## 1. Some Basic Concepts



The word 'dialetheism' was coined by Graham Priest and
Richard Routley (later Sylvan) in 1981 (Priest et al 1989, p.
*xx*). The inspiration for the name was a passage in
Wittgenstein's *Remarks on the Foundations of
Mathematics*, concerning
 Russell's paradox
 (see also below):



> 
> Why should Russell's contradiction not be conceived of as
> something supra-propositional, something that towers above the
> propositions and looks in both directions like a Janus head? The
> proposition that contradicts itself would stand like a monument (with
> a Janus head) over the propositions of logic (1978, III.59).
> 



A dialetheia is a two-way truth, facing both truth and falsity like a
Janus-headed figure. Unfortunately, Priest and Routley forgot to agree
how 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 view
being *inconsistent* and being *incoherent*. Both are
unacceptable. Since dialetheism claims that some inconsistency can be
maintained without incoherence, these two concepts must be
disentangled. For a start, dialetheism should be distinguished from
so-called *trivialism*, that claim that *all* sentences
are true, and hence all contradictions are true too (Kabay 2010).
Dialetheism is the view that *some* contradictions are true. A
trivialist must be a dialetheist, since a trivialist accepts every
claim. But since 'some' does not imply 'all',
the converse is not the case: a dialetheist typically claims that only
some (and, usually, very specific) sentences are dialetheias, not that
all of them are. Trivialism is by all accounts incoherent, and so is
minimally what counts as unacceptable even for someone who thinks that
some contradictions are true.



How to allow inconsistency without incoherence is one of the main
topics of dialetheim. A standard solution consists in subscribing to
the view that deductive logical consequence is
 *paraconsistent*.
 A logical consequence relation \(\vdash\) is *explosive* if,
according to it, a contradiction entails everything (*ex
contradictione quodlibet*: for all \(A\) and \(B\): \(A,\neg A
\vdash B\)). It is paraconsistent if and only if (iff) it is not
explosive. By adopting a paraconsistent logic, a dialetheist can
countenance some contradictions without being thereby committed to
countenancing everything and, in particular, all contradictions. The
development of paraconsistent logics has contributed to the most
recent developments (and acceptance of the legitimacy) of
dialetheism.



Dialetheism should, however, be clearly distinguished from
paraconsistency. Whereas dialetheists must embrace some paraconsistent
logic or other to avoid trivialism, paraconsistent logicians need not
be dialetheists: they may subscribe to a non-explosive view of
entailment for other reasons. Within paraconsistency, one may
distinguish (at least) four grades of paraconsistent involvement
(Beall and Restall 2006, p.80): 


1. *Gentle-strength paraconsistency* is simply the rejection
of explosion with respect to logical consequence.
2. *Full-strength paraconsistency* has it that there are
interesting or important theories that are inconsistent but not
trivial.
3. *Industrial-strength paraconsistency* has it that some
inconsistent but non-trivial theories are *possibly* true.
4. *Dialetheic paraconsistency* has it that some inconsistent
but non-trivial theories *are* true.



Most people working on paraconsistent logics have commitments at the
lower grades of the spectrum. The unifying thought behind
paraconsistency is gentle-strength: logical consequence should not
validate arbitrary conclusions following indiscriminately from
inconsistent premises. This may simply be because entailment must
preserve 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-dialetheic
interpretation of paraconsistent logic, where there may be epistemic
contradictions--inconsistencies in evidence or belief--but
no true contradictions. Or it may be that a paraconsistent logician
simply need not assume anything about truth in order to provide a
working treatment of inconsistency as it may arise in databases, legal
situations, works of fiction, theory change, belief-revision, etc.




Stepping up, full-strength paraconsistentists treat inconsistent
models, in which contradictions hold, as useful mathematical tools
without committing to them as representing *real*
possibilities. 



And stepping up again, industrial-strength paraconsistentists may hold
that, though truth at the actual world is consistent, still entailment
must preserve what holds in peculiar non-actual situations, some of
which may be inconsistent (see Berto, 2007a, Ch. 5, and Priest, Beall
and Armour-Garb, 2004, p. 6). The first three levels appear to be
independent of dialetheism. (But for reason to worry that there is a
'slippery slope' from gentle to dialetheic
paraconsistency, see Priest 2000. For particular worries about model
theory, see Asmus 2012, and in the modal case, Martin 2015.) 



At the top of the spectrum, dialethiec paraconsistency holds that some
contradictions are, in some sense, true *in the actual world*.
Even among full-fledged dialetheists, differences remain, for example,
in what is meant by "true", e.g., on whether they
subscribe to a
 deflationary theory of truth,
 a
 correspondence view,
 a view in which true contradictions are merely semantic, or something
else. We will come back to this point in
 section 6
 below.



We said at the outset that dialetheism *challenges* 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 in
versions based in e.g. the paraconsistent logic LP. Dialetheism as set
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 that are
inconsistent with it, namely, true sentences whose negations are true:
dialetheias. According to such versions of dialetheism, all
contradictions are false and some are true: dialetheism is itself a
dialetheia ('Concluding self-referential postscript' to
Priest 1979, p. 203). 



Much of the ongoing discussion about dialetheism involves not just the
LNC but its dual, the Law of Excluded Middle (LEM). (See
 section 3.2
 below.) The LEM says (informally) that for every sentence \(A\),
either \(A\) or \(\neg A\) is true (we bracket here views that
disentangle the LEM from Bivalence). Dialetheism is the thesis that
there are truth value gluts, challenging the LNC; the dual position,
challenging the LEM, is that there are truth value gaps. It is a
subtle issue, which we will not discuss here, whether a gap should
count as *lacking* any
 truth value,
 or as having a non-classical value distinct from both truth and
falsity, and similarly whether a glut has two truth values, or some
third 'glutty' value. In any case, these dual approaches
are nowadays labelled *paracomplete* versus (paraconsistent)
dialetheic theories of truth. (The paraconsistent/paracomplete
terminology is not entirely happy: see Ripley 2015a, footnote 1; but
is now standard.) Priest's logic includes the LEM and he rejects
the existence of gaps in Priest 1987/2006, ch.1. Other approaches are
both paracomplete and paraconsistent; they include a place for both
gaps and gluts. The case for and against dialetheism turns as much on
the status of the LEM as the LNC; see Beall and Ripley 2018.


## 2. Dialetheism in the History of Philosophy



While the word 'dialetheism' is relatively new, the idea
is not. In this section we note some points in the history of
philosophy where true contradictions appear salient, either explicitly
or implicitly. (Much of this scholarship is due originally to Priest
and Routley 1983, and Priest 1995/2002.) Then we address
methodological issues to do with historical interpretation, and give a
quick sketch of more recent history. For another history of
dialetheism, see Ficara 2021. (Note that Ficara, following
d'Agostini's interpretation of Hegel, takes dialetheism to be the view
that there are non-explosive true contradictions, and distinguishes
this from glut theory, the view that there can be true and false
propositions. This diverges from the way we've set things out here,
and merits further discussion in itself.)


### 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). Protagorean
relativism may be expressed by the view that man is the measure of all
things; but according to Aristotle, since "Many men hold beliefs
in 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 in
*Metaphysics*, Book \(\Gamma\). Chapter 4 of this Book contains
Aristotle's defence of the LNC. As we said above, historically
Aristotle was almost completely successful: the LNC has been orthodoxy
in Western Philosophy ever since.



