## Monday, August 09, 2004

### Scopal Ambiguity

Scopal ambiguity arises when a sentence contains different types of quantifiers, but is unclear as to their correct order. For example, the sentence "some students heard both concerts" could be interpreted as saying either:
(i) There exist some students such that each of them heard both concerts.
or (ii) Both concerts were such that each, individually, got heard by some students (but not necessarily the same ones).

The ambiguity becomes clearer if we try to formalise the sentence, as each option can find the alternative merely by swapping the order of the quantifiers:
(i) [Some x: STUDENT(x)][Both y: CONCERT(y)] HEAR(x,y)
or (ii) [Both y: CONCERT(y)][Some x: STUDENT(x)] HEAR(x,y)

Now, what I found interesting is that the ambiguity seems to disappear when the sentence is extended in a certain way. Consider: "Some students who heard both concerts were interviewed by Holmes" (new bits in italics). There is only one plausible reading of this sentence - the ambiguity seems to have disappeared entirely. Why is this?

Here is a formal representation of the sentence:
[Some x: STUDENT(x) & [Both y: CONCERT(y)] HEAR(x,y)] INTERVIEW(h,x)

A key difference here is that the 'both' quantifier is embedded within the 'some' quantifier, so they can no longer easily be swapped. But let's try to construct a (ii)-style interpretation of this sentence:

[Both y: CONCERT(y) & [Some x: STUDENT(x) & HEAR(x,y)]] INTERVIEW(h,x)
But note that this cannot work, because the 'some x' quantifier is embedded within the 'both y' one, and so the variable 'x' actually expires before we get to the INTERVIEW predicate. That is, the second argument of the predicate doesn't actually refer to anything!

We can, however, construct an un-embedded version as follows:
[Both y: CONCERT(y)][Some x: STUDENT(x) & HEAR(x,y)] INTERVIEW(h,x)
This says that for both concerts (taken individually), Holmes interviewed some students who heard that (one) concert.

So what's wrong with this? Could this, in theory, be a reasonable interpretation of the sentence "some students who heard both concerts were interviewed by Holmes"? Well, I'm not entirely sure. It certainly doesn't come naturally to us, but that is a practical, not theoretical, matter. However, I think there might be an underlying theoretical problem with it too. That is because it seems to me that to achieve the natural reading, the 'both' quantifier must be embedded within the 'some' - there is no way to make an un-embedded version which means the same thing.

Suppose we tried to simply swap the quantifiers used in the un-embedded (ii)-type attempt above:
[Some x: STUDENT(x) & HEAR(x,y)][Both y: CONCERT(y)] INTERVIEW(h,x)
This doesn't work because the variable 'y', when first used, is as yet undefined. Instead we'd need to shift the 'HEAR' predicate across to the latter quantifier:

[Some x: STUDENT(x)][Both y: CONCERT(y) & HEAR(x,y)] INTERVIEW(h,x)

But now this sentence has a different meaning. The 'some' quantifier is far less restricted than it used to be. That is, it merely restricts x to "some students". But what we really want is for x to be restricted to "some students who heard both concerts". So this unembedded version is unfaithful to the original sentence.

My (tentative) conclusion is that the ambiguity disappeared because - in the extended version of the sentence - one quantifier was embedded within the other, and this prevents the sort of 'swapping' which gives rise to ambiguities. I'm not sure how generally this applies though. Are all such embedded quantifiers non-ambiguous? Can some be reformulated into an un-embedded equivalent (which would then, presumably, be open to ambiguous readings)?