Wednesday, 16 March 2011

A Note on Hofweber's Distinction between Internal and External Quantification

ABSTRACT: Thomas Hofweber's distinction between internal and external quantification is crucial to the solution he offers to his now well-known puzzle about ontology. Here I argue that this distinction is not well motivated by the considerations he employs.

In a series of interesting papers (2000, 2005b, 2007), Thomas Hofweber has identified a puzzle about ontology and developed a novel solution. Briefly, the puzzle is that questions such as 'Do numbers exist?' seem trivial from one point of view, but highly contentious from another. On the one hand, it is obvious that, e.g., there are even numbers smaller than 6. It follows trivially from this obvious statement that there are numbers. On the other hand, it is hotly disputed among philosophers whether or not there are numbers. Hofweber himself denies their existence. Nonetheless, he agrees that, e.g., there are even numbers smaller than 6.

Hofweber's solution to this puzzle crucially involves a distinction between two kinds of quantification which he calls 'internal' and 'external'. External quantification is familiar; externally quantified statements impose conditions on a domain of objects. Hofweber plausibly argues (2000, 2005b) that we must also recognize a kind of quantification which does not impose such conditions. His strategy is to highlight a certain 'inferential role' which quantifiers play in natural language, which enables them to function as place-holders for incomplete information; suppose we knew that Fred admires Thomas Edison, but then forgot this, remembering only that there is someone Fred admires. Hofweber argues that 'this situation is completely general', and that 'the only instances of the quantifier might be things that don't exist'.1

With this distinction between internal and external quantification on board, Hofweber's solution to the puzzle about ontology is that the "trivial arguments" to the existence of contentious entities are indeed trivially valid, on the proviso that the quantifiers in their conclusions are given an internal reading. Questions about what there is, where 'what there is' is construed externally, thus remain as a non-trivial subject matter for ontology.

My object here is to show that the distinction Hofweber intends to make is not what it may appear to be at first glance, and furthermore that it cannot in fact be motivated solely by means of the considerations (indicated above) which he employs.

Let us begin with the question: why can't external quantification play the role of facilitating the expression of incomplete information? From the considerations offered, it seems that the only reason is that, as Hofweber says, the only instances of the quantifier might be things that don't exist. Thus we might think of internal quantifiers as characterised by the fact of ranging over both merely intentional objects and not-merely-intentional objects, in contrast to external quantifiers, which range over not-merely-intentional objects only. (I will call this 'the simple intentional-permissive understanding' of internal quantification.) This, however, is not how Hofweber conceives the distinction.

This becomes clear once we look at his views about arithmetical discourse with the distinction between the merely intentional and the not-merely-intentional in mind.2 You can look for a prime between 24 and 28, and thus be looking for something. However, you will not find one: in this sense, there is no such thing. Hofweber fully recognizes this distinction, while nonetheless believing all quantification in arithmetic to be internal. Thus Hofweber's distinction between internal and external quantification cannot be understood in terms of the pre-existing distinction between the intentional and the not-merely-intentional. And yet this pre-existing distinction seems a natural and sufficient basis for a notion of quantification fit to play the inferential role Hofweber identifies. Therefore his consideration of this role is not by itself a good motivation for his internal-external distinction.

Note carefully that this argument does not require that the simple intentional-permissive understanding of internal quantification be a suitable basis for a solution to the puzzle about ontology. Furthermore, it does not rule out Hofweber's using the puzzle itself as a motivation for a special reading of quantification. The point is that he has not succeeded in establishing an independent motivation for such a reading.

It might be objected that I have not made an adequate case for the possibility of the simple intentional-permissive understanding of internal quantification. But I am not seeking to establish this conclusively; only, given that Hofweber has identified an inferential role which calls for a non-external reading of the quantifiers, the simple intentional-permissive conception is prima facie a better candidate than one based on Hofweber's internal-external distinction (considered apart from any puzzle about ontology). It may seem as though I'm not being quite fair, since I haven't really made his distinction clear in its own right. But I have no idea how to do this. Hofweber wants a reading of quantification such that the following comes out true:

There is an x such that x is not a merely intentional object, and x does not exist (in the external sense).

It has not been made sufficiently clear that such a reading is available.

Finally, one might wonder how Hofweber's internalism about arithmetical discourse avoids trivializing arithmetic. For on this conception, so-called "existence statements" about merely intentional objects (e.g. the largest prime) can easily come out true. Hofweber handles this with a supplementary doctrine to the effect that quantification in arithmetic is generally restricted to statements which have instances containing number words or numerals ('one', '46', etc.). However, and as Hofweber himself acknowledges, this sort of account cannot be extended to the reals, since we do not have number words for all of them.3 This gives rise to the worry that whatever the truth is about our quantification over the reals, it may also account for our quantification over natural numbers in arithmetic, rendering internalism about arithmetic theoretically superfluous.

Tristan Haze
The University of Sydney


Hofweber, T. 2000. 'Quantification and Non-Existent Objects', in Empty Names, Fiction and the Puzzles of Non-Existence, eds. Everett, A. and Hofweber, T. CSLI Publications.

Hofweber, T. 2005a. 'Number Determiners, Numbers, and Arithmetic', The Philosophical Review 114:2.

Hofweber, T. 2005b. 'A Puzzle about Ontology', Nôus 39:2.

Hofweber, T. 2007. 'Innocent Statements and their Metaphysically Loaded Counterparts', Philosophers' Imprint 7:1, <>.

These papers are available on Hofweber's homepage:

1 (2000), p 16.
2 These views are indicated in his (2005a).
3 Thanks to Thomas Hofweber for helpful correspondence on this and related points.