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This question is in a sense a follow-up, or elaboration, of the question "What are the motivations for Dialetheism?". Reflecting on the way I phrased that question, and the way I remarked on answers, it has occurred to me that my central confusion is that I have no idea how even to begin to interpret a dialetheia. I hope to address the crux of my confusion here. Specifically, what notion logical negation do dialetheists have?

Bananas and binary distinctions

In classical logic, there are two truth values: "true" and "false". It seems to me that the role of these two concepts, and logical rules of inference, are to try and reduce the world to different clearly distinguishable states of affairs, and the way in which we distinguish them is by describing them in terms of yes/no answers to questions we may ask of it. "Is the banana yellow?" is a question which we might answer yes or no; and we associate with this the notion of "The banana is yellow" being a proposition which is true or false.

Of course, we can have nuanced states of affairs, in which small amounts of sand are only ambiguously heaplike; and bananas start off being green, slowly fade into a yellow colour, and eventually become mottled with brown and black. We may obtain coarse truths by saying "the banana is yellow", without precluding the possibility that it has some brown spots; and there is no clear point at which the banana stops "being yellow" and starts simply "being brown". To approach the question of the properties of bananas, classical logic can only proceed with more nuanced propositions describing the percentage and distribution of colouration in bananas, the shape and composition of an amount of sand, and so forth, until answers of sufficient practical reliability can be obtained.

But nuance is a question of subverting the excluded middle, rather than the law of non-contradiction. "Having some yellow" and "having some brown" are not mutually exclusive concepts, so both obtaining is no contradiction; and unless one specifies precisely which shades of yellow and brown are involved, being "simply yellow" and "simply brown" aren't necessarily distinguishable, so confusion over the shade is not a confusion over contradictory properties. For subjects such as bananas, where we have an expectation informed by experience, we tend to assume some archetypical shades of yellow and brown which normally would be mutually exclusive, when we talk about being "simply yellow" or "simply brown"; because of this, we may a posteriori assume an absence of vagueness and turn "simply yellow" and "simply brown" into mutually exclusive properties of bananas (admitting "significantly mottled" as a third case in any realistic discussion).

In classical logic, it is the exclusivity which matters most when speaking of the law of non-contradiction, and is essentially the purpose of the notion of negation. Without being able to make some distinctions between states of affairs, negation is pointless. To say that the banana is "not yellow" does not automatically entail that it is green or brown; just that the conditions for it "being yellow" are explicitly declared or decided not to obtain. (Possibly the banana has some other colour — supposing that the conditions are such that one may decide that the banana has some colour — but it is merely declared that "yellow" is not the outcome if such conditions hold.)

What does denial mean in dialetheism?

This brings me to my continued struggle with dialetheism. What is "not" supposed to mean to a dialetheist, if A & ¬A is admissible in principle? A formalist can accept this as a possible property of a logical system, if A simply means "A can be derived" and ¬A simply means "¬A can be derived". But then the derivability of two different formulae are not a priori exclusive properties, and a formalist in principle suspends judgement as to any intended meaning of the symbols. However, the inference rules of classical logic were deliberately formulated on the premise that ¬A is something which can never hold simultaneously with A, because of the priority of classical to describe incompatible binary distinctions, and the consequences of those distinctions.

I can only conclude that dialetheist logical systems have some completely different set of priorities. What are these priorities? And specifically: what meaning does the negation symbol obtain in frameworks where dialetheism makes sense?

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    You might check out this article on impossible worlds. The semantics of dialetheism isn't something I've thought a lot about, but from what I am familiar with they seem to provide a semantics similar to that of possible worlds semantics for modal logic, but adding in impossible worlds to account for the truth of contradictory statements. This obviously isn't a complete answer (hence why I'm leaving it as a comment) but maybe it's something worth looking into. Once I finish the paper I'm working on I'll try to take another look at this. – Dennis Apr 6 '13 at 4:18
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So there are at least two interesting Dialetheist responses to your question. The first is to pull apart propositional negation from negation in an Assertoric Context. The second is to say that even propositional negation has a modal sense that classical predicate logic doesn't capture effectively.

