Graham Priest in Yesterday's New York Times

So I was delighted to see my dissertation topic in yesterday's New York Times, in the form of Graham Priest's article Paradoxical Truth.

Further comments to follow.

Political Chaospet Comic

Ryan's write-up of our post-election Facebook discussion is awesome, and I say that even though I was pretty clearly on the losing end of the argument.

Definite Descriptions and Problems about the Philosophy of Time

The relational theory of time says that a moment is just a bundle of co-occuring events. 4:20 PM on April 20th of the year 420 B.C.E. was nothing above and beyond the events that occurred then. The substantival theory says that times are something like independently existing abstract entities that happen to contain various events but could have contained entirely different events, or perhaps even no events at all.

(The suggestion that there could be "empty times," where nothing changed and nothing happened is incoherent on the relational view.)

The standard Benaceraff-style objections to abstract objects would all seem to apply to substantival moments. (If they all disappeared tomorrow, how would we ever know the difference? Given that a human being having an intuition is surely ultimately a certain sort of neurological event, explainable in terms of some mixture of biological hard-wiring, socialization, and so on, and that an abstract object could play no role in the causal chains leading up to such events, why on earth should we think that our intuitions track its true properties?, and so on.) Given that, what possible argument could there be for the substantival view?

Quentin Smith has argued in many places for the substantival view on the grounds that some counterfactual statements about time seem to be true. If I arrived on time to the meeting at 4:20 PM, it still seems to be true that, for example, if I had gotten pulled over by a traffic cop on my way to the meeting, I would have been late. Put differently, if I had been pulled over by a traffic cop on my way to the meeting, the moment 4:20 PM would not have included the event of my showing up at the meeting. It would have, indeed, included entirely different events, like people asking each other why I wasn't there. Intuitively, all of this would have happened at 4:20. But wait! If 4:20 just *is* a set of co-occurring events, then it wouldn't have existed in the possible world where I didn't show up on time to the meeting. For some such counterfactuals to be true, it seems, moments need to have a separate existence above and beyond the things that happened at them.

However unattractive the idea of substantival moments might be to me, this has always seemed to me like a pretty tricky problem for the relational theory. Now I'm not so sure.

Notice, though, that if one sees time-terms ("4:20," "yesterday afternoon" and so on) as shorthand definite descriptions, like "that bundle of events which includes Event E" or even "that bundle of events that's related in such-and-such order to the bundle of events that includes Event E," the problem goes away. Smith, of course, who's written extensively on phil of language, realizes that this is a way out. I don't have the link handy--his website seems to be down at the moment--but in a short article on this a while back, he presents the objection as a dilemma between rejecting the relational theory of time and rejecting the "New Theory of Reference" (i.e. the Kripkean* theory of proper names as rigid designators). To become a full-fledged argument for a substantival view, one needs compelling arguments for the Kripkean view and against the old Russellian descriptivist view. Smith, of course, like a great many people, takes it that we have such comeplling arguments.

Fair enough, but, the more I think about this, the less convinced I am that the dilemma really exists. There's only a dilemma if one thinks that *all* proper names and proper-name-like terms rigidly designate, and one can certainly be a Kripkean about reference without making that sweeping universal claim. For example, Kripke himself never claimed it and has always explicitly disowned it, being careful to emphasize all along that he was making a general claim about most normal uses of most normal proper names and nothing more.

Of course, Kripke could be mistaken in holding back from the universal claim, but there are good reasons, quite independent of the Phil of Time issues under discussion, to think that he isn't. For one thing, taking names that don't pick out any existing object--e.g. "Santa Clause"--as rigid designators (of what?) raises all kinds of sticky problems. Of course, that itself raises all kinds of deeply controversial philosophical issues, but more banal and uncontroversial examples are plentiful. To pick an obvious one (not original to me), "Jack the Ripper" is a lot more like an archetypal ordinary proper name than "4:20 PM" is, but "Jack the Ripper" pretty clearly really is a shorthand definite description, along the lines of "whoever murdered and mutilated the bodies of those prostitutes in London." Given (a) the incompatibility of the package of the relational view and the assumption that time words rigidly designate on the one hand and the apparent truth of some temporal counterfactuals on the other hand, and (b) the obvious epistemic and ontological parsimony-based objects to the substantival view, it seems to me that (c) even if one adopts a rigid designator view of most ordinary proper names--"Quentin Smith", "Saul Kripke", etc.--it's a mistake to extend that analysis to time terms.









*Smith himself probably wouldn't call it that. He has heterodox views on history of the philosophy of language that would lead him to think of it as the Marcusian theory--he and Scott Soames have killed a lot of trees with their essays arguing back and forth on this issue--but we can put that historical dispute to one side for the moment.

The content of the body of this post is true.

The content of the title of this post is false.

XKCD

Funny

....particularly the roll-over.

Reading the Principia Mathematica

....I'm remembering that, every time I go a while without reading much Russell, I forget just how smart Russell was. (The classic example is that most people don't notice that Russell anticipated and responded to the Gettier problem as far back as Problems of Philosophy.) In any case, predictably, given my interests, the most interesting part of the early chapters, for me, is his take on the Liar. People often tend to discuss his (and Whitehead's) type theory only as a solution to the set-theoretic paradoxes, but even if one rejects it for those purposes, their type-theoretic solution to the semantic paradoxes remains independently interesting.

I think this solution is actually fairly interesting. I don't think he's right about the roots of the problem, but the solution has a lot going for it. (In some way, in fact, I find it a whole lot more attractive and plausible than some still-popular approaches to the Liar, like Kripke's.) It's extremely "unified," solving a lot of other problems at the same time, it doesn't require the rejection of any intuitively compelling logical principles, or any instances of the T-Schema (or the Capture and Release rules), it doesn't require any weird principles about certain sentences not being "contructable," it fits everything together in a tight, coherent framework, and it has interesting consequences for such apparently distant subjects as epistemic skepticism.

Again, all that said, I don't entirely buy it.

....but do expect a series of posts on Russell's Take On The Liar Paradox in the next week or two.

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.