Sunday, October 25, 2009

Absolutely Unrestricted Quantification, the Classical Principle, and Non-Existence Solutions to the Liar Paradox

Graham Priest has argued that the structure of paradoxes like the Liar (and similar paradoxes, like Russell’s Paradox for naïve set theory) is best understood in terms of his ‘Inclosure Schema.’ The three elements of the Schema are:

(1) Existence
(2) Closure
(3) Transcendence

(1) is the condition that the relevant object (like the Liar Sentence, or the Russell Set) exists. (2) is the condition that it be a member of some category (being true, being a member of itself), and (3) is the condition that it also ‘transcend’ (i.e. fail to be part of) that category (being untrue, being a non-member of itself). Whatever one thinks of the Inclosure Schema as an analysis of what’s going on with Liar-type paradoxes, and whatever one thinks of Priest’s ‘Principle of Unified Solution’ whereby all paradoxes modeled by the Inclosure Schema should be solved in the same way, it’s at least a useful way of classifying various solutions.

A type-(2) solution to the Liar Paradox would be standard truth-value gap proposals, that say that the key place where the argument from the Liar to inconsistency goes wrong is the assumption that it is either true or false. We can keep the assumption that it’s true iff it’s false, but deny that it’s either one of those things. Presto. Done!

Of course, famously, while this works for the Simple Liar:

"This sentence is false."

….it runs into problems for the Strengthened Liar:

"This sentence is not true."

A type-(3) solution to the Liar Paradox would be a denial of the relevance instance of the Biconditional Truth Schema. (For example, Gil Harman takes this line in Change Of View.) That way, we get to keep the classical assumption that it’s either true or false, but ditch the assumption that it’s true iff it’s false, and thus avoid the contradiction. A problem with this is that it seems terribly ad hoc, and, when we approach the inference from Liar to inconsistency as an argument for dialetheism, it begs the question. If all you can say in response to an argument for a position that you don’t like is that some extremely basic, obviously true-sounding conceptual principle must have exceptions, because if it applied in this case, we’d get that conclusion, you’re not going to win any arguments.

Type-(1) solutions to the Liar Paradox are a bit trickier to formulate. To see why, consider the contrast to Russell’s Paradox. The standard, orthodox mathematical and philosophical response to it is simply to abandon the assumption that sets like the Russell Set exist. And, after all, that’s sort of plausible. If the Russell Set did exist, how would we ever know? If the best reason to think that any sets exist is (a la Quine) because they’re indispensible to our best scientific theories, then the restricted universe of ZFC hierarchical set theory should be quite sufficient. Given that, all else being equal, it seems un-parsimonious to also postulate a multitude of other sets, outside of the ZFC hierarchy.

Whatever one thinks of this argument, my point here is not to defend it but to underline the contrast to the Liar Paradox. There’s no denying that sentence tokens stringing together the words “this," "sentence," "is," and "false” in the right order exist, or even that, in some cases, the intended referent of the “this” is that sentence itself. (By contrast, of course, you can point at a sentence about some empirical subject written on a chalkboard and say “this sentence is false” without any sort of paradox being in the offing.) Non-existence claims, therefore, aren’t quite literally claims that the relevant sentences don’t exist. They’re claims that the relevant truth-bearers don’t exist. If you think that the bearers of truth are propositions, the non-existence claim is the claim that “this sentence is false” fails to express a proposition. If you think that sentences are the bearers of truth (or sentences-in-contexts, or precissified sentences, or any of the many variants of sentence theory out there), then the non-existence claim is that, for whatever reason, sentences like the Liar are actually meaningless. Exactly how to motivate this claim, respond to the obvious objections (don’t such sentences seem meaningful?, etc.), get off the hook of various revenge paradoxes and so on is a terribly difficult and complicated matter, and I don’t want to get into ay of that here.

I do, however, want to note that any remotely plausible non-existence story will apply just as well to the Truth-Teller:

"This sentence is true."

