dialetheism

7/17/2019 Dialetheism

http://slidepdf.com/reader/full/dialetheism 1/31

Dialetheism

First published Fri Dec 4, 1998; substantive revision Thu Mar 28, 2013

A dialetheia is a sentence, A, such that both it and its negation, ¬ A, are true (weshall talk of sentences throughout this entry; but one could run the definition in

terms of propositions, statements, or whatever one takes as one's favourite truth-

bearer: this would make little difference in the contet!" Assuming the fairly

uncontroversial view that falsity #ust is the truth of negation, it can e$ually be

claimed that a dialetheia is a sentence which is both true and false"

Dialetheis is the view that there are dialetheias" %ne can define a contradiction as

a couple of sentences, one of which is the negation of the other, or as a con#unction

of such sentences" &herefore, dialetheism amounts to the claim that there are truecontradictions" As such, dialetheism opposes the so-called !a" o# $on%

&ontradiction ()! (sometimes also called the aw of )ontradiction!" &he aw

can, and has been, epressed in various ways, but the simplest and most

perspicuous for our purposes is probably the following: for any A, it is impossible

for both A and ¬ A to be true"

*n +ook of the Metaph'sics, Aristotle introduced (what was later to be called! the

) as the most certain of all principles. (/001b23! 4 #irissiu oniu

principioru, as the 5edieval theologians said" &he property of

being #irissiu manifests the fact that the ) has been taken as the most

indubitable and incontrovertible law of thought and being, and as the supreme

cornerstone of knowledge and science" Aristotle's defence of the ) in

the Metaph'sics was sociologically so successful that hardly any philosopher has

taken it upon herself to de#end the law afterwards" &homas 6eid put the ), in

the form 7o proposition is both true and false8, among the dictates of common

sense (together with other alleged self-evident truths, such as that every complete

sentence must have a verb, or that those thing did really happen which * distinctly

remember as having happened!"

As a challenge to the ), therefore, dialetheism flies in the face of what most

philosophers take to be common sense" Actually, that dialetheism challen(es the

) needs $ualification, since the ) is accepted as a general logical law in the

mainstream versions of the theory" +ut a dialetheist manifests her dialetheism in

accepting, together with the ), sentences that are inconsistent with it, that is,

true sentences whose negations are true: dialetheias"

*n spite of the ma#ority view, there are some dialetheists in the history of 9estern

hilosophy" 5oreover, since the development of paraconsistent logic in the second



half of the twentieth century, dialetheism has now become a live issue once more"

*n the rest of this article, /! we will start by eplaining the connection between

dialetheism and other important related concepts, such as the ones of trivialism and

paraconsistency" et, we will describe 2! the history of dialetheism and ! the

motivations for the modern dialetheic renaissance, among which the logical(semantic and set-theoretic! paradoes figure prominently, though not eclusively"

9e will then 3! indicate and discuss some of the ob#ections to dialetheism, and 1!

its connections with the notion of rationality" <inally, =! we will point at some

possible themes for further in$uiry and future philosophical research in the field,

focusing in particular on the connections between dialetheism, realism and

antirealism in metaphysics"

• /" >ome +asic )oncepts

• 2" ?ialetheism in the @istory of hilosophy

• " 5otivations for ?ialetheism

o "/ &he aradoes of >elf-6eference

o "2 A >imple )ase >tudy: the iar

o " %ther 5otivations for ?ialetheism

• 3" %b#ections to ?ialetheism

o 3"/ &he Argument from plosion

o 3"2 &he Argument from clusion

o 3" &he Argument from egation

•

1" ?ialetheism and 6ationality

o 1"/ )onsistency and %ther pistemic Birtues

o 1"2 Accepting and Asserting ?ialetheias

• =" &hemes for <urther 6esearch: ?ialetheism, 6ealism and Antirealism

• C" )onclusion

• +ibliography

http://plato.stanford.edu/archives/fall2013/entries/dialetheism/#1



http://plato.stanford.edu/archives/fall2013/entries/dialetheism/#3.1












http://plato.stanford.edu/archives/fall2013/entries/dialetheism/#Bib















http://plato.stanford.edu/archives/fall2013/entries/dialetheism/#Bib




• Academic &ools

• %ther *nternet 6esources

•

6elated ntries

1. Some Basic Concepts

&hough dialetheism is not a new view, the word itself is" *t was coined by Draham

riest and 6ichard 6outley (later >ylvan! in /EF/ (see riest, 6outley and orman,

/EFE, p" ))!" &he inspiration for the name was a passage in 9ittgenstein's *ear+s

on the Foundations o# Matheatics, where he describes the iar sentence (7&his

sentence is not true8! as a Ganus-headed figure facing both truth and falsity (/ECF,

