Monday, April 4, 2011

Singapore Talk & Revenge Problems for Dialetheism

So I just bought a roundtrip ticket to Singapore.

Three weeks from tomorrow, I'll be delivering a talk at the Philosophy Department at the National University of Singapore (NUS) entitled "Liar Paradox II: Revenge of the Liar Paradox." In lieu of Part III of the ongoing series on the same subject, here's the abstract I sent them for the talk (and then a quick explanation of where I'm going with it):

Dialetheists like Graham Priest and JC Beall conclude from the Liar Paradox that sentences like “This sentence is not true” are fact both true and untrue, and that we must therefore revise our logic to accommodate the existence of true contradictions. Similarly, “paracomplete” theorists like Hartry Field avoid the contradiction posed by the Liar Paradox by rejecting one of the central elements of classical logic, the Law of the Excluded Middle. A more conservative solution starts from the claim that sentences that attempt to attribute truth or untruth to themselves are meaningless, and therefore simply not the kinds of things we can logically symbolize or apply truth talk to without committing a nonsensical category mistake. The most common objections to this move are (1) that the “meaninglessness solution” is refuted by the existence of “revenge paradoxes” like the one revolving around the sentence “This sentence is either false or meaningless”, and that (2) the sentences involved are so obviously meaningful that it’s just not possible to take seriously the claim that they’re literally meaningless in any ordinary sense, like “Blorks geblork” or “Colorless green ideas sleep furiously,” whereas the dialetheist and paracomplete approaches have the advantages that they (1*) make room for the perfectly obvious fact that, in any language with normal expressive resources, we can construct perfectly meaningful sentences that attribute untruth to themselves, and (2*) are immune to refutation by means of “revenge paradoxes.” I will argue that (1), (2), (1*) and (2*) are all completely wrong.


On (2)/(1), of course, see Part I and Part II of the series of posts I've been doing on that here. (And, of course, stay tuned for Part III on Wednesday!) To get a sense of what I'm talking about on (1)/(2*), see (on the paracomplete side) here. On the dialetheist side, the problem, as I see it, is this*:

The dialetheist wants to argue that Disjunctive Syllogism fails to be universally truth-preserving (given true contradictions), and so it cannot be used to infer triviality after the dialetheist has embraced the contradictions entailed by the various paradoxes.

What does it mean to fail to be universally truth-preserving?

Given logical orthodoxy, the obvious answer is that an inference fails to be universally truth preserving iff there are possibilities on which the premises are all true and the conclusion is false. The dialetheist, obviously, can't conceive of it that way. Given the possibility of true contradictions, we can have possibilities where the premises of an argument are all true and the conclusion is false....but also true. On dialetheist assumptions, the mere fact that we've inferred a false conclusion from true premises is insufficient to establish that the the inference form fails to be universally truth-preserving.

Graham Priest's solution is to conceive of failures-of-truth-preservation as cases where the premises all 'relate to truth' and the conclusion fails to do so. (For technical reasons of his own, he prefers to think of truth as a relation rather than a function--so sentences 'relate' to truth or 'relate' to falsehood or both--but that's not really relevant right now. Talk of sentences relating to truth can translate to talk of them being true with no loss of nuance relevant to our discussion here.) The problem is that, obviously, it's always possible to come up with a Liar that says of itself that it fails to relate to truth. If normal Liars establish the possibility of sentences being simultaneously true and false, these anti-dialetheist revenge Liars should equally well establish the possibility of sentences simultaneously relating to truth and failing to relate to truth. Given this, establishing the possibility of all of the premises of an instance of Disjunctive Syllogism relate to truth while the conclusion fails to relate to truth should be not a single bit more relevant to showing that it fails to be universally truth-preserving than establishing the possibility of all the premises of an instance of Disjunctive Syllogism being true while the conclusion is false.

More generally, it looks like for any Status S such that the dialetheist could try to turn to in order to say "an argument fails to be universally truth preserving iff the premises are all true and the conclusion has Status S", we can always construct a revenge Liar of the form "This sentence has Status S" that will, on dialetheist assumptions, establish that true sentences can also have Status S. Once we've established this, it's not clear why Status S is a better candidate for a definition of failures-of-truth-preservation than mere falsity.

Many people have noticed the possibility of such sentences, shredding up distinctions dialetheists seem to find important--e.g. between sentences that are false-and-also-true and those that are "just" false ("This sentence is just false and not true")--and felt that in some way this was a problem for the dialetheist position on the Liar. Graham Priest, at least, has a standard response, found in multiple books and papers. He'll analyze the Liar sentence in question, show how the contradiction is derived from it, show how triviality fails to be entailed by the contradiction in his favored system of paraconsistent logic, and make some variation of a quip about how his point was never to avoid contradictions, but merely to contain them.

The problem, if I'm right about this, is that dialetheism's revenge problems don't just deliver more contradictions, they tear down the bars dialetheists want to use to contain them.

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