…as it does to the Liar. If a sentence can still be meaningful, or express a proposition, or whatever, when it isn’t trying to assert anything above and beyond its own truth, why not the same for falsity? Any non-existence solution that doesn’t treat the two cases similarly starts to look entirely too much like an ad hoc maneuver to save consistency. So a non-existence solution that even begins to be plausible must apply to all sentences that do nothing but talk about their own truth-status, or the truth-status of some other sentence (as in, say, the Yablo series), without ever "grounding out" in some other subject.

OK, but wait a second. Presumably, part of the motivation for any non-existence solution is likely a desire to defend the following principle about how truth and falsity relate to each other, which we can call the Classical Principle (CS), and, for the sake of simplicity, express in terms of “statements.” When reading “statement,” substitute in whatever your favorite candidate for the role of truth-bearers may be.

CS: "Every statement is either true or false (but not both)."

Stop. Wait. Is the statement of CS a “statement” (i.e. does the sentence express a proposition, or is it meaningful, or whatever)? It seems like it had better be, if we want to express our anti-gap, anti-glut commitments by asserting it! But if so, is it mixing use and mention in the context of talking about truth, and thus violating the requirement that truth-talk ultimately ground out in a subject other than truth? If so, it looks like, on the assumptions that got us to non-existence, it’s not eligible for truth-bearing status.

To get a grip on what’s going on here, it’s useful to start thinking about the general semantics of claims that involve quantification (some, all, etc.) For example, think about the following statements:

(1) All of the milk is gone!
(2) Everyone but John and Jack was eliminated in the first round of the game.
(3) All sentences are composed of words.
(4) Everything the Pope says is true.

In (1) and (2), interpretive charity should warn us away from seeing the domain over which the “all” and “everyone” are intended to quantify as absolutely unrestricted, including everything whatsoever, and context gives us good, solid clues about what the intended domains are. In the case of (1), if we take the domain being quantified over to be universal and unrestricted, then this is an obviously, bizarrely false claim, given the existence of full bottles and cups of milk in grocery stores and houses all around the world. Obviously, however, we take the intended domain to be something more like the objects that are inside the house of the people who are having the conversation. For parallel reasons, we shouldn’t take the “everyone” in (2) to quantify over all persons everywhere, but instead take it to quantify over the restricted domain of people who were playing the game with John and Jack on the particular occasion in question.

(3), on the other hand, seems to be a different sort of case. There’s no reason not to take the domain which the “all” is quantifying over—“for all x, if x is a sentence, x is composed of words”—to include itself. Use and mention seem to be innocently combined here.

How about (4)? Well, in the logical translation “for every x, if x is something the pope said, then x is true”, if we take the domain that “every x” is quantifying over to include questions the Pope asks, commands he gives, meaningless jumbles of word thats he murmurs in his sleep, etc, etc., then (4) is meaningless. Moreover, if he endorses Papal infallibility himself (as Popes are prone to do), then he might do so (oddly and pretentiously enough) in the third person, and thus himself utter (4). Whatever considerations led us to a non-existence solution to the Liar should rule out (4)-as-uttered-by-the-Pope-when-the-domain-over-which-he-is-quantifying-includes-his-assertion-of-(4). But it seems implausible that we should take any of these considerations to render standard statements of Papal infallibility not just false but meaningless. As such, it seems reasonable to charitably interpret the domain over which a non-Pope’s assertion of (4) quantifies as failing to include the Pope’s questions, commands, meaningless jumbles of words uttered during his holy sleep, and so on, and the domain over which a Pope’s assertion of it quantifies as also failing to include his own sentence asserting it. Which is not to say, of course, that the Pope doesn’t think his assertion of (4) is also true, but just that to express that bit of information, he’s going to need to construct a new sentence. And if he wants to express his belief in the truth of that sentence, he’s going to need to construct a new sentence to do it with, and so on.