Priest on Asserting Negations

Firstly, a couple of different ideas get caught up in the idea of talking about a proposition and "its negation". Here's how we do things classically. An atomic sentence of the form P(c) is true (/assigned the value T/1 /deemed assertable/ etc.) if, and only if, the object in the domain that our model interprets as being the referent of c is a member of the subset of the domain that our model interprets as being the extension of P. If this is not the case, the sentence is not deemed true. Now, the conditions for ¬P(c) are that it is true (...) if, and only if, P(c) is not true. So in effect, if the referent of c is not in the extension of P, then ¬P(c). Set theory is classical, being a member of P is a wholly extensional matter, and so it seems negation is entirely determinate.

Nonetheless, the classical view might be read as collapsing a distinction of value to the process of interpreting what negations are. The negation function, ¬, is a regular propositional operator - you "negate" a proposition A (i.e. ¬(A) is an instruction, rather than a term), and in the process get another that constitutes "the negative" of A, ~A. This is a process that is well defined for logical complexes, no matter what your basic logic is. But for an atomic sentence, is it always obvious what its negation amounts to? When you assert that something is not true, you presumably have to have some particular positive content in mind that backs this up - saying that the banana is not yellow ought to mean you have good reason to think it is some other colour that conflicts with its being yellow.

In this, Graham Priest in both his In Contradiction (1978) and Doubt Truth to be a Liar (2005) affirms that when it comes to Truth, we need to understand its propositional content as tied to the act of Asserting something. Specifically, Truth is supposed to be the ideal form of assertion - when you say something, you do it in the aim of its being true, and that's what being true consists in. Assertion might, we think, have some kind of deontic component - what people can and/or ought to assert is subject to some conventional norms, with the above suggestion that you can back up your claims with good evidence being one possibility.

Truth is exactly the ideal of asserting, and theories of truth are thus supposed to be theories that capture these norms. As such, the semantics of Truth are given in the norms of assertion, rather than in the ontological structure of the world. (In Maths, for instance, the semantics of our language is determined by what you can prove, and the proofs you can construct, rather than the objects out there in the world)

This move allows for so many divergent factors in an account of what negations, negating, contradiction, assertion and truth have to do with one another. We might think that for a number of cases, asserting a negation and not asserting come apart; we might say that the negating function in an assertoric context doesn't always correctly grasp a proposition's "actual" negation, we might say that you can have prima facie and ultima facie contradictions depending on whether we have an internal or external algebra of negation, we might say that you can assert a statement without failing to assert its negation etc.

I have good evidence to suggest that your banana is yellow. I also have good evidence to suggest that it's actually brown (see, look at all of those splodges and bruise marks). So, if our norms of assertion commit me to stating everything that I have good evidence to suggest, rather than hiding some of them for the sake of personal convenience or coherence with a preferred theory, I ought to really say that both of them are true, even when we might say that the banana being brown actually satisfies the assertive conditions for it not being yellow. This doesn't mean I'm committed to asserting that the moon is flat and that pigs can fly; so I should accept that my assertions here are driven by a dialethic logic that can process some contradictions without falling into trivialism.

The problem with this line as I see it is that it deliberately seems to avoid the challenge of the hardcore realist. Fine, assertions might be internally contradictory and this doesn't necessarily mean you're committed to asserting anything and everything. Dialethic logic as a way of representing and processing how people talk (or their commitments to talk) has some interesting applied use. But you only get there by talking about contrary states of affairs, that seem to have a certain amount of tension between one another (represented in assertions by the negating function).

There's nothing actually contradictory about the state of the banana: it's just both brown and yellow in certain respects. You might say in your assertions "A & ¬A". But when you say "¬A", your negating operator isn't returning the state of affairs that the realist will call ~A - it's hitting a proposition B and mislabeling it "~A".

This might be cute as a linguistic or behavioural theory, but it's not really logic with respect to the classification of the structures of facts. There aren't really any true contradictions. (I'm being deliberately harsh here for the sake of answering your question; there's merit to the thought that logic should be considered the theory of warranted psychological inference rather than ontological structure, that the semantics of logic should be captured in human and/or social language use, and that Priest's logic has a much better grasp than his classical counterparts)

Routley/Meyer Negation

To go for a more radical, metaphysical claim about the existence of true contradictions, all Priest needs to do is to say that there are determinate and ultimately dialethic principles of assertion in our basic metaphysics that correctly latch on to the world. Although I don't think he makes such an assertion explicit (he does, for instance, defend the possibility of a foundational logic, but never specifically asserts what that foundational logic is supposed to theoretically cover) he does present arguments for dialetheism in the empirical sciences, which is good news if you're a naturalist. Not, though, if you're a classical set-theoretic structuralist. You're probably just going to accept that the rules of assertion in set theory are those of classical logic.