*B"1E!" @ence a di-aletheia is a two(-way! truth" Hnfortunately, riest and 6outley

forgot to agree how to spell the 7ism8, and versions with and without the 7e8 appear

in print"

?ialetheism should be clearly distinguished from trivialis" &his is the view

that all contradictions are true (and hence, assuming that a con#unction entails its

con#uncts, it is also the view that everything is true!" &hough a trivialist must be a

dialetheist, the converse is not the case: a dialetheist typically claims that some

(and, usually, very specific! sentences are dialetheias, not that all of them are" @owone can claim the former without being committed to the latter is one of the main

topics in the dialetheic theory, since trivialism is considered by most philosophers

theoretically repugnant, if anything is (though see Iabay 20/0 for an interesting

defense of the view!" &he standard solution for the dialetheist consists in

subscribing to the view that entailment (deductively valid inference!

is paraconsistent "

A general conception of entailment (and, by etension, a logic that captures such a

conception! is e)plosive if, according to it, a contradiction entails everything (e)contradictione uodlibet : for all A and -: A,¬ A ⊢ -!" *t is paraconsistent if and

only if (iff! it is not eplosive" +y adopting a paraconsistent logic, a dialetheist can

countenance some contradictions without being thereby committed to

countenancing everything and, in particular, all contradictions" *t is likely that the

recent development of paraconsistent logics, together with the impressive

epansion of their successful applications, has contributed to the resurgence of

dialetheism"

@owever, dialetheism should also be clearly distinguished from paraconsistency(see +erto, 200Ca, )h" 1!" 9hereas a dialetheist had better embrace some

http://plato.stanford.edu/archives/fall2013/entries/dialetheism/#Aca

http://plato.stanford.edu/archives/fall2013/entries/dialetheism/#Oth

http://plato.stanford.edu/archives/fall2013/entries/dialetheism/#Rel

http://plato.stanford.edu/archives/fall2013/entries/logic-paraconsistent/

http://plato.stanford.edu/archives/fall2013/entries/dialetheism/#Aca

http://plato.stanford.edu/archives/fall2013/entries/dialetheism/#Oth

http://plato.stanford.edu/archives/fall2013/entries/dialetheism/#Rel




paraconsistent logic or other in order to avoid trivialism, a paraconsistent logician

need not be a dialetheist: she may subscribe to a non-eplosive view of entailment

for other reasons; for eample, that, though truth at the actual world, and arguably

at any logically possible world, is consistent, entailment must preserve what holds

in peculiar non-actual situations, some of which may be inconsistent; or thatentailment must preserve more than #ust truth, e"g", information content" &he core

thought behind paraconsistency is to provide logics that do not permit one to infer

anything indiscriminately from inconsistent premises" &hese may arise in

databases, counterfactual-impossible situations, inconsistent evidence presented in

a trial, works of fiction, etc", and a paraconsistent logician may not want to assume

their truth in order to provide a satisfying treatment" &his position is sometimes

called "ea+ paraconsistency in the literature, and opposed to dialetheism, taken as

a 7strongly8 paraconsistent view" 5ost relevant logicians, the +raJilian proponents

of the paraconsistent logics of formal inconsistency, and those who embrace a formof logical pluralism on the nature of entailment (see +eall and 6estall, 200=!, can

be weak paraconsistentists: they can treat inconsistent models, in which

contradictions hold, as useful mathematical tools without admitting that they

represent real possibilities"

>ometimes, a further sub-distinction is made between strong paraconsistency and

dialetheism (see riest, +eall and Armour-Darb, 2003, p" =!: the former admits

7real possibilities' in which contradictions can be true; the latter makes the final

step, and accepts true contradictions sipliciter , that is, contradictions that are trueat the actual world" ven among full-fledged dialetheists, relevant differences

remain, for eample, differences that reflect what they mean by true., e"g", on

whether they subscribe to a deflationary theory of truth, or to a robust one, like

a correspondence view" 9e will come back to this point below"