As an aside, and as mentioned above, there is an often-remarked upon
duality between truth-value gaps and truth-value gluts (Parsons 1990).
And yet, it is perhaps worth noting that in *Metaphysics*
\(\Gamma\) (Chapter 7) Aristotle also defends the dual of the LNC, the
Law of Excluded Middle, LEM, particularly in the version which has
nowadays been distinguished as the Law of Bivalence: for any \(A\), it
is 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 Philosophy
than the LNC, in spite of the numerous obvious dualities between the
two principles. Aristotle himself, at least according to one
interpretation, appears to attack the Law in *De
Interpretatione*, Chapter 9, when he comes to the famous subject
of
 future contingents.



Despite the orthodoxy about the LNC since Aristotle, during the Middle
Ages the problem of seemingly true contradictions surfaced in
connection to the paradoxes of divine omnipotence--for instance:
can God make a stone too heavy for Him to lift? (See Cotnoir 2018.) We
find St. Pier Damiani getting close to dialetheism in the *De
divina omnipotentia*, by blaming St. Girolamus for having claimed
that God cannot overturn the past and twist what happened into
something that didn't happen. Since God lives an eternal
present, denying Him power over the past equates to denying Him power
over current and future events, which is blasphemous. So God must have
the power of making what is done undone. Later on, Nicholas of Cusa
placed at the core of his book *De docta ignorantia* the idea
that God is *coincidentia oppositorum*: as a truly infinite
being, He includes all opposite and incompatible properties, therefore
being all things, and none of them: God has all properties, including
contradictory ones (Heron, 1954, I.4).



In the *Critique of Pure Reason*, Kant argues that some
contradictions, the antinomies of pure reason (concerning time, space,
and other categories), are produced by an illicit use of pure
concepts; 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 in
knowledge. Reasoning about the world as a whole, a totality which is
never given to us as such, can lead us to apparently dialetheic
conclusions: e.g. that it has a beginning in time and a limit in
space, and that it has no beginning nor limits in space, that it is
infinite in space and time. Both horns assume the opposite thesis and
seemingly perform a *reductio*. (In Kantian terms, the
"transcendental illusion" begins when we turn what should
just be a regulative ideal into a limit-object.) According to Kant, at
least in one way of resolving the antinomies, the fallacy lies in
treating the world as a whole as an object--in mistaking a
subjective "condition", as Kant says, for an objective
reality.



Now, according to Hegel such a conception has something to be said for
it as well as against it. Kant has a point in showing, via the
antinomies, that dialectics is "a necessary function of
reason", and in defending "the *necessity of the
contradiction* which belongs to the nature of thought
determinations" (Hegel, 1831, p. 56). However, Kant takes from
this only the familiar result that reason is incapable of knowing the
Absolute, that is, actual reality. A dialetheic interpretation holds
that, on the contrary, we should abandon such "tenderness for
the things of this world", and the idea that "the stain of
contradiction ought not to be in the essence of what is in the world;
it has to belong *only* to thinking reason" (Hegel, 1830,
p. 92). On a dialetheic interpretation, contrary to what Kant held,
the Kantian antinomies are not a *reductio* of the illusions of
reason; they are perfectly sound arguments, deducing the dialetheic
nature of the world. (For a reconstruction of this Kantian-Hegelian
debate, see Part II of Priest 2002.) Hegel himself has been
interpreted as being a dialetheist. Notably, d'Agostini argues that
while Hegel held that some conjunctions of contradictory sentences are
true, he did so without the further commitment that either of the two
contradictory terms are separately true; see d'Agostini 2021.


### 2.2 Dialetheism in East Asian Philosophy



In non-Western traditions, there are more overt examples of what
appears to be dialetheic thought. In ancient Indian logic/metaphysics,
there were standardly four possibilities to be considered on any
statement at issue: that it is true (only), false (only), neither true
nor false, or both true and false. Some logicians added a fifth
possibility: none of these. Both positions were, arguably, called
*catushkoti* (Priest 2002, ch.16; Deguchi et al 2008; Tillemans
2009.) The Jains went even further and advocated the possibility of
contradictory values of the kind: true (only) and both true and false
(see Smart 1964). 



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



> 
> The Way that can be spoken of is not the true Way... (Daodejing
> I)
> 



a statement which appears to be speaking about the true Way. The
Zhuangzi says: "That which makes things has no boundaries with
things, but for things to have boundaries is what we mean by saying
'the boundaries between things'. The boundaryless boundary
is the boundary without a boundary" (Mair, 1994, p. 218). When
Buddhism and Daoism fused to form Chan (or Zen, to give it its
Japanese name), a philosophy arose in which contradiction plays a
central role. The very process for reaching enlightenment (Prajna) is
a process, according to Suzuki (1969, p. 55), "which is at once
above and in the process of reasoning. This is a contradiction,
formally considered, but in truth, this contradiction is itself made
possible because of Prajna."



Recent discussions of dialetheism have centered on Buddhist thought in
particular, where some interpreters argue that texts invoke outright
contradictions. For example, in the Mahayana tradition, Nagarjuna
presents many passages similar in form to:



> 
> Everything is real and not real, both real and not real, neither real
> nor not real. This is the Lord Buddha's teaching (Garfield 1995,
> XVIII:8).
> 



On a dialetheic interpretation, advocated by Priest, Garfield, and
Deguchi (see Priest 2002 ch.16, Deguchi et al 2008), readers should
take Nagarjuna and others in the Madhyamaka school at their
word--not as invoking some sort of mysticism, but as affirming
the nature of reality. This is most salient in Nagarjuna's
doctrine of emptiness (sunyata), that all things are empty or lacking
independent existence (svabhava). Since all things are empty, it
follows that emptiness is empty too; or, more prosaically, "The
ultimate truth is that there is no ultimate truth" (Siderits
2007, p.182; Priest 2002, p.260). These utterances, if true, would
appear to be self-contradictory, and therefore dialetheias. See Priest
2018.



For work on dialetheism in Buddhism, see the collections Garfield et
al 2009 and Tanaka et al 2015. For discussion as to whether
dialetheism is the appropriate way to interpret Buddhist texts, see
the papers in Tanaka 2013. For more general difficulties see the entry
on
 comparative 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 contradictory
utterances of the philosophers in question are not really
contradictory. There are a number of standard devices that may be
employed to block a dialetheic interpretation. One is to question, in
the case of non-English sources, whether the apparently contradictory
translation is correct. Another is to claim that the contradictory
utterance 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 assertion
is ambiguous in some way, and that it is true on one disambiguation or
respect, false in another.



This last technique is called *parameterisation*: when one is
confronted with a seemingly true contradiction, \(A \wedge \neg A\),
treat the suspected dialetheia \(A\), or some of its parts, as having
different meanings, and hence as ambiguous (maybe just
*contextually* ambiguous). For instance, if one claims that
\(P(a) \wedge \neg P(a)\), parameterisation holds that one is 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 of land
animals on Earth, but not big in the context of stars and planets). In
the *Metaphysics*, Aristotle hints that a critic of the LNC is
playing with the equivocal meanings of some words: "for to each
definition there might be assigned a different word" (1006b
1-2).



