Wednesday, March 30, 2011

Some Objections to the Meaninglessness Solution to the Liar Paradox, Part II of IV

Imagine a world where the predicates 'is green' and 'is colored' were considered much more philosophically interesting than it is in the actual world, interesting enough that philosophers and logicians worried about what formal rules related these predicates. One fairly crushingly obvious rule about them would what we can call the G-out rule, allowing us to infer 'X is Colored' from any instance of 'X is Green.'

Now, imagine that there was one other big difference between that world and this world. In our world, the classical example of a syntactically correct but clearly meaningless sentence is C:

C: "Colorless green ideas sleep furiously."

In the imaginary world, this sentence is treated rather different. The humans in this world have made contact with unfathomably intelligent alien entity, capable of speaking English (perhaps with the aid of a universal translator). Every time the entity has been asked a question, and it has deigned to answer, its answer has been proven correct. Sometimes it has taken humans many years, and full-fledged scientific revolutions, to understand *how* what the entity said could have been true, but in the end, there's never been any room for serious doubt. The entity has never once been shown to have (or even been widely suspsect to have) misunderstood a question. At some point, for some strange reason, someone asks the entity about C and it points to the paper where the questioner has written down C and says "this is true."

As often happens when the alien entity says something interesting, ripples of immediate change go through entire fields of study. A few philosophers think that in this case the entity got confused and make a strange sort of category mistake--after all, as in our world, any position, no matter how odd, always has a few philosophical backers--but there's a wide consensus now that C must be true (and therefore meaningful) after all. Almost immediately, some clever theorists notice that this revelation has created a new problem, which they call the "Greenness Paradox." Pretty soon, the dialetheists in this world seize on the Greenness Paradox as an argument for the existence of true contradictions. Here's how it goes:

Start with the formalization of C, given classical logic and orthodox assumptions about how to read the existential quantifier:

1: "There exists an X such that X is green and it is not the case that X is colored and X is an idea and X sleeps furiously."

It clearly follows from 1 that:

2: "There exists an X such that X is green and it is not the case that X is colored."

Apply existential instantiation to 2 to get:

3: "P is green and it is not the case that P is colored."

Apply conjunction elimination to 3 to get:

4: "P is green."

Apply our G-out rule to 4 to get:

5: "P is colored."

Apply conjunction elimination to 3 once again to get:

6: "It is not the case that P is colored."

Apply conjunction addition to 5 & 6 and we get:

7: "P is colored and it is not the case that P is colored."

.....which, the dialetheists of this world argue, is a true contradiction! Viola.

Of course, the dialetheist take on the Greenness Paradox isn't the only game in town. For example, one would imagine that a more conservative solution to the Greenness Paradox would be to deny "the naive theory of greenness" and to restrict G-out in some way. An obvious non-classical but non-dialetheist solution would be to deny that the existential quantifier is ontologically loaded after all. Proponents of this Meinongian solution to the Greenness Paradox would argue that some things can be true of colorless green things ideas without there being colorless green ideas. The Hofweber of this world will argue that, while the existential quantifier is ontologically loaded, and classical logic and the naive theory of greenness are true, and we shouldn't be so arrogant as to reject the superior wisdom of the alien entity by denying C, truth preservation should be understood in a generic rather than universal way. Just as "bears are dangerous" can be true without every bear being dangerous, "valid logical inferences are truth preserving" can be true even if not every valid inference from a true premise preserves truth. The Greenness Paradox, Bizarro-Hofweber would argue, shows us that the universal reading of the notion of truth preservation represents an airy "ideal of validity" that has an obvious appeal, but that the paradox falsifies the ideal.

At this point, it should be pretty easy to come up with a variety of other such philosophically sophisticated solutions to the paradox and to have a pretty good idea of how the argument between proponents of various competing solutions would proceed. Inevitably, some solutions would seem to work better than others, to contain hidden inconsistencies, and so on, and everyone, including the few extreme skeptics who didn't think the unfathomably intelligent alien entity at the source of all this was on the level when it uttered the words "this is true" about C, would be able to do so perfectly easily. "You try to solve it by saying that the colorless, furiously sleeping ideas are red rather than green, but red things are just as colored as green ideas, so you haven't gotten around the original problem." "You forgot a negation sign in Step 5. Once you add it in, you can see that a contradiction is entailed later, when you say...." Etc., etc., etc.

Now, imagine that they were right, and that the entity actually had the same take on C as Chomsky and most of the rest of us citizens of the actual world. He was simply messing with the puny humans out of boredom by pointing to a meaningless sentence and saying the words "this is true." He'd never done this before--he'd always given good and helpful responses to the rest of their inane little queries--but there's a first time for everything. Certainly, from the perspective of the humans, it's understandable that they would never catch on. Having been shown so many amazing things by the entity--remember, scientific revolutions are sparked off by it's statements on a regular basis--it seems utterly plausible to them that a sentence they thought was definitely meaningless actually has a meaning that their puny monkey minds cannot fully grasp. From there, given the function of phrases like 'is green' and 'is colored' in meaningful sentences (G-out is clearly a good rule), the equivalence of 'colorless' with the negation of 'colored', and the ways that we translate into logical lingo sentences of the form "X-ish things do Y", the apparent possibility of reasoning from a contradiction to Y.

Now, assuming that we and the imaginary aliens are right about C, we now have a problem. It is, in fact, the same problem we ended Part I with. We know that nothing "follows" from C. It's meaningless, not the kind of thing we can logically symbolize or apply truth-talk to without committing a nonsensical category mistake. The idea that anything really "follows" from C is deeply confused, like saying that something 'follows' from a string of nonsense syllables, or a bit of burning candle wax. Somehow, though, we seem to be perfectly capable of 'reasoning' about it, as we've been doing for the last few paragraphs.

