Dialetheia and dialetheism
Anil Mitra, Copyright © August 2023 – December 2023

Contents

Dialetheia and dialetheism

Preface

A dialetheia is a sentence such that both the sentence and its negation are true. Dialetheism is the view that there are dialetheias.

Thus, a dialetheia is a true contradiction and dialetheism is the view that there are (some) true contradictions.

This essay begins with an account the place of dialetheism in my thought (https://www.horizons-2000.org).

Then, the document will (i) introduce dialetheism, pointing out some of its apparent problems—the apparent absurdity and the problem that in standard propositional logic, a dialetheia implies that all statements are true (and false), (ii) defuse the problems in an analysis that will suggest that all dialetheias result from suppressing distinctions or introducing distinctions when there are not any, (iii) take up a range of examples of dialetheias and uses of dialethism—as further examples to extend the analyses and to show the uses of dialetheism, (iv) raise the question of whether reality is contradictory, i.e., of whether absolute dialetheism (thought about the entire real must be dialetheic) holds, i.e., whether there are dialetheias whose dialetheic quality cannot be resolved by making sufficient distinctions (relative dialetheism will be the view that there are dialetheias, whose dialethic quality can be resolved), (v) suggest that absolute dialetheism does not hold, but that relative dialetheism does hold, and that its study may be useful (one might say that reality is relatively contradictory but that would have nowhere near the significance of absolute contradictoriness).

This is a short treatment of dialetheias and dialetheism—an adjunct to the essential way of being. For sources and further discussion of dialetheism, see the little manual (this site, has a treatment of paraconsistent logic), Dialetheism - Wikipedia, and Dialetheism (Stanford Encyclopedia of Philosophy), which have further references.

My interest in dialethism

This section may be of interest regarding dialetheia but is not essential reading for the purpose of this essay (it does shed light on the place of metaphysics in the larger context of my thought and life). The real interest of this section for this work is that of the intersection of (my) metaphysical thought and dialetheism.

When young, I found significant meaning in understanding the universe and our place in it. My story begins with my interest in science and philosophy as a way of understanding.

An aspect of my interest in philosophical thought was that it enables a broad framework and encourages awareness of the limits of science, particularly the limits of the science of any given time.

The interest in science was in (i) the fundamental theories of science, especially of physics, cosmology, and biology (ii) history and method of science. Here, it is enough to state my conclusions regarding science. It is fundamentally hypothetical—it posits entities (e.g., particles and fields) and theories. It is empirical but incompletely so—it is hypothetical to project the theories beyond observation, which is good when not too far from the empirical base, but not so in ultimate reckoning. The question, then, is what lies beyond, and what can we know of it.

Initially, I focused on what possibilities general relativity and quantum field theory held. I was aware that for a cosmos to emerge from nothing need not violate conservation of energy, for gravitational energy (negative) and matter energy (positive) can balance.

This led to the thought that the universe might emerge from nothing (ness). I had intuitions about the equivalence of nothingness and the universe, but they did not lead far until I thought to focus on the properties of the void.

There is a history of philosophical (metaphysical) thought on the void, but I did not find it sufficiently informative. Two aspects of that thought were useful (i) that we can think of the void in metaphysical, i.e., supra-scientific terms (ii) a range of detailed considerations on the void and reasoning about it.

Fundamental concerns were (i) what the void is (ii) whether it exists. I saw that the concept of the void is dual to that of the universe—the universe is all that there is (in the most inclusive sense of ‘is’) and whereas the universe contains all being, the void contains no being; the universe is the whole and the void is its complement. I came up with a number of demonstrations of the existence of the void (which I doubted and continue to doubt), particularly that the existence and nonexistence of the void are equivalent and so the void may be validly taken to exist. From the existence of the void, I developed a system of metaphysics, which can be read at https://www.horizons-2000.org, and which, in its description and analysis of the real has gone further than anything I have read. Particularly, it concludes that the universe is the realization of (logical) possibility and that this has great significance for (our understanding of) the universe and the place of ‘beings’ in it.

What is pertinent here is (i) that ‘equivalence of existence and nonexistence of the void’ or ‘the void exists and does not exist’ is dialetheic and (ii) the conceptual space into which my thought had moved was from science to metaphysics to logic. To invoke logic may seem sterile but it is not, for formal logic is sterile in what it requires but ultimately rich in what it allows (and study of formal logics is one approach to study of possible and therefore realized beings). Furthermore, (iii) these thoughts led to insights into the nature of logic and (since logic is traditionally only about inference) whether logic may encompass existence of necessary facts (I concluded that it may, in rather the same way that science is about both theories and facts). All this and more had been established by around 2010 or earlier (but continued to develop in real and metaphysical reach and sophistication).