No one would dispute that contradictory utterances are sometimes best
construed parametrically. (Again, dialetheists do not claim that
*all* contradictory statements are true.) The question is
whether parameterisation is the best approach in the case of the
philosophers we have mentioned; answering this in a dialetheic context
is a matter for detailed case-by-case consideration. A defender of the
dialetheic interpretation would say that consistent parameterisation
produces an inaccurate and distorted version of the views of the
philosopher in question. In any case, even if parameterisation is
always possible, this does not impact arguments for or against the
LNC. An *a priori* claim that contradictions *can*
always be avoided by parameterisation does not show, without begging
the question, the *correctness* of such interpretations:
sometimes parameterisation may be the best thing to do, but
independent justification should be given on each occasion.


### 2.4 Modern Dialetheism



In the second half of the twentieth century, with the rapid
development of paraconsistent logics, came the modern form of
dialetheism. In 1966, Asenjo published a short note, "A Calculus
of Antinomies", based on his 1954 PhD dissertation; the opening
line reads



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



Here, having two truth values would mean being *both true and
false*, a glut, or what Asenjo calls (following Kant) an antinomy.
Asenjo simply seems to assume without further ado that there are
antinomies, and in Asenjo and Tamburino 1975 asserts that they are
"useful logical entities", especially for dealing with
paradoxes in mathematical contexts. Born in Buenos Aires, Asenjo came
to the University of Pittsburg in 1963 as a professor in the
mathematics faculty, where he was in contact with logicians such as
Belnap, Anderson, their student Dunn (who thanks Asenjo in his 1966
PhD thesis, that also toys with true contradictions), and nearby,
Meyer. By the mid-1970s, Richard Routley (Sylvan), in collaboration
with Meyer, was developing a 'dialectical' position that
rebukes what they call "the consistency hypothesis" and
accepts some statements as both true and false (Routley and Meyer
1976; Routley 1977, 1979). In 1979, Priest's paper "The
Logic of Paradox", developed eventually into the book *In
Contradiction* (1987/2006), presented what are now the most famous
arguments for dialetheism. In the collaboration between Priest and
Routley, the contemporary dialetheic program was launched. Much of
Priest and Routley's early joint work is summarised in Priest et
al 1989 (also available as Priest and Routley 1983). More recent
developments are canvassed below. 


## 3. Motivations for Dialetheism


### 3.1 The Paradoxes of Self-Reference



Probably the master argument used by modern dialetheists invokes the
logical
 paradoxes of self-reference.
 It is customary to distinguish between two families of such
paradoxes: semantic and set-theoretic. The former family typically
involves such concepts as *truth*, *denotation*,
*definability*, etc; the latter, such notions as
*membership*, *cardinality*, etc. After
Godel's and Tarski's well known formal procedures to
obtain non-contextual self-reference in formalized languages, it is
difficult to draw a sharp line between the two families, among other
things, because of the fact that Tarskian semantics is itself framed
in set-theoretic terms. Nevertheless, the distinction is commonly
accepted within the relevant literature.



Russell's paradox
 is the most famous of the set-theoretic paradoxes; it arises when one
considers the set of all non self-membered sets, the *Russell
set*.
 Cantor's paradox
 arises in connection with the universal set. The most famous of the
semantic paradoxes is the
 Liar paradox.
 Although cases for the existence of dialetheias can be made from
almost any paradox of self-reference, we will focus only on the Liar,
given that it is the most easily understandable and its exposition
requires no particular technicalities.


### 3.2 A Simple Case Study: the Liar



In its standard version, the Liar paradox arises by reasoning on the
following 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 is the case, so it is false. Then, suppose (1) is
false: this is what it claims to be, so it is true. If we accept the
aforementioned Law of Bivalence, that is, the principle according to
which all sentences are either true or false, both alternatives lead
to a contradiction: (1) is both true and false, 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 looped
Liar:



(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. This
is as old as Buridan (his Sophism no. 9: Plato saying 'What
Socrates says is true'; Socrates replying 'What Plato
says is false'). 



Paradoxes of this kind have been known since antiquity (the standard
Liar is attributed to the Greek philosopher Eubulides, probably the
greatest paradox-producer of antiquity). But they were thrown into
prominence by developments in the foundations of mathematics around
the turn of the twentieth century. In the case of each paradox, there
appears to be a perfectly sound argument ending in a contradiction. If
the arguments are sound, then dialetheism is true. Of course, many
have argued that the soundness of such arguments is merely an
appearance, and that subtle fallacies may be diagnosed in them. Such
suggestions were made in ancient and medieval logic; but many more
have been made in modern logic--indeed, attacking the paradoxes
has been something of a *leitmotiv* of modern logic. And one
thing that appears to have come out of this is how resilient the
paradoxes are: attempts to solve them often simply succeed in
relocating the paradoxes elsewhere, as so called
'strengthened' forms of the arguments show. Let us have a
look.



A radical solution, which never won big consensus but has been
revitalised in recent times (Pleitz 2018), has it that there just
cannot be meaningful self-rererential sentences. Various works
(notably Martin 1967, van Fraassen 1968, Kripke 1975, Field 2008) have
proposed to solve the Liar paradox by dismissing Bivalence, whereby
some sentences are neither true nor false, and the Liar is one such
truth value 'gap'. Truth value gaps, and the inclusion of
the Liar among them, are differently motivated in the various
approaches. But the common core thought is the following: even though
the Liar is a sentence such that, if it were true, it would be false,
and *vice versa*, no explicit contradiction according to which
it is both true and false need follow. We can avoid the contradiction
by rejecting the idea that truth and falsity are the only two options
for a sentence: the Liar is neither. For a comparative survey of the
two kinds of approach, see Beall and Ripley 2018.



These approaches face difficulties with the so-called
'strengthened' Liars--sentences such as the
following:



(3): (3) is not true.



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



Now these sentences should be, on the gappy theorist's
non-bivalent approach, either true, or false, or neither. But, for
instance, 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; but
this is precisely what it claims to be; therefore, it is true. We seem
to have to conclude that (3) is both true and not true, contrary to
the LNC. A similar line of reasoning goes for (4).



According to Priest the strengthened Liars show that a single feature
of the semantic paradox underlies its different formulations. The
totality of sentences is divided into two subsets: the true ones, and
their 'bona fide complement'--call it the
*Rest*. Now the essence of the liar is "a particular
twisted construction which forces a sentence, if it is in the bona
fide truths, to be in the Rest (too); conversely, if it is in the
Rest, it is in the bona fide truths" (Priest 1987, p. 23). The
standard Liar, 'This sentence is false', is just a
particular instance of this, producing a contradiction within the
bivalent framework, in which the Rest is identified with the set of
the false sentences. We can try to resolve the problem by admitting
sentences that are neither true nor false, so that the false ones
become a proper subset of the Rest. However, the strengthened Liars
show that we can use the notions introduced to solve the previous
paradox to *re-describe* the Rest. Once the set of sentences is
partitioned into a trichotomy (true, false, and neither true nor
false), the disjunctive 'This sentence is false or neither true
nor false' embraces the whole Rest, i.e. the new(ly described)
complement of the set of the true sentences. What about adding more
values? If there is some fourth thing that a sentence can be, besides
true, false, and neither true nor false, we can always take the notion
*fourth 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 also
called revenge liars, and the general phenomenon we have just
witnessed has become known as *revenge*; 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 of *gap*, or
*defective sentence*, or *sentence whose truth value is
indeterminate*, can be fully expressed in the language for which
they are proposing their theory of truth. The strengthened paradoxes
then seem to force the consistent theorist to admit that the proposed
theory was formulated in a language different from, and expressively
more powerful than, the one whose semantics it was supposed to
express. This entails a limitation of the Tarskian T-schema
characterising truth, i.e., of the equivalence

 
\[
Tr\langle A\rangle \leftrightarrow A
\]


where '\(Tr\)' is the truth predicate for the relevant
language, and \(\langle A\rangle\) is the appropriate name of sentence
\(A\). This approach makes a rigid distinction between an object
language and its metalanguage. Such a distinction was introduced by
Tarski to expel the Liar paradox from formalized languages--but
Tarski himself insisted that his solution is inapplicable to natural
languages, which do not appear to depend upon some metalanguage for
their semantics. As Kripke admits at the end of *Outline of a
Theory of Truth*, "the ghost of the Tarski hierarchy is
still with us" (scil. the paracompletists: see Kripke 1975, p.
80).



