Dialetheists, notably Graham Priest and, apparently, Walt Whitman, contend we may, under certain circumstances, ascribe truth to contradictions. A dialetheia is by definition a proposition, p, that when conjoined to its negation, ~p, produces a true evaluation, such that (p & ~p) is true. Dialetheism, then, rejects the so-called law of non-contradiction. An impetus for this controversial position lies in the desire to incorporate extended liar sentences within a natural semantic framework. By including extended liars, a unified and coherent, albeit contradictory, structure may be established, which for Priest is an advantage, since a meta-language need not be introduced; to the contrary, the current object-language need only be semantically reflexive. In short, rather than, for instance, resolving liar paradoxes with a consistent meta-theoretical treatment and say they are neither true nor false, and deal with the difficulties that follow therefrom, one may permit within the object-language predicates of the form, ‘both true and false’ and, further, ‘neither true nor false’. Thus, its proponents argue dialetheism allows for a seemingly more natural, non-adhoc, solution to liar paradoxes. In fact, a principal criticism by Priest of consistent meta-language treatments of liars is that they [meta-languages] are unable to express what is quite obviously expressible.
There are, however, difficulties; many in fact, though one in particular, the peculiar truth condition of the negation, is especially problematic. Consider the following. In classical logic, each propositional parameter {p, q, etc.} is given a standard truth-value, either true (1) or false (0)- never neither, never both- and assigned an evaluation, which may be represented as ‘v’. The logical operators are functions on inputs (formulae), which when properly applied return appropriate outputs (evaluations). For instance, the negation operates in the following way:
v(~p) = 1 iff v(p) = 0, and, v(p) = 1 iff v(~p) = 0.
The use of the classical negation converts a false evaluation into a true evaluation, and back again upon subsequent applications. Therefore, if some formula of the type (~p & p) were true, then ~p would be true and not true at the same time and in the same manner, which is impossible, but would nevertheless lead dialetheism into trivialism. To avoid this criticism, a dialetheist may interpret (~p & p) as false and true, respectively, rather than not true and true. In fact, in first-degree entailment (FDE) [a dialetheic logic] a false evaluation is distinguished from a not true evaluation. To make such a distinction, an evaluation in FDE is a relation between formulae and truth-values instead of a function, like above. For instance, if some formula relates to false (0), it may not be not true, that is, it may also relate to true (1) [a purported case would be a liar sentence: ‘This sentence is false’]. Hence, the criticism is not at all definitive for it conflates a ‘not true’ interpretation with a ‘false’ interpretation, precisely what FDE does not do. (Historical note: Priest contends equating ‘not true’ with ‘false’ is a relatively recent result of modern logic, thanks to Frege and Russell & Whitehead.) Furthermore, Priest remarks, the criticism begs-the-question because the matter at hand is precisely whether true contradictions may obtain, therefore one may not say true contradictions do not obtain because they are impossible.
Whether the criticism from negation begs-the-question or not, it is not at all clear dialetheism entirely avoids the criticism. (Or, as I see it, the essential feature of the criticism.) In fact, the criticism, to the point, is that dialetheism, in order to construct a coherent semantics, must employ the classical negation; namely, it must use precisely that which it has rejected. To bring this out more fully, let us begin anew.
Isabel is a dialetheist; Anele is not. Isabel makes the following statement: Brisbane is the federal capitol of Australia (B). Anele thinks Isabel is wrong, and asserts ~B. However, as Anele is well aware, Isabel is a dialetheist, and may thus hold (B & ~B) to be a dialetheia. In other words, Isabel may hold: v(B & ~B) = 1, and, v(B & ~B) = 0. If, as is most probable, (B & ~B) were not a dialetheia, how may Isabel express as much to Anele? She could assert ~(B & ~B), translated, it is not the case that (B & ~B) is a dialetheia. However, how is Anele to be sure that Isabel does not hold ~(B & ~B) & (B & ~B) to be a dialetheia? Isabel, again, could assert, ~[~(B & ~B) & (B & ~B)]. This, though, will not do. For how is Anele to be sure Isabel does not hold ~[~(B & ~B) & (B & ~B)] & [~(B & ~B) & (B & ~B)] to be a dialetheia, and so on ad infinitum.
