Sunday, August 29, 2004

Longer than it is

Two guys wander down to the pier, and one of them says in surprise, "I thought your boat was longer than it is". The other guy raises an eyebrow and replies: "No, my boat is not longer than it is". (This example came up in our semantics class a while back. I think it originates from Bertrand Russell.)

I found it quite amusing, but what's really fun is trying to puzzle out exactly what each person is trying to say. The key seems to be the distinction between de dicto (about the words) versus de re (about reality) readings of the boat's "length". To represent these alternatives formally (using 'i' to refer to the first speaker, and 'b' as the boat), we have:

1) [The x: LENGTH(x,b)] THINK(i,[The y: LENGTH(y,b)] LONGER(y,x) )
i.e. "Call the actual (de re) length of your boat 'x'. I thought that the (de dicto) length of your boat was greater than that value x".

Compare that to the following (absurd) version:
2) [The x: LENGTH(x,b)] THINK(i, LONGER(x,x) )
i.e. "Call the actual length of your boat 'x'. I thought that this value x is greater than itself."

Similarly:
3) THINK(i, [The y: LENGTH(y,b)] LONGER (y,y) )
i.e. "I thought the length of your boat (whatever it may be) was longer than itself".

The first guy obviously intended to mean (1). The others are both absurd. I'm not sure which of (2) or (3) the second guy mistook him as saying - probably (3) fits best?

Anyway, that struck me as an interesting sort of case. It sorta reminds me of scopal ambiguity. I take it that in general, the de re reading is achieved by having the quantification occur outside of the intensional predicate (so the predicate then refers to this external reality). By contrast, quantifying inside the predicate creates the de dicto reading, as the variable is cut off from anything external, and instead merely represents the concept (or 'words') it is defined by.

What about similar cases that don't involve quantifiers?

Suppose Lois thinks Superman is in danger. Then suppose you say to Clark Kent's parents, "Lois thinks Clark is in danger". Is what you say true? (Recall that Clark is Superman, though Lois does not know this.) There seems to be an ambiguity between de re and de dicto readings here too.

I'm not sure how to represent these formally, however, as there only seems to be one option (with 'l' = Lois, and 'c' = Clark):
THINK(l, IN DANGER(c) )

This is the de dicto reading, and it falsely claims that Lois thinks the proposition "Clark is in danger". But how can we get the de re reading? I wonder if perhaps we need to indirectly refer to Clark by way of a restricted quantifier:

1) [The x: x=c] THINK(l, IN DANGER(x) )
i.e. Regarding the person who is Clark, Lois thinks that person is in danger. (This is the de re reading, and it is true.)

2) THINK(l, [The x: x=c] IN DANGER(x) )
i.e. Lois thinks that "the person who is Clark is in danger".
(This is the quantified version of the de dicto reading. It's still false, of course.)

Is that a legitimate formalisation? Is there any other way to capture the two possible readings?

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