How JC Beall's Version of Dialetheism Tries (And Fails) To Escape Triviality (And How the Account Could Be Fixed)

I've had this written up for a different purpose for a while now, so I thought I might as well stick it up here.

A couple of quick points before getting into the meat of this:

Some posts here (e.g. the anti-theist polemics, or last week's post about Zizek) are intended to be accessible to readers with a fairly minimal philosophical background. Others (e.g. my post a couple weeks ago about revenge paradoxes for paracomplete solutions to the Liar Paradox), engaging narrower issues, tend to assume intimate knowledge of the debates I'm most interested in (e.g. dialetheism, paracompleteness, logical pluralism, and so on)--in the case of those sorts of posts, I'm generally trying to make a quick point about something fairly specific, and starting with a general overview of the relevant debates each time would quickly get repetitive, and would probably be annoying for those readers most likely to be interested in the quick technical point. Sometimes I might not always clear enough about flagging which are which.

In any case, because of the nature of the specific objection to Beall I'm making here, this post is somewhere in between these two categories, but it's definitely closer to the latter than it is to the former. If that's not your cup of tea, be forewarned.

A more philosophically substantive note:

Also because of the specific nature of the criticism being lobbed here, I'm willing to just follow Beall in a lot of his philosophical machinery for these purposes. Some of that machinery (a generally disquotationalist attitude toward truth complete with unrestricted Capture and Release, a commitment to the revisability of logic, etc.) is stuff I agree with him on in any case. On the other hand, there are some important points of disagreement you could miss from all of this. One is that, at least in Spandrels of Truth, he seems to take the laws of logic as something like deep rules of language--his precise conception is often a bit unclear--while I take a more traditional approach, seeing them as laws of universal truth-preservation, meaning that one's take on which laws of logic are valid ultimately amounts to something like an overall theory of reality in general (at the level of generality and abstraction at which logical laws operate). Another important difference I have with him is that I'm deeply skeptical that one of the key distinctions that Beall makes a big fuss about when it comes to truth--"natural" properties vs. "constructed" properties--tracks any sort of real distinction whatsoever, or illuminates much of anything.

In any case, putting those larger issues aside, let's get to the part about Beall's trivialism problem:

I. The Shape of the Problem

In Spandrels of Truth, JC Beall endorses a dialetheist solution to the semantic paradoxes on the basis of a “transparent disquotationalist” account of truth. According to him, truth always plays Capture and Release. (Capture is the inference from P to Tr(P) and Release is the inference from Tr(P) to P. Given that it satisfies these intersubstitutivity rules, the truth predicate is “transparent,” letting us see through to the claims to which we are attributing truth. He signals this by continually referring to truth as "ttruth" for "transparent truth.") On the basis of familiar reasoning, this leads him to contradictions in the face of sentences like S1.

S1: This sentence is not true.

Beall is willing to pay the price of accepting true contradictions in exchange for unrestricted Capture and Release. Indeed, he regards this price as unavoidable, given his account of truth.

"Spandrels of x are inevitable, and frequently unintended, by-products of introducing x into some environment… Language has its own spandrels. This is particularly the case when a given bit of the language is introduced for a particular role, much like ttruth. The guiding metaphor, as above, has us introducing ‘ttrue’ not to name some property in the world but, rather, to enable generalizations about the world and its features….But ‘ttrue’ is a predicate, and introducing it into the grammatical environment of English yields spandrels, unintended byproducts of the device." (p. 5)

One might wonder, however, whether this result actually compromises the ‘transparency’ of his truth predicate.

"Given that the device [i.e. the truth predicate] is constructed to be entirely transparent, one expects a sort of supervenience to hold. In particular, one expects ttruth to supervene on the base-language facts—the base-language ttruths.

"The expectation of such supervenience, I think, is natural. If one were to insist on such supervenience across the board, then one would need to reject that some of the given spandrels are gluts, since such sentences are ttrue without their ttruth ‘depending’ on base-language ttruths. But…insistence seems to me to be misplaced, at least given the going conception of truth. Why insist as much when the constructed device might yield spandrels that buck supervenience? That the base language serves as a supervenient base for ttruth makes sense, I admit, for those sentences in which ttruth is eliminable via the fundamental intersubstitutivity rules—for the ‘normal’ sentences. What, though, of the inevitable spandrels? …As far as I can see, there is no reason to think as much. If anything is to determine the ttruth or tfalsity of such sentences, it’s at most the overall logic (or rules) of the language. Extending the otherwise sensible supervenience constraint to such sentences seems, as I said, to be simply misplaced." (pp. 15-16)

It is important to note that Beall’s careful formulation that the truth-values of ttruth-ineliminable sentences are determined “at most by the overall logic” (p. 4) [emphasis added] papers over a sticky problem for his account. What could possibly determine the truth-value of a non-paradoxical spandrel like S2?

S2: This sentence is true.

Certainly, the truth-value of S2 can’t supervene on any base-language fact. Moreover, unlike S1, whose gluttiness is delivered by standard Liar reasoning, Beall’s overall logic doesn’t seem to deliver the result that S2 is true, or the result that it is false, much less the result that it is both true and false. One might think that, since S2’s truth-value is not determined by either of those sources, it simply lacks a truth-value. (Indeed, an earlier version of Beall’s version of dialetheism—e.g. in his 2005 paper "Transparent Disquotationalism" (in the collection Deflationism and Paradox—he did leave room for truth-value gaps.) Unfortunately, elsewhere in Spandrels of Truth, he rules out the possibility of gaps, declaring that “negation is exhaustive.” (p. 5)

What, then, can be done about cases like S2? Initially, Beall endorses a completely unified approach to all ttruth-ineliminable sentences.

