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At the request of the moderators, I've reformulated this question to change the emphasis of the question to something perhaps a little more broad-ranging:

Question. What are the major modern motivations for Dialetheism?

Context. According to the Stanford Encyclopedia of Philosophy's article on Dialetheism:

A dialetheia is a sentence, A, such that both it and its negation, ¬A, are true [...]. Assuming the fairly uncontroversial view that falsity just is the truth of negation, it can equally be claimed that a dialetheia is a sentence which is both true and false.

Dialetheism is the view that there are dialetheias. One can define a contradiction as a couple of sentences, one of which is the negation of the other, or as a conjunction of such sentences. Therefore, dialetheism amounts to the claim that there are true contradictions.

As someone who has training in the mathematical sciences, I of course tend to adopt the policy that any contradiction is a statement about the quality of my model of the world (that it is poor), and that some assumption or method (axiom or rule of inference) is in need of improvement. Therefore I'm somewhat surprised, and incredulous, that anyone would advocate for the acceptance of a contradiction, or indeed create logics specifically to be able to accomodate "A & ¬A" being true.

The same SEP page gives historical and modern examples of apparent contradictions; however, aside from the Liar Paradox (which I would dismiss as non-signifying), they seem to concern either imprecision in language (such as equivocation or ill-defined boundary conditions), or facts of speech or belief. The original formulation of this post asked if all of the "real contradictions" were of this character.

I am hoping that someone could provide me with a stronger case for Dialetheism than I can get by reading the SEP, which leaves me unmoved. For example:

  • Can anyone provide a contradiction which could not easily be interpreted to be a matter of imprecision of language, or to concern primarily speech acts and the like, or to reduce to the Liar Paradox, and therefore plausibly to be simply accepted?

  • Can anyone provide a good reason why Dialetheism (or paraconsistent logic, in which contradictions may arise without trivializing truth) is expedient, even if one does not believe that there are statements which are in fact true at the same time as their negations are? Why would one care to avoid "explosive" logics (for which ex falso quodlibet)?

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11 Answers 11

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To follow your lead, I don't think anybody accepts true contradictions. That is, no one accepts 'P and -P' ever as 'true'. Paraconsistent logics allow in a proof, for both P and -P to be asserted separately, but the logic allows the proof of other things to continue without the entire proof to fall apart -in- the proof system. It is not a theorem (in say relevance logic) that

(P and -P) -> Q

(ex falso quodlibet) but that doesn't mean that 'P and -P' -is- a theorem.

True contradictions (or, what I think you mean, inconsistencies) aren't ever accepted as theorems, but they are 'tolerated' sometimes as long as they don't spoil something else.

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I disagree. Dialetheists do accept true contradictions, and will in fact assert P and not-P (for certain specific values of P). –  Michael Dorfman Sep 6 '11 at 20:33
    
So are advocates of paraconsistent logics merely advocating them as practical tools, when one is constrained to work within a fixed set of axioms about the world, merely so that your logical system doesn't barf if your model isn't perfectly consistent to begin with? I was under the impression that this advocacy was more on the order of intuitionism versus classical logic, where the advocates of the former really have a different notion of truth. So, not? And what does it mean for P and ~P to both be 'asserted', without (P & ~P) being a 'theorem' (i.e. derived from the axioms)? –  Niel de Beaudrap Sep 6 '11 at 20:40
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@Michael: Though the question is about dialetheism, I sidestepped and referred to paraconsistency...and these two are not the same. I think my statement holds for paraconsistency but at the same time yours holds for dialethism. –  Mitch Sep 6 '11 at 22:05
    
@Niel: for your first point, I wouldn't say 'merely' but yes, a paraconsistent logic is a proof tool which, like other proof tools, are 'constrained...so that [it] doesn't barf'. Point 2, intuition and classical logics' 'different notion[s] of truth': not -terribly- different as truth goes, not night and day but more like early and late afternoon. –  Mitch Sep 7 '11 at 2:35
    
@Niel: As to P and -P both asserted, in most any logic one infers (P & -P). But I don't think that is what I was saying (but I also think what I said is not...um...relevant. What I was trying to say is that one may have inferred, in a relevance logic proof, that P & -P, but one can't then infer anything (as in classical logic). –  Mitch Sep 7 '11 at 2:39

From the SEP article you link to, there are many justifications for dialetheism (but also many objections). But to answer your direct questions:

  • as to an example, many are (as the article gives), incompatibilities of context, either vagueness (continuous transitions), or amphiboly (a word having multiple distinct meanings), or different rule systems (legal precedents that interpret the situations differently). Their canonical example that is not of this type is the liar paradox.

