I've got a cold and am running a fever. Posting on the interwebs is contra-indicated by such a condition. Nonetheless, in my addled state I'm haunted by the following.
Assuming Trivalence (that every meaningful assertion is determinately either true, false, or neither), how does neither iterate?
To say that "P is true" is neither true nor false (NTP) is equivalent to saying that P is neither true nor false (NP).
To say that "P is false" is neither true nor false (NFP) is also equivalent to saying that P is neither true nor false (if P were false, then "P is false" would be true; if P were true, then "P is false" would be false; so P must be neither).
But what does it say about P to say that "P is neither true nor false" is neither true nor false (NNP)? I can't make sense of this, and I'm sure I'm goofing something up. Here's an argument that the notion of NNP is incoherent.
Assume that P is such that NNP holds. By trivalence P is true, false, or neither. Assume P is neither (NP). Then it is true that P is neither (TNP), but this contradicts NNP. So our assumption is wrong, and we know that from NNP it follows that P is not neither, which by trivalence entails that P is either true or false.
But if P is either true or false then we know that P is not neither true nor false which contradicts the claim that NNP.
I'm pretty sure that here last step is probably wrong, because asserting that something is not the case is the assertion that it is not true, which is consistent with it being neither.
But this still doesn't help me make sense of what is being affirmed about a sentence when we assert that the claim that it is neither true nor false is itself neither true nor false. Is it just to assert that P is either true or false?
I realize the above could be the fever. I won't be embarrassed if someone points out an obvious flaw in the above reasoning. I would be relieved. Especially if the pointing out also explained just what is going on with NNP (in particular, what this tells us about P in three valued logics).
I've got two natural deduction versions of Priest's take on Russell's paradox in a doc file HERE. The first one uses the restricted Inclosure Schema and the second uses the full Inclosure Schema (so that one can craft derivations of the semantic paradoxes with structurally analogous proofs). I've also got a proof of Berry's paradox.
I'm going to run through all the paradoxes of Priest's great 1994 Mind paper ("The Structure of the Paradoxes of Self Reference") and then use that as a resource to really closely read Beyond the Limits of Thought.
Better logicians than me (and all logicians proper are better logicians than me) almost always think it's silly to model arguments to this level of hyperspecification, but it serves me well. Sometimes you learn interesting things that everybody else has missed, and it also makes it easier for non-logicians (like myself) to follow what is going on. A couple of minor interesting points (concerning Emil Badici's criticism and also the slingshot argument) have already arisen which are noted in the document.
I just ordered another copy of Graham Priest's Beyond the Limits of Thought. He's got a couple of pages on the ontological proof and the conceivability of God. I don't remember what he says though.
His Inclosure Schema clearly works to model the argument for the paradox of the stone. It should work for the ontological proof I think if you start by assuming there exists a (possibly empty) set of entities that are such that no entity is greater than those entities. Of course if you model it that way the conclusion might be not that such an entity exists but that we were wrong to think we could form a set of that sort! That is, we don't have a concept of that which no greater can be conceived after all. For all I know, this is what Priest concludes. I read his book way too hastily years ago. I'm going to read it really carefully this time around.
Neil Tennant in conversation once showed that if the ontological proof is valid then you can in parallel fashion prove that the empty set ("that than which no lesser can be conceived") does not exist. I don't think he published this though.
I'll do a post on it this week which actually explains how the Inclosure Schema models so many different paradoxes on Russell's paradox and which at least models the proof for the paradox of the stone, but if anyone knows of anyone who (pre-Priest) has connected up Russell's Paradox to either the paradox of the stone or the ontological argument, or who has (post-Priest) either explicitly applied Priest's scheme to them or (assuming Priest does this in the book) tries to refute him, please let me know.
This may be catastrophically simple-minded and lazy to boot, but I'm just not sure that dialetheism (the view that there are true contradictions) should be thought of as leading to logical revision. [Before reading any more, I realize that I'm way out of date on all of this, and that there is an extant literature surrounding "classical recapture" that is relevant and that I haven't addressed in what follows.]