Tarski, in short, identified the cause of the semantic paradoxes to be
*semantic closure*--the fact that natural languages such
as English satisfy the T-schema. Tarski took this to mean that there
can be no consistent formalization of a semantically complete
language, only a 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 it
is semantically closed and is not hierarchically stratified, will be
inconsistent (Priest 1987, ch.1, Beall 2009, ch.1).



These two features--a claimed immunity to revenge/strengthened
liars, and dispensing with the object-language/metalanguage
distinction--are argued to give dialetheism about the paradoxes
of self-reference some of its major appeal. As Shapiro (in a critical
note) puts it, dialetheists offer that we



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



The *simplicity* of a dialetheic theory of truth, then, is
claimed as an additional further feature. The two most prominent such
theories to date are presented in Priest 1987, and Beall 2009. In the
former, the truth predicate \(Tr\) for the relevant formal language,
modelling the behaviour of truth in English, is simply characterized
by the unrestricted T-schema, \(Tr\langle A\rangle \leftrightarrow
A\), which, as stressed by many philosophers, is an overwhelmingly
intuitive--perhaps even analytic--principle concerning
truth. It is admitted that some sentences--notably, the
Liars--are truth value gluts, that is, both true and false (the
construction may also sustain sentences which are both true and not
true, although not all dialetheias need be of this kind); and no
hierarchy of metalanguages or further epicycles are needed.



Beall's 2009 theory allows a fully *transparent* truth
predicate: one such that for any sentence \(A\), \(Tr\langle
A\rangle\) and \(A\) can be replaced with each other in all
(non-opaque) contexts *salva veritate*, that is, producing
sentences logically equivalent to the sentences one started with. Then
the unrestricted T-schema follows from transparency (and the fact that
\(A \rightarrow A\) is a logical truth) as a special case. In
Beall's theory, *all* sentences \(A\) that are
dialetheias are not only true and false, i.e. (given that falsity is
truth of negation), \(Tr\langle A\rangle \wedge Tr\langle \neg
A\rangle\); they are also true and untrue, \(Tr\langle A\rangle \wedge
\neg Tr\langle A\rangle\): this again follows from the transparency of
truth. Beall's theory is based on a (relevant) paraconsistent
logic, whose modal semantics employs so-called
 non-normal worlds.
 



Overall, such paradoxes as the Liar provide some evidence for the
dialetheist's claim that some contradictions are
*provably* true, in the sense that they are entailed by plain
facts concerning natural language and our thought processes. Extended
Liar paradoxes like 'This sentence is not true' are spelt
in ordinary English. Their paradoxical characteristics, dialetheists
argue, 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 than
English; 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 transparency
of truth. For a sustained critical engagement with paraconsistent
dialetheism 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 the
semantic, dialetheism affords a treatment of the set-theoretic
paradoxes. These paradoxes arise in set theories based on an
unrestricted 'comprehension schema' for sets: for any
condition or property, including paradoxical ones like
non-self-membership, there exists a corresponding set. In particular,
inconsistent sets like Russell's are admitted; analogously to
the Liar, the Russell set is and is not a member of itself. As with
the truth schema, the set comprehension principle seems very natural
and intuitive, and such contradictions do not give rise to triviality
due to the paraconsistent logic underlying the formal theories. Though
the issue is too technical to be addressed here, and more
appropriately dealt with in the entries on
 paraconsistent logic
 and
 inconsistent mathematics,
 the reader can consult Routley 1979, Brady 1989, for inconsistent set
theories, and Weber 2012. See also Mortensen 1995. For different
approaches to dialetheic set theory, see Restall 1992 and Ripley
2015b.



Since dialetheism appears to resolve both the semantic and set
theoretic paradoxes at once and in the same sort of way (namely,
accept the contradictory outcomes as true), this has been presented as
another major strong point for dialetheism. Priest argues that the
paradoxes share an underlying structure (which he calls the Inclosure
Schema in Priest 2002). This is used in tandem with what Priest dubs
the *principle of uniform solution* ("same kind of
paradox, same kind of solution") to urge that, since all the set
theoretic and semantic paradoxes are of a kind, then dialetheism
presents a uniquely unified solution. For a detailed pursuit of a
dialetheic response to the paradoxes in general, see Weber 2021. 



A point of dispute about the dialetheic approach to the paradoxes of
self-reference concerns the
 Curry paradox.
 This is produced by a self-referential sentence claiming 'If I
am true, then \(\bot\)', where \(\bot\) is a constant (what
logicians usually call the *falsum*) which is or entails
something that is also dialetheically unacceptable, say \(\bot =\)
'Everything is true', the incoherent trivialist claim.
*Prima facie*, this does not involve negation, nor a falsity
predicate. However, many logicians think that Curry's paradox is
very similar to the Liar, and therefore by the principle of uniform
solution that it should be handled in the same way. A dialetheist,
though, cannot simply accept that the Curry sentence is both true and
false, because if it is true then \(\bot\) follows. Dialetheists need
a different treatment of Curry. A standard dialetheic strategy to deal
with the Curry paradox has consisted in exploiting paraconsistent
logics with a 'noncontractive' conditional (see again
Priest 1987, Ch, 6, Beall 2009, Ch. 2), which do not validate the
Contraction (or Absorption) Law, i.e., the rule: from \(A
\rightarrow(A \rightarrow B)\) infer \(A \rightarrow B\), or the
so-called *pseudo-modus ponens* principle \((A \wedge(A
\rightarrow B)) \rightarrow B\). Stronger forms of the paradox,
though, *validity curry*, seem to show that dropping these
principles is not enough (Beall and Murzi 2013). This leads to
contraction-free logics within the broader family of
 *substructural logics*
 (Restall 2000, Priest 2015), and indeed substructural dialetheic
approaches that drop principles other than contraction, such as
transitivity (Ripley 2012). Curry's paradox puts pressure on the
dialetheic story about uniform solution to the paradoxes. Beall
(2014a, 2014b) urges this point; Weber et al 2014 is a reply. 


### 3.4 Other Motivations for Dialetheism



Dialetheias produced by the paradoxes of self-reference are confined
to the abstract realm of notions such as *set* or to semantic
concepts such as *truth*. However, the paradoxes of
self-reference are not the only examples of dialetheias that have been
mooted. Other cases involve contradictions affecting concrete objects
and the empirical world, and include the following.