2. Dialetheism in the History of Philosophy

*n 9estern hilosophy, a number of the resocratics endorsed dialetheism" At least,

Aristotle takes them to have done it, and with apparent #ustification" <or eample,

in <ragment 3Ea, @eraclitus says: 9e step and do not step into the same rivers; we

are and we are not. (6obinson, /EFC, p" 1!" rotagorean relativism may be

epressed by the view that man is the measure of all things" According to Aristotle,

since 5any men hold beliefs in which they conflict with one another., it follows

that the same thing must be and not be. (/00Ea/0K/2!" &he resocratic views

triggered Aristotle's attack in Metaph'sics, +ook " )hapter 3 of this +ook contains

Aristotle's defence of the )" As we said above, historically this attack was

almost completely successful: the ) has been high orthodoy in 9estern

hilosophy ever since" *t is perhaps worth noting that in Metaph'sics ()hapter C!

Aristotle also defends the dual of the ), the aw of cluded 5iddle, 5,

http://plato.stanford.edu/archives/fall2013/entries/truth-deflationary/

http://plato.stanford.edu/archives/fall2013/entries/truth-correspondence/





particularly in the version which has nowadays been distinguished as the aw of

+ivalence: for any A, it is necessary for (at least! one of A and ¬ A to be true" +ut

the 5 has often had a less secure place in 9estern hilosophy than the ), in

spite of the numerous obvious dualities between the two principles" Aristotle

himself, in fact, appears to attack the aw in De .nterpretatione, )hapter E, whenhe comes to the famous sub#ect of future contingents"

?espite the orthodoy about the ), there have been a few dialetheists since

Aristotle" *t is arguably the case that some of the eoplatonists were dialetheists"

?uring the 5iddle Ages, the problem of seemingly true contradictions surfaced in

connection to the paradoes of the divine omnipotence 4 for instance: can Dod

make a stone too heavy for @im to liftL 9e find >t" ier ?amiani getting close to

dialetheism in the De divina onipotentia, by blaming >t" Dirolamus for having

claimed that Dod cannot overturn the past and twist what happened into somethingthat didn't happen" >ince Dod lives an eternal present, denying @im power over the

past e$uates to denying @im power over current and future events, which is

blasphemous" >o Dod must have the power of making what is done undone" ater

on, icholas of )usa placed at the core of his book De docta i(norantia the idea

that Dod is coincidentia oppositoru: as a truly infinite being, @e includes all the

opposite and incompatible properties, therefore being all things, and none of them:

Dod has all properties, including contradictory ones (@eron, /E13, *"3!"

According to some interpretations, 5einong, too, was a dialetheist, holding thatsome non-eistent ob#ects, such as the round s$uare, have inconsistent properties

(see 6outley, /EF0, )hapter 1!" +ut the most obvious dialetheists since the

resocratics and before the 20th century are @egel and his successors in dialectics,

such as 5ar and ngels (see riest /EE0, /EE/!" According to them, reality (in the

form of /eist for @egel, or social structures for 5ar! may be literally

inconsistent" <or eample, in the !o(ic @egel says: >omething 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 7here8, it at once is and is not.

(/F/, p" 330!" *ndeed, it is the resolution of these contradictory states that drivesthe development of the history of thought (or society! forwards" *n fact, @egel was

driven to embrace dialetheism by his assessment of Iant's achievements in

the &ritiue o# ure *eason" 9ith a little bit of massaging, even the current debate

on logical paradoes may be viewed as a ramification and formal specification of

the Iant-@egel dialectics"

Iant believed that rational antinomies were produced by an illicit use of pure

concepts; nevertheless, he also held that such an illicit use was a natural and

inevitable illusion. (Iant, /CF/, p" 00! 4 a side effect of reason's pursuit of

completeness in knowledge" Diven some phenomenon, we can be curious about its



7condition8, as Iant says" &his condition being another phenomenon, now we can

be curious in turn about its condition" t cetera" 6eason asks us to in$uire further,

but it also gives us some idea of an unconditioned totality of all conditions of a

certain realm" &he antinomies of pure reason, in particular, have their origin in such

basic concepts as be#ore, part o# , cause, depends on" As soon as children begin tomake use of reason, they start asking, 9hat is beyond thatL 9hat was before thisL

