Monday 19 August 2013

A Fallacy in Hofweber's Arguments in Ontology

[This is a draft of a paper.]

Hofweber's ontological project crucially involves inferring negative existential statements from statements of non-reference, i.e. statements that say that some term or terms do not refer. Here, after explaining the context of this move, I want to show that it is fallacious, and that this vitiates Hofweber's ontological project.

Thomas Hofweber has for several years been developing a distinctive approach to ontological and metaontological questions.

One of his starting points is the way some ontological questions in philosophy can apparently be settled with trivial arguments - for example, since mathematics has established that there are infinitely many prime numbers, it follows that there are numbers, and so there is no room for a special philosophical discipline of ontology (if it is to respect mathematics) to deal with this as a substantial question, the way ontologists of mathematics seem to try to do. Call this the puzzle about ontology.

Hofweber attempts to solve the puzzle about ontology by independently motivating a distinction between two different readings of quantifiers, or two sorts of quantification: internal and external. Internal quantificational statements, unlike external ones, do not work by placing conditions on a domain of objects. (To see that we might need something like this, consider the quantifier in 'Santa Claus doesn't exist, therefore there is something that doesn't exist'.) He then argues that the trivial arguments go through, but only when the quantifiers are given an internal reading. Give the quantifiers an external reading, and it is not clear that their premises have been established - in the case of 'There are infinitely many prime numbers', for instance, it might be that mathematics has established this on its internal reading, but not on its external reading.

Hofweber doesn't just want to solve the puzzle about ontology with his internal/external distinction, however. He also wants to use it to establish answers to certain (external) ontological questions - negative answers. This is what I call 'Hofweber's ontological project'.

Taking the number case, the project goes roughly like this. Hofweber argues that, if we can establish internalism about number-talk, including arithmetic (i.e. if we can establish that the quantifiers involved in number-talk, including arithmetic, are internal ones which do not place conditions on a domain of objects), we can show that the external question of whether numbers exist is left open by this talk, and is thus free for the taking by ontology.

Next, Hofweber argues that numerals, number words like 'four', and terms like 'the number 2' are not referring terms. I.e., that they are not in the business of referring to things. They sometimes assume the superficial grammatical position of referring terms for sophisticated linguistic reasons involving the notion of a 'focus construction' (and other considerations, depending on the kind of occurrence).

Then, on the basis that number terms don't refer, Hofweber concludes (via a principle designed to enable one to infer non-existence of things from statements of non-reference) that numbers don't exist, i.e. that there are no numbers (the quantifier here being intended externally), thus answering one of ontology's fundamental questions. I will good-naturedly call this last step 'the Howler'.

A couple of years ago, I inconclusively argued that Hofweber's distinction between internal and external quantification is ill-motivated. Here, I want to grant that distinction, and even grant that it enables Hofweber to explain the validity of the trivial arguments.

I want to make it clear that the Howler is a fallacious move, and that this vitiates Hofweber's project for answering certain ontological questions (e.g. about numbers, properties and propositions) in the negative. I will not be concerned here with whether Hofweber succeeds in establishing internalism about number-talk - my point is only that his argument from internalism to negative answers in ontology contains a fallacy.

The Howler appears in Hofweber's contribution to the influential 2009 anthology, Metametaphysics: New Essays on the Foundations of Ontology. The contribution is called 'Ambitious, Yet Modest, Metaphysics'. (I include other relevant papers in the bibliography, to help readers piece together a more detailed view of Hofweber's overall project, but he gives a good sense of it in the paper just mentioned.)

I think my criticism will be most effective if I quote the Howler along with the argument in which it appears, rather than reconstructing it and insisting that that is what Hofweber was doing. Here is the argument:

Let’s briefly reflect on what seems to be a central thesis about reference or denotation: 
(REF) If Fred exists then ‘Fred’ refers to Fred. 
Of course, I am assuming that ‘Fred’ is unambiguous, or at least used in the same way throughout. (REF) is uncontroversial, I take it, and probably a conceptual truth. Note that it implies the following: 
(REF∗) If ‘Fred’ doesn’t refer to Fred then Fred doesn’t exist. 
There are two ways for an expression not to refer. One is to aim to refer, but not to succeed. A classic case of this are empty names. Although the details of any example one might try to give of this are controversial, let’s nonetheless take ‘Sherlock’ to be an empty name of this kind. That is, suppose Sherlock is a name and thus has the semantic function of picking out an object. But it fails in carrying out that function. It thus doesn’t succeed in referring, and thus doesn’t refer. Thus Sherlock does not exist. Nothing in the world is Sherlock, no matter what in general the world contains. There could be all kinds of people, with all kinds of professions, but no matter how general properties are instantiated in the world, nothing in it is Sherlock. And nothing could be. If ‘Sherlock’ does not refer then Sherlock does not exist. This is all fairly trivial, but I go over it to make it vivid for our next case. 
Names aim to refer, but they can fail to succeed in what they aim for. The second way in which an expression might not refer is when it does not even aim to refer. Non-referential expressions, like ‘very’, don’t refer since they don’t even aim to refer. If internalism is correct about talk about numbers, properties, and propositions, then the relevant singular terms are non-referential. They do not aim to refer, and thus they do not refer. According to the above version of internalism ‘two’ is just like ‘most’. But since it doesn’t refer we know that there is no such thing as the number two. Since ‘two’ and ‘the number two’ are non-referring expressions nothing out there is (or can be) the number two. There can be all kinds of objects, abstract or concrete, they can have all kinds of properties and relations to each other. Nonetheless, none of them is (or can be) the number two. Or any of the other numbers. Internalism thus answers the ontological question.

