Adialetheia is a sentence, \(A\), such that both it and itsnegation, \(\neg A\), are true. If falsity is assumed to be the truthof negation, a dialetheia is a sentence which is both true and false.Such a sentence is, or has, what is called a truth-valueglut, in distinction to agap, a sentence that isneither true nor false. (We shall talk of sentences throughout thisentry; but one could run the definition in terms of propositions,statements, or whatever one takes as one’s favouritetruth-bearer: this would make little difference in the context.)

Dialetheism is the view that there are dialetheias. If wedefine acontradiction as a couple of sentences of which one is the negation of the other,or as a conjunction of such sentences, then dialetheism amounts to theclaim thatthere are true contradictions. As such,dialetheism opposes—contradicts—theLaw ofNon-Contradiction (LNC), sometimes also called the Law ofContradiction. The Law can be expressed in various ways; fixing theprecise formulation is itself a topic of debate (Priest et al. 2004,Part II). Thomas Reid put the LNC in the form ‘No proposition isboth true and false’. A strong (modal) statement of the LNC is:for any \(A\), it is impossible that both \(A\) and \(\neg A\) betrue.

In Book \(\Gamma\) of theMetaphysics, Aristotle introduced(what was later to be called) the LNC as “the most certain ofall principles” (1005b24)—firmissimum omniumprincipiorum, as the Medieval theologians said. Since Aristotle,there have been few sustained attempts todefend the law. TheLNC has been an (often unstated) assumption, felt to be so fundamentalto rationality that some claim itcannot be defended, e.g.David Lewis 1999. As a challenge to the LNC, therefore, dialetheismassails what most philosophers take to be unassailable common sense,calling into question the rules for what can be called into question(cf. Woods 2003, 2005; Dutilh Novaes 2008).

Since the advent ofparaconsistent logic in the second half of the twentieth century, dialetheism has beendeveloped as a view in philosophical logic, with precise formallanguage. Dialetheism has been most famously advanced as a response tological paradoxes, in tandem with a paraconsistent logic. The view has been gaining, ifnot acceptance, the respect of other parties in the debate; one criticwrites that dialetheists have shown,

as clearly as anything like this can be shown, that it is coherent tomaintain that some sentences can be true and false at the same time.… [A]nd that perhaps is a radical conclusion, and a majoradvance in our understanding of the issues. (Parsons 1990)

In this article, 1) we will start by explaining the connection betweendialetheism and other important related concepts, such as the ones oftrivialism and paraconsistency. Next, we will describe 2) the historyof dialetheism and 3) the motivations for contemporary dialetheism,among which the logical (semantic and set-theoretic) paradoxes figureprominently, though not exclusively. We will then 4) indicate anddiscuss some of the objections to dialetheism, and 5) its connectionswith the notion of rationality. Finally, 6) we will point at somepossible themes for further inquiry concerning the connections betweendialetheism, realism, and antirealism in metaphysics.

• 1. Some Basic Concepts
• 2. Dialetheism in the History of Philosophy
  • 2.1 Dialetheism in Western Philosophy
  • 2.2 Dialetheism in East Asian Philosophy
  • 2.3 On interpretation of history
  • 2.4 Modern Dialetheism
• 3. Motivations for Dialetheism
  • 3.1 The Paradoxes of Self-Reference
  • 3.2 A Simple Case Study: the Liar
  • 3.3 Other Paradoxes of Self-Reference
  • 3.4 Other Motivations for Dialetheism
• 4. Objections to Dialetheism
  • 4.1 The Argument from Explosion
  • 4.2 The Argument from Exclusion
  • 4.3 The Argument from Negation
• 5. Dialetheism and Rationality
  • 5.1 Consistency and Other Epistemic Virtues
  • 5.2 Accepting and Asserting Dialetheias
• 6. Themes for Further Research
• 7. Conclusion
• Bibliography
  • Some Basic Concepts
  • Dialetheism in the History of Philosophy
  • Motivations for Dialetheism
  • Objections to Dialetheism
  • Dialetheism and Rationality
  • Themes for Further Research
• Academic Tools
• Other Internet Resources
• Related Entries

1. Some Basic Concepts

The word ‘dialetheism’ was coined by Graham Priest andRichard Routley (later Sylvan) in 1981 (Priest et al. 1989, p.xx). The inspiration for the name was a passage inWittgenstein’sRemarks on the Foundations ofMathematics, concerningRussell’s paradox (see also below):

Why should Russell’s contradiction not be conceived of assomething supra-propositional, something that towers above thepropositions and looks in both directions like a Janus head? Theproposition that contradicts itself would stand like a monument (witha Janus head) over the propositions of logic. (1978, III.59)

A dialetheia is a two-way truth, facing both truth and falsity like aJanus-headed figure. Unfortunately, Priest and Routley forgot to agreehow to spell the ‘ism’, and versions with and without the‘e’ appear in print.

In philosophy, there tends not to be a distinction between a viewbeinginconsistent and beingincoherent. Both areunacceptable. Since dialetheism claims that some inconsistency can bemaintained without incoherence, these two concepts must bedisentangled. For a start, dialetheism should be distinguished fromso-calledtrivialism, the claim thatall sentencesare true, and hence all contradictions are true too (Kabay 2010).Dialetheism is the view thatsome contradictions are true. Atrivialist must be a dialetheist, since a trivialist accepts everyclaim. But since ‘some’ does not imply ‘all’,the converse is not the case: a dialetheist typically claims that onlysome (and, usually, very specific) sentences are dialetheias, not thatall of them are. Trivialism is by all accounts incoherent, and so isminimally what counts as unacceptable even for someone who thinks thatsome contradictions are true.

How to allow inconsistency without incoherence is one of the maintopics of dialetheism. A standard solution consists in subscribing tothe view that deductive logical consequence isparaconsistent. A logical consequence relation \(\vdash\) isexplosive if,according to it, a contradiction entails everything (excontradictione quodlibet: for all \(A\) and \(B\): \(A,\neg A\vdash B\)). It is paraconsistent if and only if (iff) it is notexplosive. By adopting a paraconsistent logic, a dialetheist cancountenance some contradictions without being thereby committed tocountenancing everything and, in particular, all contradictions. Thedevelopment of paraconsistent logics has contributed to the mostrecent developments (and acceptance of the legitimacy) ofdialetheism.

Dialetheism should, however, be clearly distinguished fromparaconsistency. Whereas dialetheists must embrace some paraconsistentlogic or other to avoid trivialism, paraconsistent logicians need notbe dialetheists: they may subscribe to a non-explosive view ofentailment for other reasons. Within paraconsistency, one maydistinguish (at least) four grades of paraconsistent involvement(Beall and Restall 2006, p. 80):

1. Gentle-strength paraconsistency is simply the rejectionof explosion with respect to logical consequence.
2. Full-strength paraconsistency has it that there areinteresting or important theories that are inconsistent but nottrivial.
3. Industrial-strength paraconsistency has it that someinconsistent but non-trivial theories arepossibly true.
4. Dialetheic paraconsistency has it that some inconsistentbut non-trivial theoriesare true.

Most people working on paraconsistent logics have commitments at thelower grades of the spectrum. The unifying thought behindparaconsistency is gentle-strength: logical consequence should notvalidate arbitrary conclusions following indiscriminately frominconsistent premises. This may simply be because entailment mustpreserve more than just truth, e.g., information content, topicality,or some meaningful connection between premises and conclusion.Carnielli and Rodrigues (2019) advance an explicitly anti-dialetheicinterpretation of paraconsistent logic, where there may be epistemiccontradictions—inconsistencies in evidence or belief—butno true contradictions. Or it may be that a paraconsistent logiciansimply need not assume anything about truth in order to provide aworking treatment of inconsistency as it may arise in databases, legalsituations, works of fiction, theory change, belief-revision, etc.

Stepping up, full-strength paraconsistentists treat inconsistentmodels, in which contradictions hold, as useful mathematical toolswithout committing to them as representingrealpossibilities.

And stepping up again, industrial-strength paraconsistentists may holdthat, though truth at the actual world is consistent, still entailmentmust preserve what holds in peculiar non-actual situations, some ofwhich may be inconsistent (see Berto 2007a, Ch. 5, and Priest, Bealland Armour-Garb 2004, p. 6). The first three levels appear to beindependent of dialetheism. (But for reason to worry that there is a‘slippery slope’ from gentle to dialetheicparaconsistency, see Priest 2000. For particular worries about modeltheory, see Asmus 2012, and in the modal case, Martin 2015.)

At the top of the spectrum, dialetheic paraconsistency holds that somecontradictions are, in some sense, truein the actual world.Even among full-fledged dialetheists, differences remain, for example,in what is meant by “true”, e.g., on whether theysubscribe to adeflationary theory of truth, acorrespondence view, a view in which true contradictions are merely semantic, or somethingelse. We will come back to this point insection 6 below.

We said at the outset that dialetheismchallenges the LNC.Now this claim needs some qualification, since forms of the LNC are,in fact, accepted as a general logical law in dialetheism, at least inversions based in e.g., the paraconsistent logic LP. Dialetheism asset out by Priest takes all instances of the schema \(\neg(A \wedge\neg A)\) to be true, as well as taking as true some sentences thatare inconsistent with it, namely, true sentences whose negations aretrue: dialetheias. According to such versions of dialetheism, allcontradictions are false and some are true: dialetheism is itself adialetheia (‘Concluding self-referential postscript’ toPriest 1979, p. 203).

Much of the ongoing discussion about dialetheism involves not just theLNC but its dual, the Law of Excluded Middle (LEM) (seesection 3.2 below). The LEM says (informally) that for every sentence \(A\),either \(A\) or \(\neg A\) is true (we bracket here views thatdisentangle the LEM from Bivalence). Dialetheism is the thesis thatthere are truth-value gluts, challenging the LNC; the dual position,challenging the LEM, is that there are truth-value gaps. It is asubtle issue, which we will not discuss here, whether a gap shouldcount aslacking anytruth value, or as having a non-classical value distinct from both truth andfalsity, and similarly whether a glut has two truth values, or somethird ‘glutty’ value. In any case, these dual approachesare nowadays labelledparacomplete versus (paraconsistent)dialetheic theories of truth. (The paraconsistent/paracompleteterminology is not entirely happy – see Ripley 2015a, footnote 1– but is now standard.) Priest’s logic includes the LEMand he rejects the existence of gaps in Priest 1987/2006, Ch. 1. Otherapproaches are both paracomplete and paraconsistent; they include aplace for both gaps and gluts. The case for and against dialetheismturns as much on the status of the LEM as the LNC; see Beall andRipley 2018.

2. Dialetheism in the History of Philosophy

While the word ‘dialetheism’ is relatively new, the ideais not. In this section we note some points in the history ofphilosophy where true contradictions appear salient, either explicitlyor implicitly. (Much of this scholarship is due originally to Priestand Routley 1983, and Priest 1995/2002.) Then we addressmethodological issues to do with historical interpretation, and give aquick sketch of more recent history. For another history ofdialetheism, see Ficara 2021. (Note that Ficara, followingd’Agostini’s interpretation of Hegel, takes dialetheism tobe the view that there are non-explosive true contradictions, anddistinguishes this from glut theory, the view that there can bepropositions that are both true and false. This diverges from the waywe’ve set things out here, and merits further discussion initself.)