(1) *Transition states*: when I exit the room, I am inside the
room at one time, and outside of it at another. Given the continuity
of motion, there must be a precise instant in time, call it \(t\), at
which 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) I
am both; and (d) I am neither. There is a strong intuition that (a)
and (b) are ruled out by symmetry considerations: choosing either
would be completely arbitrary. (This intuition is not at all unique to
dialetheists: see the article on
 boundaries
 in general.) As for (d): if I am neither inside not outside the room,
then I am not inside and not-not inside; therefore, I am either inside
and not inside (option (c)), or not inside and not-not inside (which
follows from option (d)); in both cases, a dialetheic situation. Or so
it has been argued. For a description of inconsistent boundaries using
formal mereology, see Weber and Cotnoir 2015.



(2) Some of *Zeno's paradoxes* concerning a
particular--though, perhaps, the most basic--kind of
transition, that is, *local motion*: a moving arrow is both
where it is, and where it is not. In any given instant, argues Zeno,
it cannot move to where it is, since it is already there, and it
cannot move to somewhere else, because there isn't time for it
to get there. The orthodox way out of the paradoxical situation, as
formulated, e.g., by Russell 1903, has it that motion is the mere
occupation of different places at different times. But one could argue
that this is denial of the phenomenon itself, that is, of the
actuality of motion: Russell's solution entails that motion is
not an *intrinsic* state of the (allegedly) moving thing, for,
at each instant, the arrow is not moving at all. Can a going-somewhere
be composed by an (even continuum-sized) infinity of going-nowheres?
An alternative, dialetheic account of motion, which takes at face
value the aforementioned Hegelian idea that "Something moves,
not because at one moment it is here and another there, but because at
one and the same moment it is here and not here, because in this
'here', it at once is and is not", is exposed in
Priest 1987, Ch. 12.



(3) *Borderline cases of vague predicates*. Popular approaches
to
 vagueness
 and the
 sorites paradox,
 such as the ones based on many-valued logics, or supervaluations,
require some under-determinacy of reference, and/or the rejection of
Bivalence: if an adolescent, \(m\), is a borderline case of adultness,
\(A\), then \(A(m)\) may turn out to have an intermediate truth value
between truth and falsity, or no truth value at all. But it may be
conjectured that a borderline object like \(m\), instead of satisfying
neither a vague predicate nor its negation, satisfies them both: an
adolescent both is and is not an adult. Given the obvious dualities
between the LEM and the Law of Bivalence on the one side, and
(respectively, syntactic and semantic formulations of) the LNC on the
other, it is not too difficult to envisage a
'sub-valuational' semantic approach, dual to the
supervaluation strategy. Sub-valuational paraconsistent semantics have
been proposed by Hyde (1997), and Varzi (1997). Other
'glutty' approaches to vagueness have been proposed by
Colyvan (2009), Weber (2010), Priest (2010), Ripley (2013), and
Cobreros et al (2010, 2015), motivated on both theoretical and
empirical grounds. If inconsistencies due to vague predicates and
borderline objects are taken to be due to merely semantic under- and
over- determination of ordinary language (*de dicto*), then
this is further evidence for dialetheism about semantics. If the
aforementioned phenomena are taken to be not just about words but
things (*de re*), then actually inconsistent objects are
admitted, together with vague objects. And this spreads inconsistency
all over the empirical world: if borderline cases can be inconsistent,
inconsistent objects are more or less everywhere, given how pervasive
the phenomenon of vagueness notoriously is: adolescents, borderline
bald men, etc. Inconsistency is *observable* (Beall and Colyvan
2001). Beall argues against the approach in Ch. 5 of Beall 2009, and
Beall 2014a.



(4) *Multi-criterial predicates*. We may assume that the
semantics of a predicate is specified by means of its criteria of
application. 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 entail
that object \(m\) (e.g., a political party) is in the extension of the
predicate, some others (promoting nationalism, oppressing immigrants),
that \(m\) is in its anti-extension, or negative extension. Criteria
can in some cases be encoded by such things as meaning postulates (or
other similar, albeit more sophisticated, semantic devices); but
conflicting meaning postulates may be embedded in our standard
linguistic practices, and difficult to detect and identify. If the
extensions of our ordinary predicates are constrained by our
intuitions, and such intuitions turn out to be inconsistent, a good
semantic account of the situation may well have to reflect this fact
(accepting, in our example, that a party counts as both left and right
wing), instead of destroying it by means of some regimentation (e.g.
via the usual parameterisation, or distinction of respects). As Priest
1987, pp 67-9, clarifies, it is also hard to analyse such
situations as cases of vagueness (so that a party has to be understood
as being left wing to some degree, and right wing to some degree).



(5) Certain *legal situations*, such as inconsistent bodies of
law. Suppose, for instance, that some norm states that a marriage
performed by the captain of a ship counts as a legal marriage only if
the ship was in open water throughout the ceremony. It turns out,
then, that some other law has established that such a marriage is
valid 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 to
be both a married man and not a married man. (Since it does not appear
to follow that he is not a man anymore, or both a man and not a man,
we have another putative counterexample to *ex contradictione
quodlibet*). If one accepts the plausible view that statements
concerning legal rights, obligations, and statuses, can be truth-value
apt, we seem to have a dialetheia. Of course, legal systems sometimes
have mechanisms that can be used to remove such inconsistencies (e.g.,
by ordering different kinds of laws in a hierarchy from customary
laws, to established jurisprudence, to ordinary legislation, to
constitutional norms, etc.; or via the *lex posterior*
principle, giving priority to the more recent norm in case of
conflict). But this is not always the case: the inconsistent laws may
be of the same rank, enacted at the same time, etc. For a recent
discussion, see Beall 2016.



(6) In *post-Kantian metaphysics*, a movement sometimes dubbed
'new realism' or 'speculative realism' has
developed, in various work by Quentin Meillassoux, Graham Harman,
Markus Gabriel, and others. The idea is that much of philosophy since
Kant, especially in the continential tradition, has been lost in
postmodernist concern with thought and language, and that a return to
the "Great Outdoors" is overdue. The return to realism
though is speculative in the sense that it continues to take seriously
epistemic concerns: in Harman's case at least, mind-independent
objects will be radically resistant to knowledge and almost impossibly
removed from us. If such a realism involves specifying objects or
making knowledge claims about them that the theory itself seems to
preclude, then, much as with the paradoxes at the limit of thought
(presented in section 2.1 above), this may crucially involve
contradiction. Some practitioners in this area have explicitly
endorsed dialetheism, e.g. Morton 2012. Cogburn has identified in this
approach what he dubs the 'object oriented ontology'
paradox (as fitting Priest's inclosure schema): "To the
extent that object-oriented ontologists really are offering
speculative systems of metaphysics, they seem to be trying to do what
they themselves take to be impossible" (Cogburn 2017, chapter 3;
see 3.3 above).



(7) It has long been recognized that *theology* may involve
paradoxes. In traditional Western monotheism, various divine
attributes appear to be logically inconsistent. Famously, the question
of whether an omnipotent God can create a stone so heavy He Himself
cannot lift seems to lead to a contradiction--something that is
both possible and not possible for God. Cotnoir 2018 explores whether
dialetheism and the tools of paraconsistent logic provide a plausible
response to these paradoxes, namely, to accept their conclusions.
Beall 2021 focuses more specifically on Christianity and the apparent
contradiction that Jesus Christ was both mortal (and limited) but also
divine (and so immortal and unlimited), with the suggestion that this
can best be accommodated as a truth glut. Weber 2019 expresses
disagreement with theological applications of dialetheism. 