In the beginning of Part I, I argued that the diquotationalist "nothing above and beyond" principle about truth--"to say that 'P' is true is to say nothing above and beyond P", or to put it differently, "to prefix a quoted sentence with the words 'it is true that' has the semantic effect of simply removing the quotation marks" (the claim, remember, from which the word "disquotationalism" is derived)-is best explained by a general view that the truth predicate/operator is only parasitically meaningful. Of course, the original claim is about sentences that ascribe truth to sentences quoted within them and my claim broadens this to all ascriptions of truth, but I would argue that the former claim, in the absence of the latter, has some awkward consequences. For example, consider the following three sentences*:

(11) "It's true that 'snow is white.'"
(12) Sentence (13) is true.
(13) Snow is white.

I'd submit that there's something a bit strange about arguing that (11) and (12) have distinct meanings. If one asserts meaning-parasiticalness for sentences that ascribe truth to an internally quoted sentence and rejects meaning-parasiticalness for sentences that ascribe truth to other sentences in other ways, one has to explain what substantive difference the *method* of applying the predicate/operator to the claim to which truth is being ascribed makes. Moreover, the obvious explanations of *why* the "nothing above and beyond" principle would be true--most obviously, general philosophical stories like "the word 'true' doesn't pick out some substantive feature of the world, but rather functions as a time-saving way of saying other things, especially useful for cases where we aren't entirely sure *what* we're saying (i.e. blind endorsements)"--would seem to apply equally well, to sentences like (11), sentences like (12), to sentences like "everything John just said is true", to the one-word exclamation "true!" uttered in response to something one's friend has just said, and so on. The syntactic form the truth-ascription takes seems to make no difference. All sentences that do nothing but ascribe truth to a sentence inherit their meanings from the meanings of the sentences to which they ascribe truth. If a sentence S1 tries to ascribe truth to another sentence S2 that has no meaning, S1 will have no meaning either. It has nowhere to get it.

A happy consequence of this view is that, given some other plausible assumptions (e.g. that adding the word "not" to a meaningless sentence does not convert it into a meaningful one), it entails that sentences like "this sentence is not true" are meaningless. This lets us solve the Liar Paradox without having to give up on "the naive theory of truth"--a unitary truth predicate obeying all the standard rules about truth, etc.--or the unrestricted power of classical logic, or much of anything else except many people's initial intuition that the sentences involved are meaningless. At the end of Part I, though, we confronted what sounds like a serious problem:

Someone like me, who says that Liars are meaningless, has presumably been convinced of it by prolonged reflection on the paradox. In the course of this, they've sifted through various possible diagnoses of the sentences in question, thinking about consequences of various approaches, objections to failed solutions and so on. Right? Well, then, wait a damn second. Doesn't all of this involve reasoning about what does and doesn't follow from these supposedly meaningless sentences, in conjunction with various other claims. For example, to embrace the meaninglessness analysis is to reject the analysis that says that Liar sentences are meaningful but that they don't express propositions. Presumably, in explaining why the meaninglessness analysis is superior, its partisans want to bring up "revenge paradoxes" like (8). (At any rate, I certainly want to bring it up!)

(8) The sentence marked as (8) does not express a true proposition.

If (8) doesn't express a proposition, it doesn't express a true one, just as if a cat isn't a dog, it isn't a black dog. And anyone who endorsed the meaningful-but-not-expressing-a-proposition analysis presumably doesn't think a sentence can be true without expressing a true proposition--after all, if truth can exist without propositions, why clutter one's ontology with them? Thus, the solution under consideration collapses into contradiction.

Now, while I tend to lean skeptical on the subject, I'm officially agnostic about the existence of propositions. I take its neutrality on this topic to be a big selling point of my preferred approach. (For the sake of simplicity, I usually talk about "sentences", but wherever I talk about "sentences" being true or false, an enthusiast for propositions can always mentally subsitute some phrase about the propositions expressed by those sentences being true or false...and, of course, presumably, if propositions exist at all, only meaningful sentences can express them, so if I'm right that Liars are meaningless, it follows that they don't express propositions any more than bits of burning candlewax express propositions.) If, however, I abandoned my agnosticism in favor of a full-throated embrace of propositions, I'd presumably be forced to classify (8) as meaningless as well. (If I abandoned it in the opposite direction, matters would be quite different. After all, if there are no such things as propositions, it's true of every sentence that it doesn't express one!) Certainly, I view more common revenge paradoxes, like (9):

(9) The sentence marked as (9) has some status other than 'true.'

....or the familiar anti-dialetheist revenge paradox (10):

(10) This sentence is just false, rather than being both true and false. being meaningless, and still deploy them against the approaches to the paradoxes that I reject, using standard Liar reasoning, like everyone else does. Doesn't the fact that I'm able to play this game as well as anyone else, that we all understand and can use the rules against each other, proof that the sentences are meaningful, that, after all, we all understand what they mean?

Now, there's a lot more to be said about all this--particularly about the thorny question of what sort of mistake someone can be accused of when they 'reason' about something meaningless in a 'bad' way and 'contradict' themselves about it, above and beyond the original sin of treating the something in question as if it were meaningful--but I take the example at the beginning of this post to pretty definitively answer the question I ended the last post with in the negative. Someone who (as we would all agree here in the actual world, correctly) characterized C as meaningless would be faced with precisely the same problem that a pardadox-solver who takes the Liar to be meaningless is faced with in our world. Although it's still somewhat unclear *why* the objection doesn't work in either case--we'll say more about that--it's failure in the closely parallel imaginary Greenness Paradox case would seem to show that it fails when it comes to the actual Liar Paradox as well.

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