My Take on the Liar Paradox (Part I of IV)

I've certainly spent enough time in the last few years taking pot shots at other people's approaches to the Liar. Now that I've finished the degree and all that, it's probably time to present my own in detail. Devastating objections go in the comment box. Have fun!



JC Beall calls his view of truth “transparent disquotationalism.” His idea is that the truth predicate is a “transparent device” introduced into our language in order to make certain sorts of generalizations, blind endorsements and so on. (For example, “everything the Pope says is true.”) The device is exhaustively defined by the two rules that govern its behavior—Capture (the inference from ‘P’ to ‘Tr(P)’) and Release (the inference from ‘Tr(P)’ to ‘P’). It is ‘transparent’ because it allows us to see through to the ‘base language’ facts we use it to describe. (The ‘base language’ here is the fragment of the language in which the truth predicate is not used. It may, of course, be mentioned, as it is in the base-language sentence which came just before this one.) In Beall’s story, despite the truth predicate being introduced into the language as a way of talking about base language facts, this isn’t the only thing it can be used to do. After all, while normal sentences in which ‘true’ is used can be converted into sentences in which it is not used by applications of Release, no such procedure is available to convert a Liar sentence like “this sentence is false” (or its Truth-Telling brother “this sentence is true”) into a base-language sentence.

Thus, Beall’s use of the phrase “transparent disquotationalism” is a bit misleading. For Beall, as we’ve just seen, transparency is a merely contingent feature of typical uses of ‘true,’ not something necessary or universal. There are failures of transparency, applications of the truth predicate that don’t function to let us see through to some base-language fact. He’d be better off calling his view “mostly transparent disquotationalism.”

This creates tricky problems for Beall when it comes to how to think about the truth or falsity of non-transparent sentences. In Spandrels of Truth, he rejects the possibility of truth-value gaps, which leaves him with three values—true, false, and both. Non-triviality dictates that Curry sentences like “if this sentence is true, everything is true” get the value “false”, and as a dialetheist, Beall takes standard Liar reasoning to guarantee that (non-Curry) paradoxical sentences get the value “both”, but what to do with all the non-paradoxical and also non-transparent sentences? Beall decrees that they’re all “both” as well, but gives no indication whatsoever of any sort of principled motivation for this above and beyond the fact that it’s simpler to assume that it’s the case.

In fact, it seems terribly difficult to see what sort of principled reason anyone could ever give for either asserting or denying that any such sentence is true. “’Snow is white’ is true” is true and “’snow is green’ is true” is false. Both of these facts are functions of the actual color of snow. Clearly, however, no base-language fact will come to the rescue of a “transparent disquotationalist” trying to figure out the truth-value of the sentence “this sentence is true,” nor can they use the sort of reasoning they use to assign “both” to Liar sentences. Moreover, throwing up one’s hands in agnostic confusion wouldn’t be a principled move here, since the issue here clearly has nothing to do with some limit of our epistemic access to any relevant information. What ingredient do we not have, such that if we had it, the truth-value of such sentences would become clear?
I think that where Beall goes wrong is in assuming truth to be merely typically transparent. Let’s think about a classic statement of disquotationalism, Quine’s “nothing above and beyond” principle. In In Pursuit of Truth, he says this:

“To ascribe truth to the sentence [‘Snow is white.’] is to ascribe whiteness to snow…Ascription of truth just cancels out the quotation marks. Truth is disquotation.”

So, for any base-language sentence P, “’P’ is true” and “P” have, literally, exactly the same meaning. No additional content whatsoever is added to P by the application of the truth predicate.

My proposal, which, for contrast to Beall’s formulation, we can call “reductive disquotationalism,” is that the truth predicate does not suddenly become more substantive or play by different rules when we construct a sentence like “this sentence is true.” If the application of the truth predicate to a sentence doesn’t add any content to a sentence, a sentence with no content other than the use of the truth predicate has no content at all. Consider Sentences 1-3:

Sentence 1: Sentence 2 is true.
Sentence 2: Sentence 3 is true.
Sentence 3: Snow is white.