It was about four years ago, in 2019, that I encountered the thought of Graham Priest and others on dialetheia and dialetheism. I began to read on the matter, my understanding began to grow, I saw dialetheism as useful in my thinking, and, finally, I thought it would be useful to my thought to write up what I knew about dialetheism. This work is the result of my initial writeup of August 2023, which has now been rewritten and edited a number of times.

In this work readers will see overlap with and divergence from ‘standard’ works on dialetheism. Some of the examples are mine. My ‘void’ is Priest’s ‘nothing’ and my ‘universe’ is his ‘everything’. A fundamental divergence from Priest’s (and Hegel’s) work is that I do not think the universe to be contradictory. Hegel seems to think that reality is truly contradictory, but, though I do not dismiss what he says, I do think that his language is a metaphorical attempt to capture features of the ultimate. Priest does not think that the universe—reality itself—is contradictory; rather, he thinks that it is a property of reality that our descriptions of entire reality must be contradictory. In contrast, I think that dialetheias arise in using language that is not sufficiently precise, I suspect that the language of contra-diction can be overcome with sufficient precision, and I am not as enthusiastic as Priest concerning the use of formal systems such as predicate calculus and axiomatic mereology in describing the dialethic aspect of reality. Instead, I think careful attention to thought and description in relation to the real is the way—an effective way—to go. However, I do not at all discount dialetheism; rather, I think that dialetheic description of the world is powerful and exciting (though not ultimately necessary); and I do not discount the usefulness of formal systems in (i) precision of thought (ii) in generating insight, generally.

The reference to Priest’s writing is significant because he is perhaps the foremost current dialetheist. However, this work is not intended as a review or criticism—positive or negative—of Priest’s work.

Dialetheism – an introduction

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 dialetheias (note that dialetheism is not the view that all contradictions are true).

Definitions of course do not entail existence. Indeed, a dialetheia would (at least) seem to be a violation of our common sense (rather obviously) and of logical principle, particularly lnc—the law of non-contradiction, e.g., the assertion that ‘no proposition and its negation are both true’, which has been and remains the majority position at least since Aristotle. Indeed, there is good reason to hold lnc, over and above the seeming absurdity of ‘true contradictions’, for in standard two-valued propositional calculus, the truth of a contradiction, implies that all statements are true (and false), i.e., the standard calculus is ‘explosive’ as follows.

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. 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.

However, consider ‘It is 10 am,’ and ‘It is not 10 am.’ The apparent contradiction might be resolved by clarifying ‘It is 10 am in Kolkata but not 10 am in Beijing.’

‘It is 10 am’ usually means that it is 10 am in some particular place and thus there is no real contradiction above.

However, what if we do not know where we are on Earth? There might then be meaning and truth to both ‘It is 10 am’ and ‘It is not 10 am’, i.e., to ‘It is 10 am and not 10 am.’

Thus, while the example is not a real dialetheia, it might be useful to treat it as one. Dialetheism, even if it does not hold, might be useful. Whether there are dialetheia or not, both eastern and western philosophy, ancient through recent, have tendered many apparent dialetheia (see articles linked in the preface).

What we saw in this example is that usage allows three meanings to ‘It is 10 am and not 10 am’. In one case it is ’It is 10 am here but not 10 am elsewhere’, which is not a contradiction at all. In the second, it is ‘It is 10 am here and not 10 am here,’ which is a contradiction, but (can) not (be) true, and therefore, not a dialetheia. In the third case it is ‘It is 10 am somewhere and not 10 am somewhere’ which is (has the form of) a dialetheia.

Two observations about the third case are (i) the dialetheia arises by being imprecise in meaning (ii) while it may be useful to be imprecise, the contradiction—and sense of absurdity—can be removed by being more precise.

This raises the question whether all dialetheias are indeed true, only because what they assert is imprecise in that the mode of descriptions suppresses distinctions (e.g., between it is 10 am here and 10 am somewhere else). That is, (i) if we define absolute dialetheism as the position that there are dialetheias that cannot be converted into non-dialetheias by making sufficient distinctions (ii) the question raised now, is whether there are absolute dialetheias. It is now natural to introduce relative dialetheism as the existence of dialetheias whose dialetheic property can be removed by making relevant distinctions. The previous question then becomes, “Are all dialetheias relative?”