  • as to expediency, I don't think that dialetheism is offered as a system to substantiate contradictions, but merely to recognize that they are possible utterances, for which it would be good to be able to manipulate them, deal with them in a coherent way.

  • as to why one would want to avoid 'explosive' logics, one example is, in a mechanical verification system that needs to deal with non-monotonic situations (atomic facts are propositions about the real world which can change (the light red one moment then green the next)), it may be the case that in a transition from knowing 'red light' to knowing 'green light' they are both in the system at the same time, and so a classical logic might then start to make multiple random inferences from that pair (from a contradiction anything follows), that is, 'exploding' with a bunch of irrelevant propositions, before the 'red light' fact is removed. This is just one narrow application. (this is also another case, not mentioned explicitly, of a benefit of a paraconsistent logic).

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Given the extended exchange of comments between myself and @Michael Dorfman, I suspect that he (and if I am to take the SEP article quite seriously, at least some other Dialetheists) would disagree with your second point; Michael at least seems to think that contradictions are more a feature of the world than an artifact of the language we use to describe it (i.e. that there exist instances of apparent contradictions, which are accurate representations of the state of affairs). –  Niel de Beaudrap Sep 7 '11 at 15:59
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@Niel: I suppose I am projecting what I think -should- be the case in point 2, given Michael's point, and the article's comments about modern western logic being 'superficial' in having consistency being so central. Maybe it'd be better to say that dialetheism is expedient just to recognize that inconsistencies are things that can be thought about rather than just plain avoided. –  Mitch Sep 7 '11 at 17:07

My favorite example is one that Graham Priest and Jay Garfield identify in the thought of Nāgārjuna, which they call Nāgārjuna's paradox; it's described in their joint article, Nāgārjuna and the Limits of Thought.

The schematic version is as follows (quoting from the aforementioned paper):

If Nāgārjuna is correct in his critique of essence, and if it thus turns out that all things lack fundamental natures, it turns out that they all have the same nature, that is, emptiness, and hence both have and lack that very nature. This is a direct consequence of the purely negative character of the property of emptiness, a property Nāgārjuna first fully characterizes, and the centrality of which to philosophy he first demonstrates.

Obviously, this is far from the mainstream in terms of most Western philosophy, but Nāgārjuna forms the philosophical basis of pretty much all Mahāyāna Buddhism, so this is actually quite an orthodox position (in some quarters).

EDIT: Due to the rephrasing of the question, and the conversation in the comments, I will try to elaborate here a bit more on the issue more generally.

Since the original questioner refers to a mathematical background, I'll try to stick with mathematical examples.

Let's begin with a trivial paradox. We know, of course, that there are integers that are not prime numbers. In fact, there seem to be a good number of them. And yet, we also know that there exactly as many prime numbers as there are integers. We have here a simple paradox; two contradictory statements which are both true.

Similarly, we can look at Russell's paradox, which seems to point at problems concerning the nature of sets, or the Burali-Forti paradox.

A previous version of the question referred to Heraclitus's dictum that one cannot step into the same river twice. This is not merely an issue of "imprecision in language", but rather, gets to the heart of what is meant by the notion of identity.

And, as Tarski has pointed out, any language which has a truth-function is going to be subject to the Liar paradox. The fact that the questioner chooses to consider this as "non-signifying" is interesting, as it raises the question of by what rigorous criteria one could choose to exclude it (and other similar propositions).

Each of these represents a real paradox; none are due to imprecision of language, equivocation, or ill-defined boundary conditions. There are some parts of the world which are, sad to say, paradoxical-- and if one remains committed to explosive logics, one is faced with the prospect of either a) attempting to satisfactorily resolve all of these paradoxes (and many others), or b) forsaking reason altogether (since now everything and and nothing is provable.)

I tend to view the presence of paradoxes not as an indication that the quality of the model is poor, but rather, the opposite; any model which does not contain paradoxes is most likely too simple a model to accurately model our world (and based on insufficiently subtle axioms).