2.1 Dialetheism in Western Philosophy

Aristotle takes a number of the Presocratics to endorse dialetheism,and with apparent justification. For example, in Fragment 49a,Heraclitus says: “We step and do not step into the same rivers;we are and we are not” (Robinson 1987, p. 35). Protagoreanrelativism may be expressed by the view that man is the measure of allthings; but according to Aristotle, since “Many men hold beliefsin which they conflict with one another”, it follows that“the same thing must be and not be” (1009a10–12).The Presocratic views triggered Aristotle’s attack inMetaphysics, Book \(\Gamma\). Chapter 4 of this Book containsAristotle’s defence of the LNC. As we said above, historicallyAristotle was almost completely successful: the LNC has been orthodoxyin Western Philosophy ever since.

As an aside, and as mentioned above, there is an often-remarked uponduality between truth-value gaps and truth-value gluts (Parsons 1990).And yet, it is perhaps worth noting that inMetaphysics\(\Gamma\) (Chapter 7) Aristotle also defends the dual of the LNC, theLaw of Excluded Middle, LEM, particularly in the version which hasnowadays been distinguished as the Law of Bivalence: for any \(A\), itis necessary for (at least) one of \(A\) and \(\neg A\) to be true.But the LEM has often had a less secure place in Western Philosophythan the LNC, in spite of the numerous obvious dualities between thetwo principles. Aristotle himself, at least according to oneinterpretation, appears to attack the Law inDeInterpretatione, Chapter 9, when he comes to the famous subjectoffuture contingents.

Despite the orthodoxy about the LNC since Aristotle, during the MiddleAges the problem of seemingly true contradictions surfaced inconnection to the paradoxes of divine omnipotence—for instance:can God make a stone too heavy for Him to lift? (see Cotnoir 2018). Wefind St. Pier Damiani getting close to dialetheism in theDedivina omnipotentia, by blaming St. Girolamus for having claimedthat God cannot overturn the past and twist what happened intosomething that didn’t happen. Since God lives in the eternalpresent, denying Him power over the past equates to denying Him powerover current and future events, which is blasphemous. So God must havethe power of making what is done undone. Later on, Nicholas of Cusaplaced at the core of his bookDe docta ignorantia the ideathat God iscoincidentia oppositorum: as a truly infinitebeing, He includes all opposite and incompatible properties, thereforebeing all things, and none of them: God has all properties, includingcontradictory ones (Heron 1954, I.4).

In theCritique of Pure Reason, Kant argues that somecontradictions, the antinomies of pure reason (concerning time, space,and other categories), are produced by an illicit use of pureconcepts; nevertheless, he also holds such an illicit use to be a“natural and inevitable illusion” (Kant 1781, p.300)—a side effect of reason’s pursuit of completeness inknowledge. Reasoning about the world as a whole, a totality which isnever given to us as such, can lead us to apparently dialetheicconclusions: e.g., that it has a beginning in time and a limit inspace, and that it has no beginning nor limits in space, that it isinfinite in space and time. Both horns assume the opposite thesis andseemingly perform areductio. (In Kantian terms, the“transcendental illusion” begins when we turn what shouldjust be a regulative ideal into a limit-object.) According to Kant, atleast in one way of resolving the antinomies, the fallacy lies intreating the world as a whole as an object—in mistaking asubjective “condition”, as Kant says, for an objectivereality.

Now, according to Hegel such a conception has something to be said forit as well as against it. Kant has a point in showing, via theantinomies, that dialectics is “a necessary function ofreason”, and in defending “thenecessity of thecontradiction which belongs to the nature of thoughtdeterminations” (Hegel 1831, p. 56). However, Kant takes fromthis only the familiar result that reason is incapable of knowing theAbsolute, that is, actual reality. A dialetheic interpretation holdsthat, on the contrary, we should abandon such “tenderness forthe things of this world”, and the idea that “the stain ofcontradiction ought not to be in the essence of what is in the world;it has to belongonly to thinking reason” (Hegel 1830,p. 92). On a dialetheic interpretation, contrary to what Kant held,the Kantian antinomies are not areductio of the illusions ofreason; they are perfectly sound arguments, deducing the dialetheicnature of the world. (For a reconstruction of this Kantian-Hegeliandebate, see Part II of Priest 2002.) Hegel himself has beeninterpreted as being a dialetheist. Notably, d’Agostini arguesthat while Hegel held that some conjunctions of contradictorysentences are true, he did so without the further commitment thateither of the two contradictory terms are separately true; seed’Agostini 2021.

2.2 Dialetheism in East Asian Philosophy

In non-Western traditions, there are more overt examples of whatappears to be dialetheic thought. In ancient Indian logic/metaphysics,there were standardly four possibilities to be considered on anystatement at issue: that it is true (only), false (only), neither truenor false, or both true and false. Some logicians added a fifthpossibility: none of these. Both positions were, arguably, calledcatushkoti (Priest 2002, Ch. 16; Deguchi et al. 2008;Tillemans 2009.) The Jains went even further and advocated thepossibility of contradictory values of the kind: true (only) and bothtrue and false (see Smart 1964).

Contradictory utterances are a commonplace in Daoism, perhaps mostfamously in Laozi (Lao-Tsu): on one translation,

The Way that can be spoken of is not the true Way… (DaodejingI)

a statement which appears to be speaking about the true Way. TheZhuangzi says: “That which makes things has no boundaries withthings, but for things to have boundaries is what we mean by saying‘the boundaries between things’. The boundaryless boundaryis the boundary without a boundary” (Mair 1994, p. 218). WhenBuddhism and Daoism fused to form Chan (or Zen, to give it itsJapanese name), a philosophy arose in which contradiction plays acentral role. The very process for reaching enlightenment (Prajna) isa process, according to Suzuki (1969, p. 55), “which is at onceabove and in the process of reasoning. This is a contradiction,formally considered, but in truth, this contradiction is itself madepossible because of Prajna.”

Recent discussions of dialetheism have centered on Buddhist thought inparticular, where some interpreters argue that texts invoke outrightcontradictions. For example, in the Mahayana tradition, Nagarjunapresents many passages similar in form to:

Everything is real and not real, both real and not real, neither realnor not real. This is the Lord Buddha’s teaching. (Garfield1995, XVIII: 8)

On a dialetheic interpretation, advocated by Priest, Garfield, andDeguchi (see Priest 2002 Ch. 16, Deguchi et al. 2008), readers shouldtake Nagarjuna and others in the Madhyamaka school at theirword—not as invoking some sort of mysticism, but as affirmingthe nature of reality. This is most salient in Nagarjuna’sdoctrine of emptiness (sunyata), that all things are empty or lackingindependent existence (svabhava). Since all things are empty, itfollows that emptiness is empty too; or, more prosaically, “Theultimate truth is that there is no ultimate truth” (Siderits2007, p. 182; Priest 2002, p. 260). These utterances, if true, wouldappear to be self-contradictory, and therefore dialetheias (see Priest2018).

For work on dialetheism in Buddhism, see the collections Garfield etal 2009 and Tanaka et al. 2015. For discussion as to whetherdialetheism is the appropriate way to interpret Buddhist texts, seethe papers in Tanaka 2013. For more general difficulties see the entryoncomparative philosophy: Chinese and Western.

2.3 On interpretation of history

Interpreting the philosophers we have mentioned is a sensitive issue.Many commentators have suggested that the seemingly contradictoryutterances of the philosophers in question are not reallycontradictory. There are a number of standard devices that may beemployed to block a dialetheic interpretation. One is to question, inthe case of non-English sources, whether the apparently contradictorytranslation is correct. Another is to claim that the contradictoryutterance is to be taken as having some non-literal form of meaning,e.g., that it is a metaphor, or a way of pointing towards some higher,ineffable truth. Another is to claim that the contradictory assertionis ambiguous in some way, and that it is true on one disambiguation orrespect, false on another.

This last technique is calledparameterisation: when one isconfronted with a seemingly true contradiction, \(A \wedge \neg A\),treat the suspected dialetheia, or some of its parts, as havingdifferent meanings in \(A\) and in \(\neg A\), and hence as ambiguous(maybe justcontextually ambiguous). For instance, if oneclaims that \(P(a) \wedge \neg P(a)\), parameterisation holds that oneis in effect claiming that \(P_1 (a) \wedge \neg P_2 (a)\) (e.g.,elephants are big and not big, because they are big in the context ofland animals on Earth, but not big in the context of stars andplanets). In theMetaphysics, Aristotle hints that a criticof the LNC is playing with the equivocal meanings of some words:“for to each definition there might be assigned a differentword” (1006b 1–2).

No one would dispute that contradictory utterances are sometimes bestconstrued parametrically. (Again, dialetheists do not claim thatall contradictory statements are true.) The question iswhether parameterisation is the best approach in the case of thephilosophers we have mentioned; answering this in a dialetheic contextis a matter for detailed case-by-case consideration. A defender of thedialetheic interpretation would say that consistent parameterisationproduces an inaccurate and distorted version of the views of thephilosopher in question. In any case, even if parameterisation isalways possible, this does not impact arguments for or against theLNC. Ana priori claim that contradictionscanalways be avoided by parameterisation does not show, without beggingthe question, thecorrectness of such interpretations:sometimes parameterisation may be the best thing to do, butindependent justification should be given on each occasion.

2.4 Modern Dialetheism

In the second half of the twentieth century, with the rapiddevelopment of paraconsistent logics, came the modern form ofdialetheism. In 1966, Asenjo published a short note, “A Calculusof Antinomies”, based on his 1954 PhD dissertation; the openingline reads

Let us assume that atomic propositions have either one or two truthvalues. (Asenjo 1966, p. 103)

Here, having two truth values would mean beingboth true andfalse, a glut, or what Asenjo calls (following Kant) an antinomy.Asenjo simply seems to assume without further ado that there areantinomies, and in Asenjo and Tamburino 1975 asserts that they are“useful logical entities”, especially for dealing withparadoxes in mathematical contexts. Born in Buenos Aires, Asenjo cameto the University of Pittsburg in 1963 as a professor in themathematics faculty, where he was in contact with logicians such asBelnap, Anderson, their student Dunn (who thanks Asenjo in his 1966Ph.D. thesis, that also toys with true contradictions), and nearby,Meyer. By the mid-1970s, Richard Routley (Sylvan), in collaborationwith Meyer, was developing a ‘dialectical’ position thatrebukes what they call “the consistency hypothesis” andaccepts some statements as both true and false (Routley and Meyer1976; Routley 1977, 1979). In 1979, Priest’s paper “TheLogic of Paradox”, developed eventually into the bookInContradiction (1987/2006), presented what are now the most famousarguments for dialetheism. In the collaboration between Priest andRoutley, the contemporary dialetheic program was launched. Much ofPriest and Routley’s early joint work is summarised in Priest etal 1989 (also available as Priest and Routley 1983). More recentdevelopments are canvassed below.

3. Motivations for Dialetheism

3.1 The Paradoxes of Self-Reference

Probably the master argument used by modern dialetheists invokes thelogicalparadoxes of self-reference. It is customary to distinguish between two families of suchparadoxes: semantic and set-theoretic. The former family typicallyinvolves such concepts astruth,denotation,definability, etc; the latter, such notions asmembership,cardinality, etc. AfterGödel’s and Tarski’s well-known formal procedures toobtain non-contextual self-reference in formalized languages, it isdifficult to draw a sharp line between the two families, among otherthings, because of the fact that Tarskian semantics is itself framedin set-theoretic terms. Nevertheless, the distinction is commonlyaccepted within the relevant literature.

