Wednesday, June 02, 2010

Every F is G, with Fewer G's than F's

I notice at Dale Tuggy's "Trinities" blog that the claim came up that "if every F is a G, then there can’t be fewer Gs than Fs." This claim goes back to Richard Cartwright's old paper on the Trinity, and is often presented as if it were self-evident.

It's notable, though, that it's not difficult to find apparent counterexamples. Suppose I have a list:

1. Mary Ann Evans
2. Samuel Clemens
3. George Eliot
4. Mark Twain

If you ask me how many things are itemized on my list, it's entirely reasonable to say that there are 4 things itemized on my list. I can also say, "Everything itemized on my list is a real person." But how many real people are on my list? Only two; each one shows up twice. The F's here are things itemized on my list; the G's are real people; every F is a G; but there are fewer G's than F's.

Let's take an example that's closer to home. On a typical statistics counter for a webpage or blog, there is a counter for what are called "Unique Visitors." Unique visitors are counted by determining what number of visits (which are different from the number of views) has been made by way of that particular computer. Now it's true that every unique visitor is someone visiting your site, at least, if one sets aside bots (which many statistics counters do). But the number of unique visitors has no particular relation to the number of people visiting your site; if a single person visits your site from two different computers, they will be counted as two unique visitors. On the other hand several different people using the same computer to visit your site will register as the same unique visitor. The former characteristic, the ability of one person to be more than one unique visitor, is the one of interest here. Setting aside bots again, every unique visitor is a person visiting your website. But you cannot conclude from this that there can't be fewer F's than G's.

When faced by this people will usually reply by saying something like, "Well, look, the four things on the list are really not people; they are names. And the unique visitors are really not people; they are identifiable computers." This is an attempt to remove all intensional and modal factors in counting; but it needs to be noted that this is already a restriction of the original claim. In practice we do not treat counting in purely extensional terms; how we describe or know things is important. I know my list counts people; but I may not know that it fails to count each person uniquely. And it is entirely possible, as with the unique visitor count, to count things in ways that do not count them uniquely. Indeed, there are many cases where one might do this quite deliberately. Setting aside a few odd cases, every airline passenger is a person, but the same person can get counted by airlines as different passengers. This makes a lot of sense, because the thing that primarily matters for airlines is not the number of people but the number of passengers.

Thus the original claim really boils down to the claim, "When every uniquely identifiable and separate F is a uniquely identifiable and separate G, then there cannot be fewer G's than F's." Every dog is a mammal; but, more than this, this can be glossed in straightforwardly extensional terms: each separate, individual thing that is a dog is exactly the same as a separate, individual thing that is a mammal. Given this, there cannot be fewer mammals in a room than there are dogs, because each separate dog will count as a separate mammal. In such a case the claim really is self-evident. But once we are in territory where different ways of counting have different intensional and modal features, it not only is no longer self-evident; it's provably false. The claim can only be saved by restricting it to cases where there is either only one way of counting or, if there are several ways of counting, if their results are directly convertible into each other (i.e., their results map onto each other perfectly). But there are plenty of areas of life where we are interested in how ways of counting that are not directly convertible relate to each other.

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