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"if it thus turns out that all things lack fundamental natures, it turns out that they all have the same nature, that is, emptiness"... but isn't this just reacting to the realisation of the vacuity of an initial concept of 'nature', by immediately redefining the word to mean something which is not vacuous (albeit in practise trivial)? That is, reacting by changing one's linguistic model? The problem then arises from equivocating between the previous model in which nature is 'vacuous', and the new one in which 'nature' is not vacuous. –  Niel de Beaudrap Sep 6 '11 at 16:04
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I don't think so; perhaps it is easier if we omit the term "emptiness", which is a complex term-of-art in these discourses, and go with a famous paraphrase by Mark Siderits: "The ultimate truth is that there is no ultimate truth." The essential essencelessness of all things is, itself, an essence-- an essence of essencelessness. –  Michael Dorfman Sep 6 '11 at 18:56
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Ah, but the argument (of course) is that the essenceless is an essence; it is (in fact) the essential defining characteristic of things. (In the Buddhist literature, this notion is known as "The Emptiness of Emptiness"). All things partake of the same essence, the essence of having no essence-- or, put in other terms, have the nature of having no nature. Priest's 1995 book "Beyond the Limits of Thought" deals more with the Western philosophical tradition, btw-- that is probably a better general reference for most people. –  Michael Dorfman Sep 6 '11 at 19:45
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I'll defer to you on the mathematical issues. In a real-world (non-mathematical) setting, you're going to have quite a time distinguishing between "objects" and "sets". re: Heraclitus, I don't find it to be equivocation at all-- I think that it points out that "identity" is a much trickier issue than most people credit. Similarly, the Liar (in all of its various forms) seems to me to be as well-defined as anything else. In closing, I'm certainly not opposed to resolving contradictions where possible; however, I do not have any reason to believe that all contradictions found are resolvable. –  Michael Dorfman Sep 7 '11 at 13:00
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@Niel: I agree (and Buddhist doctrine agrees) that one should abandon the desire to speak of fundamental natures once one has observed that they don't exist; on the other hand, all discourse (and praxis) is based on provisionally accepting such essences as expedient means. I heartily recommend the Priest and Garfield paper I linked above; I think you will appreciate the mathematization that Priest proposes (the "inclosure schema"). –  Michael Dorfman Sep 8 '11 at 19:26

Several answers have pointed out the affirmative reasons why dialetheism might be worth considering, implying that the chief motivation of dialetheism lies in the applicability to certain situations (whether they be logical or material) where the only correct description involves a dialetheia and which is otherwise intractable or has to be side-stepped and avoided.

However, it seems the chief motivation is of a negative kind: a (small) number of philosophers and logicians through the history of philosophy have found the original defense of the Law of Non-Contradiction (LNC) by Aristotle wanting.

As it was Aristotle who first introduced LNC, their first step is to reverse the burden of proof; it is a task of the defenders of LNC to give a theoretical justification - not for the unconvinced philosophers to justify their opposition to LNC. The 'opposition' in this step is simply the recognition that there is insufficient justification for holding LNC to be necessarily true.

Aristotle on LNC

Simply put, it is not clear what Aristotle exactly speaks about in Met.III when defending LNC. He mixes ontological, pragmatic, semantic and syntactic versions of LNC together. (Since there is not LateX support, I will just write the interpretations.)

1) Ontological:

 It is not possible that the same object both possesses and lacks the same property.

2) Pragmatic:

No (rational) agent can simultaneously accept and reject the same sentence.

3) Semantic:

No sentence is both true and not true.
No sentence is both true and false.
A sentence and its negation cannot both be true.

4) Syntactic:

¬(a∧¬a)

Aristotle holds at one point or another that all these versions are transcendentally necessary and he ties them together as one principle. This SEP entry gives an overview on how Aristotle tried to tie these versions together and use them as a necessary condition for his ontological essentialism (i.e. his account of essence through the distinction between necessary and accidental properties).

His line of defense is the famous elenctic method. As the opponent who doubts LNC is not committed to non-contradiction, showing the opponent to be contradicting himself is not really a viable strategy. Instead, Aristotle tries to trick the opponent in showing the he accepts at least one instance of "x is F and is not at the same time not F", i.e. Aristotle's aim is to show that the opponent is committed to at least on thing that is not contradictory. He is thus arguing against trivialism, not modern dialetheism (which is not committed to the view that all contradictions are true, but only that some are).

