Thursday, June 16, 2011

Sensual quantification

I used to think that belief ascriptions – things like ‘Lois thinks Superman can fly’ and ‘Lois thinks Clark can fly’ – were very interesting and important, and that if we had a good account of them then we’d understand a lot of other things better. I still think puzzles involving coreferring names are interesting and important, but nowadays I tend to think their apparent failure of substitutivity in belief contexts is a bit of a red herring. Nonetheless, this post is about belief ascriptions.

In Ted Sider’s discussion of Mark Richard’s theory of belief ascriptions, he raises a problem. He considers these two inferences:

A1:      (i) Twain is a famous author and Odile believes that Twain is dead.
            (ii) Therefore, x(x is a famous author and Odile believes that x is dead)

A2:      (i’) Twain is a famous author, and Odile does not believe that Twain is dead.
            (ii’) Therefore, x(x is a famous author and Odile does not believe that x is dead)

Both inferences look okay to the untutored eye, and Sider says we ought to allow that either both are valid or neither is. Richard’s semantics for belief ascriptions validates A1 but not A2. I’m not quite as down on this as Sider is, but I agree it looks bad. The reason Richard gets this result is he wants to say that Odile can believe that Clemens is dead without believing Twain is dead, if she doesn’t know ‘Clemens’ and ‘Twain’ are two names for the same person. A lot of people want that result. It leads to trouble when quantifying in though, because ‘x’ in (ii’) takes an object as its value rather than a name. Twain and Clemens are the same object, so if Clemens satisfies ‘Odile believes x is dead’ then Twain does. One way of looking at this is that even if names have sense as well as reference, variables don’t. The sense of ‘Twain’ is needed to make (i’) true, and since it’s lost when you existentially generalise it as (ii’), (ii’) isn’t true.

One response is to say that A2 is valid, but if Odile believes Clemens is dead then (i’) isn’t true after all, because names don’t have sense any more than variables do. Believing Clemens is dead just is believing Twain is dead, because Clemens being dead just is Twain being dead. It does look a bit struthious, but plenty of people go that way. I’m sympathetic to it myself.

Another response says that A1 and A2 both commit a fallacy of equivocation. Following Frege, we say that in belief contexts a name doesn’t refer to its bearer; it refers to what is normally its sense. So in each of (i) and (i’) ‘Twain’ appears once referring to Twain and once referring to the sense of ‘Twain’. The inference is as bad as this:

A3:      (i’’) I cashed my cheques as the bank and then spent all afternoon fishing at the bank.
            (ii’’) Therefore, x(I cashed my cheques at x and then spent all afternoon fishing at x)

These two responses agree with Sider that A1 and A2 are equally valid. It’d be nice though if there was a way of making them both valid and allowing that Odile can believe that Clemens but not Twain is dead. I won’t do that, but I’ll try to go one worse, and to let (i) and (i’) entail these respectively:

(iii) There is a famous author who Odile believes is dead.
(iii’) There is a famous author who Odile does not believe is dead.

Those sentences, or sentences like them, are presumably the ones driving the intuitions anyway. To make this go through, we follow Frege on the references of names in belief contexts, and introduce a quantifier s ranging over senses, and a reference function R taking senses to the referents they determine. Now we analyse (iii) and (iii’) as:

(iv)  sx(R(x) is a famous author and Odile believes that x is dead)
(iv’) sx(R(x) is a famous author and Odile does not believe that x is dead)

Now we’d like to make the inferences logically valid, rather than valid in whatever sense this inference is:

A4:      (i’’’) Twain is dead.
            (ii’’’) Therefore, x(x is a name with five letters and the referent of x is dead)

For all we’ve said so far the connection between the sense of the first occurrence and reference of the second occurrence of ‘Twain’ in (i) is purely a pragmatic phenomenon, so the inference to (iii) is like A4. That isn’t what we’re after. We could complicate the semantics by taking the sense as the semantic value and covertly applying R to it outside of belief contexts but not inside them. That’d make the inferences logically valid, but the strategy wouldn’t generalise to iterated belief contexts like in ‘Odile believes that Elodie believes that Twain is dead’. (That’s because these contexts demand a hierarchy of senses going up, whereas iterations of R only allow a hierarchy of references going down.) So I’ve tried to go one worse but I’ve actually gone two worse. I’ve analysed (iii) and (iii’) differently from the way Sider and Richard did, and I’ve made the inference pragmatic instead of logical. This probably means you don’t need a primitive sensual quantifier with its own inference rules; you just define it as an objectual quantifier restricted to senses. This doesn’t constitute a grand success. Perhaps it’s a good thing I don’t think belief ascriptions are a big deal anymore.

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