Russell’s paradox is the most famous of the set-theoretic paradoxes; it arises when oneconsiders the set of all non-self-membered sets, theRussellset.Cantor’s paradox arises in connection with the universal set. The most famous of thesemantic paradoxes is theLiar paradox. Although cases for the existence of dialetheias can be made fromalmost any paradox of self-reference, we will focus only on the Liar,given that it is the most easily understandable and its expositionrequires no particular technicalities.

3.2 A Simple Case Study: the Liar

In its standard version, the Liar paradox arises by reasoning on thefollowing sentence:

(1): (1) is false

where the number to the left is the name of the sentence to the right.As we can see, (1) refers to itself and tells us something about (1)itself. Its truth value? Let us reason by cases. Suppose (1) is true:then what it says, namely that (1) is false, is the case, so it isfalse. Then, suppose (1) is false: this is what it claims to be, so itis true. If we accept the aforementioned Law of Bivalence, that is,the principle according to which all sentences are either true orfalse, both alternatives lead to a contradiction: (1) is both true andfalse, that is, a dialetheia, contrary to the LNC.

The paradox can also be produced without any direct self-reference,but via a short-circuit of sentences. For instance, here is a loopedLiar:

(2a): (2b) is true

(2b): (2a) is false

If what (2a) says is true, then (2b) is true. However, (2b) says that(2a) is false …. And so on: we are in a paradoxical loop. Thisis as old as Buridan (his Sophism no. 9: Plato saying ‘WhatSocrates says is true’; Socrates replying ‘What Plato saysis false’).

Paradoxes of this kind have been known since antiquity (the standardLiar is attributed to the Greek philosopher Eubulides, probably thegreatest paradox-producer of antiquity). But they were thrown intoprominence by developments in the foundations of mathematics aroundthe turn of the twentieth century. In the case of each paradox, thereappears to be a perfectly sound argument ending in a contradiction. Ifthe arguments are sound, then dialetheism is true. Of course, manyhave argued that the soundness of such arguments is merely anappearance, and that subtle fallacies may be diagnosed in them. Suchsuggestions were made in ancient and medieval logic; but many morehave been made in modern logic—indeed, attacking the paradoxeshas been something of aleitmotiv of modern logic. And onething that appears to have come out of this is how resilient theparadoxes are: attempts to solve them often simply succeed inrelocating the paradoxes elsewhere, as so called‘strengthened’ forms of the arguments show. Let us have alook.

A radical solution, which never won big consensus but has beenrevitalised in recent times (Pleitz 2018), has it that there justcannot be meaningful self-rererential sentences. Various works(notably Martin 1967, van Fraassen 1968, Kripke 1975, Field 2008) haveproposed to solve the Liar paradox by dismissing Bivalence, wherebysome sentences are neither true nor false, and the Liar is one suchtruth-value ‘gap’. Truth-value gaps, and the inclusion ofthe Liar among them, are differently motivated in the variousapproaches. But the common core thought is the following: even thoughthe Liar is a sentence such that, if it were true, it would be false,andvice versa, no explicit contradiction according to whichit is both true and false need follow. We can avoid the contradictionby rejecting the idea that truth and falsity are the only two optionsfor a sentence: the Liar is neither. For a comparative survey of thetwo kinds of approach, see Beall and Ripley 2018.

These approaches face difficulties with the so-called‘strengthened’ Liars—sentences such as thefollowing:

(3): (3) is not true.

(4): (4) is false or neither true nor false.

Now these sentences should be, on the gappy theorist’snon-bivalent approach, either true, or false, or neither. But, forinstance, if (3) is true, then things are as it claims they are;therefore, (3) is not true (either false or truth-valueless). If (3)is false, or neither true nor false, in both cases it is not true; butthis is precisely what it claims to be; therefore, it is true. We seemto have to conclude that (3) is both true and not true, contrary tothe LNC. A similar line of reasoning goes for (4).

According to Priest the strengthened Liars show that a single featureof the semantic paradox underlies its different formulations. Thetotality of sentences is divided into two subsets: the true ones, andtheir ‘bona fide complement’—call it theRest. Now the essence of the liar is “a particulartwisted construction which forces a sentence, if it is in the bonafide truths, to be in the Rest (too); conversely, if it is in theRest, it is in the bona fide truths” (Priest 1987, p. 23). Thestandard Liar, ‘This sentence is false’, is just aparticular instance of this, producing a contradiction within thebivalent framework, in which the Rest is identified with the set ofthe false sentences. We can try to resolve the problem by admittingsentences that are neither true nor false, so that the false onesbecome a proper subset of the Rest. However, the strengthened Liarsshow that we can use the notions introduced to solve the previousparadox tore-describe the Rest. Once the set of sentences ispartitioned into a trichotomy (true, false, and neither true norfalse), the disjunctive ‘This sentence is false or neither truenor false’ embraces the whole Rest, i.e., the new(ly described)complement of the set of the true sentences. What about adding morevalues? If there is some fourth thing that a sentence can be, besidestrue, false, and neither true nor false, we can always take the notionfourth thing and produce another strengthened Liar:

(5): (5) is false, or neither true nor false, or the fourth thing.

(See Kirkham 1992, pp. 293–4.) These strengthened liars are alsocalled revenge liars, and the general phenomenon we have justwitnessed has become known asrevenge; see Beall 2007, e.g.the introduction, and Cook 2007, among other essays.

There is no generally-agreed-upon solution to the semantic paradoxes.One typical way out attempted by the supporters of truth-value gaps,for instance, consists in denying that the notion ofgap, ordefective sentence, orsentence whose truth value isindeterminate, can be fully expressed in the language for whichthey are proposing their theory of truth. The strengthened paradoxesthen seem to force the consistent theorist to admit that the proposedtheory was formulated in a language different from, and expressivelymore powerful than, the one whose semantics it was supposed toexpress. This entails a limitation of the Tarskian T-schemacharacterising truth, i.e., of the equivalence

where ‘\(Tr\)’ is the truth predicate for the relevantlanguage, and \(\langle A\rangle\) is the appropriate name of sentence\(A\). This approach makes a rigid distinction between an objectlanguage and its metalanguage. Such a distinction was introduced byTarski to expel the Liar paradox from formalized languages—butTarski himself insisted that his solution is inapplicable to naturallanguages, which do not appear to depend upon some metalanguage fortheir semantics. As Kripke admits at the end ofOutline of aTheory of Truth, “the ghost of the Tarski hierarchy isstill with us” (scil. the paracompletists: see Kripke 1975, p.80).

Tarski, in short, identified the cause of the semantic paradoxes to besemantic closure—the fact that natural languages suchas English satisfy the T-schema. Tarski took this to mean that therecan be no consistent formalization of a semantically completelanguage, only an approximation thereof stratified into metalanguages.Dialetheists agree, but draw the conclusion in the other direction:the appropriate formalization of a language such as ours, because itis semantically closed and is not hierarchically stratified, will beinconsistent (Priest 1987, Ch. 1, Beall 2009, Ch. 1).

These two features—a claimed immunity to revenge/strengthenedliars, and dispensing with the object-language/metalanguagedistinction—are argued to give dialetheism about the paradoxesof self-reference some of its major appeal. As Shapiro (in a criticalnote) puts it, dialetheists offer that we

do not need to keep running through richer and richer meta-languagesin order to chase our semantic tails…. We embrace somecontradictions in the semantics, and get it all from the start.(Shapiro 2002, p. 818)

Thesimplicity of a dialetheic theory of truth, then, isclaimed as an additional further feature. The two most prominent suchtheories to date are presented in Priest 1987 and Beall 2009. In theformer, the truth predicate \(Tr\) for the relevant formal language,modelling the behaviour of truth in English, is simply characterizedby the unrestricted T-schema, \(Tr\langle A\rangle \leftrightarrowA\), which, as stressed by many philosophers, is an overwhelminglyintuitive—perhaps even analytic—principle concerningtruth. It is admitted that some sentences—notably, theLiars—are truth-value gluts, that is, both true and false (theconstruction may also sustain sentences which are both true and nottrue, although not all dialetheias need be of this kind); and nohierarchy of metalanguages or further epicycles are needed.

Beall’s 2009 theory allows a fullytransparent truthpredicate: one such that for any sentence \(A\), \(Tr\langleA\rangle\) and \(A\) can be replaced with each other in all(non-opaque) contextssalva veritate, that is, producingsentences logically equivalent to the sentences one started with. Thenthe unrestricted T-schema follows from transparency (and the fact that\(A \rightarrow A\) is a logical truth) as a special case. InBeall’s theory,all sentences \(A\) that aredialetheias are not only true and false, i.e., (given that falsity istruth of negation), \(Tr\langle A\rangle \wedge Tr\langle \negA\rangle\); they are also true and untrue, \(Tr\langle A\rangle \wedge\neg Tr\langle A\rangle\): this again follows from the transparency oftruth. Beall’s theory is based on a (relevant) paraconsistentlogic, whose modal semantics employs so-callednon-normal worlds.

Overall, such paradoxes as the Liar provide some evidence for thedialetheist’s claim that some contradictions areprovably true, in the sense that they are entailed by plainfacts concerning natural language and our thought processes. ExtendedLiar paradoxes like ‘This sentence is not true’ are speltin ordinary English. Their paradoxical characteristics, dialetheistsargue, are due exactly to the intuitive features of ordinary language:unavoidable self-reference; the failure of metalinguistic hierarchies,which only produce languages that are expressively weaker thanEnglish; and the obvious presence of a truth predicate for English,‘is true’, which is characterized, at least extensionally,by either the Tarskian T-schema or rules amounting to the transparencyof truth. For a sustained critical engagement with paraconsistentdialetheism as a solution to the semantic paradoxes, see Field 2008,part 5 (chapters 23–26).

3.3 Other Paradoxes of Self-Reference

In the larger family of paradoxes of self-reference beyond thesemantic, dialetheism affords a treatment of the set-theoreticparadoxes. These paradoxes arise in set theories based on anunrestricted ‘comprehension schema’ for sets: for anycondition or property, including paradoxical ones likenon-self-membership, there exists a corresponding set. In particular,inconsistent sets like Russell’s are admitted; analogously tothe Liar, the Russell set is and is not a member of itself. As withthe truth schema, the set comprehension principle seems very naturaland intuitive, and such contradictions do not give rise to trivialitydue to the paraconsistent logic underlying the formal theories. Thoughthe issue is too technical to be addressed here, and moreappropriately dealt with in the entries onparaconsistent logic andinconsistent mathematics, the reader can consult Routley 1979, Brady 1989, and Weber 2012, forinconsistent set theories (see also Mortensen 1995; for differentapproaches to dialetheic set theory, see Restall 1992 and Ripley2015b).

Since dialetheism appears to resolve both the semantic andset-theoretic paradoxes at once and in the same sort of way (namely,accept the contradictory outcomes as true), this has been presented asanother major strong point for dialetheism. Priest argues that theparadoxes share an underlying structure (which he calls the InclosureSchema in Priest 2002). This is used in tandem with what Priest dubstheprinciple of uniform solution (“same kind ofparadox, same kind of solution”) to urge that, since all theset-theoretic and semantic paradoxes are of a kind, dialetheismpresents a uniquely unified solution. For a detailed pursuit of adialetheic response to the paradoxes in general, see Weber 2021.