And the $uestion can be iterated 4 9hat is beyond that , thenL )uriosity is good

4 makes us human" &he transcendental illusion. begins when we turn what

should #ust be a regulative ideal into a limit-ob#ect" egitimate inferences on the

world as a whole (a totality which is never given to us as such! can lead us to

dialetheic conclusions: 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" +oth

horns assume the opposite thesis and seemingly perform a reductio" According to

Iant (at least in one way of resolving the antinomies!, the fallacy lies in treatingthe world as a whole as an ob#ect 4 in mistaking a sub#ective condition for an

ob#ective reality"

ow, according to @egel such a conception has something to be said for it as well

as against it" Iant has a point in showing, via the antinomies, that dialectics is a

necessary function of reason.; in defending the necessit' o# the

contradiction which belongs to the nature of thought determinations. (@egel,

/F/, p" 1="! @owever, Iant mistakenly imputes ob#ectification, as an error, to

reason: the result is only the familiar one that reason is incapable of knowing theAbsolute, that is, actual reality" %n 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 onl' to thinking reason. (@egel, /F0, p" E2"! )ontrary to what Iant held,

the Iantian antinomies are not a reductio of the illusions of reason" &hey are

perfectly sound arguments, deducing the dialetheic nature of the world (for a

reconstruction of this Iantian-@egelian debate, see art ** of riest /EE1!"

?ialetheism appears to be a much more common and recurrent view in asternhilosophy than in the 9est" *n ancient *ndian logicMmetaphysics, 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" +uddhist

logicians sometimes added a fifth possibility: none of these" (+oth positions were

called the catush+oti"! &he Gains went even further and advocated the possibility of

contradictory values of the kind: true (only! and both true and false" (>mart, /E=3,

has a discussion of the above issues"!

)ontradictory utterances are a commonplace in &aoism" <or eample, the )huang

&su says: &hat which makes things has no boundaries with things, but for things



to have boundaries is what we mean by saying 7the boundaries between things'"

&he boundaryless boundary is the boundary without a boundary. (5air, /EE3, p"

2/F!" 9hen +uddhism and &aoism fused to form )han (or Nen, to give it its

Gapanese name!, a philosophy arose in which contradiction plays a central role" &he

very process for reaching enlightenment (ra#na! is a process, according to >uJuki(/E=E, p" 11!, which is at once above and in the process of reasoning" &his is a

contradiction, formally considered, but in truth, this contradiction is itself made

possible because of ra#na".

%f course, interpreting the philosophers we have mentioned is a sensitive issue;

and many commentators, especially 9estern ones who have wanted to make sense

of their chosen philosopher whilst subscribing to the ), have suggested that the

contradictory utterances of the philosopher in $uestion are not really contradictory"

&here are a number of standard devices that may be employed here" %ne is to claimthat the contradictory utterance is to be taken as having some non-literal form of

meaning, e"g", that it is a metaphor" Another is to claim that the contradictory

assertion is ambiguous in some way, and that it is true on one disambiguation (or in

one respect! and false in another" &his techni$ue is called paraeterisation and is

adopted $uite generally: when one is confronted with a seemingly true

contradiction, A O ¬ A, it is a common strategy to treat the suspected dialetheia A,

or some of its parts, as having different meanings, and hence as ambiguous (maybe

#ustconte)tuall' ambiguous!" <or instance, if one claims that (a! O ¬ (a!,

parameterisation holds that one is in effect claiming that a is and is not underdifferent parameters or in different respects 4 say, r1 and r2" &o the etent that

one's claim shows no sign of such parameters, it is tempting to ascribe

inconsistency to the claim" +ut this can be resolved by clarifying that r1(a! O

¬ r2(a! (Guliette +inoche is and is not a star, but she is a star in the sense that she is

a great actress, not a star in the sense of Alpha )entauri!" *n

the Metaph'sics Aristotle also hints that a critic of the ) does not get the point

insofar as he plays with the e$uivocal meanings of some words: for to each

formula there might be assigned a different word. (/00=b /K2!"