Now, imagine an infinite chain of sentences where each sentence ascribes truth to the next one in the series:

Sentence 4: Sentence 5 is true.
Sentence 5: Sentence 6 is true.
Sentence 6: Sentence 7 is true.
…and so on into eternity.

If, by the Quinean principle just mentioned, the meaning of Sentence 1 is entirely inherited from the meaning of Sentence 3, with nothing left over, where does Sentence 4 get its meaning?

My answer would be that it doesn’t get one. There is no base-language sentence from which it could inherit its meaning, so it is literally meaningless. Similarly for a simple case like “this sentence is true.” And, of course, adding the word ‘not’ into the mix changes nothing. As the philosophers of the Vienna Circle were so fond of pointing out, the negation of nonsense is nonsense.

We can summarize reductive disquotationalism as the claim that all meaningful sentences in which the truth predicate (or, equivalently, the truth operator) is used can, in principle, be accurately paraphrased without the use of the truth predicate (or operator). That “in principle” is important, because, for example, we might not have enough information to supply the paraphrase (as in “what John just said is true” where we don’t remember what John said), or the paraphrase could be infinitely long. If one finds the “in principle” suspect, we can express the principle by saying that, for every meaningful sentence in which ‘true’ is used, there is a set of sentences such that (a) none of them use ‘true’ and (b) they collectively mean exactly the same thing as the ‘true’-using sentence which they paraphrase. They capture its entire content.

Thus, sentences like “this sentence is true” and “this sentence is false,” although the may seem meaningful, are actually meaningless. Nor is this some desperate ad hoc maneuver to save consistency. It’s a fairly-clear cut application of a principle (“nothing above and beyond”) which is widely held for independent reasons, which has obvious advantages in terms of simplicity and so on, and which is closely linked to principles held by some significant figures on the other side of the dialetheism debate (e.g. Beall). Happily, though, it allows us to save not only consistency but classical logic. No rules of inference are being sacrificed, no additional values are being added to truth-tables, nothing revisionary, in fact, is going at all, except that those with the initial intuition that such sentences are meaningful are shown, for non-question-begging reasons, to be mistaken.

Once, after sketching out this solution to a friendly acquaintance who works in the same area, he politely told me that it was intuitively appealing but he’d have to “wait to see the formal details.” This is very funny to me, as an illustration of how deeply entrenched the notion that solutions to the Liar need to be logically revisionary has become in the era of Kripke, Field and the rest. A solution that saves classical logic, non-regimented natural language, the ‘naïve theory of truth’ complete with Capture and Release, etc., is one that can’t have ‘formal details.’ There’s no new syntactic machinery to show off, to prove the consistency of, to play with and test the limits of and so on. Everything is being done on the semantic, ‘informal’ side. Which is not to say, of course, that there isn’t plenty of tricky detail-level work to be done, as we’ll see when we address the various revenge paradoxes…it’s just that this work is all happening on the ‘informal,’ semantic side.

Now, dear reader, you may be getting pretty frustrated at this point, impatiently wondering how I could be so confused as to think that it’s even possible for such sentences to be literally meaningless when ordinary, competent speakers of the language take themselves to grasp their meaning. I must be either talking nonsense or using ‘meaningless’ in some strange non-standard sense of the term, right? And anyway, surely meaningless sentences aren’t true, so how am I getting around the Strengethened Liar? And even if I had some way around that, surely the solution must collapse on contact with sentences like “this sentence is either false or meaningless”?

And even if some plausible way can be found around that, then, surely, the Universal Law Of Liar Paradox Solutions (ULOLPS) still dictates that, for any claimed consistent solution to the paradoxes, a ‘revenge paradox’ can be formulated using the clever new terminology of the solution. Perhaps, these days, Hartry Field has convinced a few people that the ULOLPS breaks down in the case of a few fantastically formally sophisticated solutions, much the same way that normal rules about velocity and simultaneity break down as we approach the speed of light, but even those who grant this remain largely convinced that the mighty ULOLPS remains in full force down here at the lower reaches where a few fools still try to reason about the paradoxes with nothing more than simple juggling of intuitions and counter-examples.

If you have any of these worries all I can say is, well, stay tuned for the next few posts!