A dialetheist must for obvious reasons bring this regress to an end, and it is at this point we again encounter the distinction Priest would like to make between denying a proposition and asserting the negation of a proposition, that is, a proposition being ‘false’ and ‘not true’, respectively. We see this distinction in the semantics of FDE, where formulae are related to: True and false (b), neither true nor false (n), true and not false (1), or false and not true (0). In denying a formula, B, one claims B relates to (0) but may also relate B to (1), so in denying the truth of B, one could be asserting B is both true and false. Similarly, in negating B, though it does not relate to (1), it need not also relate to (0), hence B may be neither true nor false; so, to negate B is not deny B. The semantics are thus far easy enough. However, a dialetheist is at pains to assert B is true and not also false (likewise for ‘B is false and not also true’). For it is at least consistent with dialetheism that B may, per above, be true and false (or neither true nor false). It is not enough for a dialetheist to assert B is true only because it is true and only true; such a move would beg-the-question. Rather, the dialetheist must employ some consistent negation that excludes a true interpretation from one that is false, or else s/he would be unable to construct a coherent semantics; what is more troubling, dialetheism would result in trivialism. On the other hand, if a dialetheist were to employ a consistent negation, then the entire dialetheist program would fastly become suspect.
P.S. I never did like Whitman.
Your post raises a number of interesting issues, but you’re running some of them together.
A few points of clarification should be made at the outset:
(1) Dialetheias probably shouldn’t be defined in terms of propositions or any other specific truth-bearers, rather they should be defined in terms of the truth-bearers, whatever they may be. We don’t want to give the impression that dialetheists are committed to the existence of propositions.
(2) It also might be better not to define dialetheias in terms of conjunctions of the form ‘p & ~p’. Presumably, someone might claim that a logic without conjunction was a dialetheically acceptable logic (if a sentence and its negation could be both be true).
(3) It’s misleading to call anything a “dialetheic logic” since dialetheism is a metaphysical position. One could be a dialetheist and a classical logician, it would just mean that one was a trivialist. It’s also important to draw this distinction because many proponents of FDE are not dialetheists. I take it that what you mean to suggest is that FDE is a paraconsistent logic that might be thought suitable for a non-trivialist dialetheist to adopt. It’s important to draw this distinction since many paraconsistent logics aren’t appropriate for dialetheism.
(4) We also need to be careful about saying that dialetheists reject the Law of Non-contradiction, since, e.g., the LNC (~(p & ~p)) is a logical truth in LP, Priest’s favored logic. In fact the set of logical truths of LP is the same as the set of logical truths in classical logic. FDE on the other hand, has no logical truths.
(5) It’s also perhaps better to talk simply in terms of the metaphysical claim that some statements are both true and false, or in terms of some contradictions being true, than to say that “we may, under certain circumstances, ascribe truth to contradictions” since the talk of ascriptions might suggest that we are incorrect but somehow pragmatically entitled to call a contradictions true in some cases.
(6) In the body of your post it seems that you’re mixing Priest’s claims about speech acts with semantic issues. Denial and assertion are, for Priest, sui generis speech acts. Acceptance is, roughly, having something in your belief box and denial is something like excluding something from your belief box. Frege equated denial with assertion of negation, and it is this claim that Priest and many others want to call into question.
Now, on to more substantive discussion.
At various points your post seemed to conflate speech acts, communication, disagreement, semantics, and charges of trivialism. It’s possible that some of what seems confused is just badly presented. I suggest reformulating the argument(s) you wish to make to make it clear exactly what you are and aren’t claiming. And to make clear how exactly you think untruth/falsity, etc. relate to denial/assertion, etc.
I take it that your central worries are that a dialetheist cannot communicate disagreement and that the dialetheist faces certain expressive limitations.
Taking the charges one at a time:
There are at least two ways that a dialetheist might be able to communicate disagreement:
(i) By using standard Gricean maxims.
(ii) By appealing to the speech act of denial.
Recall, denial is sui generis speech act here.
Regarding (i), there might be worries that we’d have to alter the Gricean to story to accommodate the dialetheist, but there doesn’t seem to be an in principle objection to the possibility of doing so in a way that will serve dialetheic needs.