A point of dispute about the dialetheic approach to the paradoxes ofself-reference concerns theCurry paradox. This is produced by a self-referential sentence claiming ‘If Iam true, then \(\bot\)’, where \(\bot\) is a constant (whatlogicians usually call thefalsum) which is or entailssomething that is also dialetheically unacceptable, say \(\bot =\)‘Everything is true’, the incoherent trivialist claim.Prima facie, this does not involve negation, nor a falsitypredicate. However, many logicians think that Curry’s paradox isvery similar to the Liar and, therefore, by the principle of uniformsolution, that it should be handled in the same way. A dialetheist,though, cannot simply accept that the Curry sentence is both true andfalse, because if it is true then \(\bot\) follows. Dialetheists needa different treatment of Curry. A standard dialetheic strategy to dealwith the Curry paradox has consisted in exploiting paraconsistentlogics with a ‘noncontractive’ conditional (see againPriest 1987, Ch, 6, Beall 2009, Ch. 2), which do not validate theContraction (or Absorption) Law, i.e., the rule: from \(A\rightarrow(A \rightarrow B)\) infer \(A \rightarrow B\), or theso-calledpseudo-modus ponens principle \((A \wedge(A\rightarrow B)) \rightarrow B\). Stronger forms of the paradox,though,validity curry, seem to show that dropping theseprinciples is not enough (Beall and Murzi 2013). This leads tocontraction-free logics within the broader family ofsubstructural logics (Schroeder-Heister and Došen 1993, Restall 2000, Priest 2015),and indeed substructural dialetheic approaches that drop principlesother than contraction, such as transitivity (Ripley 2012).Curry’s paradox puts pressure on the dialetheic story aboutuniform solution to the paradoxes. Beall (2014a, 2014b) urges thispoint; Weber et al. 2014 is a reply.

3.4 Other Motivations for Dialetheism

Dialetheias produced by the paradoxes of self-reference are confinedto the abstract realm of notions such asset or to semanticconcepts such astruth. However, the paradoxes ofself-reference are not the only examples of dialetheias that have beenmooted. Other cases involve contradictions affecting concrete objectsand the empirical world, and include the following.

(1)Transition states: when I exit the room, I am inside theroom at one time, and outside of it at another. Given the continuityof motion, there must be a precise instant in time, call it \(t\), atwhich I leave the room. Am I inside the room or outside at time \(t\)?Four answers are available: (a) I am inside; (b) I am outside; (c) Iam both; and (d) I am neither. There is a strong intuition that (a)and (b) are ruled out by symmetry considerations: choosing eitherwould be completely arbitrary. (This intuition is not at all unique todialetheists: see the article onboundaries in general.) As for (d): if I am neither inside nor outside the room,then I am not inside and not-not inside; therefore, I am either insideand not inside (option (c)), or not inside and not-not inside (whichfollows from option (d)); in both cases, a dialetheic situation– or so it has been argued. For a description of inconsistentboundaries using formal mereology, see Weber and Cotnoir 2015.

(2) Some ofZeno’s paradoxes concerning aparticular—though, perhaps, the most basic—kind oftransition, namely,local motion: a moving arrow is bothwhere it is and where it is not. In any given instant, argues Zeno, itcannot move to where it is, since it is already there, and it cannotmove to somewhere else, because there isn’t time for it to getthere. The orthodox way out of the paradoxical situation, asformulated, e.g., by Russell 1903, has it that motion is the mereoccupation of different places at different times. But one could arguethat this is a denial of the phenomenon itself, that is, of theactuality of motion: Russell’s solution entails that motion isnot anintrinsic state of the (allegedly) moving thing, for,at each instant, the arrow is not moving at all. Can a going-somewherebe composed of an (even continuum-sized) infinity of going-nowheres?An alternative, dialetheic account of motion, which takes at facevalue the aforementioned Hegelian idea that “Something moves,not because at one moment it is here and another there, but because atone and the same moment it is here and not here, because in this‘here’, it at once is and is not”, is exposed inPriest 1987, Ch. 12.

(3)Borderline cases of vague predicates. Popular approachestovagueness and thesorites paradox, such as the ones based on many-valued logics, or supervaluations,require some under-determinacy of reference, and/or the rejection ofBivalence: if an adolescent, \(m\), is a borderline case of adultness,\(A\), then \(A(m)\) may turn out to have an intermediate truth valuebetween truth and falsity, or no truth value at all. But it may beconjectured that a borderline object like \(m\), instead of satisfyingneither a vague predicate nor its negation, satisfies them both: anadolescent both is and is not an adult. Given the obvious dualitiesbetween the LEM and the Law of Bivalence on the one side, and(respectively, syntactic and semantic formulations of) the LNC on theother, it is not too difficult to envisage a‘sub-valuational’ semantic approach, dual to thesupervaluation strategy. Sub-valuational paraconsistent semantics havebeen proposed by Hyde (1997), and Varzi (1997). Other‘glutty’ approaches to vagueness have been proposed byColyvan (2009), Weber (2010), Priest (2010), Ripley (2013), andCobreros et al. (2010, 2015), motivated on both theoretical andempirical grounds. If inconsistencies due to vague predicates andborderline objects are taken to be due to merely semantic under- andover-determination of ordinary language (de dicto), then thisis further evidence for dialetheism about semantics. If theaforementioned phenomena are taken to be not just about words butthings (de re), then actually inconsistent objects areadmitted, together with vague objects. And this spreads inconsistencyall over the empirical world: if borderline cases can be inconsistent,inconsistent objects are more or less everywhere, given how pervasivethe phenomenon of vagueness notoriously is: adolescents, borderlinebald men, etc. Inconsistency isobservable (Beall and Colyvan2001). Beall argues against the approach in Ch. 5 of Beall 2009, andBeall 2014a.

(4)Multi-criterial predicates. We may assume that thesemantics of a predicate is specified by means of its criteria ofapplication. Now ordinary language hosts predicates with different,and occasionally conflicting, criteria of application (e.g.,‘left wing’): some criteria for applying a predicate \(P\)(e.g., caring about social welfare, raising minimum wages) may entailthat object \(m\) (e.g., a political party) is in the extension of thepredicate, some others (promoting nationalism, oppressing immigrants),that \(m\) is in its anti-extension, or negative extension. Criteriacan in some cases be encoded by such things as meaning postulates (orother similar, albeit more sophisticated, semantic devices); butconflicting meaning postulates may be embedded in our standardlinguistic practices, and difficult to detect and identify. If theextensions of our ordinary predicates are constrained by ourintuitions, and such intuitions turn out to be inconsistent, a goodsemantic account of the situation may well have to reflect this fact(accepting, in our example, that a party counts as both left and rightwing), instead of destroying it by means of some regimentation (e.g.via the usual parameterisation, or distinction of respects). As Priest1987, pp 67–9, clarifies, it is also hard to analyse suchsituations as cases of vagueness (so that a party has to be understoodas being left wing to some degree, and right wing to some degree).

(5) Certainlegal situations, such as inconsistent bodies oflaw. Suppose, for instance, that some norm states that a marriageperformed by the captain of a ship counts as a legal marriage only ifthe ship was in open water throughout the ceremony. It turns out,then, that some other law has established that such a marriage isvalid also if the ceremony has only begun with the ship in open water,but has ended with the ship in the port. Then someone may turn out tobe both a married man and not a married man. (Since it does not appearto follow that he is not a man anymore, or both a man and not a man,we have another putative counterexample toex contradictionequodlibet.) If one accepts the plausible view that statementsconcerning legal rights, obligations, and statuses, can be truth-valueapt, we seem to have a dialetheia. Of course, legal systems sometimeshave mechanisms that can be used to remove such inconsistencies (e.g.,by ordering different kinds of laws in a hierarchy from customarylaws, to established jurisprudence, to ordinary legislation, toconstitutional norms, etc.; or via thelex posteriorprinciple, giving priority to the more recent norm in case ofconflict). But this is not always the case: the inconsistent laws maybe of the same rank, enacted at the same time, etc. For a recentdiscussion, see Beall 2016.

(6) Inpost-Kantian metaphysics, a movement sometimes dubbed‘new realism’ or ‘speculative realism’ hasdeveloped, in various work by Quentin Meillassoux, Graham Harman,Markus Gabriel, and others. The idea is that much of philosophy sinceKant, especially in the continental tradition, has been lost inpostmodernist concern with thought and language, and that a return tothe “Great Outdoors” is overdue. The return to realismthough is speculative in the sense that it continues to take seriouslyepistemic concerns: in Harman’s case at least, mind-independentobjects will be radically resistant to knowledge and almost impossiblyremoved from us. If such a realism involves specifying objects ormaking knowledge claims about them that the theory itself seems topreclude, then, much as with the paradoxes at the limit of thought(presented in section 2.1 above), this may crucially involvecontradiction. Some practitioners in this area have explicitlyendorsed dialetheism, e.g., Morton 2012. Cogburn has identified inthis approach what he dubs the ‘object-oriented ontology’paradox (as fitting Priest’s inclosure schema): “To theextent that object-oriented ontologists really are offeringspeculative systems of metaphysics, they seem to be trying to do whatthey themselves take to be impossible” (Cogburn 2017, chapter 3;see 3.3 above).

(7) It has long been recognized thattheology may involveparadoxes. In traditional Western monotheism, various divineattributes appear to be logically inconsistent. Famously, the questionof whether an omnipotent God can create a stone so heavy He Himselfcannot lift seems to lead to a contradiction—something that isboth possible and not possible for God. Cotnoir 2018 explores whetherdialetheism and the tools of paraconsistent logic provide a plausibleresponse to these paradoxes, namely, to accept their conclusions.Beall 2021 focuses more specifically on Christianity and the apparentcontradiction that Jesus Christ was both mortal (and limited) but alsodivine (and so immortal and unlimited), with the suggestion that thiscan best be accommodated as a truth glut. Weber 2019 expressesdisagreement with theological applications of dialetheism.

Each of the above topics undoubtedly calls for further development(see Priest 1987 or Ficara 2014). This list is also not exhaustive.For example, someconnexive logics are contra-classical (they validate classically invalid arguments)and are even contradictory in their propositional fragment. That is,whereas most motivations for dialetheism come from non-logicalsources, connexive logic is dialetheic at the most basic logical level(see Omori and Wansing 2019). Or, for both a motivation andapplication of dialetheism, Casati 2021 argues for a dialetheicinterpretation of the late Heidegger in which Being itself is both anentity and not an entity, and that this can be made sense of throughmereology and metaphysical grounding.

4. Objections to Dialetheism

We now turn to arguments against dialetheism. The most prominentsustained defence of the LNC in the history of philosophy is, asmentioned,that given by Aristotle in Chapter 4 ofMetaphysics, \(\Gamma\). A critical analysisof Aristotle’s arguments is given point for point by Priest(1998b/2006 Ch. 1), who finds them to often conflate dialetheism withtrivialism (that is, conflating the claim that some contradictions aretrue with the one that all contradictions are). We will look at somemore modern arguments against dialetheism.

4.1 The Argument from Explosion

A standard argument against dialetheism is to invoke the logicalprinciple of Explosion, in virtue of which dialetheism would entailtrivialism. Granted that trivialism is absurd (though see Priest2000a, Priest 2006, Ch. 3, and Kabay 2010), dialetheism must berejected. Since this argument assumes that Explosion is logicallyvalid, it will carry no weight against a dialetheicparaconsistentist.

Interestingly enough, while Aristotle’s defence of the LNC mayslide between attacking dialetheism and trivialism, Aristoteliansyllogistic—the first formally articulated logic in Westernphilosophy—is not explosive. Aristotle held that some syllogismswith inconsistent premises are valid, whereas others are not (An. Pr.64a 15). Just consider the inference:

(P1) Some logicians are intuitionists.