Each of the above topics undoubtedly call for further development. See
Priest 1987 or Ficara 2014. This list is also not exhaustive. For
example, some
 connexive 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-logical
sources, connexive logic is dialetheic at the most basic logical
level; see Omori and Wansing 2019. Or, for both a motivation and
application of dialetheism, Casati 2021 argues for a dialetheic
interpretation of the late Heidegger in which Being itself is both an
entity and not an entity, and that this can be made sense of through
mereology and metaphysical grounding. 


## 4. Objections to Dialetheism



We now turn to arguments against dialetheism. The most prominent
sustained defence of the LNC in the history of philosophy is, as
mentioned,
 that given by Aristotle
 in Chapter 4 of *Metaphysics*, \(\Gamma\). A critical analysis
of Aristotle's arguments is given point-for-point by Priest
(1998b/2006 ch.1), who finds them to often conflate dialetheism with
trivialism (that is, conflating the claim that some contradictions are
true, and the one that all contradictions are). We will look at some
more modern arguments against dialetheism.


### 4.1 The Argument from Explosion



A standard argument against dialetheism is to invoke the logical
principle of Explosion, in virtue of which dialetheism would entail
trivialism. Granted that trivialism is absurd (though see Priest
2000a, Priest 2006, Ch. 3, and Kabay 2010), dialetheism must be
rejected. Since this argument assumes that explosion is logically
valid, it will carry no weight against a dialetheic
paraconsistentist.



Interestingly enough, while Aristotle's defence of the LNC may
slide between attacking dialetheism and trivialism, Aristotelian
syllogistic--the first formally articulated logic in Western
philosophy--is not explosive. Aristotle held that some syllogisms
with 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 are
inconsistent. The principle of Explosion had a certain tenure at
places and times in Medieval logic, but it became well-established
mainly with the Fregean and post-Fregean development of what is now
called 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 of
*exclusion*. The very rough idea is that dialetheists have let
*in* more truths than non-dialetheists, but subsequently now
have trouble keeping enough falisity *out*. This objection, in
one form or another, has been a recurring theme in criticisms of
dialetheism. 



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 the
LNC fails, \(A\) does not rule out \(\neg A\), or, *a
fortiori*, anything else. Hence meaningful language presupposes
the LNC.



There are problems one may find with this argument. One is that even
though a dialetheia does not rule out its negation, it still may rule
out several other things. A bigger trouble is that the first premise
looks to be simply false. Consider again the sentence
'Everything is true'. This entails everything, and so
rules out nothing. Yet it is meaningful. It is something that
everyone, except a trivialist, rejects.



One might attempt a more sophisticated explanation of the notion of
*ruling out*, for instance in terms of information theory, or
perhaps possible worlds. One may claim that a statement 'rules
out' something insofar as there are situations, or worlds, at
which it fails. In this sense, 'Everything is true' does
rule something out. But now, it is this account of propositional
meaning which can be challenged in general. If mathematical truths
have a strictly necessary status (which may safely be assumed here),
Fermat's Last Theorem rules out nothing: being a necessary
truth, it holds at all possible worlds. But it is perfectly
meaningful; people wondered whether it was true or false for
centuries; and its proof by Andrew Wiles was a substantial
discovery.



One argument from exclusion, with a more *ad hominem* twist,
claims that the dialetheist has trouble with *expressing
disagreement* with rival positions in debates (see Parsons 1990,
Shapiro 2004, Littman and Simmons 2004). For when the dialetheist
utters '\(\neg A\)', this is in itself insufficient to
rule out that \(A\) is the case, given that, in a dialetheic world, it
may well be that both \(A\) and \(\neg A\). Similarly, '\(A\) is
false' and even '\(A\) is not true' might not do the
trick, since for the dialetheist some \(A\)'s being false, or
untrue, 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 think
> that he had disagreed with you. But then you remember that Priest is a
> dialetheist, and it occurs to you that he might very well agree with
> you after all--since he might think that \(A\) and \(\neg A\) are
> *both* true. How can he indicate that he genuinely disagrees
> with you? The natural choice is for him to say '\(A\) is not
> true'. However, the truth of this assertion is also consistent
> with \(A\)'s being true--for a dialetheist, anyway (Parsons
> 1990, p. 345).
> 



Elaborations and repetitions of this line of thought have come to be
known in the literature as the 'just true' or 'true
only' problem, e.g. Rossberg 2013. (Young 2015, however, argues
that the exclusion problem is distinct from the just-true problem.)




To this, the dialetheist has various replies. Priest has argued that
the logical operation of negation should be distinguished from the
speech acts of denial and the cognitive state of rejection. So, contra
Frege, assertion of a negation is not the same as denial. The idea is
that exclusion is expressed via a primitive notion of
*rejection*: to reject \(A\) is to positively refuse to believe
that \(A\). That the notion is taken as primitive means, in
particular, that it is not reducible to the acceptance of negation: it
is a *sui generis* act. The linguistic counterpart of rejection
is 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 the
assertion of \(\neg A\) (Priest 2006, Ch. 6). More simply, one can
often 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 be
reducible 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, a
dialetheist 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 dialetheic
exclusion-expressing device in all cases. One reason is due to
side-effects of the aforementioned Curry paradox. Another is that
'arrow-*falsum*' is very strong; a mundane sentence
like 'Alice had spaghetti for dinner' may be false-only
(she didn't), but if it were true too, that wouldn't be
*absurd*. Cf. Beall 2013, proposing exclusion via non-logical,
theory-specific *rules* of the form \(A,\neg A \vdash \bot\)
for each sentence \(A\) we take to be consistent; Scharp 2018 suggests
possible problems for this approach. 



The exclusion problem gets its strength from some background
assumptions. Specifically, it seems to presuppose that the content of
a proposition is given in terms of splitting all possible situations,
or worlds, into those where the proposition holds and those where it
does not. Even accepting this presupposition, though, would not affect
a dialetheic challenge to the LNC. For a given \(A\) to be a
dialetheia, putting things in these terms, it is sufficient that there
be overlap between the worlds where \(A\) holds, and those where its
negation holds. And this is compatible with the idea of propositional
content as splitting the totality of worlds--but now with
overlap. The tenability of such an overlap, though, requires
discussing the account of negation from classical logic, to which we
now turn.


### 4.3 The Argument from Negation



Arguments against dialetheism are often focused on the concept of
logical negation. The main one goes as follows. The truth conditions
for 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 for
negation employed here are contentious. An alternative view has it
that \(\neg A\) is true iff \(A\) is false, and \(\neg A\) is false
iff \(A\) is true--and in the semantics of many paraconsistent
logics (for instance, the logic of First Degree Entailment), truth and
falsity may overlap. Such an account preserves our intuition that
negation is the operator which (truth-functionally) switches truth and
falsity. It also preserves our intuition on contradictoriness, in the
form: \(A\) and \(B\) are contradictories iff, if \(A\) is true, \(B\)
is false, and if \(A\) is false, \(B\) is true. What has to go, on
this proposal, is the assumption that truth and falsity are exclusive
in all cases: there exist dialetheias, that is, sentences falling
simultaneously under both categories.