The real worry about both of these strategies is that they don’t embed. The dialetheist should have a way to entertain the hypothesis that someone is mistaken, etc.
Another, wackier thought, is that the dialetheist should be pushed into thinking acceptance and rejection can be overlapping states, but it seems that they can resist this move.
This brings us to the second worry, that the dialetheist can’t express notions like true only and false only or even not true. This is still a concern, but it is muted somewhat by the above considerations.
Priest’s standard reply to this kind of charge is to say that of course he can express such notions, and in the very words anyone else would use. Let ‘’ be a name forming device of some kind, then Priest’s system allows him to express:
~T
T & ~F
F & ~T
(T=true, F=false) What the dialetheist can’t do is somehow guarantee that these notions behave consistently, but as Priest is fond of pointing out: nobody can guarantee consistency!
Perhaps the worry is that these notions don’t really behave like we would like them to. If one accepts that if something is false then it is not true, that is:
If F then ~T
We can see that anything that is false will also be simply false and vice versa, i.e., falsehood is equivalent to simple or sole falsehood. However, Priest doesn’t accept the above principle (though he does accept ‘If ~T then F’). So, in Priest’s system, there is a distinction between being solely false and false, since they aren’t logically equivalent notions. Although, even for Priest, some sentences, like the strengthened liar, will be both false and solely false.
Essentially the same issues arise when we try to find a way for the dialetheist to express that something is or is not a dialetheia. The obvious ways to say it: ~(p & ~p), ~(T & ~T), ~(T & F) all suffer from problems. The first two probably don’t make any distinction and the last doesn’t unless we go with Priest and drop the false to untruth principle. So Priest does have a way to express that something is not a dialetheia, but it is still true that some dialetheias will also not be dialetheias. This goes back to Priest’s point about the impossibility of enforcing consistency.
So we’ve seen that the obvious candidates for expressing some obvious notions collapse for the dialetheist if they accept certain principles concerning truth and falsity. But Priest himself denies one of the crucial principles and thus has a non-vacuous way to express some needed notions.
However, there’s still a nagging problem. It seems like many of the notions we’ve been discussing shouldn’t overlap at all. We want certain notions to exclude other notions. But what is exclusion? Perhaps it’s something that we can assert that makes it impossible to also accept p for a given p? What could this be? The trivialist thinks that everything is true, including whatever statement we suggest for the role we’re considering.
The best that can be done is to have something that if accepted along with p will lead to the acceptance of everything. The classical logician has this with ~p. But, as Priest has pointed out, the dialetheist has sentences that work like this as well:
If p then everything is true
Or:
If T then bottom
Or:
If p then bottom
Etc. Where bottom is absurdity; it entails everything. Acceptance of one of the above along with p leads to triviality. So it seems like once again the dialetheist has an escape route. Perhaps this is the right strategy for the dialetheist to use in expressing the notions of sole falsity or false only?
However, Hartry Field has pointed out that this kind of move doesn’t work in general. The statements above are a bit too strong to work like an intuitive notion of sole falsity, because of the existence of Curry sentences. There are weaker notions, like:
If p then bottom V If p then (if p then bottom)
But the problem obviously recurs. Generally, the dialetheist can approximate a notion of sole falsity using a sequence of operators defined as above, but it seems that they won’t be able to perfectly emulate the intuitive notion of sole falsity. I’m ignoring some technicalities here, but roughly this seems to be the state of play.
Perhaps it’s open to the dialetheist to claim that there isn’t a coherent notion of sole falsity that works exactly as we might have expected it to work? I don’t know how satisfying I find such a move, but others may disagree.
I’ve ignored a number of relevant issues in the above in an effort to keep this reply relatively brief. I should perhaps point out that I’m not a dialetheist. I just think that dialetheism doesn’t die easily! I’m sometimes inclined to agree with David Lewis here, but at other times I think that maybe some line like those we’ve been discussing can be made to work.
Jared
It looks like my name forming device, , and the sentences inside of it, didn’t show up in my post. Hopefully it’s still possible to follow the arguments.
Basically, every time that there’s a ‘T’ or ‘F’ read it as T for appropriate choice of p.