ow, it is certainly the case that contradictory utterances that one sometimes hears

are best construed in some such way" 9hether this is so in the case of the

philosophers we have mentioned, is a matter for detailed case-by-case

consideration" *n most of these cases, it may be argued, such interpretations

produce a manifestly inaccurate and distorted version of the views of the

philosopher in $uestion" *n any case, parameterisation as such is hardly an

argument a(ainst the opponent of the )" An a priori claim that contradictions

can always be avoided by parameterisation begs the $uestion against the

dialetheist: sometimes parameterisation may be the best thing to do, butindependent #ustification is re$uired on each occasion"



3. Motivations for Dialetheism

&urning now to contemporary philosophy, the second half of the twentieth century

saw a resurgence of dialetheism, driven by largely new considerations" robably

the ma#or argument used by modern dialetheists invokes the logical paradoes ofself-reference"

3.1 The Paradoxes of Self!eference

*t is customary to distinguish between two families of such paradoes: the

semantic and set-theoretic" &he former family typically involves such concepts

as truth, denotation, de#inabilit', etc" &he latter, such notions

as ebership,cardinalit', etc" After DPdel's and &arski's well known formal

procedures to obtain non-contetual self-reference in formaliJed languages, it is

difficult to draw a sharp line between the two families (among other things,

because of the fact that &arskian semantics is itself framed in set-theoretic terms!"

evertheless, the distinction is commonly accepted within the relevant literature"

6ussell's parado is prominent among the set-theoretic paradoes (it arises when

one considers the set of all non self-membered sets!, and )antor's (which arises in

connection with the universal set, which can be taken as the set of all sets, or also

as the set of everything, depending on one's favourite version of set theory!"

rominent among the semantic paradoes is the so-called iar parado" Although

cases for the eistence of dialetheias can be derived from almost any parado of

self-reference, we will focus only on the iar, given that it is the most easily

understandable and its eposition re$uires no particular technicalities"

3.2 " Simple Case St#dy$ the %iar

*n its standard version, the iar parado arises by reasoning on the following

sentence:

(/! (/! is false"

As we can see, (/! refers to itself and tells us something about (/! itself" *ts truth

valueL et us reason by cases" >uppose (/! is true: then what it says is the case, so

it is false" &hen, suppose (/! is false: this is what it claims to be, so it is true" *f we

accept the aforementioned aw of +ivalence, that is, the principle according to

which all sentences are either true or false, both alternatives lead to a contradiction:

(/! is both true and false, that is, a dialetheia, contrary to the )"

&he parado can also be produced without any direct self-reference, but via a

short-circuit of sentences" <or instance, here is a looped iar:

http://plato.stanford.edu/archives/fall2013/entries/self-reference/



http://plato.stanford.edu/archives/fall2013/entries/russell-paradox/

http://plato.stanford.edu/archives/fall2013/entries/liar-paradox/







(2a! (2b! is true

(2b! (2a! is false"

&his is as old as +uridan (his >ophism no" E: lato saying 79hat >ocrates says istrue8; >ocrates replying 79hat lato says is false8!" *f what (2a! says is true, then

(2b! is true" @owever, (2b! says that (2a! is false Q" And so on: we are in a

paradoical loop"

aradoes of this kind have been known since anti$uity (for instance, the standard

iar is attributed to the Dreek philosopher ubulides, probably the greatest

parado-producer of anti$uity!" +ut they were thrown into prominence by

developments in the foundations of mathematics around the turn of the twentieth

century" *n the case of each parado, there appears to be a perfectly sound

argument ending in a contradiction" *f the arguments are sound, then dialetheism is

true" %f course, many have argued that the soundness of such arguments is merely

an appearance, and that subtle fallacies may be diagnosed in them" >uch

suggestions were made in ancient and medieval logic; but many more have been

made in modern logic 4 indeed, attacking the paradoes has been something of

a leitotiv of modern logic" And one thing that appears to have come out of this is

how resilient the paradoes are: attempts to solve them often simply succeed in

relocating the paradoes elsewhere, as so called 7strengthened8 forms of the

arguments show" et us have a look"

