Suppose I’ve just passed by a colleague’s office, and I see denoting phrases on the board there. That puts me in the mood to write denoting phrases of my own, and so I enter an adjacent room, and write on the board the following expressions:The question is, what is X when X = pi + 6 + X, and the obvious answer is “infinity”. Possible answers are omega, aleph-null, aleph-one, and so forth. There are lots of possible answers, so the “the” at the start of the third expression is a little deceptive. It is like asking for the name of the king of France; the obvious reply is, which king of France?
......the sum of the numbers denoted by expressions on the board in room 213.
Now I am in fact in room 213, though I believe that room 213 is my colleague's office. I set you the task of providing the denotations of these expressions.
......But, what if “numbers” in the third expression was replaced with “finite numbers”? The expression “the present king of France” denotes nobody. But, if the third expression with “finite” does not denote anything, then the sum of the numbers denoted by expressions on the board would be pi + 6, so the third expression ought to denote pi + 6. And then it should denote twice that, whence it should denote thrice, and so forth. If it denotes anything, it does it inconsistently. So, it denotes nothing consistently if it denotes anything, but if it denotes nothing then it denotes pi + 6.
......The light on the horizon is that reference is in general a matter of degree. To why, imagine a man staggering through a desert and seeing a mirage, which he takes to be a pool. Coincidentally there is a pool just where he takes one to be, but it is obscured from his view by the mirage. As he staggers on he is constantly thinking “that pool looks cool” rather obsessively. And as he nears the pool its image gradually replaces the illusory one without him noticing. The referent of “that pool” gradually changes to the pool, and so he will at some point have referred to the pool only as much as not.
......What did “that pool” denote when the man was referring to the pool only as much as not? I would say that it denoted the pool, but only vaguely. And similarly, the third expression with “finite” refers us to pi + 6 insofar as it does not, so it refers us to pi + 6 as much as not. It denotes pi + 6, but only vaguely.