition solely about the parts.

One might worry that there is more than one candidate for the domain of the fundamental quantifier. I think that Richard Swinburne accepts this claim. (At least, he thinks that there is more than one conceptually scheme in which the whole truth about the world can be expressed.) This won't necessarily make all of ontology a matter of convention, for this view does assume that there are objective truths and that they are sometimes expressed by plain language. But it does mean that some seemingly different metaphysical systems may turn out not to differ after all.

Posted by Kenny at June 21, 2010 5:18 PM
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