(P2) No intuitionist is a logician.

(C) Therefore, all logicians are logicians.

This is not a valid syllogism, despite the fact that its premises areinconsistent. The principle of Explosion had a certain tenure atplaces and times in Medieval logic, but it became well-establishedmainly with the Fregean and post-Fregean development of what is nowcalled classical logic. For the logical aspects of denying Explosion,see the article on paraconsistent logic.

4.2 The Argument from Exclusion

There are several objections to dialetheism involving the notion ofexclusion. The very rough idea is that dialetheists have letin more truths than non-dialetheists, but subsequently nowhave trouble keeping enough falsityout. This objection, inone form or another, has been a recurring theme in criticisms ofdialetheism.

A version of the argument from exclusion against dialetheism found,for instance, in McTaggart 1922 (see Berto 2006, 2012) is as follows.A sentence is meaningful only if it rules something out. But if theLNC fails, \(A\) does not rule out \(\neg A\), or,afortiori, anything else. Hence meaningful language presupposesthe LNC.

There are problems one may find with this argument. One is that eventhough a dialetheia does not rule out its negation, it still may ruleout several other things. A bigger trouble is that the first premiselooks to be simply false. Consider again the sentence‘Everything is true’. This entails everything, and sorules out nothing. Yet it is meaningful. It is something thateveryone, except a trivialist, rejects.

One might attempt a more sophisticated explanation of the notion ofruling out, for instance in terms of information theory, orperhaps possible worlds. One may claim that a statement ‘rulesout’ something insofar as there are situations, or worlds, atwhich it fails. In this sense, ‘Everything is true’ doesrule something out. But now, it is this account of propositionalmeaning which can be challenged in general. If mathematical truthshave a strictly necessary status (which may safely be assumed here),Fermat’s Last Theorem rules out nothing: being a necessarytruth, it holds at all possible worlds. But it is perfectlymeaningful; people wondered whether it was true or false forcenturies; and its proof by Andrew Wiles was a substantialdiscovery.

One argument from exclusion, with a moread hominem twist,claims that the dialetheist has trouble withexpressingdisagreement with rival positions in debates (see Parsons 1990,Shapiro 2004, Littman and Simmons 2004). For when the dialetheistutters ‘\(\neg A\)’, this is in itself insufficient torule out that \(A\) is the case, given that, in a dialetheic world, itmay well be that both \(A\) and \(\neg A\). Similarly, ‘\(A\) isfalse’ and even ‘\(A\) is not true’ might not do thetrick, since for the dialetheist some \(A\)’s being false, oruntrue, does not rule out its being true. As Parsons puts it,

Suppose that you say ‘\(A\)’ and Priest replies‘\(\neg A\)’. Under ordinary circumstances you would thinkthat he had disagreed with you. But then you remember that Priest is adialetheist, and it occurs to you that he might very well agree withyou after all—since he might think that \(A\) and \(\neg A\) areboth true. How can he indicate that he genuinely disagreeswith you? The natural choice is for him to say ‘\(A\) is nottrue.’ However, the truth of this assertion is also consistentwith \(A\)’s being true—for a dialetheist, anyway.(Parsons 1990, p. 345)

Elaborations and repetitions of this line of thought have come to beknown in the literature as the ‘just true’ or ‘trueonly’ problem, e.g., Rossberg 2013. (Young 2015, however, arguesthat the exclusion problem is distinct from the just-trueproblem.)

To this, the dialetheist has various replies. Priest has argued thatthe logical operation of negation should be distinguished from thespeech acts of denial and the cognitive state of rejection. So, contraFrege, assertion of a negation is not the same as denial. The idea isthat exclusion is expressed via a primitive notion ofrejection: to reject \(A\) is to positively refuse to believethat \(A\). That the notion is taken as primitive means, inparticular, that it is not reducible to the acceptance of negation: itis asui generis act. The linguistic counterpart of rejectionis the speech act of denial. Then the dialetheist can rule out that\(A\) is the case by denying \(A\); and this does not amount to theassertion of \(\neg A\) (Priest 2006, Ch. 6). More simply, one canoften express denials by uttering ordinary language negations:‘not’ is, in this sense,pragmatically ambiguous.We will come back below to how, and why, rejection-denial may not bereducible to the acceptance-assertion of any negation.

Another way in which the dialetheist may express the exclusion that\(A\) is the case is by uttering ‘\(A \rightarrow \bot\)’,where again \(\bot\) is ‘Everything is true’. Similarly, adialetheist may try to fix that \(A\) is consistent by saying‘if \(A\) is both true and false, then \(\bot\)’.Considerations by e.g., Field (2008, Ch. 27), Murzi and Carrara(2013), and Berto (2014), however, cast doubt on whether‘arrow-falsum’ can work as a dialetheicexclusion-expressing device in all cases. One reason is due toside-effects of the aforementioned Curry paradox. Another is that‘arrow-falsum’ is very strong; a mundane sentencelike ‘Alice had spaghetti for dinner’ may be false-only(she didn’t), but if it were true too, that wouldn’t beabsurd (cf. Beall 2013, proposing exclusion via non-logical,theory-specificrules of the form \(A,\neg A \vdash \bot\)for each sentence \(A\) we take to be consistent; Scharp 2018 suggestspossible problems for this approach).

The exclusion problem gets its strength from some backgroundassumptions. Specifically, it seems to presuppose that the content ofa proposition is given in terms of splitting all possible situations,or worlds, into those where the proposition holds and those where itdoes not. Even accepting this presupposition, though, would not affecta dialetheic challenge to the LNC. For a given \(A\) to be adialetheia, putting things in these terms, it is sufficient that therebe overlap between the worlds where \(A\) holds and those where itsnegation holds. And this is compatible with the idea of propositionalcontent as splitting the totality of worlds—but now withoverlap. The tenability of such an overlap, though, requiresdiscussing the account of negation from classical logic, to which wenow turn.

4.3 The Argument from Negation

Arguments against dialetheism are often focused on the concept oflogical negation. The main one goes as follows. The truth conditionsfor negation are: \(\neg A\) is true iff \(A\) is not true. Hence, if\(A\) and \(\neg A\) were true, \(A\) would be both true and not true,which is impossible.

What to make of this argument? First, the truth conditions fornegation employed here are contentious. An alternative view has itthat \(\neg A\) is true iff \(A\) is false, and \(\neg A\) is falseiff \(A\) is true—and in the semantics of many paraconsistentlogics (for instance, the logic of First Degree Entailment), truth andfalsity may overlap. Such an account preserves our intuition thatnegation is the operator which (truth-functionally) switches truth andfalsity. It also preserves our intuition on contradictoriness, in theform: \(A\) and \(B\) are contradictories iff, if \(A\) is true, \(B\)is false, and if \(A\) is false, \(B\) is true. What has to go, onthis proposal, is the assumption that truth and falsity are mutuallyexclusive in all cases: there exist dialetheias, that is, sentencesfalling simultaneously under both categories.

Secondly, a dialetheist can point out that the argument againstdialetheism based on the truth conditions for (classical) negationbegs the question at its last step: why should we assume that it isimpossible for \(A\) to be both true and not true? Well, because itwould be a contradiction. But that is precisely what is at issue; thecritic was supposed to be arguing for the impossibility of anycontradiction holding to begin with. In fact, the dialetheist may evenaccept a characterisation of the truth conditions fornegation as: ‘\(\neg A\) is true iff \(A\) is not true’.For if the ‘metalanguage’ in which the characterisation isexpressed can be inconsistent in its turn, as a thoroughgoingdialetheist is likely to allow (as in Weber et al. 2016), then thereis no guarantee that the ‘not’ in that clause behavesconsistently. So the debate comes back to the basic question ofwhether or not consistency can be presupposed.

A variant on the anti-dialetheic argument from negation comes from aQuinean conception of logical vocabulary. It goes as follows. Evengranting that there is an operator, say, \(*\), which behaves asdialetheists claim (namely, such that in particular in some cases\(A\) is true together with \(*A)\), it is still perfectly possible todefine a negation with all the properties of classicalnegation; in particular, the property of being explosive. And sinceclassical negation is the standard operator in logic, it is misleadingto translate anything non-classical as ‘not’: such atranslation risks simply calling ‘negation’ somethingdifferent. A change in logical vocabulary is a “change ofsubject”, as the Quinean slogan goes. A version of thisobjection to dialetheism is due to Slater (1995); see also Restall1993.

One line of reply to the Quinean objection available to thedialetheist is that the objection is confused between a logical theoryand what the theory is a theory of. There are many different andwell-worked-out logical theories of negation (minimal negation,intuitionistic negation, De Morgan negation, etc.). Insofar as eachone of them characterizes its own theoretical object, there is norivalry between logics. Rivalry begins when we wonder whether someaccount or other captures the meaning and functioning of negation asit is used in the vernacular. An applied account of negation is atheoryof something, and the theoretical object has to fitthe real object. Now, to assume beforehand that the classical accountof negation is the correct one, in the sense that it captures hownegation works in the vernacular, again begs the question against thedialetheist (and, indeed, against most non-classical logicians): it iscircular just toassume that classical negation gets itright. People who propose a treatment of negation alternative to theclassical one are not thereby proposing to revisenegation,but to revise an account of it, which they consider incorrect.

If it is agreed that a fair discussion between dialetheists andnon-dialetheists cannot presuppose one account of negation over theother, then this suggests a simpler reply from dialetheists to thejust-true/exclusion problems from 4.2 above. The reply now is to demurfrom the presuppositions of the just-true/exclusion objections.Without begging any questions one way or the other, this reply goes,all agree that for a sentence to be just-true amounts to it being trueand not false, and just-false being false and not true. Then as Priestputs it,

A dialetheist can express the claim that something, \(A\), is nottrue—in those very words, \(\neg Tr\langle A \rangle\). What shecannot do is ensure that the words she utters behave consistently:even if \(\neg Tr\langle A\rangle\) holds, \(A\) and \(\neg Tr\langleA\rangle\) may yet hold. (Priest 1987 [2006, 291])

In the exclusion objection, it appears that the dialetheist is beingasked to ensure that the words she utters behaveconsistently—which is exactly what is at issue (see Restall 2010and the reply in Beall et al. 2011 for one round of this sort ofdiscussion). A dialetheist goes on to question why one should thinkthat commitment to an explosive logic is, in itself, a guard againstcontradiction, and therefore why dialetheists are supposed to be inspecial difficulty with respect to the exclusion question any morethan non-dialetheists. The way one fills in the details here willvary, but the reply, in short, is that for a dialetheist, as withanyone else, ‘just-true’ is just ‘true’ (Beall2009, p. 54). For elaboration of a dialetheic solution to the‘just true’ problem see Omori and Weber 2019.

One final way this same sort of objection is expressed is by askingdialetheists for some way to sort the true-only sentences from thetrue-and-false sentences:

I often find myself being asked the following question: ‘Sinceyou believe some contradictions, but not all, you must have acriterion for deciding between those that are true and those that arenot. What is it?’ In reply I usually point out that thequestioner believes some things are true, but not all, and ask themwhat criterion they have for deciding between those things that aretrue and those that are not. The answer, I think, is the same in bothcases. Nice as it would be to have a criterion of truth, to expect onewould seem utopian. One has to treat each case on its merits, whetherthe proposition concerned is a contradiction or some other thing.(Priest 2006 p. 56)

Priest’s suggestion is that dialetheists are working with truthand falsity, just like non-dialetheists. Accounts of negation differ,but this on its own does not create any special obligations on thepart of dialetheists.