Defusing dialetheism

To defuse the problematicity of dialetheism, two issues need to be addressed, (i) the problem of explosion in standard propositional calculus and (ii) the seeming absurdity of a ‘true contradiction’.

One way to resolve the problem of explosion is to see that standard two-valued logic need not be the logic for all propositional contexts (i.e., those in which the structure of the proposition does not figure). Let us motivate this consideration.

In ordinary English, assertions are regularly but tacitly held to be either true or false, but not both, and, while this is questionable and questioned, it informs our common thinking about logic. That is, it has been commonly held that—Assertions are either true or false.

However, consider the assertions, ‘Earth’s fifth moon is green’ and ‘Plato’s philosophy is ferpiculous.’ Those assertions are arguably neither true nor false, the first, since Earth does not have a fifth moon, and the second, because it does not have (adequate) meaning. So let us modify the opening statement above as follows.

Assertions that are meaningful and about the world, i.e., about something real, are either true or false (and many semantic and logical paradoxes may be seen as the result of admitting sentences that do not satisfy the stated condition).

So, ‘Assertions are either true or false,’ may be modified as follows. Assertions are either true or false or neither.

But can assertions be both true and false? That is, can contradictory assertions be true? We already seen that they can be true. Thus, standard propositional calculus cannot be a logic for dialetheism. This ought not to be a concern, for we have seen that the standard calculus is inadequate to all propositional forms. 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; syntactic consistency is not enough. We now recognize that there are dialetheias, but they are not in the universe of validity of standard propositional calculus. In retrospect, of course, this ought to be obvious.

But it would be absurd if all contradictory assertions could be true. What would the status of a logic for dialetheia be? There would be true propositions, false propositions, and the rest, which might have no truth value or both (true and false). For dialetheism there is an obvious need for ‘both’ but no obvious need for ‘neither’. There is a three valued logic that can incorporate dialetheia, as in the little manual on this site and in Paraconsistent Logic (Stanford Encyclopedia of Philosophy). In this logic, statements can have one of three truth values, ‘true’, ‘false’, and ‘both true and false’, designated, t, f, and b, respectively. If the logic is applied to a universe of statements that have no truth values b, it reduces to standard the two valued propositional calculus.

Let us now turn to the problem of absurdity. We will resolve the seeming absurdity with reference to the theory of meaning of the essential way of being, which, in on meaning, was shown to have necessity and adequacy for a good theory of linguistic meaning (the point was made there in part because it has been remarked in the written literature that there are no adequate theories of linguistic meaning). The essence of the theory is that a meaning is a referential concept (e.g., a proposition) and an object (to what the concept refers).

To resolve the issue of contradictions being ‘real’, first, see that ‘contradiction’ unpacks in two ways—(i) as contra-diction, i.e., the word or syntactic form is that of a contradiction, (ii) contra-real, which means ‘reality contradicting reality, i.e., not being itself’. Now, it is quite possible to see multitudes within reality as one and therefore an apparent contra-real but in fact, there is no contra-real; and the contra-diction and the apparent contra-real are the result of conflating the multitudes; which may be an error of description or a convenient way to talk of the multitudes for the purpose of communicating a maxim or a problem of grasping the real (and which may be expressed as contra-diction or metaphor).

Now, return to the example of (*) ‘It is 10 am and not 10 am.’ It is obviously a contra-diction but not a contra-real. And if we want to resolve the apparent contradiction, we refine our description and say, (a) ‘It is 10 am in Kolkata and not 10 am in Beijing,’ or (b) ‘It 10 am in Kolkata and not 10 am in Kolkata’ (where the times and place are precise). The original statement, *, is a contra-diction, but in the meaning (a) it is a dialetheia and there is no contra-real (b) there is no contra-real (because there is no real corresponding to b) and there is no dialetheia.

Liar paradox

Consider ‘This sentence is false.’ The sentence is true if and only if it is false. This is one form of the liar paradox. Dialetheism has been promoted as a resolution.

But what does ‘This sentence is false’ mean? Consider, instead, ‘This sentence is true.’ It is true if true and false if false. It refers to its truth value without it being the case that it has a truth value. Similarly, ‘This sentence is false’ refers to something that does not exist. It is neither true nor false.

