Dialetheia and dialetheism
Anil Mitra, © October 2023

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Introduction | The law of non-contradiction | Dialetheia | A trivial example | A non-trivial example | A truly non-trivial example | Reckoning with dialetheia | Logic for dialetheism | Is reality contradictory?

Dialetheia and dialetheism

Introduction

This is a short treatment. It is adjunct to the essential way of being. For further discussion of dialetheism, see the little manual (this site) and dialetheism (Stanford Encyclopedia of Philosophy, SEP, which has further references).

This essay may be improved, especially with regard to treatment of the void and the universe as contradictory (‘contra-dictory’) beings.

The law of non-contradiction

The law of non-contradiction (lnc) asserts that it is impossible for a sentence (or fact, statement, assertion, or proposition) to be both true and false. In the west, lnc is a time-honored principle (law of logic), at least since Aristotle, which seems at least intuitively obvious (for example, that it should be 10 AM here and now and not 10 AM here and now, is patently absurd—at least, seemingly so).

Indeed, violation of lnc is more than seemingly absurd. Let A be any statement and let A and its negation -A be true. Let B be any (other) statement. Then (A or B) is true, but since A is false, B must be true (and since B is any statement, it is not necessary to repeat the argument for -B). Thus, in classical sentence calculus, truth and falsity of given statement implies truth and falsity of all statements. This is the well-known principle of explosion of classical logic, from falsehood, anything follows (i.e., from the falsehood of a truth).

These have been strong arguments that lnc holds. Indeed, even though there are good recent arguments against lnc in recent (philosophical) logic and which (as of 2023) are gaining a foothold in philosophy, the majority of philosophers and logicians still hold lnc to be a true and fundamental principle of logic.

Dialetheia

A dialetheia is defined as a sentence (proposition) such that both the sentence and its negation are true. dialetheism is the view that there are dialetheia.

Definitions of course do not entail existence. Indeed, a dialetheia would (at least) seem to be a violation of our sense of logical principle, particularly lnc—the law of non-contradiction.

Whether there are dialetheia or not, eastern and western philosophy, ancient through recent, has tendered many apparent dialetheia (see the SEP and Wikipedia articles).

A trivial example

It is 10 am and not 10 am. How could that be? It is 10 am in Kolkata but not 10 am in London. It is of course a trivial example. Since time of day depends on place, it is understood that saying it is 10 am, presumes some particular place. Therefore, ‘it is 10 am’ means ‘it is 10 am here’ or ‘it is 10 am in Mumbai’ and so on. ‘It is 10 am and not 10 am’ is not a dialetheia.

However, what if we do not know where we are on Earth. There might be some sense to ‘It is 10 am and not 10 am’. Dialetheism might therefore be useful (even though not necessary in this case). Are there essential dialetheia?

A non-trivial example

Consider the Thomson lamp paradox. At time 0s (seconds), a lamp with a switch is on. At s, it is switched off, at s, on, at s, off again; switching continues as it approaches 1s. Thomson asks At 1s, is the lamp on or off? He argues that it is both on and off (the paradox), which is impossible (by lnc), and therefore, per Thomson, his lamp and its process cannot exist (Thomson was investigating ‘supertasks’ that proceed at an infinite rate and his conclusion was that supertasks are impossible).

But it is not a contradiction, per the following argument. It would be a contradiction if there were a lamp state ‘on and off’. But there is not. What obtains is that infinitely many lamp states occur at time 1s, but they are distinct states. They are simultaneous states, and we normally think of simultaneous states as singular (one or other but not both), and therefor ‘both on and off at time 1s’ as contradictory. However, in seeing it as contradictory, we were assuming one situation corresponds to one time, which, however, is not logically necessary. That is, we are considering the lamp from a mathematical or logical perspective, not a physical one (but perhaps there are other cosmoses with laws of physics that allow infinitely fast processes).

Thus ‘the lamp is on and off at the same time’ seems like a dialetheia but is not a true dialetheia. On the other hand, we do not seem to have the mathematics to describe infinitely many states for a system occurring at an instant and therefore, even if there are no true dialetheias, dialetheism may well be useful.