An alternative, direct account of negation in a semantic dialethic setting is presented by J. C. Beall in his Spandrels of Truth (2008), where he explicitly invokes ideas found in Relevance logics. Beall borrows from Routley/Meyer's semantics, arguing that negation is centrally modal and that in order to understand the negation of a particular proposition at a possible world, we must look at some other well-specified possible world that represents opposing states of affairs.

The Routley-Meyer semantics is a generalization of Kripke frames. Routley's Star operator takes each world in the domain to a dual world, such that w = w**. The semantics for negation are defined in terms of this star operator:

w ⊨ ¬A if, and only if, w* !⊨ A

The actual substance of the dual worlds aspect is not really fully fleshed out by Beall - he thinks the value of the semantics is mostly to give an account of a properly deflationary truth predicate, and interpreted semantics is the job of theory builders rather than logicians. But he hints at possible ways of reading it, and thinks the idea of a Truthmaker is a valuable starting point.

The Truthmaker line is to say, as we did above, that there needs to be something in virtue of which a given proposition is true. This is true of negative propositions as much as it is of positive atomic basics - it can't be the absence of a truthmaker that is responsible for a negated atomic sentence being true, unless we want to reify the notion of truth-making gaps. Beall's suggestion is that we might read the negation operator intensionally, because we understand what it means for a negation of an atomic proposition to be true by considering the scope of possibilities in which that proposition would not be true.

In his semantics, then, the duals of worlds are related in terms of the facts that they posit. We say that in Normal worlds, we have consistency by noting that the only things that there are are positive states of affairs. Their dual worlds, too, are consistent, but where Normal worlds consist of fully compatible states of affairs, their duals contain richer collections of states of affairs that are in some sense incompatible with one another from our perspective in a Normal world.

A speculative reading of the construction might be this: if you take Tarski's point about the impossibility of consistently defining a theory of Truth without a stronger metatheory at its face value, and you suggest that the accessible distance between our world and a world at which a full notion of truth is definable can be realized as a modal claim, you get what Beall is trying to do. Negation is essentially modal, to understand negation we move to our world's metatheoretically richer clone (whose own clone, apparently, is our non-enriched version), and when we do that, we wind up with some basic semantic contradictions, but nothing that overrules the consistency of truth at the level of the states of affairs we posit to constitute our actual world.

It's a bit of a hack, I reckon. But it does seem to match with what a lot of the maths is pointing at! If a Tarski-like axiomatic theory is how mathematical practitioners operate, and we take the essential richness of something like proper class theory to be strictly necessary for practice but ontologically undesirable, the suggestion that it exists as an extra-theoretical posit that we appeal to when considering the interpretation of our theories has definite advantages.

  • "When you assert that something is not true, you presumably have to have some particular positive content in mind that backs this up [...] what people can and/or ought to assert is subject to some conventional norms, with the above suggestion that you can back up your claims with good evidence being one possibility." This I think I grasp; but it doesn't allow me to understand how A & ¬A can be accepted by a dialethieist except if she interprets ¬ differently to me. Does this not simply mean that one ought not assert ¬A, but rather some B which one supposes is incompatible with A? – Niel de Beaudrap Apr 26 '13 at 17:19
  • @NieldeBeaudrap, the key here is the supplement I give at the start of the following section - the "ultima facie" dialetheist takes ideal assertion containing some conflicting statements to be mirrored in metaphysical reality. We say that ¬ here is the "right" negation function because there actually are some facts that are in contradiction to one another, and our language properly represents that. Classically this won't make sense, but then on this account, the classical logician is always going to miss when they try to speak logically about the world. – Paul Ross Apr 26 '13 at 17:42
  • So then, by contradiction, you might mean simply that there are 'facts', such as "The banana is yellow" and "The banana is [name of a colour taken to be distinct from yellow]", and that by virtue of their form — rather than how one evaluates the extension of these phrases (for there are many things we might mean by "is yellow") — they are contradictory despite being compatible? – Niel de Beaudrap Apr 26 '13 at 17:59
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The dialetheist wants A to mean "A is true," and ¬A to mean "A is false." But the dialetheist thinks some statements are both true and false. "This statement is false" is both true and false; the Russell set contains and doesn't contain itself. It's not that the Russell set sort of contains and sort of doesn't contain itself; it completely, 100% contains itself, and completely, 100% doesn't contain itself.

