Saturday, October 28, 2006

The Number of Zits on My Face

Hi all.

I'm currently reading a paper by Thomas Hofweber called "Innocent Statements and Their Metaphysically Loaded Counterparts". (It is available on his website for anyone that is interested.) In his paper, Hofweber is concerned with the transition between sentences like (A) "I have four zits on my face" and (B) "The number of zits on my face is four". (Hofweber uses different examples.) Hofweber claims that (A) and (B) seem truth-conditionally equivalent and that we can infer either from the other in natural language, but (B) apparently entails the existence of numbers whereas (A) does not. His paper is an attempt to determine what is going on here.

Anyway, in Section 3.1 of his paper, Hofweber draws attention to a puzzle for those who think that "The number of zits on my face" in (B) functions as a singular definite description. He notes that, in general, a sentence containing a singular definite description "the F" entails the corresponding sentence containing the indefinite description "a F". However, Hofweber says, attempting to apply this general rule to (B) yields:
(C) A number of zits on my face is four.
And (C) seems very awkward.

So, my question to you guys is: Any idea what is going on here? It certainly looks to me like "The number of zits on my face" functions as a singular definite description in (B). So why does (C) sound so strange if a sentence containing a singular definite description entails the corresponding sentence containing the indefinite description?

7 Comments:

Blogger Joshua said...

Here is the idea that I was trying to get at when I talked to you and Andrew this morning. This idea is not meant to be an explanation, but merely an observation that may help in a later explanation. Also, I may be making some horrible mistake here and encourage you to tell me if that is so.

With normal definite descriptions, such as *The apple in the door is red*, even though it may in fact be true that there is only one apple in the door, it is possible that there is more than one apple in the door.

On the other hand, the corresponding claim for definite descriptions with numbers seems false. That is, not only is it true that there is one and only one number that numbers the moons of jupiter, it is not even possible that there is more than one number that numbers the moons of jupiter.

Perhaps this difference in the two cases can be used to explain the strange inference to (C) in the original post.

If I am correct that this is a difference between the two cases, then a good way to test whether or not this difference plausibly plays a role in explaining our resistence to the inferred claim (C) is to find cases that are like the number cases and unlike the apple cases in the very respect that I have indicated and see if the corresponding indefinite claim in the new cases sounds as stangely as the one in the number case.

2:00 PM  
Blogger Andrew Cullison said...

This comment has been removed by a blog administrator.

2:42 PM  
Blogger Andrew Cullison said...

Andrew Cullison said...

Another reasons (C) sounds odd is that we sometimes use the locution "A number" as a plural term.

For example:

(D) A number of cars were destroyed.


If you're used to hearing sentences like (D), then (C)sounds ungrammatical.

The weirdness may be due to a contingent feature of our language...

2:43 PM  
Blogger Andrew Cullison said...

p.s. Is there an RSS feed for the comments here?

2:44 PM  
Blogger rock* said...

I'm inclined to agree with Joshua that something like the observation he makes is true. That is, it seems right that there could be more than one apple in the door, but not that there could be more than one number that numbers the moons of Jupiter. I'm not sure if this will help in explaining the fact that (C) seems awkward. Consider:
(D) The person who is identical to me is Greg.
and
(E) A person who is identical to me is Greg.
There couldn't be more than one person who is identical to me, but (E) doesn't sound as weird as (C).

Another point to note is that there are other cases in which a definite description is on the left-hand side of an identity sign where the inferred sentence containing the indefinite description sounds weird. For instance, (F) sounds fine to me, but (G) sounds a little weird, although not as weird as (C):
(F) The man hiding behind the tree in our backyard is Jones.
(G) A man hiding behind the tree in our backyard is Jones.

I would also like to say that I think that something like what Andy says seems right to me. We have a general convention in English in which "A number of Fs" can be used to mean something like "Some of the Fs", and so needs to take the plural. And I think that this probably explains at least some of (C)'s weirdness.

4:56 PM  
Blogger Chris Tillman said...

In addition, it is strange to say (1)'a biggest ball of twine in Minnesota is worth taking a road trip to see' even though it's supposed to be entailed by the perfectly fine (2)'the biggest ball of twine in Minnesota is worth taking a road trip to see'.

Also, all of these indefinites sound better to me when translated into the quantifier-variable idiom. This seems as it should be, since the entailments are perhaps more apparent in FOPL than they are in natural languages. So the Russellized (2) entails the fine-sounding (1r): 'there is an x such that x is the biggest ball of twine in MN and x is worth taking a road trip to see'.

Likewise, it does not sound weird to me at all to infer from 'the number of zits on my ass is 4' that 'there exists an x such that x numbers the zits on my ass & x = 4'.

Maybe another potential source of the weirdness then is just good ol' implicature--using the indefinite implicates non-uniqueness.

6:59 PM  
Blogger Chris Tillman said...

um, for (1r) I meant 'Ex Biggest-ball-of-twine(x) . . .' rather than 'Ex the-biggest-ball-of-twine(x).

7:02 PM  

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