Secondly, a dialetheist can point out that the argument against
dialetheism based on the truth conditions for (classical) negation
begs the question at its last step: why should we assume that it is
impossible for \(A\) to be both true and not true? Well, because it is
a contradiction. But that is precisely what is at issue; the critic
was supposed to be arguing for the impossibility of any contradiction
holding to begin with. In fact, the dialetheist may even
*accept* a characterisation of the truth conditions for
negation as: '\(\neg A\) is true iff \(A\) is not true'.
For if the 'metalanguage' in which the characterisation is
expressed can be inconsistent in its turn, as a thoroughgoing
dialetheist is likely to allow (as in Weber et al 2016), then there is
no guarantee that the 'not' in that clause behaves
consistently. So the debate comes back to the basic question of
whether or not consistency can be presupposed.



A variant on the anti-dialetheic argument from negation comes from a
Quinean conception of logical vocabulary. It goes as follows. Even
granting that there is an operator, say, \(\*\), which behaves as
dialetheists claim (namely, such that in particular in some cases
\(A\) is true together with \(\*A)\), it is still perfectly possible to
*define* a negation with all the properties of classical
negation; in particular, the property of being explosive. And since
classical negation is the standard operator in logic, it is misleading
to translate anything non-classical as 'not': such a
translation risks simply calling 'negation' something
different. A change in logical vocabulary is a "change of
subject", as the Quinean slogan goes. A version of this
objection to dialetheism is due to Slater (1995); see also Restall
1993.



One line of reply to the Quinean objection available to the
dialetheist is that the objection is confused between a logical theory
and what the theory is a theory of. There are many different and
well-worked out logical theories of negation (minimal negation,
intuitionistic negation, De Morgan negation, etc.). Insofar as each
one of them characterizes its own theoretical object, there is no
rivalry between logics. Rivalry begins when we wonder whether some
account or other captures the meaning and functioning of negation as
it is used in the vernacular. An applied account of negation is a
theory *of* something, and the theoretical object has to fit
the real object. Now, to assume beforehand that the classical account
of negation is the correct one, in the sense that it captures how
negation works in the vernacular, again begs the question against the
dialetheist (and, indeed, against most non-classical logicians): it is
circular just to *assume* that classical negation gets it
right. People who propose a treatment of negation alternative to the
classical one are not thereby proposing to revise *negation*,
but to revise an account of it, which they consider incorrect.



If it is agreed that a fair discussion between dialetheists and
non-dialetheists cannot presuppose one account of negation over the
other, then this suggests a simpler reply from dialetheists to the
just-true/exclusion problems from 4.2 above. The reply now is to demur
from 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 true
and not false, and just-false being false and not true. Then as Priest
puts it (Priest 1987, p.291 of 2006 edition),



> 
> A dialetheist can express the claim that something, \(A\), is not
> true--in those very words, \(\neg Tr\langle A \rangle\). What she
> cannot do is ensure that the words she utters behave consistently:
> even if \(\neg Tr\langle A\rangle\) holds, \(A\) and \(\neg Tr\langle
> A\rangle\) may yet hold.
> 



In the exclusion objection, it appears that the dialetheist is being
asked to ensure that the words she utters behave
consistently--which is exactly what is at issue. (See Restall
2010 and the reply in Beall et al 2011 for one round of this sort of
discussion.) A dialetheist goes on to question why one should think
that commitment to an explosive logic is, in itself, a guard against
contradiction, and therefore why dialetheists are supposed to be in
special difficulty with respect to the exclusion question any more
than a non-dialetheist. Filling in the details here will vary, but the
reply, in short, is for a dialetheist as with anyone else,
'just-true' is just 'true' (Beall 2009, 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 asking
dialetheists for some way to sort the true-only sentences from the
true-and-false sentences:



> 
> I often find myself being asked the following question: 'Since
> you believe some contradictions, but not all, you must have a
> criterion for deciding between those that are true and those that are
> not. What is it?' In reply I usually point out that the
> questioner believes some things are true, but not all, and ask them
> what criterion they have for deciding between those things that are
> true and those that are not. The answer, I think, is the same in both
> cases. Nice as it would be to have a criterion of truth, to expect one
> would seem utopian. One has to treat each case on its merits, whether
> the proposition concerned is a contradiction or some other thing
> (Priest 2006 p. 56).
> 



Priest's suggestion is that dialetheists are working with truth
and falsity, just like non-dialetheists. Accounts of negation differ,
but this on its own does not create any special obligations on the
part of dialetheists.


## 5. Dialetheism and Rationality

 Let us turn from some basic objections to dialetheism to
concerns that arise in relation to pragmatics and rationality.

### 5.1 Consistency and Other Epistemic Virtues



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



It is a matter of ongoing debate what the conditions are under which
it is rational to accept something. Nevertheless, it is commonly
agreed that, as Hume put it, the wise person "proportions his
beliefs to the evidence" (1955, p. 118). If this is right, then
if a sufficient case can be made out for a contradiction, it will be
rational to believe it. And sometimes this does seem possible. We have
seen that a seemingly compelling argument can be made in favour of the
truth of the strengthened Liar sentence, 'This sentence is not
true'. Whether or not one takes the argument in question to be
completely persuasive, it suggests that there is nothing in principle
impossible about the existence of good arguments for true
contradictions. Of course, if there were conclusive evidence for the
LNC, then no case for a contradiction could be strong enough. But
conclusive evidence for any philosophical position is difficult to
achieve.



A more persuasive worry about dialetheism, relating to rationality, is
the claim that if a person could legitimately accept a contradiction,
then no one could be forced, rationally, to *abandon* any view
held. For if a person accepts \(A\) then, when an argument for \(\neg
A\) 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 its
merits. While a case can be made for the claim that the Liar sentence
is both true and false, but this in no way shows that a case can also
be made for Brisbane being and not being in Australia. (Of course, if
one subscribes to the claim that entailment is explosive, a case for
one contradiction 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, many
different considerations that speak for or against the rational
acceptability of a theory or a view. Among the epistemic virtues of a
theory are: its adequacy to the data; its simplicity, cleanness and
elegance; its unity and freedom from *ad hoc* hypotheses; its
explanatory and predictive power; etc. Not only do these (and other)
criteria come in degrees, but they may also be orthogonal to each
other. In the end, the rational evaluation of a view must balance it
against 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 lacking
the virtue of consistency may still overcomes its rivals in all or
most of the other respects. According to dialetheists, this is
actually the case with the dialetheic account of the semantics of
ordinary language, whose advantages with respect to consistent
accounts have already been shortly suggested above. And conversely, of
course, an inconsistent theory may well be trumped by a consistent
theory, all things considered. So it may be rational to reject an
inconsistent position, even if it is logically possible that it is
true. Rationality considerations are dealt with at length in Priest
2006. 


### 5.2 Accepting and Asserting Dialetheias



If some contradictions are true, it is natural to expect that a
dialetheist will sometimes accept, or believe in, contradictions, and
assert them. Priest (2006, p. 109) adopts the following Rationality
Principle:



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



Belief, acceptance, and assertion have a *point*: when we
believe and assert, what we aim at is believing and asserting what is
the case or, equivalently, the truth. Therefore, the dialetheist will
accept and, sometimes, assert both \(A\) and \(\neg A\), if she has
evidence that \(A\) is a dialetheia.