Barious authors (notably 5artin, /E=C, van <raassen, /E=F, Iripke, /EC1, <ield,

200F! have proposed to solve the iar parado by dismissing +ivalence, that is, by

admitting that some sentences are neither true nor false, and that the iar is one

such truth value 7gap8 (it is a subtle issue, which we will not discuss here, whether

being a gap should count as lac+in( any truth value, or as having a non-classical

value distinct from both truth and falsity!" &hese approaches are nowadays often

labelled as paracoplete, and they are natural duals of the (paraconsistent!

dialetheic theories of truth to be described below (for a comparative survey of the

two kinds of approach, see +eall and 6ipley (forthcoming!!" &he admission of truth

value gaps, and the inclusion of the iar among them, is differently motivated in

the various approaches (and some motivations appear to be decidedly ad hoc!" +ut

the common core thought is the following: even though the iar is a sentence such

that, if it were true, it would be false, and vice versa, no eplicit contradiction

according to which it is both true and false need follow" 9e can avoid the

contradiction by re#ecting the idea that truth and falsity are the only two options for

a sentence, and maintain that the iar is neither"

&hese approaches face difficulties with the so-called 7strengthened8 iars 4sentences such as the following:



(! (! is not true"

(3! (3! is false or neither true nor false"

ow these sentences should be, on the gappy theorist's non-bivalent approach,either true, or false, or neither" +ut, for instance, if (! is true, then things are as it

claims they are; therefore, (! is not true (either false or truth-valueless!" *f (! 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" 9e seem to have to conclude that (! is both

true and not true, contrary to the )" A similar line of reasoning goes for (3!"

According to riest the strengthened iars show that a single feature of the

semantic parado underlies its different formulations" &he totality of sentences is

divided into two subsets: the true ones, and their 7bona fide complement8 4 call it

the *est " ow 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 6est (too!; conversely,

if it is in the 6est, it is in the bona fide truths. (riest, /EFC, p" 2!" &he standard

iar, 7&his sentence is false8, is #ust a particular instance of this, producing a

contradiction within the bivalent framework, in which the 6est is identified with

the set of the false sentences" ow, 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 6est" @owever, the strengthened iars show that we can use the

notions introduced to solve the previous parado to re%describe the 6est" *n a

framework in which the set of sentences is partitioned in terms of a trichotomy

(true, false, and neither true nor false!, the dis#unctive nature of 7&his sentence is

false or neither true nor false8 means that it embraces the whole 6est, i"e" the

new(ly described! complement of the set of the true sentences" Adding more values

is, of course, useless" *f there is some fourth thing that a sentence can be, besides

true, false, and neither true nor false, we can always take the notion #ourth

thin( and produce another strengthened iar:

(1! (1! is false, or neither true nor false, or the fourth thing"

(>ee Iirkham, /EE2, pp" 2EK3!"

*t comes as no surprise, then, that there is no generally agreed upon solution to the

semantic paradoes" %ne typical way out attempted by the supporters of truth value

gaps, for instance, consists in denying that the notion of (ap, or de#ective sentence,

or sentence "hose truth value is indeterinate, can be fully epressed in the

language for which they are proposing their theory of truth" &he strengthened

paradoes then seem to force the consistent theorist to admit that the proposed

theory was formulated in a language different from, and epressively more powerful than, the one whose semantics it was supposed to epress" &his entails a



limitation of the &arskian &-schema characterising truth, i"e", of the

e$uivalenceTr ⟨ A⟩ R A, where 7Tr 8 is the truth predicate for the relevant language,

and ⟨ A⟩ is the appropriate name of sentence A; and a retreat to a rigid distinction

between an ob#ect language and its metalanguage" >uch a distinction, though

introduced by &arski to epel the iar parado from formaliJed languages, wasdoomed by &arski itself as inapplicable to natural languages, which do not appear

to depend upon some (ineffableL! metalanguage for their semantics" As Iripke has

admitted at the end of utline o# a Theor' o# Truth, the ghost of the &arski

hierarchy is still with us. (scil" the paracompletists: see Iripke, /EC1, p" F0!"