Though non-trivial, Thomson’s lamp does not yield a dialetheia or true paradox (supertasks are logically possible, but perhaps we have no number system capable of describing ordinary and supertasks in a unified framework).

Note that intuiting a switch on and off at the same time may be helped by considering a switch on for one tenth of a second, off for a tenth, on for a tenth, and so on. Then ask, Is the switch on or off for any interval a second in length? The answer, It is on and off, is now clearly not paradoxical or contradictory. But now instead of a tenth of a second, consider an interval so small that it is below the ability of any physical instrument to measure. Such an instrument would register on and off even though it is never in an on and off situation. Analogously, with a lamp switching on and off with infinite frequency, there is no on-off situation even though there is on-off at any given time. The real number system is inadequate to describing switching—processes—with infinite frequency (perhaps Robinson’s theory of infinitesimals or the surreal numbers are adequate, i.e., perhaps there is a number system that is a unified framework for ordinary and supertasks).

As an aside, note that from mathematical theories of infinitesimals and infinities, there is a hierarchy of super-slow to super-fast, and tasks are super-slow or super-fast relative to other tasks. Also, there are super-super and super-super-super tasks and so on—see the Wikipedia article, supertasks.

A truly non-trivial example

This example is from the essential way of being and it is assumed that the reader has read that essay through the section on metaphysics.

In the essential way, it was argued that the existence and nonexistence of the void were equivalent—i.e., it is valid to assert that the void exists and does not exist.

This is a true dialetheia. Dialetheism holds (that is, there are dialetheia, but it is not asserted that all contradictions are dialetheia).

Reckoning with dialetheia

A simple reckoning is as follows. Our sense of the term ‘contradiction’ unpacks in two ways. The first is ‘contra-diction’ in which a sentence asserts something true and false. The second sense is what the sentence entails of reality. An example is ‘It is 10 am and not 10 am’, which cannot be true and is a ‘contra-real’. Thus, it is 10 am and not 10 am, is simply impossible—a contra-real, not a dialetheia. However, as we have seen ‘the void exists and does not exist’ is a contra-diction but does not entail a contra-real (i.e., the unhyphenated form, contradiction).

Logic for dialetheism

A simple logic is as follows. Since Bertrand Russell’s famous paradox regarding sets that do not contain themselves (they contain themselves if and only if they do not), we have learned that to have valid application, that is, to avoid semantic invalidity, systems of language and logic, may need to have a restricted ‘universe’ of application. We now recognize that there are dialetheia, but they are not in the universe of validity of standard propositional calculus.

What would be a formal logic for dialetheia? There are three valued logics that can incorporate dialetheia (as in the little manual on this site and paraconsistent logic, SEP).

Is reality contradictory?

Some thinkers have argued that reality itself is contradictory. It is apparent from discussion so far that there are and cannot be contra-reals, i.e., reality is and cannot be contradictory.

A refinement of the claim that reality is contradictory is the claim that reality is such that there will be contradictions in describing the real. This does indeed seem to be a substantial claim, for, if we did not unpack the term ‘contradiction’ it would say that paradox is unavoidable. But we have unpacked the term into ‘contra-diction’ and ‘contra-real’ which defuses paradox as essential (thinking this way, it is not base reality that is paradoxical but—perhaps—reality mapping reality; this, however, may be simply the fact that mis-takes can be made, rather than essentially paradoxical meta-reality).

But can we not then say that reality is such that we must find contra-diction? We do find contra-diction, but it seems that many contra-dictions can be defused by more careful descriptions. But in existing that it does not exist is a property of the void which is not a contra-real even though it is a contra-diction and certainly seems like a contra-real.