5. Dialetheism and Rationality

5.1 Consistency and Other Epistemic Virtues

Some have felt that what is wrong with dialetheism is not so muchviolation of the LNC itself, as that an acceptance of the LNC is aprecondition for rationality. For example, it is often suggested thatit could not possibly be rational to accept a contradiction.

It is a matter of ongoing debate what the conditions are under whichit is rational to accept something. Nevertheless, it is commonlyagreed that, as Hume put it, the wise person “proportions hisbeliefs to the evidence” (1955, p. 118). If this is right, thenif a sufficient case can be made out for a contradiction, it will berational to believe it. And sometimes this does seem possible. We haveseen that a seemingly compelling argument can be made in favour of thetruth of the strengthened Liar sentence, ‘This sentence is nottrue’. Whether or not one takes the argument in question to becompletely persuasive, it suggests that there is nothing in principleimpossible about the existence of good arguments for truecontradictions. Of course, if there were conclusive evidence for theLNC, then no case for a contradiction could be strong enough. Butconclusive evidence for any philosophical position is difficult toachieve.

A more persuasive worry about dialetheism, relating to rationality, isthe claim that if a person could legitimately accept a contradiction,then no one could be forced, rationally, toabandon any viewheld. For if a person accepts \(A\) then, when an argument for \(\negA\) is put up, they could simply accept both \(A\) and \(\neg A\).

A dialetheist can reply that, again, not all contradictions are equal.Each sentence, including each contradiction, is evaluated on itsmerits. While a case can be made for the claim that the Liar sentenceis both true and false, this in no way shows that a case can also bemade for Brisbane being and not being in Australia. (Of course, if onesubscribes to the claim that entailment is explosive, a case for onecontradiction is a case for all; but if entailment is paraconsistent,this argument is of no use.)

As orthodox philosophy of science indicates, there are, in fact, manydifferent considerations that speak for or against the rationalacceptability of a theory or a view. Among the epistemic virtues of atheory are: its adequacy to the data; its simplicity, cleanness andelegance; its unity and freedom fromad hoc hypotheses; itsexplanatory and predictive power; etc. Not only do these (and other)criteria come in degrees, but they may also be orthogonal to eachother. In the end, the rational evaluation of a view must balance itagainst all criteria of this kind (of which consistency is, arguably,one), each, on its own, being defeasible.

Dialetheism asks us to consider the possibility that a theory lackingthe virtue of consistency may still overcomes its rivals in all ormost of the other respects. According to dialetheists, this isactually the case with the dialetheic account of the semantics ofordinary language, whose advantages with respect to consistentaccounts have already been shortly suggested above. And conversely, ofcourse, an inconsistent theory may well be trumped by a consistenttheory, all things considered. So it may be rational to reject aninconsistent position, even if it is logically possible that it istrue. Rationality considerations are dealt with at length in Priest2006.

5.2 Accepting and Asserting Dialetheias

If some contradictions are true, it is natural to expect that adialetheist will sometimes accept, or believe in, contradictions, andassert them. Priest (2006, p. 109) adopts the following RationalityPrinciple:

(RP) If you have good evidence for the truth of \(A\), you ought toaccept \(A\).

Belief, acceptance, and assertion have apoint: when webelieve and assert, what we aim at is believing and asserting what isthe case or, equivalently, the truth. Therefore, the dialetheist willaccept, and sometimes assert, both \(A\) and \(\neg A\), if she hasevidence that \(A\) is a dialetheia.

Notice that this need not entail that the dialetheist both acceptsand rejects \(A\) at the same time. We now come back to theissue flagged in Section 4.2, on the irreducibility of rejection tonegation. That rejecting \(A\) is tantamount to accepting its negationis a common view, famously endorsed and defended (more precisely interms of the corresponding speech acts of assertion and denial) byFrege and Peter Geach. But dialetheists have argued that this fusionis a confusion (see Berto 2008 on this issue). The point can be madeindependently of the issue of dialetheism: a paracompletist may wellwant to deny \(A\), but it would be unfair to take such a denial asequivalent to the assertion of \(\neg A\), since if \(A\) istruth-valueless, \(\neg A\) is normally considered truth-valueless,too, not a truth, and so not to be asserted. A dual position can holdfor dialetheism: given that accepting \(\neg A\) is different fromrejecting \(A\), a dialetheist can do the former and not thelatter—exactly when she thinks that \(A\) is a dialetheia.

Does this show that dialetheism is compatible with rationality? Thestory about assertion and denial, acceptance and rejection has beenchallenged from several directions. We will just mention a few.

Restall (2015) argues that the acceptance/rejection issue makes gappyand glutty approaches symmetrical: gap theorists cannot assert sometrue claims about paradoxes, but glut theorists cannot reject somefalse claims about non-paradoxes (see also Restall 2013, Jenny 2017).In a note, Laura Goodship (1996) suggests that the proposal ofseparating denial from assertion of negation runs into problemsrelated, again, to Curry’s paradox. Her proposal is, in effect,that it would be more natural for dialetheists to both acceptand reject things after all. Focusing on the phenomenon ofdisagreement in natural language, Ripley (2015a) argues thatdialetheists (and paracompletists) should rejoin classical logiciansin taking negation to embed denial, and denial to expressdisagreement. Ripley proposes that dialetheists simply admit that (1)agreement and disagreement areincompatible(“it’s incoherent to do both”, p. 306), but (2) insome cases, assert and deny the same thing. The result would be whatRipley callsparacoherentism, which (in echo ofparaconsistency) would try to allow local incoherence without globalincoherence. How this could be done is an open question. Ripleysuggests dropping the transitivity of logical consequence. Goodshipherself ultimately recommends dropping modus ponens, a proposal thathas been dubbed the ‘Goodship project’ (Beall 2015, Omori2016, Priest 2017).

There certainly are various other arguments against dialetheism in thephilosophical market (see papers in Priest et al. 2004). For example,Zalta (2004, p. 432) argues that preserving “our pretheoreticunderstanding of what it is to exemplify or instantiate aproperty” requires us to preserve the LNC. This entry haspresented only some of the most immediate issues that arise inobjections and replies.

6. Themes for Further Research

Since dialetheism is simply a claim about truth, it can play a role inany area—say in traditional or mainstream philosophy—wheretruth is involved. Among such topics, a prominent one is the debatebetweenrealists andanti-realists (for example, idealists or constructivists) in metaphysics. Veryroughly, to be a realistabout entities of some kind is tomaintain that such entities objectively exist apart from, andantecedently to, anyone’s thought of them; and, therefore, thatour thoughts, beliefs and theories concerning such entities are madeobjectively true or objectively false by them, apart from what wethink of them (more refined definitions of realism and anti-realismare certainly available; but this characterization will suffice forour purposes).

It has been claimed (Priest 2000b, Priest 2006, Ch. 2) thatdialetheism is not by itself committed to a specific conception oftruth (deflationist, semantic, correspondentist, coherentist,constructivist, etc.).

If something is true, there must be something that makes it so. Callthis the world. If some contradictions are true, then the world mustbe such as to make this the case. In this sense, the world iscontradictory. What it is in the world that makes something true isanother matter. (Priest 2006, p. 299)

Nevertheless, if we accept even a mild form of realism, the truth ofsome contradictions entails the existence of inconsistent objectsand/or states of affairs: those that make the contradictions true(Berto 2007b). One may claim that it makes no sense to talk ofinconsistent objects, situations, or states of affairs. The world isall there, all together: how could some pieces of itcontradict some other pieces? Consistency and inconsistencymight be taken as properties of sentences, or theories (sets ofsentences closed under logical consequence), or propositions (whatsentences express), or maybe thoughts, or (sets of) beliefs, etc.Contradiction (Widerspruch, the Latincontradictio)has to do with discourse (diction,sprechen,dicere). The world, with its non-mental and non-linguisticinhabitants—armchairs, trees, people—is not the rightkind of thing that can be consistent or inconsistent, andascribing such properties to (a part of) the world is, to use GilbertRyle’s terminology, a category mistake.

These considerations might drive dialetheism towards an anti-realistinterpretation of the claim that there are dialetheias, truecontradictions; and anti-realist dialetheic theories of truth have, infact, been proposed (see e.g., Beall’s ‘constructivemethodological deflationism’, in Beall 2004). But other optionsare available to a dialetheist who wants to embrace some form ofmetaphysically robust realism about truth. For instance, she canstress that consistency and inconsistency can be ascribed to (piecesof) the world in aderived sense: to say that the world is(locally) inconsistent just is to say that some true purelydescriptive sentences about the world have true negations.Consequently, and not accidentally, it is quite common in the currentliterature both for and against dialetheism to straightforwardly speakof inconsistent objects, states of affairs, and entire inconsistentworlds. A dialetheic correspondence theory of truth might becommitted, in particular, to negative facts (requiring thesimultaneous existence of truth-makers both for \(A\) and for itsnegation, when \(A\) is a dialetheia); but these may be not toodifficult to handle (see e.g., Priest 2006, pp. 51–3).

There are further intermediate positions. One is ‘semanticdialetheism’ which accepts true contradictions withoutinconsistent objects or states of affairs as their truth-makers. Thisposition has been explored in the literature (Kroon 2004, Mares 2004).Beall’s view, expressed in his transparent truth theory in Beall2009, may also be seen as a form of semantic dialetheism. Transparencycan be naturally paired with adeflationary view of truth. For suppose the truth predicate is a merely semantic device, coined,as Quine famously stressed, for expressive,‘disquotational’ purposes. Then dialetheias such as theLiar(s) may well be semantic side-effects (‘spandrels’, inBeall’s terminology) of the introduction of such device, notinvolving any metaphysically committing contradiction in a language-and mind-independent world. Woodbridge and Armour-Garb (2013) arguethat a deflationary view of truth is best understood in terms ofsemantic pretense (a hermeneutic fictionalist perspective), and onthat basis offer a pretense account of the semantic paradoxes. Wansing(2024) advances a view calleddimathematism, where there maybe contradictions in information (information being “what isleft from knowledge when you subtract justification, truth, andbelief” as Dunn [2001, p. 423] puts it) again distinct fromcontradictions in truth.

Debates on realism and anti-realism quickly spill over into questionsconcerning the nature of reality in general. Thus the dialetheicprogramme looks to metaphysical issues: if reality is dialetheic, howshould the ontology of a dialetheic world be spelt out? If metaphysicsshould be placed (once again) at the very core of philosophy, thedebate on the possibility of dialetheias occupies a central place inthe core. This was, after all, Aristotle’s view, too: he decidedto speak on behalf of the unconditional validity of the LNC, not inhisOrganon (his writings on the subject of logic), but intheMetaphysics, for this was for him an issue to beaddressed ontologically, not (only) via formal logical tools. For workin dialetheic metaphysics see Priest 2014, which grounds the veryunity of objects in an inconsistent theory of parthood and boundary,and applies it to classic problems such as the One and the Many, andthe instantiation of universals.