*t is these facts that give dialetheism about the paradoes of self-reference one of

its ma#or appeals" *t is not the only one, though: the simplicity of a dialetheic

theory of truth is another" &he two most prominent such theories to date are

presented in riest, /EFC, and +eall, 200E" *n the former, the truth predicate Tr forthe relevant formal language, modelling the behaviour of truth in nglish, is

simply characteriJed by the unrestricted &-schema, which, as stressed by many

philosophers, is an overwhelmingly intuitive 4 one may dare say, 7analytic8 4

principle concerning truth" *t is admitted that some sentences 4 notably, the iars

4 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 artificial hierarchy of metalanguages is needed 4 not to

speak about the further epicycles of the (allegedly! consistent solutions to the iar

paradoes"

G) +eall's 200E theory is based on a (relevant! paraconsistent logic, whose modal

semantics employs so-called non-normal worlds" *t allows a fully transparent truth

predicate: one such that for any sentence A, Tr ⟨ A⟩ and A can be replaced with each

other in all (non-opa$ue! contets salva veritate, that is, producing sentences

logically e$uivalent to the sentences one started with" &hen the unrestricted &-

schema, Tr ⟨ A⟩ R A, follows from transparency (and the fact that A S A is a

logical truth! as a special case" *n +eall's theory, all sentences A that are dialetheias

are not only true and false, i"e" (given that falsity is truth of negation!, Tr ⟨ A⟩ ∧ Tr ⟨¬ A⟩; they are also true and untrue, Tr ⟨ A⟩ ∧ ¬ Tr ⟨ A⟩: this again follows from

the transparency of truth"

%verall, such paradoes as the iar provide some evidence for the dialetheist's

claim that some contradictions are provabl'true, in the sense that they are entailed

by plain facts concerning natural language and our thought processes" tended

iar paradoes like 7&his sentence is not true8 are spelt in ordinary nglish" &heir

paradoical characteristics, as dialetheists stress, are due eactly to the intuitive

features of ordinary language: unavoidable self-reference; the failure of

metalinguistic hierarchies, which only produce languages that are epressively

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weaker than nglish; and the obvious presence of a truth predicate for nglish, 7is

true8, which is characteriJed (at least etensionally! by the &arskian &-schema"

9e conclude our discussion of the semantic paradoes by briefly mentioning one,

which is properly dealt with in another entry, on the )urry parado" &his is produced by a self-referential sentence claiming 7*f * am true, then ⊥8, where ⊥ is

a constant (what logicians usually call the #alsu! which is or entails something

that is also dialetheically unacceptable, say ⊥ T 7verything is true8, the trivialist

claim" ria #acie, this does not involve negation, nor a falsity predicate"

@owever, it re$uires a careful dialetheic treatment: from the )urry sentence we can

deduce ⊥, therefore that everything is true, using logical principles that do not

involve negation, such as as the so-called )ontraction (or Absorption! aw, i"e", the

rule: from AS ( A S -! infer A S -, or the so-called pseudo%odus ponens, the

principle ( A ∧ ( A S -!! S -" &he standard dialetheic strategy to deal with the)urry parado has consisted in eploiting paraconsistent logics with a

7noncontractive8 conditional (see again riest, /EFC, )h, =, +eall, 200E, )h" 2; for

recent work on the topic, +eall and 5urJi (forthcoming!; and for contraction-free

logics within the broader family of substructural logics, 6estall, 2000!"

?ialetheism also affords a treatment of the set-theoretic paradoes via set theories

based on an unrestricted comprehension schema. for sets: for any condition or

property, including paradoical ones like non-self-membership, there eists a

corresponding set" *n particular, inconsistent sets like 6ussell's, which is and is nota member of itself, are admitted" Again, such contradictions do not give rise to

triviality due to the paraconsistent logic underlying the relevant theories" &hough

the issue is too technical to be addressed here, and more appropriately dealt with in

the entries on paraconsistent logic andinconsistent mathematics, the reader can

consult 6outley, /ECE, +rady, /EFE, for classical inconsistent set theories, and

9eber, 20/0b, 20/2, for important recent results in the field"

3.3 &ther Motivations for Dialetheism

?ialetheias produced by the paradoes of self-reference have a limited range, being

confined to the realm of such abstract notions as the notion of set, or to semantic

concepts 4 though very basic ones, such as the concept of truth" @owever, the

paradoes of self-reference are not the only eamples of dialetheias that have been

mooted" %ther cases involve contradictions affecting concrete ob#ects and the

empirical world, and include the following"