Note first that Hofweber says that '[a]ccording to the above version of internalism "two" is just like "most"'. But what do we get if we substitute 'most' for 'Fred' in Hofweber's (REF*) principle?:

(REF*-Most) If 'most' doesn't refer to most then most doesn't exist.

But this seems like ungrammatical nonsense. Furthermore, it doesn't seem that 'Does most exist?' or 'Is there such a thing as most?' are substantial, sensible questions. It may be argued that 'Does most exist? No.' is not complete gibberish, if it is construed as a kind of metalinguistic point - it's not true to say 'Most exists'. This question-and-answer does not appear to be about whether the domain of our external quantifiers meets certain genuine conditions (and not simply metalinguistic conditions such as 'being referred to by the word "most"').

So if 'two' really is just like 'most' in all relevant respects, Hofweber has a problem. There is, of course, an important difference. Consider:

(REF*-Two) If 'the number two' doesn't refer to the number two then the number two doesn't exist.

Unlike (REF*-Most), (REF*-Two) is superficially grammatical. It even appears not to be nonsense (if we consider it independently of Hofweber's views). Do either of these two differences help?

Superficial grammaticality doesn't help; consider the nonsensical but superficially grammatical question 'Is there a rock of eggs?'. This doesn't seem to turn on whether a domain of objects meets some genuine condition, and 'Is there a rock of eggs? No.', like the 'most' case, seems to be a metalinguistic point at best.

The appearance of sense doesn't help either, for Hofweber has no way of explaining it except in terms of internalism; he explains occurrences of number-expressions always in terms of their being non-referring terms that appear in the syntactic guise of referring terms (for sophisticated linguistic reasons). And it is not at all clear what these could possibly be doing in an external quantificational context.

In general, the point might be captured by the following principle: non-existence of something only follows by semantic descent from non-reference when the non-referring term plays the semantic role of referring. Otherwise you can't semantically descend to a well-formed, sensical proposition.

(This is a necessary condition for the non-existence of something following by semantic descent from non-reference, but it may not be sufficient. I say 'follows by semantic descent' rather than simply 'follows' because '"X" does not refer' may be argued to always imply 'The referent of "X" does not exist' - but there is no semantic descent there, as there is in Hofweber's arguments.)

This seems like the natural view, in lieu of some special story, and Hofweber hasn't given any such story.

In a forthcoming post, I will argue that Hofweber's ontological project is impossible, for a different (though related) reason: internalism at the strength he requires it is inconsistent with the thesis that there is a substantial ontological question about numbers left open by arithmetic and other non-metaphysical number-talk, since such a question would constitute a counterexample to internalism. This mistake is, I think, more profound than the one exposed here - making it involves a kind of sawing-off of the branch one is sitting on.


Main reference: 

Hofweber, T. 2009. 'Ambitious, yet modest, metaphysics', in David John Chalmers, David Manley & Ryan Wasserman (eds.), Metametaphysics: New Essays on the Foundations of Ontology. Oxford University Press.

Background reading for Hofweber's project:

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. Apologies if the following is a naive question.

    I never understood the need to invoke the internal/external quantifier distinction to deal with the trivial deduction "there are infinitely many prime numbers, therefore there are numbers".

    All proofs of the infinitude of prime numbers quite explicitly ASSUME the existence of numbers. Euclid's original proof shows that it is possible to enlarge any given finite (non-empty) collection of primes; the missing ingredient --- that there exist non-empty finite collections of primes at all --- is taken as obvious, not even worth mentioning.

    Mathematics more generally only gets off the ground by taking the natural numbers as given. Thus all deductions of the trivial form above are circular, and uninformative for that reason.

    1. Hi Pietro, thanks for the question!

      'All proofs of the infinitude of prime numbers quite explicitly ASSUME the existence of numbers.'

      There are really two things going on here, I think: inside math, the existence of numbers is more basic than the infinitude of primes. So it's a bit weird inferring their existence from the infinitude of primes, if all quantifiers are given the same reading, and you've picked up on this weirdness.

      The other thing is that you want clarification about why Hofweber uses an internal/external quantifier distinction.

      Regarding the weirdness of inferring existence of numbers from infinitude of primes, when all quantification is read uniformly: yeah, it is weird from that point of view, but doesn't mean that the existence of numbers doesn't follow logically from the statement that there are infinitely many primes. The choice of infinitude of primes as a premise, rather than some more basic mathematical proposition which seems to imply that numbers exist, is just used for dramatic effect, and to emphasize the non-trivial nature of mathematics (at least, that's my understanding).

      But this isn't essential. Another sort of puzzling argument Hofweber would want the int/ext quantifier distinction for is:

      (P): There are numbers between 2 and 5. (establlished by elementary arithmetic)
      Therefore (C): Numbers exist. (apparent answer to difficult ontological question)

      Hofweber invokes the internal/external quantifier distinction here in order to explain how, while this argument is valid when all quantifiers are given a uniform reading, it is not valid if the premise's quantifier is internal, the conclusion's external.

      His position is that the trivial arguments are valid, but we cannot use them to infer answers to ontological questions from claims established by mathematics; Math establishes quantificational claims about numbers, but the quantifiers (according to him) are internal.

      So inside math as it were, you can say 'There exist numbers with such and such properties', but Hofweber will insist that, insofar as this is something math has established, its quantifier is internal, so it doesn't imply 'Numbers exist' when this is construed externally, with what he calls a 'domain conditions reading'.