Partially defined predicates

Is cutting one head off a two-headed person a case of beheading?

Examples like this are normally used as illustrations of vagueness. It’s natural to think of cases like this as ones where we have a predicate defined over a domain and being applied outside it. Thus, “is being beheaded” is defined over n-headed animals that are being deprived of all heads or of no heads.

I don’t like vagueness. So let’s put aside the vagueness option. What else can we say?

First, we could say that somehow there are deep facts about the language and/or the world that determine the extension of the predicate outside of the domain where we thought we had defined it. Thus, perhaps, n-headed people are beheaded when all heads are cut off, or when one head is cut off, or when the number of heads cut off is sufficient to kill. But I would rather not suppose a slew of facts about what words mean that are rather mysterious.

Second, we could deny that sentences using predicates outside of their domain lack truth value. But that leads to a non-classical logic. Let’s put that aside.

I want to consider two other options. The first, and simplest, is to take the predicates to never apply outside of their domain of definition. Thus,

1. False: Cutting one head off Dikefalos (who is two headed) is a beheading.

2. True: Cutting one head off Dikefalos is not a beheading

3. False: Cutting one head off Dikefalos is a non-beheading.

4. True: Cutting one head off Dikefalos is not a non-beheading.

(Since non-beheading is defined over the same domain as beheading). If a pre-scientific English-speaking people never encountered whales, then in their language:

5. False: Whales are fish.

6. True: Whales are not fish.

7. False: Whales are non-fish.

8. True: Whales are not non-fish.

The second approach is a way modeled after Russell’s account of definite descriptors: A sentence using a predicate includes the claim that the predicate is being used in its domain of definition and, thus, all of the eight sentences exhibited above are false.

I don’t like the Russellian way, because it is difficult to see how to naturally extend it to cases where the predicate is applied to a variable in the scope of a quantifier. On the other hand, the approach of taking the undefined predicates to be false is very straightforward:

9. False: Every marine mammal is a fish.

10: False: Every marine mammal is a non-fish.

This leads to a “very strict and nitpicky” way of taking language. I kind of like it.

"John and John"

I just sent out an email to two philosophers whose first name was "John" and the email's first line said "Dear John and John". After I sent the email, I wondered to myself: Is there a fact of the matter as to which token of "John" referred to whom?

Normally, if I write an email to two people, I think about the issue of which order to list their names in, and I typically proceed alphabetically. But in this case, I didn't think about the order of names I was writing down. It is possible that I thought about the one while typing the first "John" and then about the other while typing the second "John". Would that be enough to determine which token refers to whom? Maybe. But I don't know if I did anything like that, and we may suppose I didn't.

Now:

1. John and John are philosophers.

This is true. But I didn't think of a particular one of the two while typing a particular "John" token. It seems unlikely that there be a fact of the matter as to which "John" refers to whom. But the sentence is, nonetheless, true, and hence meaningful.

Is the sentence ambiguous in its speaker meaning? If so, that's a hyperintensional ambiguity, because necessarily "x and y are Fs" and "y and x are Fs" have the same truth value. I am hesitant to say that (1) is ambiguous in its speaker meaning. (I will leave its lexical meaning alone, not to complicate things.)

Suppose that there is no ambiguity in speaker meaning, or at least none arising from the issue of which token refers to whom (maybe "philosopher" is ambiguous). Then this rather complicates compositional semantics on which the content of a whole arises from the content of the parts. For if either token of "John" in (1) has a content, the other token has the same content, since they are on par. But if the content is the same, we're not going to get out of this a sentence that means the same thing as (1) does. Suppose, for instance, the content of each token of "John" is the same as that of of "x or y", where "x" and "y" are unambiguous names for the two philosophers. Then we would have to say that (1) is equivalent to:

2. (x or y) and (x or y) are philosophers,

but in fact (1) and (2) are not equivalent—all that (2) needs for its truth is that one of x and y be a philosopher.

Maybe the solution is this. Neither "John" in (1) refers. But "John and John" is the name of a plurality. I think not, though. Here's why. Suppose instead I said: "John and the most productive member of my Department and John are all philosophers." Well, "John and the most productive member of my Department and John" is not a name, as it does not refer rigidly.

I am just a dilettante on semantics, and it would not surprise me if this was exhaustively discussed in the literature.