None of this limits the Pope’s expressive power. He can express any bit of information that strikes his holy fancy. He can’t, however, do it all at once—even if he was infallible, he still wouldn't be capable of the miracle of expressing all of his beliefs with one short sentence.

The road from this to a plausible claim about the CS and non-existence solutions should be starting to get pretty clear. Before we get there, though, one reminder of the stakes.

When the antecedent and the consequent of a conditional are both true, the conditional itself is true. (Material conditionals work that way, anyway. The question of whether the material conditional is the right analysis of the conditional of ordinary language is actually going to be irrelevant to the problem I'm getting at…all I’ll need is the assumption that the conditional we end up with is at least false.) Now, say that someone has the false belief that McCain won the 2008 election. (Perhaps they just woke up from a coma they’ve been in since before the election, and the last polling data they saw showed McCain in the lead.) Given that they have that belief, and standard classical beliefs about the semantics of conditionals, and the plausible belief that “P” and “”P’ is true” are universally intersubstitutable, they (validly) conclude the following McCain Principle (MP):

MP: "For every statement, if that statement is true, then McCain won the election."

If you take the domain over which “every statement” quantifies to include the statement of the MP itself, then it asserts (among other things) that if it is true, then McCain won the election. As anyone who’s familiar with the Curry Paradox knows, that claim is one that has some very strange properties (if it's meaningful). Given the Biconditional Truth Schema and any logic strong enough to include Modus Ponens and Conditional Proof, the truth of its consequent can be derived from the simple statement of its truth conditions. If we want the MP to be false (and thus meaningful, truth-evaluable, etc.), and we don’t want to believe that it's also true*, then it might be a good idea to take the domain over which the MP quantifies as one that does not include itself. (Or, rather, to say that it's only meaningful if the domain over which it is supposed to be quantifying fails to include itself, and that when we say that it is false, we're reading it in this way.) Of course, this isn’t to say that the person who mistakenly believes McCain to have won the election doesn’t believe about the statement of the MP itself that if that statement is true, then McCain won the election, or that they can’t or shouldn't express this belief of theirs. They will, however, have to construct a new statement to express it with. It’s not always possible to say everything you want to say all at once.

Similarly for the CS. If, for example, the gap theorist expresses their view by saying that “all statements are either true, false or neither” and we (as non-existence theorists) take this as a meaningful, truth-bearing statement (since we think it’s false), we won’t take the domain over which the gap theorist’s statement is quantifying to include itself. Thus, our counter-statement that “every sentence is either true or false” will be presumed to quantify over exactly the same domain, and we’ve succeeded in disagreeing with each other. Moreover, when someone asks about the truth-status of the sentence we constructed to express our disagreement with the gap theorist, we can attribute truth to it, and when someone asks about the truth-status of the sentence attributing truth to that sentence, we can attribute truth to it, and so on. Again, expressive power is unlimited, but we can’t necessarily express all of our relevant beliefs all at once. Of course, as we've seen, we have excellent independent reasons to think that we'll often be unable to do so.

Now, note that I’m not actually asserting the non-existence solution here, or defending it against the multitude of other obvious objections that one could make to it. My point is a much narrower one, which is this:

There’s no special objection to the non-existence solution lurking in a non-existence advocate uttering the sentence “every sentence is true or false (but not both),” or in the fact that they are using that sentence to express their disagreement with gap and glut theorists despite the fact that it doesn't quantify over itself. Given that they wouldn’t be asserting it if they didn’t take it to be true, we have really excellent third person justification for believing that their attitude to the truth-status of that sentence isn’t an exception to the general pattern of their attitudes about the truth-status of every other sentence. However, this extra belief that we should so obviously attribute to the non-existence theorist who claims that "every sentence is either true or false (but not both)" isn't expressed by that very sentence. That’s OK. It doesn’t mean they can’t express that belief as well. It just means that they can’t say everything that they have to say all at once.

*....and, moreover, that every other false statement is also true, which could be shown with the production of exactly similar sentences for them.

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