Do you think that all the versions above are equivalent? That all can be defended in the same way? That one of the versions is analytically contained in another version? Aristotle did, and this was the status quo, including his arguments, until the early 20th century.

Goodbye Aristotle

It seems to me not so difficult to imagine that some philosophers, starting with Jan Łukasiewicz, were not really impressed by this argument with heavy premises (Aristotelian essentialism!) and messy formulations. And, since logic was not seen anymore as laws of thought, and also not as correspondence with some metaphysical truth about how the world is, they started to think about how to deal with a logical possibility in which LNC doesn't necessarily hold (as Aristotle thought it would). At this point there are several possibilities to formulate weaker or stronger positions, and for the dialetheist the affirmative reasons above kick in, which lead them to take dialetheia seriously.

Allow me to draw a parallelism to the discovery of non-euclidean geometries. For centuries philosophers assumed this to be the only possible geometry. They adduced transcendental proofs (Kant tried to show that euclidean space is the "condition of possibility" to conceive of space), physical proofs (the physical space is just structured that way) and logical reductio ad absurdum-proofs (no other consistent geometry is possible). It was this last aim that actually got mathematicians like Saccheri to formulate, without intending to do so, non-euclidean geometries:

The intent of Saccheri's work was ostensibly to establish the validity of Euclid by means of a reductio ad absurdum proof of any alternative to Euclid's parallel postulate. To do this he assumed that the parallel postulate was false, and attempted to derive a contradiction. Since Euclid's postulate is equivalent to the statement that the sum of the internal angles of a triangle is 180°, he considered both the hypothesis that the angles add up to more or less than 180°.

The first led to the conclusion that straight lines are finite, contradicting Euclid's second postulate. So Saccheri correctly rejected it. However, today this principle is accepted as the basis of elliptic geometry, where both the second and fifth postulates are rejected.

The second possibility turned out to be harder to refute. In fact he was unable to derive a logical contradiction and instead derived many non-intuitive results; for example that triangles have a maximum finite area and that there is an absolute unit of length. He finally concluded that: "the hypothesis of the acute angle is absolutely false; because it is repugnant to the nature of straight lines". Today, his results are theorems of hyperbolic geometry.

... and that found some "nice applications" (though one could certainly argue that there are no logical reasons that compelled physicists to abandon euclidean geometry and we could have stuck with LET instead of SRT).

If you find this comparison misleading, there may be a more apt parallelism with the rise of multi-valued logics by giving up the law of bivalence.

The same happened with LNC. It was considered ontologically, pragmatically and logically necessary. Then it occurred, very late, that one could in fact construct logics weakening or abandoning LNC. From there these logics found some interesting applications in vagueness, paradoxa, etc. - an application which not everyone, as you show, finds compelling enough, because these applications are not logically compelling interpretations, and it is always possible to interpret them by maintaining LNC.

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Your argument would have appeal, except I cannot conceive of what it should mean for something and its negation each to be simultaneously true (or rather for the logical conjunction to be true, which is more to the point but might not be equivalent in a given logic). The argument made by Aristotle is itself a negative: that it cannot mean anything for a proposition and its negation to be true. A dialetheist can retort most effectively by presenting such a meaning, by presenting a robust intellectual difficulty which can essentially only resolved by suspending LNC. –  Niel de Beaudrap Mar 10 '12 at 16:40
    
@Niel I understand your main concern, which I haven't answered actually. My point was tangential: To give a counterexample is just one of the possibilities to expose a non sequitur. I simply pointed out that the original motivation was the recognition that from Aristotle's arguments the adoption of LNC doesn't necessarily follow, i.e. that its negative arguments fail - without the logical necessity of formulating a counterexample. My aim was to illuminate how the skepsis regarding LNC came about. I am aware that this doesn't fulfil your request for an interesting counterexample. –  DBK Mar 10 '12 at 17:22
    
My problem is not that you don't present an interesting counterexample, per se. Your position is that LNC doesn't hold a priori; my position is that if it doesn't hold a priori, there must be some reasonable semantics behind the logical conjunction of a proposition and its negation being true; where exhibiting a fundamental dialethia would be perhaps only one way of hinting at those semantics. But I don't know what those semantics would be, in any logic which to me seems coherent. Without exhibiting a specific paradox, can you present me with meaningful semantics for a dialethia? –  Niel de Beaudrap Mar 11 '12 at 19:53
    