Jared
Okay, still not showing up. I thought it might have happened because I pasted from word.
Just think T(p) where “()” is the needed device.
Jared
One quick correction.
The relevant part of (6) should say:
“Acceptance is, roughly, having something in your belief box and rejection is something like excluding something from your belief box. Assertion and denial are the related speech acts.”
Jared,
I apologize for my belated reply. Granting I was remiss in regards to (1) – (6), I would like to address a two concerns, (a) Gricean implicature and (b) the claim that the distinction between denial and assertion of negation can help the dialetheist.
(a)
Since I know Priest accepts the existence of true contradictions, and if I utter P (and mean P and not also ~P) and he reposts ~P, I will rightly wonder whether he disagrees with me. If I knew he was using Gricean implacture to express disagreement, then I would know that if he meant (P & ~P) he would have said as much. However, I may still doubt that the dialetheist is actually disagreeing with me.
If, for instance, he proved to himself that ~P, but (a la the intuitionist logician [the analogy here is superficial]) he has not yet proved to himself that not ~P, then his assertion of ~P would not express the strongest information. Hence, ~P would in itself not be adequate to express disagreement, even under the auspices of Gricean conversational maxims.
Thus, we would be right to require of him a formulation of some statement that expresses: ‘false and not also true.’ Let us suppose the dialetheist devises one, S, such that S = ‘simply false’. Unfortunately, we now have a re-newed liar sentence:
L: This sentence is simply false.
In which case L is also true. Thus, L may be simply false and simply true. I think this poses problems for the dialetheist.
(b)
Because he distinguishes denial from assertion of negation:
(1) Priest may assert the negation of a liar sentence without denying it.
(2) And he may do so because, as a matter of fact, he does negate the sentence but also accepts it. In other words, he accepts that the liar is both true and false.
Reasoning semantically from (1) and (2), it follows that denying and accepting are logically incompatible: if one denies an utterance, then one can’t accept it. But this begs a question I am not confidant Priest can answer adequately: why can’t one at the same time both deny and accept an utterance? Surely the answer(s) will call upon notions of consistency, no?
At the end of the day, I agree with Lewis. The means by which we are to conduct the debate are so much less certain than that which is being debated. Why not simply refrain from participating?
The “that not ~P” in, “If, for instance, he proved to himself that ~P, but (a la the intuitionist logician [the analogy here is superficial]) he has not yet proved to himself that not ~P, then his assertion of ~P would not express the strongest information,” should be “that P”.
Hi Aaron,
On the two topics raised in your comment:
(a) Gricean Conversational Implicature
Let’s talk of acceptance and rejection here rather than what we “mean”, to avoid sliding into muddles. Say you accept P and reject ~P (let the variables range over the appropriate items) and you utter u, performing the speech act of asserting that P. The dialetheist then performs the speech act of asserting that ~P. Via the standard Gricean story, you conclude that the dialetheist does not accept P, or else they would have violated a Gricean maxim.
P is in your belief box and it is not in the dialetheist’s belief box, and this fact has been successfully communicated, though nothing actually said by the dialetheist entails the non-acceptance of P. This much is common ground.
I don’t think your argument here is clear (it’s helped a but by the correction), but I think what’s worrying you is this (let me know if I have you wrong): The fact that the dialetheist doesn’t accept P doesn’t mean that the dialetheist rejects P; perhaps they simply have no opinion. Rejection isn’t simply not having something in your belief box; it’s stronger than that. The thought is that the dialetheist needs to do more because just asserting ~P allows that they are unsure about P, thus they might not *really* disagree with you.
This is one of the things I was alluding to in my first comment when I wrote:
“Regarding (i), there might be worries that we’d have to alter the Gricean story to accommodate the dialetheist, but there doesn’t seem to be an in principle objection to the possibility of doing so in a way that will serve dialetheic needs.”
(I cleaned up an infelicity in the above, but otherwise it’s as posted)
The thought is that disagreement is when one party accepts A and the other party rejects A. So, the standard conversational maxim would have to be altered. Perhaps the dialetheist would have to signal non-disagreement in order to avoid the presumption of disagreement in such cases and this signaling would be a requirement of the new maxim. That’s the kind of story the dialetheist will want to tell; I used to think that there could be a problem in telling this story coherently, but I’m less inclined to think so now for various reasons.