Absence of truth or falsity is also seen in ‘Earth’s fifth moon is green.’ This case is interesting for another reason—we might say that since the fifth moon is not not-green, it is therefore green and is therefore dialetheic. But that confuses not having a truth value with having the truth value not-true.

Going back to the liar, from ‘it has no truth value’ we might read that it is not true and that it is not not-true, and therefore a dialetheia. And that is also confused.

In the case of the liar paradox, apparent dialetheism arises from assigning meaning where there is no meaning, i.e., in assuming there is an object, where there is no object of the kind assumed. Why defuse a paradox that is not. Of course, there are good reasons—to prevent other paradoxes of the kind from emerging. But, still, the Liar paradox is not a dialetheia.

Philosophy and metaphor

In ‘philosophical thought’ there are seeming contradictions that are commonly given as cases of dialetheias that are not serious examples as dialetheias (except, however, that the logic of dialetheism may contribute to the meanings of what is said). Here are some common examples—

“No man ever steps in the same river twice” (Heraclitus). The literal meaning is that the river does not remain the same, which is true but hardly a dialetheia (Heraclitus’ intended meaning was the constancy of change in reality and in the life of a human being, which may be interesting but is not dialetheic).

Being is ineffable (this Heideggerian saying says something about being, which is therefore effable if it is known to be ineffable). The effability is at a level that stands above the ineffability, which defuses the possible dialetheia.

Whereof one cannot speak, thereof one must remain silent (Wittgenstein). Wittgenstein was making a serious statement about a range of topics in philosophy; however, it was based on a model of the real—the world is all the facts—and Wittgenstein did not remain silent, by talking about the model, i.e., recourse to a more inclusive but informal language).

Such kinds of dialetheia-by-shifting-level-of-discourse are interesting, but hardly significant as dialetheias.

Thomson lamp paradox

Consider the Thomson lamp paradox. To begin, a lamp with a switch is off. After (1/2) minutes it is turned on; after another (1/4) minutes off, (1/8) minutes on, and so on, without end to the number of switchings. The time intervals sum to 1 minute, at which time the time intervals are zero, so the rate of switching is infinite. Thomson asks At 1 minute, 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 1 minute, 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 1 minute’ 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 (there are perhaps ways around the question of physical possibility, e.g., diminishing amplitude of switching or alternate physics, but that is not what we are looking at here).

Thus ‘the lamp is on and off at the same time’ seems like a dialetheia but is not a 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 dialetheias, dialetheism may well be useful.

Though not altogether 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. Depending on the construction, such an instrument might register a flicker or a range, which might be interpreted as on and off, even though the lamp is never in an on-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—for rates that are infinite or infinitesimal relative to one another).

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.

The void and the universe

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

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

Is this a dialetheia? It is. But is it an absolute dialetheia, i.e., one whose contradiction can be removed by careful discrimination, without removing its truth? The contradiction can indeed be removed by noting that for the void, nonexistence is existence. There is the question of whether we ought to allow this for perhaps it is no more than a verbal construction. On the other hand, the void cannot disallow existing things to emerge from it, so its nonexistence is equivalent to existence of the (limitless) universe and, therefore, to existence of the void itself. We might say, then, that while existence and nonexistence are contradictory for most objects, they are not contradictory for the void.

A second example arises when the fundamental principle of metaphysics is accepted. The principle states that the universe is the realization of all possibility. It follows that for any region and state in extension (spacetime etc) of the universe, neighboring states are not at all determinate (that is, even if there are probabilities, there are no certainties); this is absolute indeterminism. On the other hand, whatever state a being is in, it will be in all other states that are possible for it. This is absolute determinism. In other words, both absolute determinism and absolute indeterminism hold. This is only superficially dialetheic because absolute determinism and absolute indeterminism are not contradictory in their conception.

A third example is from Everything and Nothing (2022) by Graham Priest and Markus Gabriel. Everything and nothing in that essay are the same as the universe and the void, respectively, in my essay, the essential way of being. Priest’s everything contains all objects, even abstract ones, so is it really the same as the universe as I conceive it? Yes, for my universe is all being, and my being is that which is the object of concept and so contains concrete, abstract, and as if objects (and non-existent objects as well, if you wish). As an aside, I note that, in my conception, abstract objects are in the universe without requiring specification of a (Platonic) universe of abstract things, for my concept of abstract being is formed by abstraction from the universe, perhaps by removing causal, spatial, and temporal characteristics (as alternative to positing abstract objects linguistically, whether in terms of natural or axiomatic language).