7. Conclusion

Since Aristotle, the assumption that consistency is a requirement fortruth, validity, meaning, and rationality, has gone largelyunchallenged. Modern investigations into dialetheism, in pressing thepossibility of inconsistent theories that are nevertheless meaningful,valid, rational, and true, call that assumption into question. Ifconsistency does turn out to be a necessary condition for any of thesenotions, dialetheism prompts us to articulate why; just by pushingphilosophers to find arguments for what previously were undisputedbeliefs it renders a valuable service (Scharp 2007, p. 544). And ifconsistency turns out not to be an essential requirement for alltheories, then the way is open for the rational exploration of areasin philosophy and the sciences that have traditionally been closedoff.

Bibliography

We break up the references into sections corresponding to those of thetext. Where a reference is not explicitly referred to in the text, weadd a sentence concerning its relevance.

Some Basic Concepts

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• Kabay, P., 2010,On the Plenitude of Truth: A Defense ofTrivialism, Saarbrücken: Lambert Academic Publishing.
• Lewis, D., 1999, “Letters to Beall and Priest”, inPriest et al. 2004, pp. 176–177.
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• Pleitz. M., 2018,Logic, Language, and the Liar Paradox,Münster: Mentis.
• Priest, G., 2000, “Motivations for Paraconsistency: theSlippery Slope from Classical Logic to Dialetheism”, in D.Batens et al. (eds.),Frontiers of Paraconsistent Logic,Baldock, UK: Research Studies Press.
• Priest, G., JC Beall, and B. Armour-Garb (eds.), 2004,The Lawof Non-Contradiction. New Philosophical Essays, Oxford: OxfordUniversity Press.
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• Woods, J., 2003,Paradox and Paraconsistency, Cambridge:Cambridge University Press.
• –––, 2005, “Dialectical Considerations onthe Logic of Contradiction: Part I”,Logic Journal of theIGPL, 13: 231–60.

Dialetheism in the History of Philosophy

• Aristotle,The Complete Works, J. Barnes (ed.),Princeton: Princeton University Press.
• Cusanus, Nicholas, 1440,Of Learned Ignorance, G. Heron(trans.), London: Routledge and Kegan Paul, 1954.
• Deguchi, Y., J.L. Garfield and G. Priest, 2008, “The Way ofthe Dialetheist: Contradictions in Buddhism”,PhilosophyEast and West, 58: 395–402.
• Dunn, J., 1966,The Algebra of Intensional Logics, Ph.D.Thesis, University of Pittsburgh; reprinted, inLogic Ph.D.s(Volume 2), London: College Publications, 2019.
• Ficara, E., 2013, “Dialectic and Dialetheism”,History and Philosophy of Logic, 34(1): 35–52.
• –––, 2021, “The Birth ofDialetheism”,History and Philosophy of Logic, 42(3):281–296.
• Garfield, J. (translation and commentary), 1995,TheFundamental Wisdom of the Middle Way: Nagarjuna’sMulamadhyamakakarika, Oxford: Oxford University Press.
• Garfield, J., T. Tillmans and M. D’Amato (eds.), 2009,Pointing at the Moon: Buddhism, Logic, Analytic Philosophy,Oxford: Oxford University Press.
• Hegel, G.W.F., 1830,Enzyklopädie der philosophischenWissenschaften in Grundrisse, inWerke im zwanzigBänden, hrg. von E. Moldenhauer und K.M. Michel, Bände8–10, Suhrkamp, 1970; page references are to the Englishtranslation,The Encyclopaedia Logic (with the Zusätze),Indianapolis: Hackett, 1991.
• –––, 1831,Wissenschaft der Logik,1831, vols. 11 and 12 ofGesammelte Werke, in Verbindung mitder Deutschen Forschungsgemeinschaft, hrg. von derRheinisch-Westfälischen Akademie der Wissenschaften, Meiner,1968ff; page references are to the English translation,Hegel’s Science of Logic, New York: Humanity Books,1969.
• Kant, I., 1781,Kritik der reinen Vernunft, 1781, vols. 3and 4 ofGesammelte Schriften, Berlin: De Gruyter & Co.,1969; page references are to the English translation,Critique ofPure Reason, New York: Palgrave Macmillan, 2003.
• Priest, G., 1990, “Dialectic and Dialetheic”,Science and Society, 53: 388–415.
• –––, 1991, “Was Marx aDialetheist?”,Science and Society, 54:468–75.
• –––, 1995,Beyond the Limits ofThought, Cambridge: Cambridge University Press, 2nd expandededition, Oxford: Oxford University Press, 2002.
• –––, 2018,The Fifth Corner of Four,Oxford: Oxford University Press.
• Priest, G., and R. Routley, 1989a, “The History ofParaconsistent Logic”, Chapter 1 of Priest, Routley and Norman,1989 (above).
• Robinson, T.M., 1987,Heraclitus: Fragments, Toronto:University of Toronto Press.
• Routley, R., 1980,Exploring Meinong’s Jungle andBeyond, Canberra: Australian National University.
• Siderits, M., 2007,Buddhism as Philosophy: anintroduction, Aldershot: Ashgate.
• Smart, N., 1964,Doctrine and Argument in IndianPhilosophy, London: Allen and Unwin.
• Tanaka, K. (ed.), 2013, “Buddhism and Contradiction”,Philosophy East and West (Special Issue), 63(3).
• Tanaka, K., Y. Deguchi, J. Garfield, and G. Priest (eds.), 2015,The Moon Points Back, Oxford: Oxford University Press.
• Tillmans, T., 2009, “How do Madhyamikas Think?: Notes on JayGarfield, Graham Priest, and Paraconsistency,” in Garfield, etal. 2009, pp. 83–99.
• Suzuki, D.T., 1969,The Zen Doctrine of No Mind, London:Rider and Co.
• Zhuangzi,Wandering on the Way: Early Taoist Tales andParables of Chuang Tzu, V. H. Mair (trans.), New York: BantamBooks, 1994.

Motivations for Dialetheism

• Beall, Jc (ed.), 2007,Revenge of the Liar, Oxford:Oxford University Press.
• Beall, Jc, 2009,Spandrels of Truth, Oxford: OxfordUniversity Press.
• –––, 2014a, “Finding Tolerance WithoutGluts”,Mind, 123(491):791–811.
• –––, 2014b, “End of Inclosure”,Mind, 123(491): 829–849.
• –––, 2016, “Do Laws Deliver Gluts?”,inLaw and the New Logics, Glenn and Smith (eds.), Cambridge:Cambridge University Press, 199–207.
• –––, 2021,The Contradictory Christ,Oxford: Oxford University Press.
• Beall, Jc and J. Murzi, 2013, “Two Flavors of Curry’sParadox”,Journal of Philosophy, 100(3):143–165.
• Beall, Jc and D. Ripley, 2018, “Non-Classical Theories ofTruth”, in M. Glanzberg (ed.),The Oxford Handbook ofTruth, Oxford: Oxford University Press.
• Brady, R., 1989, “The Non-Triviality of Dialectical SetTheory”, in Priest, Routley and Norman (above), pp.437–71.
• Cobreros, P., P. Egre, D. Ripley, and R. van Rooij, 2012,“Tolerant, Classical, Strict”, 41(2): 347–385.
• –––, 2015, “Pragmatic Interpretations ofVague Expressions: Strongest Meaning and NonmonotonicConsequence”,Journal of Philosophical Logic, 44(4):375–393.
• Cogburn, J., 2017,Garcian Meditations: The Dialectics ofPersistence in Form and Object, Edinburgh: Edinburgh UniversityPress.
• Colyvan, M., 2009, “Vagueness and Truth”, in H. Dyke(ed.),From Truth to Reality: New Essays in Logic andMetaphysics, Oxford: Routledge, 2009, pp. 29–40.
• Cook, R., 2007, “Embracing Revenge: On the IndefiniteExtensibility of Language”, in Beall 2007, p. 31–52.
• Cotnoir, A., 2018, “Theism and Dialetheism”,Australasian Journal of Philosophy, 96(3):592–609.
• Ficara, E., editor, 2014,Contradictions: Logic, History,Actuality, Berlin: de Gruyter.
• Field, H., 2008,Saving Truth from Paradox, Oxford:Oxford University Press.
• Hyde, D., 1997, “From Heaps and Gaps to Heaps ofGluts”,Mind, 106: 640–60.
• Kirkham, R.L., 1992,Theories of Truth. A CriticalIntroduction, Cambridge, Mass: MIT Press.
• Kripke, S., 1975,“Outline of a Theory of Truth”,Journal of Philosophy, 72: 690–716. Reprinted in R.M.Martin (ed.),Recent Essays on Truth and the Liar Paradox,Oxford: Oxford University Press, 1984, pp. 53–81.
• Martin, R.L., 1967, “Towards a Solution to the LiarParadox”,Philosophical Review, 76: 279–311.
• Mortensen, C., 1995,Inconsistent Mathematics, Dordrecht:Kluwer Academic Publishers.
• Morton, T., 2012, “An Object-Oriented Defense ofPoetry”,New Literary History, 43(2):205–224.
• Omori, H., and H. Wansing, 2019, “Connexive Logics: AnOverview and Current Trends”,Logic and LogicalPhilosophy, 28: 371–387.
• Priest, G., 1987,In Contradiction, Dordrecht: MartinusNijhoff. 2nd expanded edition, Oxford: Oxford University Press,2006.
• –––, 2010, “Inclosures, Vagueness, andSelf-Reference”,Notre Dame Journal of Formal Logic,51: 69–84.
• –––, 2015, “Fusion and Confusion”,Topoi, 34: 55–61.
• Priest, G., and R. Routley, 1983,On Paraconsistency,Research Report #13, Research School of Social Sciences, AustralianNational University; reprinted in Priest, Routley and Norman (eds.)1989. [Priest & Routley 1983 available online]
• –––, 1989b, “Applications ofParaconsistent Logic”, Chapter 13 of Priest, Routley and Norman,1989.
• –––, 1989c, “The PhilosophicalSignificance and Inevitability of Paraconsistency”, Chapter 18of Priest, Routley and Norman, 1989 (above).
• Restall, G., 1992, “A Note on Naive Set Theory in LP”,Notre Dame Journal of Formal Logic, 33(3):442–432.
• –––, 2000,An Introduction to SubstructuralLogics, London-New York: Routledge.
• Ripley, D., 2012, “Paradoxes and Failures of Cut”,Australasian Journal of Philosophy, 91: 139–64.
• –––, 2013, “Sorting Out theSorites”, in K. Tanaka, F. Berto, E. Mares and F. Paoli (eds.),Paraconsistency: Logic and Applications, Dordrecht: Springer,pp. 327–45.
• –––, 2015b, “Naive Set Theory andNontransitive Logic”,Review of Symbolic Logic, 8(3):553–571.
• Routley, R., 1979, “Dialectical Logic, Semantics andMetamathematics”,Erkenntnis, 14: 301–31. (Adefence of a dialetheic account of the paradoxes ofself-reference.)
• Routley, R., and R.K. Meyer, 1976, “Dialectical Logic,Classical Logic, and the Consistency of the World”,Studiesin Soviet Thought, 16: 1–25.
• Russell, B., 1903,Principles of Mathematics, Cambridge:Cambridge University Press.
• Schroeder-Heister, P. and K. Došen (eds.), 1993,Substructural Logics, Oxford: Oxford University Press.
• van Fraassen, B., 1968, “Presuppositions, Implication andSelf-Reference”,Journal ofPhilosophy, 65: 136–51.
• Varzi, A., 1997, “Inconsistency withoutContradiction”,Notre Dame Journal of Formal Logic, 38:621–39.
• Weber, Z., 2010, “A Paraconsistent Model ofVagueness”,Mind, 119: 1026–45.
• –––, 2012, “Transfinite Cardinals inParaconsistent Set Theory”,Review of Symbolic Logic,5(2):269–293.
• –––, 2019, “Atheism and Dialetheism; or,Why I am not a (paraconsistent) Christian”,AustralasianJournal of Philosophy, 97(2): 401–407.
• –––, 2021,Paradoxes and InconsistentMathematics, Cambridge: Cambridge University Press.
• Weber, Z., and A. Cotnoir, 2015, “InconsistentBoundaries”,Synthese, 192: 1267–1294.
• Weber, Z., D. Ripley, G. Priest, D. Hyde, and M. Colyvan, 2014,“Tolerating Gluts”,Mind, 123(491):813–828.