@Niel Just to be clear: It isn't my position that LNC doesn't hold a priori. (My view: There probably is a basic form of exlusive negation that even dialethists do/have to accept. And if there is a valuable contribution to logic, I think it lies in our more sophisticated understanding of negation.) ... –  DBK Mar 13 '12 at 14:31
    
@Niel ... I was trying to convey some more motivations that got some logicians to think of dialetheism as a possible option first, and then as an expedient option in some cases. To appreciate the expediency, one has first to admit the possibility and this came about because Aristotle's defense mixed several versions of LNC (syntactic, semantic, pragmatic, ontological) together - I'll try to expand my answer on this point. However, if the sum of these motivations and examples do not "move" you, I doubt that there is something else that will :) –  DBK Mar 13 '12 at 14:31

I cannot see why ex contradictione quodlibet should be thought a problem, since no contradiction can be true. So what if everything follows from a true contradiction? There aren't any true contradictions. There can't be any. (To think otherwise betrays a failure to understand negation.) So you'll never get actual explosion.

Ex falso quodlibet is more of an apparent problem, since then "If the moon is made of green cheese, then tiny purple unicorns prance on Mars" becomes a true conditional. One wants to ask what the moon's being made of green cheese could possibly have to do with tiny purple unicorns' prancing on Mars. But it's the conditional that is true when the antecedent is false; nothing says that one can somehow deduce the consequent from the antecedent. Still, if this bothers one, he may always switch to a presuppositional conditional--one which simply has no truth-value when its antecedent is false. In fact, the perceived problem arises from treating the material conditional as though it were presuppositional and thinking, "Gee, if 'if the moon is made of green cheese, then tiny purple unicorns prance on Mars' is true, that means that were it really true that the moon was made of green cheese it would also be really true that tiny purple unicorns pranced on Mars!" But the truth-functional material conditional is different from the presuppositional conditional. One might think this a fault with interpreting the ordinary, everyday conditional as the material conditional, of course.

I've been trying to find purported examples of true contradictions in order to show that they are not really true or are not really contradictions. If people send you examples, I'd love to know what they are.

In the consecutive-sevens-in-the-expansion-of-pi example, the word "know" is being equivocated upon. "We know there are a million consecutive sevens" is defensible as likely to be true, under the conditions given, so "know" only means "know beyond a reasonable doubt." "We don't know there are a million consecutive sevens" is defensible as false only if "know" means "know beyond any doubt whatsoever."

"The Way that can be named is not the Way" is not a contradiction. It does not say "The Way is not the Way." It says something like "The-Way-that-can-be-named is not the-Way-that-I-have-in-mind." The former referent differs from the latter referent.

"This six-foot professional basketball player is very tall" is true if "tall" means "taller than five-foot ten inches" but false if "tall" means "taller than six-foot two inches." Whether or not it is a true sentence depends on what one means by his words. And that's always the case: a sentence isn't true or false on its own, but is true or false under an interpretation. We try to speak clearly enough that we all give sentences the same interpretations, but sometimes we don't, and then we may end up thinking we disagree when we agree or that we agree when we disagree, just because we're interpreting the same sentence in different ways. We must always stipulate a fixed meaning and then assign a truth-value, if the sentence has one under that fixed interpretation.

In the Nagarjuna example: "If Nāgārjuna is correct in his critique of essence, and if it thus turns out that all things lack fundamental natures, it turns out that they all have the same nature, that is, emptiness, and hence both have and lack that very nature. This is a direct consequence of the purely negative character of the property of emptiness, a property Nāgārjuna first fully characterizes, and the centrality of which to philosophy he first demonstrates." But either all things lack fundamental nature and therefore emptiness is not their nature, or all things have the same fundamental nature of emptiness. If by "emptiness" is meant "the lack of a fundamental nature," then it is true both that all things lack fundamental nature and that all things "have emptiness," i.e., lack fundamental nature. It is not a contradiction but rather a tautology. Only by denying all things fundamental nature and then treating emptiness as though it were a fundamental nature does the apparent contradiction arise.

Of course, it's possible that some human beings hold contradictory beliefs--but that's far different from contradictory beliefs' being simultaneously true.