Gabriel argued that objects are what they are by being part of a context that he (Gabriel) calls a field of sense which is its network of relations to other objects. An object is a sub-part or ‘proper part’ of its field of sense (that is, it is a part but not the whole), i.e., if o is an object, and f(o) its field of sense, then o < f(o) (where < is a symbol for sub-part). Now, Graham considers the object e or ‘everything’. Since o < f(o) is general, e < f(e). But, since e is everything, f(e) must be part of e, and since e and f(e) are distinct, f(e) must be a sub-part of e, i.e., f(e) < e. That is,

…e < f(e) < e < f(e)…

Gabriel argues that this is impossible and therefore the world in the sense of ‘everything’ does not exist. On the other hand, Priest argues, with examples, that this is possible, and therefore ‘everything’ is contradictory and that …e < f(e) < e < f(e)… is a dialetheia. However, it does not in fact follow from the examples, in which e < f(e) makes good sense, that it is true for all objects, particularly the universe. Indeed, what Gabriel’s argument amounts to is that many things are what they are as parts of a field of sense relations (true), this necessarily extends to the universe (not demonstrated). This is taken up later, in Is reality contradictory?

Priest has also argued from mereology that nothing is contradictory in being an object and not an object (a formal proof can be given under a particular mereology). I do not see this as a conclusion about the world. Rather, the conclusion about the void as part of the world was given above and what Priest has shown is that a particular mereological system fits the world (better than some other mereologies do).

A question related to ‘everything’ is whether an individual can know everything. A logical rather than empirical problem with knowing everything is that if I know everything then I know myself knowing everything, which means that I know myself knowing myself knowing everything and so on. Here are three ways in which that it is possible. In the first, I know an abstract of everything. This is of course, not truly knowing everything. In the second, I am an infinite being of the same order of the infinity of the universe and therefore, even though the latter is larger in containing things that I do not, there is a 1-1 or even many-1 mapping from my mind to the universe. In the third, the universe is limitless which implies (as seen in the way of being) that I can and will know everything (‘limitless’ is greater than any definite infinity except that infinity which is absolute in all possible ways). The first item and third ways are perhaps ‘wisdom’ for limited and limitless beings, respectively (regarding limitless beings see the way of being). Knowing everything is neither contradictory, nor necessarily a source of absolute dialetheia.

Mereology

In a final example of dialetheia for this work, let us take up another 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 or at a window, to look at a tree. I raise two issues (i) what the parts of the tree are (ii) what the oneness of the tree is, over and above the collection of parts. What the parts are is not definite—one may choose what one sees: leaves, branches, etc; one may choose the ‘biological elements’, the cells, the capillary action and so on; the physical elements, elementary particles-and-fields (or both); one may of choose random combinations, which is not particularly interesting in this example. What the oneness may be is also indefinite—eternal, a mere coming together, or somewhere in between. Now note, that from these indefinitenesses, there seems to be no essential tree, part, or oneness. Experience suggests there is ‘tree-hood’; temporal explanations are useful; ultimate explanations are not difficult; but the present analysis suggests that none of these point to a tree – i.e., a tree with its oneness.

Given the constitution of the tree, as described, there is a good point of view, derived from positing the parts in such a way that the oneness is manifest in the parts, in which there is nothing—no oneness—over and above the naming and the fact of the tree. That is, we are bound to find the oneness contradictory if we grant its existence over and above the fact. 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 (i.e., a contra-real). 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, for gluons are only needed on (i) assigning more than a mere coming together of a mere coming together or (ii) inadequate identification of parts in the case of something that is more than a mere coming together. In the case of a tree, for example, if we choose roots, trunk, branches, leaves and so on (or particles) and their interactions (fields) as the parts, there would be no need for a gluon; but if we omitted the interactions (fields), there is an artificial need for a gluon. That is, the need for oneness over and above the parts arises in some but not all ways of describing (the tree). 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. At this point, one might add the need for another kind of oneness—not the mere coming together, and not the physical binding, but something metaphysical, e.g., having to do with form. We might say it is the gluon or perhaps the symmetry; but we have discounted the need for gluons, and the symmetry is immanent in object (tree) for it is abstracted from the object. Which is to say there is no particular need for metaphysical or formal oneness beyond, the mere coming together. This ought not to imply that gluons are not objects worthy of study.

