Here’s a little argument that was inspired by some things Williamson says in chapter 3 of “The Philosophy of Philosophy”. It’s not at all the way Williamson intended his arguments to be used I guess.
- Any logical truth is true in virtue of meaning facts alone.
- Timothy Williamson is a philosopher is not true in virtue of meaning facts alone.
- Any disjunction with exactly one true disjunct is true in virtue of whatever the true disjunct is true in virtue of.
- So, Timothy Williamson is a philosopher or Timothy Williamson is not a philosopher is not a logical truth.
The premises could use being tidied up a little bit, but I think there’s something close to this in Williamson. Of course, he rejects (4), so he’s more interested in the argument from (2), (3) and the negation of (4) to the negation of (1). (Not that he would be quite as cavalier in the formulation of the argument as I’ve been.) Still, I think it’s a pretty interesting argument this way.
When I first saw this in Williamson, I thought, wow there’s a nice argument against the law of excluded middle. But now I’m worried that a structurally similar argument could, in principle, be run against the law of non-contradiction. I’ll leave it as an exercise for the reader to figure out the best way such an argument would go. I’m leaving it as an exercise in part because I’m not quite happy with any of my attempts, and in part because I’m too lazy. But unless I’m confident that no such argument could be used to reject LNC, I’m not going to be using this argument against LEM. And as of now, I’m certainly not confident of that.