thoughts on yachts

there is a sentence about a yacht that I am thinking about.

(1) I thought that your yacht was larger than it is.

As is well known (1) has two readings.

(1a) I thought that the size of your yacht was greater than the size of your yacht.
(1b) The size that I thought your yacht was is greater than the size that your yacht is.

In order to highlight the logical form of the two readings, (1a) and (1b) can be represented semi-formally using the following scheme of abbreviation and symbolization.

B: I thought that a
S: the size of your yacht
>: a is greater than b

(1a*) B [S > S]
(1b*) Ex [(B(x = S)) ^ x > S]

There is another sentence about a yacht that has a similar type of ambiguity. I wonder if it can be given a similar treatment.

(2) Your yacht could be larger than it is.

I think (2) has the following two readings.

(2a) It could be that the size of your yacht is greater than the size of your yacht.
(2b) The size that your yacht could be is greater than the size that your yacht is.

These readings parallel the treatment of (1) as is clear from the following formalization.

M: it is possible that a
S: the size of your yacht
>: a is greater than b

(2a*) M [S >S]
(2b*) Ex [(M (S = x)) ^ (x > S)

But (2b*) seems to have two readings:

(i) There is a size that the size of your yacht could be equal to, namely x, and x > S
(ii) There is a size that could be equal to the size of your yacht, namely x, and x > S


Spelled out a bit more these become:

(i`) There is a size such that: (it is possible that: the size of your yacht is equal to that size) and (that size is greater than the size of your yacht)

(ii`) There is a size such that: (it is possible that: that size is equal to the size of your yacht) and (that size is greater than the size of your yacht)

Aren’t these the same? Yes. But there is a subtle difference which the different orderings (almost) bring out. This is what I have in mind.

First lets look at (i`): There is a size, say 20’, such that, (it is possible that, the size of my yacht is equal to that size, i.e. 20’), and (that size, i.e. 20’, is greater than the size of my yacht, say 15’). Roughly put, it is possible that the size of my yacht be 20’ instead of 15’.

Turning to (ii`): There is a size, say 20’, such that, (it is possible that, that size, i.e. 20’, is equal to the size of my yacht, say 15’), and (that size, i.e. 20’, is greater than the size of my yacht, i.e. 15’). Roughly put, it is possible that 20 be equal to 15’ instead of being more than 15’.

The reading we want, of course, is (i`).

Putting an ‘actual’ in seems to make it clearer.

(2A) Your yacht could be larger than it actually is.
(2bA) The size that your yacht could be is greater than the size that your yacht actually is.

Adding “actually” to the scheme (A: actually), we get the following symbolization.

(2b*A) Ex [(M (S = x)) ^ (x > AS)

Which reads something like this:

(iA) There is a size such that: (it is possible that: the size of your yacht is equal to that size) and (that size is greater than the actual size of your yacht)

Or should it be this:

(2b*A`) Ex [M (S = x) ^ A(x > S)]

(iA`) There is a size such that: (it is possible that: the size of your yacht is equal to that size) an (actually: that size is greater than the size of your yacht)

I am not sure how best to represent it. We want the definite description represented by ‘S’ in the left conjunct to be non-rigid but we want the ‘S’ in the right conjunct to be rigidified. The actuality operator achieves this but it might be redundant. I keep getting a funny reading like (ii`) and think I must have gone wrong somewhere. Quantifying over sizes is a bit weird. Its not quite right for (1), since as Kripke has pointed out, there is not an exact size you thought my yacht was. That type of problem shouldn’t affect the modal sentence, since there is an exact size (well, many exact sizes) that my yacht could be. I'm not sure....

my thoughts have drifted off of yachts.