"For present purposes, I shall follow the simplest approach: I treat all such sentences as gluts." (pp. 14-15)

This approach, however, would lead to triviality if it were applied to ‘Curry sentences’ like S3, i.e. conditionals whose antecedents say of the whole sentence that it is true.

S3: If this sentence is true, JC Beall is fourteen feet tall.

Beall solves Curry in the usual dialetheist way, with conditionals too weak to enable Contraction (the inference from "if A, then if A, then B" to "if A, then B.") His criteria for “suitable conditionals” are that they obey Modus Ponens, that they honor Identity (the universal truth of "if A, then A"), and that they do not deliver triviality. However, for obvious reasons, another component of his view is that such sentences are (just) false. “On my account, Curry sentences are false; I reject that they’re ttrue.” (p. 33) In light of this position, Beall qualifies his earlier simple approach.

"In Chapter 1, I noted my openness to an asymmetric treatment of such sentences (e.g., treating some ttruth-ineliminable sentences as gluts, some classically), but officially embraced the simple approach according to which all such sentences are gluts—transparently true with transparently true negations. This position remains, but only for the conditional-free fragment." (p. 34)

This is Beall’s final, considered position in Spandrels of Truth. Unfortunately, while it may take care of Curry, it can be shown that this position still generates triviality. Consider S4.

S4: This sentence is true, and the moon is made of green cheese.

S4 is ttruth-ineliminable. The truth predicate can’t be removed from it through application of the intersubstitutivity rules, since the first conjunct refers to the whole conjunction, not just the (ttruth-eliminable) second conjunct. As such, it follows from Beall’s policy of treating all ttruth-ineliminable sentences in the conditional-free fragment of his language as gluts that S5 is both true and false, which means that it is true, which means that the second conjunct is true and that the moon is made of green cheese.

II. The Persistence of the Problem

Having shown that Beall’s account (as stated) leads to triviality, one might wonder if this problem can be solved through some extension of the move Beall employs to get around Curry. Perhaps he could just say that all ttruth-ineliminable sentences in the conditional-free and conjunction-free fragment of his language are gluts, and that, like Curry sentences, ttruth-ineliminable conjunctions are (just) false.

Unfortunately, given S5, this won’t work.

S5: This sentence is false, and it is not true.

S5 fails to be conjunction-free, but Beall’s overall logic will still get us the result that it is a glut by means of familiar Liar reasoning. Fishing around for a principled, unified policy that could help Beall out of these difficulties, one might postulate that all ttruth-ineliminable complex statements with ttruth-eliminable atomic statements within them are (just) false, whereas all sentences that are ttruth-ineliminable “straight through” are gluts. One advantage of this approach would be that it would simultaneously save him from S4-generated trivialization and from Curry-generated trivialization, without having to make his current unqualified (and thus untenable) claim that Curry sentences are (just) false. To see why this claim is untenable, consider S6.

S6: If this sentence is true, then it is true.

Remember that one of Beall’s criteria for “suitable conditionals” is that they validate every instance of Identity. S6 is an instance of Identity. Thus, there is at least one Curry sentence whose truth-value is determined by Beall’s overall logic to be true. On the proposal we are considering for patching up Beall’s views, however, S6 is not a problem. If the only spandrels that get classified as glutty are those that are ttruth-ineliminable “straight through,” S6 is true, S5 is glutty, and a simple, unified policy prevents us from claiming that any triviality-generating conditionals or conjunctions are true.

Unfortunately, even this version of Beall’s account doesn’t work. After all, consider cases like S7 and S8.

S7: Either this sentence is false or it is meaningless.

S8: Either this sentence is false or Hitler won World War II.

Given the historical facts, and Beall’s assumption that the spandrels are meaningful, his overall logic will deliver the result that both S7 and S8 are gluts. Moreover, it would be strange if it failed to deliver that result, given the prominence of examples like S7 and S8 in dialetheist arguments against various consistent solutions to the semantic paradoxes.

So, to sum up, (a) Beall’s official position generates triviality, and (b) the most obvious ways of getting around this problem (i.e. the ones closest to Beall’s own approach to getting around Curry) fail to respect Beall’s foundational position that, when the overall logic delivers gluts, we shouldn’t quarrel with the result.

III. A Better Solution

Fortunately, there is a fairly simple, principled way to patch up Beall’s account. He could simply say that the mechanisms he lists for determining the truth-values of sentences in the passage quoted above—(i) supervenience on base-language facts, and (ii) the unaided results of the overall logic—are jointly exhaustive of the possible ways in which sentences can become true. Given this move, and Beall’s principle that “negation is exhaustive,” it follows that all sentences that don’t become true in one of these two ways are (just) false. Thus, S6 is made true by (ii), S1, S5, S7 and S8 are all made true (and false) by (ii), S2, S3 and S4 remain (just) false, and triviality is avoided.

Of course, even on this patched-up version of Beall’s account, the only reason the “overall logic” doesn’t generate triviality as a result of sentences like S3 (despite their falsity) is that traditional (contracting) conditionals have been rejected in favor of “suitable conditionals.” One might wonder why, if such otherwise unmotivated tinkering with the formal rules is an acceptable way of preserving non-triviality in the face of Curry, similar tinkering wouldn’t be an acceptable way of preserving consistency in the face of the Liar. (With even weaker conditionals, we could, for example, continue to accept that (1) is either true or false, continue to accept that if it’s true, it’s false, and continue to accept that if it’s false, it’s true, but reject the conclusion that it’s both true and false.) If this would be an acceptable solution to the Liar, we have no need to postulate true contradictions. If it wouldn’t be an acceptable solution to the Liar, one might wonder why it counts as an acceptable solution to Curry.

If one wondered about these things, I would wonder with them. In terms of Beall’s more immediate problem, however, assigning the truth-values in the way I suggest should plug the hole and save his account from triviality.