This is a post on the Liar Paradox. The Liar Paradox arises from sentences such as (l).
(l): (l) is not true.
Let (~l) be the negation of (l), i.e. (~l) is supposed to contradict (l). The paradox ensues from this kind of reasoning: Assume (l). Then, (l) is not true. So, (~l). Thus, (l) and (~l). This reduces our original assumption to absurdity. So, we reject our assumption and conclude (~l). But, then (l) is not true, from which (l) follows. And, we’ve derived a contradiction.
I’ve been reviewing a fairly recent (advanced) introduction on truth by Alexis G. Burgess and John P. Burgess. They suggest that the most common initial reaction to the paradox is to deny that (l) expresses a proposition. This would appear to block even the initial assumption of (l); (l) would not even be a candidate for assumption.
As Burgess and Burgess suggest, this common sense reply is problematic. Even if it succeeds in blocking the paradoxical reasoning above, there’s the “revenge of the liar.” So consider (l*).
(l*): (l*) does not express a true proposition.
Note that this sentence is distinct from (l), which basically amounts to (l’).
(l’): (l’) expresses a not-true proposition.
Because (l) is equivalent to (l’), it seems like one can block the paradox by denying that (l) expresses a proposition. But, the standard line is that this can’t help with (l*). Here’s how Burgess and Burgess explain it (in square brackets):
[The would-be vindicator of the intuitive notion of truth who says of (l*) what was said of (l)/(l'), that it does not express a proposition, faces the reply, "So, a forteriori (l*) does not express a true proposition, and since that is just what it says, it is true." The would-be vindicator may then insist, "No, (l*) does not say that; (l*) does not 'say' anything; (l*) does not express a proposition. So, a forteriori (l*) does not express a true proposition, and contrary to what you say, (l*) is not true." But then the would-be vindicator will be in the position of having said something (namely (l*), which is to say, "(l*) does not express a true proposition"), and also having said that it is not true. The problem, sometimes called that of ineffability, is that the would-be vindicator's theory cannot be enunciated, at least not without saying something that, according to that theory itself, is untrue.]
I’m interested in the step of this reasoning that I’ve set in bold. I guess I’m not sure why the person pushing the common sense line would want to make that move, i.e. go from ‘(l*) does not express a proposition’ to ‘(l*) does not express a true proposition’. The later sentence is just (l*) itself. Surely, if this sentence doesn’t express a proposition, then it isn’t a good candidate for the conclusion of an argument.
Of course, the move in question seems to be innocuous. It seems to be the conclusion of a valid argument form. But, of course, on first acquaintance (l*) seems to be a perfectly meaningful sentence that we might well expect to express a proposition. It seems to be at least a slight departure from common sense to think that sentences that have the right form to express propositions—in the sense that meaningful constituents are properly combined into something that should be a well formed thought—sometimes don’t express propositions. If this is the common sense position though, then I don’t see how it should be a further affront to common sense to think that there are sequences of sentences that have the right form to express valid arguments but don’t. That just seems to be a natural consequence of taking up the position that form isn’t always a good guide to propositional content once the concept of truth and its affiliates have been introduced.
Of course, Burgess and Burgess are right that this leads to a kind of ineffability. But, we should anticipate this. The common sense line appears to be that, in certain cases, there is literally nothing to say even though the ordinary rules for composing propositions out of concepts would suggest otherwise. The ineffability, then, isn’t because there’s some thought that we can’t find a way to voice; it’s because there isn’t a thought at all. I’m not really sure whether this is plausible in the end, but it’s never been entirely obvious to me that it isn’t either.