Objections to Dialetheism

• Armour-Garb, B. and J. Woodbridge, 2006, “Dialetheism,Semantic Pathology, and the Open Pair”,Australasian Journalof Philosophy, 84: 395–416. (An objection to dialetheismbased on the notion of pathological sentence.)
• Beall, Jc, 2013, “Shrieking Against Gluts: The Solution tothe ‘Just-True’ Problem”,Analysis, 73(3):438–445.
• –––, 2015, “Free of Detachment: Logic,Rationality, and Gluts”,Noûs, 49(2):410–423.
• Beall, Jc, and G. Priest, 2007, “Not So Deep Inconsistency:A Reply to Eklund”,Australasian Journal of Logic, 5:74–84. (A reply to Eklund 2002.)
• Beall, Jc, G. Priest, and Z. Weber, 2011, “CanU DoThat?”,Analysis, 71(2): 280–285.
• Berto, F., 2006, “Meaning, Metaphysics, andContradiction”,American Philosophical Quarterly, 43:283–97.
• –––, 2012, “How to Rule Out Things withWords”, in G. Restall and G. Russell (eds.),New Waves inPhilosophical Logic, New York: Palgrave Macmillan, pp.169–89.
• –––, 2014, “Absolute Contradiction,Dialetheism, and Revenge”,Review of Symbolic Logic,7(2):193–207.
• Denyer, N., 1989, “Dialetheism and Trivialisation”,Mind, 98: 259–63. (A critique of a dialetheic accountof the paradoxes of self-reference.)
• Eklund, 2002, “Deep Inconsistency”,AustralasianJournal of Philosophy, 80: 321–31. (Another critique of adialetheic account of the paradoxes of self-reference.)
• Irvine, A.D., 1992, “Gaps, Gluts and Paradox”,Canadian Journal of Philosophy, 18 (Supplementary Volume):273–99. (A critique of a dialetheic account of the paradoxes ofself-reference.)
• Littman, G., and K. Simmons, 2004, “A Critique ofDialetheism”, in Priest, Beall and Armour-Garb (eds.) 2004, pp.314–35.
• McTaggart, J.M.E., 1922,Studies in the HegelianDialectic, 2nd edition, Cambridge: Cambridge UniversityPress.
• Murzi, J., and M. Carrara, 2015, “Denial andDisagreement”,Topoi, 34(1): 109–119.
• Omori, H., and Weber, Z., 2019, “Just True? On theMetatheory for Paraconsistent Truth”,Logique etAnalyse, 248: 415–433.
• Parsons, T., 1990, “True Contradictions”,CanadianJournal of Philosophy, 20: 335–53. (A critique of adialetheic account of the paradoxes of self-reference.)
• Priest, G., 1989, “Denyer’s $ Not Backed by SterlingArguments”,Mind, 98: 265–8. (A reply to Denyer,1989.)
• –––, 1995, “Gaps and Gluts: Reply toParsons”,Canadian Journal of Philosophy, 25:57–66. (A reply to Parsons, 1990.)
• –––, 1998a, “What’s So Bad AboutContradictions?”,Journal of Philosophy, 95:410–26. Reprinted in Priest, Beall and Armour-Garb (eds.) 2004,Ch. 1. (A detailed discussion of some modern objections todialetheism.)
• –––, 1998b, “To Be and Not to Be: That Isthe Answer. On Aristotle on the Law of Non-Contradiction”,Philosophiegeschichte und Logische Analyse, 1: 91–130.Reprinted as Chapter 1 of Priest 2006.
• –––, 2003, “Inconsistent Arithmetic:Issues Technical and Philosophical”, in V. F. Hendricks and J.Malinowski (eds.),Trends in Logic, Dordrecht: KluwerAcademic Publishers, pp. 273–99. Reprinted as Chapter 17 of the2nd edition of Priest 1987. (A discussion of inconsistent arithmetics,including a reply to Shapiro 2002.)
• –––, 2017, “What If? The Exploration of anIdea”,Australasian Journal of Logic, 14 (SpecialIssue:Non-classicality: logic, mathematics, philosophy),edited by Z. Weber, P. Girard, and M. McKubre-Jordens.
• Priest, G., and T. Smiley, 1993,“Can Contradictions beTrue?”,Proceedings of the Aristotelian Society, 68(Supplement): 17–54. (A debate on the issue ofdialetheism.)
• Restall, G., 1993, “Deviant Logic and the Paradoxes ofSelf-Reference”,Philosophical Studies, 70:279–303. (Includes a discussion of Quinean objections tonon-classical accounts of negation.)
• –––, 2010, “Ont anduand What They Can do”,Analysis, 70(4):673–76.
• Rossberg, M., 2013, “Too Good to be ‘JustTrue’”,Thought, 2: 1–8.
• Scharp, K., 2007, “Review ofDoubt Truth to Be aLiar by Graham Priest”,Bulletin of SymbolicLogic, 13(4): 541–544.
• –––, 2018, “Shrieking in the Face ofVengeance”,Analysis, to appear.
• Shapiro, S., 2002, “Incompleteness and Inconsistency”,Mind, 111: 817–32. (A critique of the possibility ofinconsistent arithmetic.)
• –––, 2004, “Simple Truth, Contradictionand Consistency”, in Priest, Beall and Armour-Garb (eds.) 2004,pp. 336–54.
• Slater, H., 1995, “Paraconsistent Logics?”,Journal of Philosophical Logic, 24: 451–4.
• Weber, Z., G. Badia, and P. Girard, 2016, “What is anInconsistent Truth Table?”,Australasian Journal ofPhilosophy, 94(3): 533–548.
• Young, G., 2015, “Shrieking, Just False andExclusion”,Thought, 4(4): 269–276.
• Zalta, E., 2004, “In Defense of the Law ofNon-Contradiction”, in Priest, Beall and Armour-Garb (eds.)2004, pp. 416–36.

Dialetheism and Rationality

• Andreas, H. and P. Verdée, 2016,Logical Studies ofParaconsistent Reasoning in Science and Mathematics (Trends inLogic 45), Dordrecht: Springer. doi:10.1007/978-3-319-40220-8
• Beall, Jc and M. Colyvan, 2001, “Looking forContradictions”,Australasian Journal of Philosophy,79: 564–9. (On the spread of dialetheias in the empiricalworld.)
• Berto, F., 2008, “Adynaton and MaterialExclusion”,Australasian Journal of Philosophy, 86:165–90.
• Bremer, M., 2008, “Why and How to Be a Dialetheist”,Studia Philosophica Estonica, 1: 208–27 (A discussionof the conditions on the rational believability of dialetheism.)
• Dutilh Novaes, C., 2008, “Contradiction: the Real Challengefor Paraconsistent Logic”, in J.Y. Béziau, W. Carnielli,and D. Gabbay (eds.),Handbook of Paraconsistency, London:College Publications. (A specification of the conditions for anon-question-begging debate between dialetheists and supporters of theLNC.)
• Girard, Patrick and Koji Tanaka, 2016, “ParaconsistentDynamics”,Synthese, 193(1): 1–14.doi:10.1007/s11229-015-0740-2
• Hume, D., 1748,An Inquiry Concerning HumanUnderstanding, C.W. Hendel (ed.), Indianapolis: Bobbs-MerrilCompany Inc., 1955.
• Jenny, M., 2017, “Classicality Lost: K3 and LP after theFall”,Thought, 6(1): 43–53.
• Omori, H., 2016, “From Paraconsistent Logic to DialetheicLogic”, in Andreas and Verdee, pp. 111–134.
• Priest, G., 2000a, “Could Everything Be True?”,Australasian Journal of Philosophy, 78: 189–95.Reprinted as Chapter 3 of Priest 2006.
• –––, 2006,Doubt Truth to Be a Liar,Oxford: Oxford University Press.
• Restall, G., 2013, “Assertion, Denial, and Non-classicalTheories”, in Tanaka et al. 2013, pp. 81–100.
• –––, 2015, “Assertion, Denial, Accepting,Rejecting, Symmetry and Paradox”, in C. Caret and O. Hjortland(eds.),Foundations of Logical Consequence, Oxford: OxfordUniversity Press, pp. 310–321.
• Ripley, D., 2015a, “Embedding denial”, in C. Caret andO. Hjortland (eds.),Foundations of Logical Consequence,Oxford: Oxford University Press, pp. 289–309.

Themes for Further Research

• Beall, Jc, 2000, “On Truthmakers for Negative Truths”,Australasian Journal of Philosophy, 78: 264–8. (Adiscussion of the connections between dialetheism, correspondencetheory, and negative facts.)
• –––, 2004, “True and False – AsIf”, in Priest, Beall and Armour-Garb (eds.) 2004,197–216.
• Berto, F., 2007b, “Is Dialetheism an Idealism?”,Dialectica, 61: 235–63.
• Casati, F., 2021,Heidegger and the Contradiction of Being: AnAnalytic Interpretation of the Late Heidegger, Routledge.
• Dunn, J., 2001, “The Concept of Information and theDevelopment of Modern Logic”, in W. Stelzner and M.Stöckler (eds.),Zwischen traditioneller und modernerLogik, Paderborn: Mentis, pp. 423–447.
• Grim, P., 2004, “What is a Contradiction”, in Priest,Beall and Armour-Garb (eds.) 2004, 49–72.
• Kroon, F., 2004, “Realism and Dialetheism”, in Priest,Beall and Armour-Garb (eds.) 2004, 245–63.
• Mares, E., 2004, “Semantic Dialetheism”, in Priest,Beall and Armour-Garb (eds.) 2004, 264–75.
• Priest, G., 2000b, “Truth and Contradiction”,Philosophical Quarterly, 50: 305–19. Reprinted asChapter 2 of Priest 2006.
• –––, 2014,One: Being an Investigation intothe Unity of Reality and of its Parts, including the Singular Objectwhich is Nothingness, Oxford: Oxford University Press.
• Tahko, T., 2009, “The Law of Non-Contradiction as aMetaphysical Principle”,Australasian Journal of Logic,7,available online. (A defense of the Law of Non-Contradiction as a metaphysical –as opposed to logical or semantic – principle.)
• Wansing, H., 2024, “One Heresy and One Orthodoxy: OnDialetheism, Dimathematism, and the Non-normativity of Logic”.Erkenntnis, 89: 181–205. [Wansing 2024 available online]
• Woodbridge, R., and B. Armour-Garb, 2013, “SemanticDefectiveness and the Liar”,Philosophical Studies,164(3): 845–863.

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Acknowledgments

The authors would like to thank Jc Beall, Max Carrara, David Ripley,Koji Tanaka, and three anonymous referees, for providing helpfulcomments and suggestions. The editors would like to thank Christophervon Bülow for pointing out a number of infelicities, includingtypos, formatting issues, etc.