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Of course, if we suppose that a contradiction can never be derived (ie. we restrict to logically consistent theories — and set aside ones such as ZFC for which consistency proofs are pipe-dreams), the whole question of explosive logic and dialetheism is counterfactual. You adopt a position (as I do) that contradiction indicates falsehood. But this sheds no light on the motivations of those with different priorities in their pursuit of logic, which is the whole point of my inquiry. So, -1. –  Niel de Beaudrap Nov 20 '12 at 1:22

Here is another motivation for dialethism - inconsistent set theory:

  1. It allows for a formalisation of naive set theory with the naive expectation that any predicate determines a set. That is, it's another solution to Russell's paradox apart from the theory of types or ZFC.

  2. So it has a universal set, and Cantor's paradox is now a theorem.

  3. This theory proves the axiom of choice, and disproves the continuum hypothesis.

  4. It disarms both of Gödel's theorems that derailed Hilbert's programme, so that programme can be revived and completed.

  5. Tarski showed that the truth-predicate is not definable in ZFC. In paraconsistent foundations an inconsistent truth-predicate is shown to be definable.

These seem like fairly remarkable achievements to me.

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This is perhaps the first interesting motivation that I have heard: expediency in set-theoretic foundations. But how does it prove AC and disprove CH? Do you have convenient references for these technical ramifications? –  Niel de Beaudrap Jun 21 '13 at 11:16
    
Its also worth noting I think, given a previous explanatory point about intuitionistic logic, is that there is a broad class of paraconsistent logics that are dual to it in some way. –  Mozibur Ullah Jun 21 '13 at 16:16
    
Expediency is a bit strong, it is another argument for logical pluralism in foundations. –  Mozibur Ullah Jun 21 '13 at 16:31
    
@deBeaudrap: I have no idea. The SEP points towards Weber, Z., 2012 “Transfinite Cardinals in Paraconsistent Set Theory”, Review of Symbolic Logic, 5(2): 269–293 for a fuller explanation. If you have access to it (its not online - so I don't) perhaps you can post a brief explanation? –  Mozibur Ullah Jun 21 '13 at 16:54
    
It's an argument for pluralism inasmuch as it is considered about as worthy a logic as others; what your answer seems to indicate is that what may make it worthy is that it admits a simple and in some senses more definite foundation for set theory, i.e. it does more with less (hence "expedient"). –  Niel de Beaudrap Jun 21 '13 at 18:33
  • Newtonian physics/causal determinism vs Quantum physics (indeterminacy)
  • Understanding the definitions of things and yet being unable to describe the essence of anything truly objectively (similar to Nagarjuna's paradox as Michael brings up but I was thinking more from an Aristotelian perspective)
  • Multiple theories of time travel and space-time involve contradictions, such as going back in time and doing something that would end up preventing yourself from being born, traveling faster than the speed of light (and having your mass increase infinitely), trying to conceive of a start or end point in time when our very notion of time presupposes that things always came before and can always come after, and so on.

Similar to what Mitch wrote, but on a different vein: I'm not sure you'll ever be satisfied with any example we can provide, because—if you are looking for "true" contradictions where two statements/ideas are correct and contradict each other (as opposed to one statement being false and the whole thing not being a true contradiction in the first place)—many contradictions we name are contradictions only because we have deemed them outside the scope of human understanding. That is, they may not intrinsically be contradictions, but based on our limited observational ability and primitive intellects, they appear to conflict. I (and I'm sure others can too) can provide you with countless examples of these, but like I said, I'm not sure we (human beings on planet Earth) can provide you with "true" contradictions as you ask.

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I don't see a "logical contradiction" in Newtonian vs Quantum physics; and people have considered the hypothesis that time-travel must lead to stable world-histories (at least in a probabilistic sense) seriously enough to characterize its power as a computational resource, so that it's not clear that even hypothetical time-travel leads necessarily to a hypothetical real-world paradox. Setting these aside, the rest of what you say expresses what I would suppose myself: that apparent contradictions say more about our understanding than they do of the world. –  Niel de Beaudrap Sep 6 '11 at 20:31