Mereology is the study of parts and wholes. The analysis so far suggests that while there are formal mereologies, their significance is questionable (in the sense of “what does a formal mereology capture?” For example, if we point to some random things, e.g., the sun – my left little toe – Plato’s Republic, do they constitute a ‘real’ whole or one? I.e., are they a one or a mere congeries (Priest remarks on the question of congeries)?

In fact, we might say “there are no congeries”. It might be replied, “How so, for your example about the sun etc is one such example?” To which we may respond that it would not ‘be’ if it had not been pointed at. That is, there are no mere congeries and, importantly, the oneness and reality of the congeries consists in that it was pointed at. Which in turn may be criticized, “But the oneness is external to the ‘one’ and therefore it is not one.” Which is a valid thing to say.

So, now the question arises, “When, if at all, is the oneness that is internal to the one, to be discovered or simply known rather than pointed at, i.e., part of the one without being something over and above the parts?” That is, “What constitutes natural oneness?” Or “Are there natural mereological descriptions, i.e., are there natural part-wholes, i.e., for which the one-ness or whole-ness is part of the whole?” Yes, here is one, “For any given whole or one, a natural system of parts and wholes is the void and the whole (and nothing else)!”

Are there other such natural descriptions? To answer, first note that while many objects that we see or experience as a whole may be so on as the result of a temporal process, it is at the time one, on account of something atemporal, e.g., a form, recognizable as such from, e.g., symmetry, which may be physical, biological, or other. That is, noting that such descriptions need not be unique, there are natural descriptions of oneness that require no object over and above the parts (e.g., abstraction to symmetry).

The issue of natural mereology is taken further in natural mereology.

Is reality contradictory?

Some thinkers, e.g., Hegel have argued that reality is contradictory. One approach to establishing that reality is contradictory is to argue that reality or a part of it is not what it is. That would be truly contradictory. However, the argument invariably appears to be that reality is one way in some cases or one point of view, another way in other cases or another point of view, but that is not a contradiction. It focuses on distinct parts, perhaps at separate times, places, and sub-universes (e.g., an abstract sub-universe)… or different points of view and then sees the differences as contradictions.

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 (and this is what some writers mean by ‘reality is contradictory’; however, Hegel’s claim is the stronger one that it is reality and not just descriptions that are contradictory). This does indeed seem to be a substantial claim, for, if we did not unpack the term ‘contradiction’, we might conclude 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 seems like a contra-real. It may be objected, “but ‘in existing, that it does not exist’ is or at least seems dialethic”. The reason it is not dialethic is that the meaning of ‘to exist’ for the void has difference from its meaning for manifest beings. In the language of computer science, the word ‘existence’ is overloaded (here, with more than one meaning).

In all cases so far, we have defused contra-diction.

In this section we continue to review the status of absolute dialetheism, the existence of true contra-dictions whose contra-dictory status cannot be removed, even by beings with limitless powers of description and discrimination.

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 ten things (chairs, people, …) in a room and all (ten) are removed there are now zero things (of the given kind in the room). But if there are zero things and all (zero) are removed, there are now (still) zero 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 zero but is X when X is zero; this is of course trivial for the usual number systems, but a similar argument regarding objects 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 (note that while ‘removing’ objects is intuitive, the argument may be made rigorous in terms of Mereology—Stanford Encyclopedia of Philosophy). 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 way of being for definition of the term ‘universe’). But this apparently contradictory character would emerge from limited discrimination (under-specification of the infinities).

The existence of the void derives from equivalence of its existence and nonexistence. Can we further 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, the of meaning discussed earlier, 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.

This suggests the following analysis. The concept of absolute nothingness is contra-dictory because the maintenance of eternal nothingness, i.e., non-emergence of manifest being, is something, i.e., not nothing. Therefore, there may be local patches of nothingness (in, e.g., space and time or, rather, at a level of description above spacetime, from which spacetime emerges), but no more. That is, real absolute nothingness would be truly contradictory, but not a dialetheia because, while there is an as if object, it has no corresponding being—and that is on account of logic, not, say, physics. On the other hand, there is, even on the basis of logic alone, a concept of patchy nothingness, which refers to a being (of course existing). This suggests, in turn, that there are no absolute dialetheia but, rather, that dialetheia and their analysis in terms of paraconsistent logic, will be useful when we are not able to or choose not to discriminate (it is not clear that we are ever unable to discriminate). Note, (i) though these thoughts are not final, ‘patchy nothingness’ is a more robustly realized concept than ‘absolute nothingness’ (ii) there is an analogy (at least) between real (patchy) nothingness and the quantum vacuum.

