Motivations for dialetheism?

Question

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)?

Answer 1

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.

Answer 2

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.

Answer 3

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).

Answer 4

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).

Answer 5

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.

Answer 6

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.

Answer 7

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.

Answer 8

• 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.

Answer 9

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.

Answer 10

If you accept dialetheism, you have to develop some new logical systems to make it work out. We don’t need people running around like, “The Liar’s Paradox! Therefore, it follows that NASCAR is a sport!” See how that could get out of hand quickly? Obviously no one wants to make every contradiction true. So we have this hilarious image, an impetuous child angrily stomping her foot and screeching, “Contradictions CAN’T be true! They just can’t!” If you’re a Randian, you are perhaps stomping your foot here and just insisting that contradictions can’t be true, and that’s all there is to it.

Now, to review, I think that the problem is not as weak as 'we revise our theories', or 'we have false beliefs', or 'we make mistakes'. No, -- the problem is the claim that some contradictions really are true. And, my favorite example of the relevance of this is non-Euclidean geometry, which had people seriously scrambling for a way to reason about inconsistent information without lapsing into absurdity. People were arguing that mathematical obsession with non-euclidean geometry was a waste of time. Today we would say that of course, non-euclidean geometry is perfectly legitimate. But, the rise of non-Euclidean geometries, in the minds of some, was a war on Euclidean geometry’s claim to the shape of space.

It was certainly a shock to Frege who would claim that either Euclidean geometry is true or Non-Euclidean geometry is true, but not both. Frege, no less! Committed himself to this view!

I don't think anybody accepts true contradictions. Maybe that's just me quibbling about semantics, though, because I'll also say that contradictions are apparently 'tolerated' sometimes as long as they don't spoil something else. It actually happens all the time, and again is not news at all, in the world of axioms, definitions, postulates, and proofs of propositions from these three things.

Ambiguity causes confusion in logic, and the issue is larger than it may seem. Suppose that I say 'it is raining', and then I say 'it is not raining'. Well, I've contradicted myself, and both statements can't be true. But then I say, 'well, but it is raining in Los Angeles, California and it is not raining in Phoenix, Arizona, I meant'. That's fine. But it's not always clear what people mean. And there is a whole world, in mathematics, logic, and formal systems, of so-called primitive notions, which are undefined concepts, and in particular, a primitive notion is not defined in terms of previously defined concepts. They are only motivated informally, usually by an appeal to intuition and everyday experience. The concept of the set is an example of a primitive notion, in set theory. In Euclidean geometry, under Hilbert's axiom system, the primitive notions are point, line, plane, congruence, betweeness, and incidence. So okay, we employ the expressions without explaining their meanings. Maybe you think I am joshing you. But it's true, and in such cases, there will be true contradictions, or somesuch, due to the ambiguity.

It can be hard to distinguish between definitions, versus mere attempts at explication of something which is being given the status of a primitive, undefined term. Again: in Peano arithmetic, the successor function and the number zero are primitive notions. And this point about definitions, requires that we supply a limited context within which something can be true, without being logically true. Such as 'it is raining', for a simple example, but most any true judgment is only true in a certain context, actually. Beyond 'A is A', you really can't even do math without -- well, put it this way, there are the ways in which logical systems must be extended to permit the derivation of arithmetical truths. To extend a formal system of predicate logic with nothing but logically valid axioms so that it does capture arithmetic, we must add axioms which are not logically valid. Arithmetical truths are not logically valid. And these are commonplaces in metamathematics. Again: that arithmetic expressions are satisfiable at best, and not logically valid, is common knowledge in metamathematics.

So this dialetheism is not just for punks and hippies, although I would start by reviewing what that term means that I just used: 'satisfiable'.

Answer 11

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.

Answer 12

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.