For another perspective on what might motivate things here, suppose we have a logic which has the whole continuum of truth values in [0, 1] for its truth set. A statement with truth value of 1 qualifies as true. So, it seems reasonably to infer that a statement with truth value of .999 qualifies as true. It also seems reasonable to infer that a statement with truth value of .998 qualifies as true, and to think that changing the truth value of a statement by .001 won't change it from true to false. But, this immediately implies contradictions (statements with very low truth value) as true. Now we could reject that changing the truth value of a statement by .001 (this could get made smaller, of course) won't change a statement from true to false, but some don't think this rational. One might feel tempted to think "we could just throw out the notion of statement as true or false, and instead assign a degree of truth to them." But, of course, not many people talk about (or seem to want to talk about) statements as "very true, somewhat true, exceptionally true, etc." So, with this sort of reasoning, accepting true contradictions makes sense.

Due to the demonstrated utility of fuzzy expert systems in engineering, and given an interpretation as accepting any statement with truth value in (0, 1) as accepting a contradiction, the expediency of using a logic with contradictions seems easy to demonstrate.

Also, consider a statement such as "this 6 foot professional basketball player is very tall." Now according to the perspective of classical logic, this sentence either comes as true or false, or does not qualify as a proposition when we know who "this" refers to. It's not like a contingent statement "both p and q". I simply don't see a reasonable way to deny the statement about the basketball player as a proposition. But, if we take it as either true or take it as false, then either way we can infer a falsity, since the statement also has the other truth value. So, from the perspective of classical logic, it ends up as a contradiction.

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+1 for mentioning fuzzy logic (which I would interpret as simply modelling qualities as a matter of degree rather than by sharp transitions) as an alternative to classical logic; but it's not clear whether these are strictly speaking "Dialetheistic", as in most formulations the value of A & ¬A will be at most 0.5 — granted, not zero, but bounded well away from 1. (And accepting any truth-value between 0 and 1 as a contradiction, without accepting either 0 or 1 as being so, is not exactly a fuzzy approach itself!) –  Niel de Beaudrap Sep 8 '11 at 19:18
    
You'd still take 0 as a contradiction, but you'd also take values say in (0, .5) as a contradiction. –  Doug Spoonwood Sep 8 '11 at 20:08
    
Perhaps the degree to which a statement A is deemed to give rise to a contradiction scales like the value of (A & ¬A) — in which case one might say that fuzzy logic does not allow full-blown contradictions, but only half-hearted ones. –  Niel de Beaudrap Sep 8 '11 at 21:49
    
@Niel Not full-blown contradictions sure. But, to what degree of contradictions you have, I think, depends on the fuzzy logic at hand. Some might have statements with .999999 degree of contradictoriness, I think. –  Doug Spoonwood Sep 9 '11 at 0:59
    
one can certainly devise such a "logic", although it would almost certainly be non-monotonic (conjunctions having truth values no greater than either of the conjuncts); this I thought was something people wanted to have in logics generally. Note that I'm not concerned with statements which might merely be 0.000001 true, on the axioms; I'm concerned with those which a classical logic would deem the opposite of a tautology, like (A & ¬A), but which are evaluated to be not-false (or fuzzily, quite substantially true). –  Niel de Beaudrap Sep 9 '11 at 11:14

The Tao, has a line 'The Way that can be named is not the Way'. This to me looks like a contradiction. We have already named it as the 'Way', but it is then denied that it is. But I find the statement true/meaningful.

The simplest paraphrase I can think of is, the truth that can be formalised is not the truth. Truth escapes our ever increasing ability to encompass it. It always exceeds our grasp. A mathematical analogy would be with Godels Theorem where it is shown that a formal system can express truths that are not provably true.

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I find that utterance meaningful as well, but only in the sense that I recognise that it conforms to a style which requires me to unpack its equivocation, much like replacing a reductio ad absurdum in a mathematical proof with something more direct to get a piece of constructive reasoning. In this case, after this step of pre-processing, I might arrive at the less mysterious (for good and for ill) proposition: "The Dao cannot be adequately expressed in language alone", or perhaps more provocatively, "The Dao in all its detail is uncomputable". This is a perfectly non-dialethial assertion. –  Niel de Beaudrap Jul 13 '12 at 8:24
    
As for Godel: of course, the (provable!) inability to prove everything which you might wish to observe about your mode of reasoning just echoes the uncomputability of the totality of knowledge -- as distinct from the idea that somehow there are things which are both true and false. –  Niel de Beaudrap Jul 13 '12 at 8:32
    