Usually the dialetheist doesn't think that this happens in any "ordinary" situations. In particular, if A is provable in classical set theory (ZFC or whatever), the dialetheist usually wants it to be true that their system doesn't prove ¬A. So while we think it's OK if our axiom system proves that the Russell set contains and doesn't contain itself, we don't think it's OK if our axiom system proves that 2+2=3. It's only in self-referential contexts, and other contexts that only arise when you build an inconsistent set theory or a theory with full truth predicates, that we get paradoxes. Ordinary math is business as usual, or at least it's supposed to be.

OK, but I still haven't even slightly addressed the central thrust of your question. What does it mean, on an intuitive level, to say that the Russell set contains and doesn't contain itself? Having spent the last year researching inconsistent mathematics, I can say that I still don't have any serious intuition that I can convey in words about that. I am sincerely sorry.

I can, however, explain how I (myself, personally) arrived at the intuition that some things are both true and false. In what follows I do not speak for other dialetheists. In reading mystical literature (Hindu literature, the poetry of Aleister Crowley, etc.) I came across the idea of nondualism. Merriam-Webster defines it nicely:

"A doctrine of classic Brahmanism holding that the essential unity of all is real whereas duality and plurality are phenomenal illusion and that matter is materialized energy which in turn is the temporal manifestation of an incorporeal spiritual eternal essence constituting the innermost self of all things."

Some of my spiritual experiences convinced me that nondualism was true. Nondualism says that all is one --- everything is the same. It follows that a table is a chair, a hat is a handbasket, etc. It follows, in essence, that every statement is true and false. I would be willing to say that nondualism is the assertion that everything is true and false. (I realize this doesn't appear consistent with the Merriam-Webster definition; but that is the sort of difficulty one runs into in studying nondualism. Every definition is inadequate and wrong.)

I wanted some math to back up the idea of nondualism. Of course it's easy and perfectly valid to build an axiom system where everything is true and false, but that's not very interesting from a mathematical standpoint, nor is it much of a talking point in arguments. It is somehow much more convincing to build an axiom system that agrees with everything we already believe, but also includes paradoxes. This sort of provides a "way in," where it becomes apparent that the idea of logical contradictions might not be irrational.

So that's how I (myself, personally) got to dialetheism. If you are truly to determined to get an intuition about what it might mean for a paradox to be true, I might suggest meditating on it, and having a spiritual experience. Anyway that's what works for me. Best of luck, and thank you for reading!

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    The crux of my problem is that "true" and "false" are also not really fundamental semantically, when describing models containing objects. The objects have "properties" from satisfying "predicates", which as I described are themselves tools for describing distinctions between possible worlds. At best, I suppose that your answer claims that distinctions are illusory; why then should we experience them in the first place? – Niel de Beaudrap Apr 7 '13 at 15:21
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    As for duality being an illusion, should I conclude that you consider paraconsistent logic to be misguided, or that there is a semantics for it which ought to be understood in more subtle terms than "truth" and "falsehood"? And in any case, what meaning does the negation symbol take? – Niel de Beaudrap Apr 7 '13 at 15:25
  • Thanks for replying! Let me see if I understand your issue, re: true and false are not fundamental semantically. Since we are describing models containing objects, what is really fundamental is not the truth of statements, but the properties of the objects. And so you would like to know what it means for an object to have and not have a property. Is that correct? – Nick Thomas Apr 7 '13 at 18:52
  • Re: your second comment. If you press me on it, I'm not 100% certain that nondualism is true. I have no evidence, so it is just a conjecture. But supposing it is true, I would answer that yes, paraconsistent logic is misguided on a basic level, in which case its philosophical interest (in my view) is basically as an elaborate mathematical thought experiment exploring the idea that contradictions might be true. That makes room for the idea that in fact every contradiction might be true. – Nick Thomas Apr 7 '13 at 18:57
  • Re: the meaning of the negation symbol. Before I answer, I want to clarify: are you asking for the formal semantics, or the philosophical interpretation? – Nick Thomas Apr 7 '13 at 18:59

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