The argument that dialetheism entails revising classical inference simply notes that in normal classical logic a contradiction entails everything (ex falso quodlibet, P, ~P |- Q). This is actually a really hard principle to give up because any logic that entails disjunctive syllogism entails ex falso quodlibet [Assume you've got a contradiction P and ~P. From P by disjunction introduction you've got (P v Q) and ~P. But then by disjunctive syllogism you can conclude Q. So from any P and ~P, any Q follows.] So if I want to say with Graham Priest that paradoxical sentences such as "I am not true" are both true and false, and I don't want to be committed to the truth of everything, I better adopt a paraconsistent logic that does not license ex falso quodlibet.
But couldn't a true contradiction work as a logical/pragmatic stop-sign? The thought would be that if you derive a contradiction within something assumed for further discharge such as negation introduction or disjunction elimination you go about your business and continue to use classical logic, but if you derive it outside of these contexts it tells you to stop using the logic.
This works really well for true contradictions that are generated outside of paradoxes involving self-reference (I don't know any other philosophers besides me, in my essay in The Law of Non-Contradiction, who have argued for such). Say that our only notion of ethical truth is truth-relative-to-a-moral-tradition such as in MacIntyre's historicist days (and that the T-Schema is valid for each relativized truth predicate). These traditions will each be independently plausible, but inconsistent with one another. Given the T schema you can generate contradictions from the traditions' joint inconsistency.
And people are pretty good at bouncing back and forth between different moral theories, combining them when it's safe. But sometimes we mess up and apply both in situations where they contradict one another (maybe think of Kantian and Millian ethics on whether a given lie is morally permissible). If the best we can do at the limit of reflective equilibrium is getting a set of good mutually inconsistent (but each internally consistent) theories (something I think holds for ethical, philosophical, and aesthetic theories generally), then a true contradiction is just a place where two of the theories at the limit of reflective equilibrium disagree.
But at least that kind of true contradiction should not lead you to abandon classical logic. In subvaluational "glut semantics" a sentence is supertrue in a model if there is at least one valuation in that model that makes it true. True contradictions happen when a sentence is made true on one valuation and false on another.
On the account of entailment where you say gamma entails alpha if and only if alpha is subtrue when all of gamma are you get a horrifically messy non-classical entailment relation (in Williamson's book on vagueness this is called "wide entailment," albeit there in respect to gap supervaluational semantics for vagueness, where a sentence is true in a model if it is true at *all* relevant interpretations). But such semantics also give you the option of just keeping classical entailment, gamma entails alpha if and only if there is no valuation (remember that models are sets of such valuations) that makes all of gamma true and alpha false. Williamson argues that this form of narrow entailment should not the relevant one for vagueness for the supervaluationist, but I remember finding his arguments here less than persuasive. One would need to examine them in this context.
But the model is compelling. Each one of the interpretations making up the glut semantics model is analogous to a correct theory in the limit of investigation after reflective equilibrium is considered. And narrow entailment can be understood as legitimizing classical logic as long as we in some sense stay within one of the correct theories in the limit of investigation after reflective equilibrium is considered. Since we're in the limit of investigation all a true contradiction tells you is that your bringing together of two of the theories into the same context has run into trouble. It just tells you not to do that in this context.
The big if is whether such a view can be argued to apply to self referential paradoxes. I'm not at all sure that it can. Consider the paradox of the stone. What would the analog to the two interpretations be? I'm going to look at Priest's brilliant 1994 Mind paper "The Structure of the Paradoxes of Self-Reference" (still brilliant even if one agrees with the conclusion of Badici's "The Liar Paradox and the Inclosure Schema," Australasian, 86, 2008) really carefully over the next two weeks and see if there is any hope at all for the idea true contradictions generated from paradoxes can be argued to be consistent with classical logic.
I'm not that optimistic, because you'd have to say something like a new subvaluational interpretation is generated every time Priest's Transcendence is applied. To the extent that it might work, one could probably piggy back on extant attempted solutions to the paradoxes that involve indexing in various ways. But that literature is a mess and one of the primary benefits of dialetheism is that you were supposed to be able to circumvent it (albeit at the cost of a messy logic).