Monday, March 1, 2010

The Rejection-Liar: A Revenge Paradox For Paradox-Solvers Who Make A Big Deal About Rejection

In recent literature about the semantic paradoxes, a big deal is made in certain quarters about the distinction between negation and rejection. For example, for reasons of revenge-paradox-avoidance, paracomplete theorists scrupulously speak of "rejecting" the claim that the Liar sentence is true and "rejecting" the claim that it is false, rather than asserting the negation of those claims, or asserting them to be neither true nor false or what have you.

In "Spandrels of Truth," JC Beall grants that "just true" can't express any information that "true" doesn't--"'just true' is just 'true'"--but suggests that saying of certain statements that they're "just true" (while technically devoid of any information about whether they're also false) signals, in a conversational implicature sort of way, the speaker's intention to reject the claim that the statements are false.

Now, all of this acceptance/rejection talk crucially trades on the assumption that--whatever one thinks about the relationship between truth and falsity--as a matter of psychological fact, it's impossible for an agent to simultaneously accept and reject a claim. Whatever one thinks about that claim, however, I think that the sort of use to which its put in literature about paradox only makes sense if one takes the point of acceptance/rejection talk not to be mere self-reports about the actual psychological states of the paradox-solver making the claim, but a normative claim. For a paracomplete logician, claims about the truth of paradoxical sentences are the kinds of things that, rationally, we should reject. For the non-trivialist dialetheist struggling with the difficulties with "just true," the point is that sentences like "it's not true that 'grass is green'" are the sorts of things that one should reject.

When, in "Saving Truth From Paradox," Hartry Field talks about rejecting infinite chains of Excluded Middle instances originating in problematic sentences, he presumably shouldn't be taken as self-reporting his psychological atittudes towards each and every member of the infinite series. Even as a dispositional thing, it seems ludicrously implausible on that level. He couldn't hear a complicated, not-obviously-a-member member of that infinite series and be tricked or slip up and take the wrong atittude? What about members too long to be processed by human brains?

So, if I'm right about the exegetical issue (at least as a matter of charitable interpretation), the dominant assumption of those who throw around acceptance/rejection talk to help them navigate their way around paradoxes is that there's not just not overlap between the claims one does accept and the claims one does reject (at a given time), but also no overlap between the claims one should accept and the claims one should reject.

Presumably, everyone thinks that we should accept claims that are (just) true, and dialetheists think we should also accept claims that are both true and false. (We should accept, for example, that the Russell Set is a member of itself.) Everyone thinks we should reject claims that are (just) false, and paracomplete theorists thinks we should also reject claims that...well, the subject of puzzling out exactly what we can say about the claims they go into the "to be rejected" slot on their account is an immensely complicated one, but let's just say that there are some claims about which assert that they're false (although they'd certainly put their truth-value at less than 1) that they want to reject, and leave it at that.

Taking all that into account, I suggest the following as a revenge problem for paradox-solvers who take (correct) acceptance and (correct) rejection to be mutually exclusive. Provided that they take Liar sentences to be meaningful, standard Liar reasoning to be correct, etc., I'm hard-pressed to see what a defender of the mutual-exclusivity claim could say about this.

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Let’s call the following sentence the Rejection-Liar:

“This sentence would be rejected by someone who accepts all true sentences, who rejects all sentences he doesn’t accept, and who doesn’t act on any rules mandating acceptance or rejection except for the ones listed in this sentence.”

If the Rejection-Liar is true, then it’s both true and a sentence that would be rejected by someone who accepts all true sentences, hence it’s a sentence that would both be rejected and accepted by someone with absolutely perfectly rational acceptance/rejection behavior.

If the Rejection-Liar is both true and false, then it’s both true and a sentence that would be rejected by someone who accepts all true sentences, hence it’s a sentence that would both be rejected and accepted by someone with absolutely perfectly rational acceptance/rejection behavior.

If the Rejection-Liar is (just) false, then it would be rejected by someone who accepts all true sentences, who rejects all sentences he doesn’t accept, and who doesn’t act on any rules mandating acceptance or rejection except for those two, which is exactly what the Rejection-Liar says, so it’s true, hence it’s a sentence that would both be rejected and accepted by someone with absolutely perfectly rational acceptance/rejection behavior.

If the Rejection-Liar is neither true nor false, then it would be rejected by someone who accepts all true sentences, who rejects all sentences he doesn’t accept, and who doesn’t act on any rules mandating acceptance or rejection except for those two, which is exactly what it says, so it’s true, hence it’s a sentence that would both be rejected and accepted by someone with absolutely perfectly rational acceptance/rejection behavior.

2 comments:

lumpy pea coat said...

You said:

"there's not just not overlap between the claims one does accept and the claims one does reject (at a given time), but also no overlap between the claims one should accept and the claims one should reject."

Graham Priest denies the exclusiveness (no overlap) of what one ought to accept/reject. See "Doubt Truth to be a Liar" (sections 6.5 and 6.6) where he grants we have conflicting rational obligations (e.g. toward a strengthened liar) on plausible assumptions (that he accepts) concerning what we ought to accept/reject. That's why he has to argue that the actual psychological states of accepting and rejecting a given proposition are exclusive. It supports his claim that, e.g., dialetheists can express disagreement with others. (I imagine he grants that assertions (the speech acts) express acceptances (the propositional attitudes) and denials express rejections.)

Ben said...

Lumpy,

Right, in my post I granted that the actual psychological states don't overlap. I argue, however, that this doesn't much help the dialetheist--the purposes for which Beall in particular wants to apply rejection simply won't be served by psychological self-reports as opposed to assertions about what one *should* reject.

In an earlier stage of his career (as of the first edition of In Contradiction) Priest was on Beall's side here, but as you note, he's switched positions (in the discussion in Doubt Truth To Be A Liar and in the autocommentary on the second edition of In Contradiction). Unfortunately, he gives no indication of where that leaves him in terms of e.g. expressing the distinction between moderate dialetheism and trivialism, given that "just false" won't get the job done.