I find the utterance meaningful, as well as the liar's paradox, but both (in different ways, I don't consider problematic for logic. For me, the taoist line depends on context, it is metaphorical (if you think you can name it, then you really don't have it yet). The liar's paradox (and Goedel's reformulation) just needs a 'larger' technical universe to map our intuitions about it so that it is formalized. –  Mitch Jul 14 '12 at 23:05

This is tangential to the main thrust of your question. Intuitionistic logic doesn't dispense with the law of non-contradiction, but it does deny the law of the excluded middle. This means that there are more than 2 truth values and these form a poset. Now, lets reinterpret the law of non-contradiction to mean that every statement can have only one truth value. Then this is manifestly false, a statement can have two truth values where the second is comparable to the first in the poset of truth-values. Of course, this does mean that one truth value is redundant. But I think this angle is pretty interesting nonetheless.

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"A statement can have two truth values", to me, contradicts the meaning of the noun 'value' as it pertains to mathematics (for the same reason that in my field of work, describing a quantum system as being in 'two states at once' is impossible by virtue of the meaning of the word 'state'). Given an appropriate formal system, I might admit that the truth value might be constrained by a set of possible truth values without being able to say precisely which; but one can then develop an alternate notion of truth based on the ensembles rather than the individual elements. –  Niel de Beaudrap Jul 12 '12 at 11:15
    
@beaudrap: But doesn't that mean that you are committed to a bivalent system of truth and that you're looking at the world through bivalent glasses? That it can be made to work, for me, at least,doesn't neccessarily imply the truth of it. –  Mozibur Ullah Jul 13 '12 at 1:00
    
@beaudrap: one of the interpretations of intuitionistic truth is justification. To then say a statement has two truth values t1<t2, is to say that you have two justifications for the statement, and that one justification is better than the other. This seems eminently reasonable. Of course, justification is not truth; but then again I'm interpreting the logic. –  Mozibur Ullah Jul 13 '12 at 1:06
    
A start would be to abandon a notion of 'the' truth value of a proposition if it may have more than one truth value; or at least have a good hard think of what purpose definite articles and unique identifiers play if your logic isn't definite enough to require singly-valued propositions. As for justifications, wouldn't the stronger proof represent the proper truth value -- or, if you had two incomparable proofs, might not their direct sum, in some sense, represent the truth value? Or a characterisation in linear logic, of the resources required for any proof? That's still single valued. –  Niel de Beaudrap Jul 13 '12 at 8:13
    
@beaudrap: in a sense yes, since one can deduce the lesser truth value from the greater; and the same goes for the direct sum. However, I think this is still missing something. Summing over to return to a single-valued truth is to look at the world in a particular way. A direct sum can always be decomposed to its summands, but a single proof may not be. –  Mozibur Ullah Jul 17 '12 at 22:31

There are a million consecutive sevens somewhere in the decimal expansion of Pi.

Suppose we can prove the digits are evenly distributed (and meet a few other conditions) and can also prove that the only way to know where they are is to find them, and we have not found them. In this case, "we know there are a million consecutive sevens in the expansion" is defensible as true and "we do not know there are a million consecutive sevens in the expansion" is also defensible as true. And it's not because of any vagueness in the terms, it's because of the vagueness of the expansion.

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I am not sure of the motivation of your statement. Are you proposing that the truth-value of this is somewhat ambiguous, in the sense that we (probably) don't know at the moment whether there is such a subsequence, and that therefore it ought to be regarded as both true and false simultaneously? –  Niel de Beaudrap Sep 20 '11 at 16:11
    
That doesn't sound like a contradiction. Possibly hard to know, but not a contradiction (pair of statements of opposite truth value). –  Mitch Sep 20 '11 at 18:40
    
It's not just something we don't know at the moment. It's entirely possible we could prove that we can never know that there are not a million consecutive sevens (say we can prove exhaustive search is the only method to find them). Yet if we can prove the digits are evenly distributed, we may 'know' that they must be there somewhere. –  David Schwartz Sep 20 '11 at 22:10
    
So what you are saying is that this is a statement which we may never know constructively, but which we may come to know via the probabilistic method; making its unprovability a matter of which logic you adopt, but still not in any obvious way both true and false using any particular logic. –  Niel de Beaudrap Sep 21 '11 at 9:14

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