Notice that this need not entail that the dialetheist both accepts
*and rejects* \(A\) at the same time. We now come back to the
issue flagged in Section 4.2, on the irreducibility of rejection to
negation. That rejecting \(A\) is tantamount to accepting its negation
is a common view, famously endorsed and defended (more precisely in
terms of the corresponding speech acts of assertion and denial) by
Frege and Peter Geach. But dialetheists have argued that this fusion
is a confusion (see Berto 2008 on this issue). The point can be made
independently of the issue of dialetheism: a paracompletist may well
want to deny \(A\), but it would be unfair to take such a denial as
equivalent to the assertion of \(\neg A\), since if \(A\) is
truth-valueless, \(\neg A\) is normally considered truth-valueless,
too, not a truth, and so not to be asserted. A dual position can hold
for dialetheism: given that accepting \(\neg A\) is different from
rejecting \(A\), a dialetheist can do the former and not the
latter--exactly when she thinks that \(A\) is a dialetheia.



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



Restall (2015) argues that the the acceptance/rejection issue makes
gappy and glutty approaches symmterical: gap theorists cannot assert
some true claims about paradoxes, but glut theorists cannot reject
some false claims about non-paradoxes; see also Restall 2013, Jenny
2017. In a note by Laura Goodship (1996), she suggests that the
proposal of separating denial from assertion of negation runs into
problems related, again, to Curry's paradox. Her proposal is, in
effect, that it would be more natural for dialetheists to both accept
*and* reject things after all. Focusing on the phenomenon of
disagreement in natural language, Ripley (2015a) argues that
dialetheists (and paracompleteists) should rejoin classical logicians
in taking negation to embed denial, and denial to express
disagreement. Ripley proposes that dialetheists simply admit that (1)
agreement and disagreement are *incompatible*
("it's incoherent to do both", p. 306), but (2) in
some cases, assert and deny the same thing. The result would be what
Ripley calls *paracoherentism*, which (in echo of
paraconsistency) would try to allow local incoherence without global
incoherence. How this could be done is an open question. Ripley
suggests dropping the transitivity of logical consequence. Goodship
herself ultimately recommends dropping modus ponens, a proposal that
has been dubbed the 'Goodship project' (Beall 2015, Omori
2016, Priest 2017). 



There certainly are various other arguments against dialetheism in the
philosophical market (see papers in Priest et al. 2004). For example,
Zalta (2004, p. 432) argues that preserving "our pretheoretic
understanding of what it is to exemplify or instantiate a
property" requires us to preserve the LNC. This entry has
presented only some of the most immediate issues that arise in
objections and replies.


## 6. Themes for Further Research



Since dialetheism is simply a claim about truth, it can play a role in
any area--say in traditional or mainstream philosophy--where 
truth is involved. Among such topics, a prominent one is the
debate between
 realists
 and
 anti-realists
 (for example, idealists and constructivists) in metaphysics. Very
roughly, to be a realist *about* entities of some kind is to
maintain that such entities objectively exist apart from, and
antecedently to, anyone's thought of them; and, therefore, that
our thoughts, beliefs and theories concerning such entities are made
objectively true or objectively false by them, apart from what we
think of them (more refined definitions of realism and anti-realism
are certainly available; but this characterization will suffice for
our purposes).



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



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



Nevertheless, if we accept even a mild form of realism,
the truth of some contradictions entails the existence of inconsistent
objects and/or states of affairs: those that make the contradictions
true (Berto 2007b). One may claim that it makes no sense to talk of
inconsistent objects, situations, or states of affairs. The world is
all there, all together: how could some pieces of it
*contradict* some other pieces? Consistency and inconsistency
might be taken as properties of sentences, or theories (sets of
sentences closed under logical consequence), or propositions (what
sentences express), or maybe thoughts, or (sets of) beliefs, etc.
Contradiction (*Widerspruch*, the Latin *contradictio*)
has to do with discourse (diction, *sprechen*,
*dicere*). The world, with its non-mental and non-linguistic
inhabitants--armchairs, trees, people--is not the right
*kind* of thing that can be consistent or inconsistent, and
ascribing such properties to (a part of) the world is, to use Gilbert
Ryle's terminology, a category mistake.




These considerations might drive dialetheism towards an anti-realist
interpretation of the claim that there are dialetheias, true
contradictions; and anti-realist dialetheic theories of truth have, in
fact, been proposed (see e.g. Beall's 'constructive
methodological deflationism', in Beall 2004). But other options
are available to a dialetheist who wants to embrace some form of
metaphysically robust realism about truth. For instance, she can
stress that consistency and inconsistency can be ascribed to (pieces
of) the world in a *derived* sense: to say that the world is
(locally) inconsistent just is to say that some true purely
descriptive sentences about the world have true negations.
Consequently, and not accidentally, it is quite common in the current
literature both for and against dialetheism to straightforwardly speak
of inconsistent objects, states of affairs, and entire inconsistent
worlds. A dialetheic correspondence theory of truth might be
committed, in particular, to negative facts (requiring the
simultaneous existence of truth-makers both for \(A\) and for its
negation, when \(A\) is a dialetheia); but these may be not too
difficult to handle (see e.g. Priest 2006, pp. 51-3).



There are further intermediate positions. One is 'semantic
dialetheism' which accepts true contradictions without
inconsistent objects or states of affairs as their truth-makers. This
position has been explored in the literature (Kroon 2004, Mares 2004).
Beall's view, expressed in his transparent truth theory in Beall
2009, may also be seen as a form of semantic dialetheism. Transparency
can be naturally paired with a
 deflationary 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 the
Liar(s) may well be semantic side-effects ('spandrels', in
Beall's terminology) of the introduction of such device, not
involving any metaphysically committing contradiction in a language-
and mind- independent world. Woodbridge and Armour-Garb (2013) argue
that a deflationary view of truth is best understood in terms of
semantic pretense (a hermeneutic fictionalist perspective), and on
that basis offered a pretense account of the semantic paradoxes.
Wansing (2022) advances a view called *dimathematism*, where
there may be contradictions in information (information being
"what is left from knowledge when you subtract, justification,
truth, and belief" as Dunn puts it) again distinct from
contradictions in truth. 



Debates on realism and anti-realism quickly spill over into questions
concerning the nature of reality in general. Thus the dialetheic
programme looks to metaphysical issues: if reality is dialetheic, how
should the ontology of a dialetheic world be spelt out? If metaphysics
should be placed (once again) at the very core of philosophy, the
debate on the possibility of dialetheias occupies a central place in
the core. This was, after all, Aristotle's view, too: he decided
to speak on behalf of the unconditional validity of the LNC, not in
his *Organon* (his writings on the subject of logic), but in
the *Metaphysics*, for this was for him an issue to be
addressed ontologically, not (only) via formal logical tools. For work
in dialetheic metaphysics see Priest 2014, which grounds the very
unity of objects in an inconsistent theory of parthood and boundary,
and applies it to classic problems such as the One and the Many, and
the instantiation of universals.


## 7. Conclusion



Since Aristotle, the assumption that consistency is a requirement for
truth, validity, meaning, and rationality, has gone largely
unchallenged. Modern investigations into dialetheism, in pressing the
possibility of inconsistent theories that are nevertheless meaningful,
valid, rational, and true, call that assumption into question. If
consistency does turn out to be a necessary condition for any of these
notions, dialetheism prompts us to articulate why; just by pushing
philosophers to find arguments for what previously were undisputed
beliefs renders a valuable service (Scharp 2007, p. 544). And if
consistency turns out not to be an essential requirement for all
theories, then the way is open for the rational exploration of areas
in philosophy and the sciences that have traditionally been closed
off.