Let us consider everything and nothing again. We saw that in general an object was a proper part of its field of sense. Therefore, everything is a proper part of its field of sense which, in turn, is part of everything, which is part of its field of sense and so on… a dialethic contradiction. However, perhaps the conclusion that all objects are proper parts of their fields of sense was an over-generalization. An electron is what it is by being part of its network of relations other particles and fields, but the universe is not part of a network—it is all parts and all networks (as far as that mode of description is real). Therefore o < f(o) does not generalize to e < f(e). Rather, e = f(e) since everything ought to contain its field of sense. That is, o £ f(o), which is reasonable, since And this would defuse the apparent contradiction and therefore the dialetheia. But does not the universe require to be known and described and therefore is it not part of a field of sense? Though there is good reason to think in terms of fields of sense, there is also a vagueness to the idea. The vagueness may be removed by referring to the universe (U) as a field of experiential being (foe) as in ‘universe as a field of experience’ where the field is not a concept invented to explain what things are (even though the concept may be real) but is reality—with its system of networks among things—itself. Here, it becomes clear that the universe is the field, U = FOE, or u = f(u).

‘Knowing everything’, is a source of many dialetheias. However, none of them would seem to be an absolute dialetheia; rather, they would all be dialetheic relative to some mode of description.

In mereology, we saw that the dialethic aspects of mereological description, particularly, that of the one and the many, are resolved by appropriate (‘natural’, but perhaps there are others) description.

Because I am not sure that essentially all possible dialetheias have been considered or that the reasoning for those considered is perfect, and so, here the issue of absolute dialetheism must remain open.

Conclusion

But there are significant conclusions—

There are dialetheias, of which some are trivial but others of some depth.

Many dialetheias are the result of how language is used—under or over assigning meaning or using a restricted level of language—and not that reality is such that there must be contradictions in describing it. It seems likely that all dialetheias are of this kind, i.e., that relative dialetheism holds while absolute dialetheism does not.

On the other hand, it is not fully clear that inadequate language can or should be overcome or patched up in all cases and therefore study of dialetheism and use of paraconsistent logic in dialetheic analysis is at least useful. Some dialetheias, though they are ‘resolvable’ by more precise description, may be more effective in expressing some truths, especially regarding the meaning of our being or in presenting the essence of some metaphysical truths. We might coin the term aesthetic dialethism to express the effectiveness of some dialetheias relative to detailed non-dialethic descriptions.

Some apparent cases of dialetheias, e.g., the liar paradox, are the result of assigning meaning where there is no meaning.

What is the significance and future of dialetheism? For my thought, the significance is that it is one more direction in which my metaphysical system may advance in sophistication and insight. The root of that metaphysics can be seen as dialethic in the relative sense, even though that was not its explicit origin.

In general thought, I think that dialetheism will become better accepted, once it is generally realized that contra-dictions do not imply contra-reals, and that dialetheic description is insightful in metaphysics and the nature of being, especially human being. Even if dialetheism is essentially relative, it has value in that it shakes us free of the dogma of lnc, promotes further freedoms, e.g., from empiricism as dogma, and projection to ultimate human limits from the experience of proximate limits. Note that overcoming dogmatic attitudes to principles does not mean rejecting such principles altogether.

There are already significant insights, not due to dialetheism alone, but at the convergence of dialetheism with metaphysics, as in my writing and the writing of others. I will not attempt to predict its future. Metaphysics and its logic have many threads; dialetheism is one. No doubt, as the threads intertwine, knowledge and insight will grow. Intimations of infinity and limitlessness will abound (assuming that we survive and flourish). Will the thread of dialetheism continue as it is, or will it fuse with other threads? I do not know, but I do think there will be influence, perhaps considerable influence.

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In my interest in dialethism, I said “When young, I found significant meaning in understanding the universe and our place in it.” Today, my interest has gone beyond understanding or even understanding-and-application. Rather, I see thought and realization as essentially intertwined. That is, I do not see philosophy as just an endeavor that enhances human understanding. Today I see that it is an essential essence of philosophy—of knowledge and thought—that it is part of our being on the way to ultimate being.