As I said in my original response: perhaps the biggest worry about using the Gricean strategy is that it doesn’t act on embedded sentences. And I think that this, as opposed to the above line of thought, is the best way to push a dialetheist on the need for the expressibility of notions like false-only.
And here the state of play seems to be as layed out in my first comment: A dialetheist can express standard notions of sole or simple falsehood using the truth predicate. However, the notions so expressed won’t behave as we might expect. If the dialetheist accepts the standard definition of falsity and also certain conditionals linking falsity to untruth then the notions of, e.g., falsehood and sole falsity collapse entirely. Priest can express the notions without collapse but even on his view there will be overlap. You reiterate the overlap point with the renewed liar example, but (i) you move from truth to sole truth without comment and (ii) it isn’t clear which problems you have in mind (there are several ways to go). Is your worry the one mentioned in my post, viz., the notion doesn’t “behave” properly or intuitively?
On the other hand, if the dialetheist wants a notion of the sole falsity of A that explodes with A, then they can get close but not all the way to the finish line (because of the Curry problem). This is another potential pressure point to push against.
I think we agree in general outline here but not in detail.
(b) Acceptance/Rejection & Assertion/Denial
I’m not sure what the argument in your comment is doing, but Priest clearly takes acceptance and rejection to be exclusive but non-exhaustive states. This seems like a reasonable position, but you question whether a dialetheist like Priest is entitled to the position. This is the “wackier thought” that I briefly mentioned in my original post.
We should note that utterances aren’t likely to be the occupants of belief boxes. So it’s probably better to say that someone performs a speech act of denial by making a particular utterance, or something in that ballpark. Still, I take your meaning to be clear: can we accept and reject the same thing at the same time, etc.? If not, why not?
Priest’s answer to charges like this is simple: he doesn’t believe that the behavior patterns associated with accepting P and rejecting P can hold simultaneously. He doesn’t think it’s possible to truly do both in the same way/time/etc. It would be like both catching and missing a bus in the literal sense. Surely this is open to Priest if we grant him that accepting some contradictions is to be distinguished from accepting all contradictions. One important question might be: does Priest’s answer here commit him to a kind of behaviorism?
Priest’s response isn’t about logical consistency, and we should be also clear about which notion of “logical incompatibility” is on offer. When we are discussing different logics with different understandings of negation, etc. it’s best to be fully explicit about which concepts we’re using or assuming.
It’s also important to note that Priest isn’t asking for special treatment here, he thinks that pretty much everyone agrees that the Fregean position on assertion/denial is wrong, once they think about it.
So I think Priest can answer the challenge you pose, as long as we’re fair to him. Although, once again I want to note that the denial/assertion strategy doesn’t act on embedded sentences, so it doesn’t solve all of the dialetheist’s problems.
On the general issues:
As mentioned in my initial comment, I sometimes think it may be possible to push a critical line like those that we’ve been discussing in a way that gets traction against a dialetheist. However, two points that Priest has made illustrate some of the difficulties nicely (other illustrations could obviously be constructed):
(i) Someone might argue that irony makes agreement impossible for similar reasons. You assert P and I say: “Yes, P.” It seems I agree, but perhaps I was being ironical. Anything I add about not having been being ironical doesn’t help, since it might just be more irony. Yet, I take it that we all agree that the existence of irony and the ability to express agreement are compatible.
(ii) Anything said by anyone is compatible with that person being a trivialist. So the classical logician, with a fully standard understanding of negation, can say nothing that magically differentiates him from the trivialist. Yet, we all seem able to communicate the fact that we are not trivialists to each other. Don’t we?
There are many potential ways to attack dialetheism; it seems to me that quite a few of them fail to get off the ground. The types of concerns we’ve been discussing look promising, but the dialetheist has more potentially plausible responses than it initially seems.
I’ve mainly been critical in my comments, but I’m probably more sympathetic to the spirit of your concerns than you may suspect. However, formulating the worries correctly takes care and a serious concern with potential responses.
Jared