The void is a prime example of a contradictory object. It is argued to be contradictory in that it is an object, but it is also what remains after all objects have been removed, and therefore not an object. But though this is a contra-diction it was argued above that it is not a contra-real. Let us think in terms of a simple analogy. Suppose there are 10 things (chairs, people, …) in a room and all (10) are removed there are now 0 things (of the given kind in the room). But if there are 0 things and all (0) are removed, there are now (still) 0 things. That is, if there are X things, with X finite, and X are removed, what is left is not X when X is not 0 but is X when X is 0; this is of course trivial for the usual number systems, but we will now make a similar argument regarding objects that is non-trivial in its meaning and consequences. Similarly, if a standard object is removed from a collection, the object does not remain—for it to remain would be a contra-real; but removing the void object leaves ‘the’ void object—it has contra-diction but is not a contra-real. An interesting aside is to consider what happens when an infinite number of things is removed from an infinite number of things. Without further specificity on the precise infinity, the remaining number of things is indefinite. What is interesting about this is that it suggests that everything—the universe—is also contradictory and, further, that the nature of its contradictoriness is not identical to that of the void (see the essential.html way of being for definition of the term ‘universe’).

The existence of the void derives from equivalence of its existence and nonexistence. Can we unpack “in existing it does not exist”? Perhaps in that the void is eternal and in that in eternity there is an eternity of potency and an eternity of the absence of potency—a parallel to the thought that having no laws is lawlike but not contradictory if it is only a sub-eternity in which the potency of no laws is manifest.

Isn’t that a clumsy formulation though? It is—but there is a resolution. The void does not require the emergence of all possibilities; rather, it does not prevent it: the possibility of emergence is there and without balance between emergence and non-emergence, nothingness would not be nothingness.

Here, a good theory of meaning is invaluable. Given the concept ‘the void’, there is an object. However, in the sense of the void as absolute nothingness there is an object but, it seems, no being; but there may be void objects that are local in terms of accessibility for which there are also beings. This is a good resolution of the problem of the void. On the other hand, perhaps there are other concepts of the void, e.g., minimal being, that are workable—

Tentative definition—the void is the minimal being that remains after all other beings are removed. An explanation that makes the definition clearer is—the void is (a) a being that remains after all other beings are removed and (b) the minimal such being.

Before closing this work, there is an interesting issue arising from dialethic studies from Plato to Graham Priest. It is the problem of the one (and the many). The analysis that follows is suggested by Priest’s work but does not follow and is rather neutral toward his analysis. Focus on the word ‘tree’. You are likely to visualize a tree and, if you are outdoors, to look at a tree. The tree has parts. What are they? It is really rather indefinite—are they the elementary particles, the interactions, the leaves, and the branches (and more, or combinations of the same)? That is a problem in itself for we do not know whether those parts are fundamental. Indeed, we may ask “Though there are things we see as parts do things have ultimate parts at all?” Let’s pass over that issue except to note that if there are no true parts, perhaps there is no true one or oneness of the tree. What I want to look at is What constitutes the oneness of the tree? Particularly, Is there an intrinsic oneness of the tree, over and above its appearance as an entity and the fact that all the parts, whatever they are, are together at this time? Note, by the way, that this is not the question of what holds the parts together, for those things are the interactions or fields or forces which are parts as much as are the particles. Let us say there is something that constitutes the oneness, which is over and above the mere fact of the tree and over and above its naming (“the tree”) and the fact that the parts are hanging together. We grant that something being (existence).

But clearly, given the constitution of the tree, as described, there is a good point of view, in which there is nothing over and above the naming and the fact. That is, we are bound to find the oneness contradictory if we grant its existence. On the other hand, there is no necessary reason to grant being to the oneness. And if we do not, there is no contradiction—just as the concept of a square-circle is a contra-diction but not a contradiction (in the sense given earlier). This suggests that there is no oneness; i.e., we may conceive it, but it does not define or specify a being. Now the one has been regarded as dialethic, but this argument would defuse the dialetheia (this would also deny reality to Priest’s notion of ‘gluon’ as constituting the oneness of things). This is yet another reason to think that there are no real dialetheia but there can be as if dialetheia, which may be a result of a choice or limit of some language forms, and which it may be useful to treat as dialetheia and according to a system of paraconsistent logic.

The argument regarding the void needs to be refined and until that is done and the argument about ‘the one’ sharpened, the issue must here be regarded as open.

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Paradoxical or not, the void—the real—is most interesting (and it is perhaps here that any indeterminacy of quantum theory has its source).