(/! Transition states: when * eit the room, * am inside the room at one time, and

outside of it at another" Diven the continuity of motion, there must be a precise

instant in time, call it t , at which * leave the room" Am * inside the room or outsideat time t L <our answers are available: (a! * am inside; (b! * am outside; (c! * am

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both; and (d! * am neither" (a! and (b! are ruled out by symmetry: choosing either

would be completely arbitrary" As for (d!: if * am neither inside not outside the

room, then * am not inside and not-not inside; therefore, * 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"

(2! >ome of enos parado)es concerning a particular 4 though, perhaps, the most

basic 4 kind of transition, that is, local otion: the moving arrow is both where it

is, and where it is not" &he orthodo way out of the paradoical situation, as

formulated, e"g", by 6ussell, /E0, has it that motion is the mere occupation of

different places at different times (this is, clearly, another case of attempted

parameterisation!" +ut this seems to imply a denial of the phenomenon itself, that

is, of the actuality of motion: it entails that motion is not an intrinsic state of the

(allegedly! moving thing, for, at each instant, the arrow is not moving at all" venif time is dense, a continuum of states each one of which is indistinguishable from

a state of rest, one may argue, is not motion" )an a going-somewhere be composed

by an (even more-than-denumerable! infinity of going-nowheresL An alternative,

dialetheic account of motion, which takes at face value the aforementioned

@egelian idea that >omething 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 7here8, it at once is and is not., is eposed in riest, /EFC, )h" /2"

(! -orderline cases o# va(ue predicates" 9ith the eception of the so-calledepistemicist solutions, the main approaches to vagueness (such as the ones based

on many-valued logics, or supervaluations! re$uire some under-determinacy of

reference, andMor the re#ection of +ivalence: if an adolescent, , is a borderline

case of adultness, A, then A(! may turn out to have an intermediate truth value

between truth and falsity, or no truth value at all" +ut it may be con#ectured that a

borderline ob#ect like , instead of satisfying neither a vague predicate nor its

negation, satisfies them both: an adolescent both is and is not an adult" Diven the

obvious dualities between the 5 and the aw of +ivalence on the one side, and

(respectively, syntactic and semantic formulations of! the ) on the other, it isnot too difficult to envisage a 7sub-valuational8 semantic approach, dual to the

supervaluation strategy" >ub-valuational paraconsistent semantics have been

proposed by @yde, /EEC, and BarJi, /EEC" %ther 7glutty8 approaches to vagueness

have recently been proposed by )olyvan, 200E, 9eber, 20/0a, riest, 20/0, and

6ipley, 20/2a" &o be sure, it is an open option to assume that the inconsistencies

due to vague predicates and borderline ob#ects are, as a matter of fact, only de

dicto, due to merely semantic under- and over- determination of ordinary language"

+ut if the aforementioned phenomena have a de re reading, then actually

inconsistent ob#ects are admitted, together with vague ob#ects" And this spreadsinconsistency all over the empirical world: if borderline cases can be inconsistent,



inconsistent ob#ects are more or less everywhere, given how pervasive the

phenomenon of vagueness notoriously is: adolescents, borderline bald men, etc" *t

is fair to say, though, that there is a certain schism in the dialetheic community on

the issue whether vagueness ought to be given a dialetheic treatment: G) +eall has

argued against the approach in )h" 1 of +eall, 200E, and in +eall (forthcoming!"

(3! Multi%criterial predicates" 9e may assume that the semantics of a predicate is

specified by means of its criteria of application" ow ordinary language hosts

predicates with different, and occasionally conflicting, criteria of application: some

criteria for applying ( ! may entail that ob#ect is in the etension of the

predicate, some others, that is in its anti-etension, or negative etension"

)riteria 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 todetect and identify" *f the etensions 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, instead of destroying it

by means of some regimentation (e"g" via the usual parameterisation, or distinction

of respects!"

(1! )ertain le(al situations, such as inconsistent bodies of law" >uppose, 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 theceremony" *t 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" &hen someone may turn out to be both a

married man and a bachelor, therefore, given the meaning of 7bachelor8, both a

married man and not a married man (and, of course, nobody would infer from this

that he is not a man anymore, or both a man and not a man, etc"; so we have

another countereample toe) contradictione uodlibet !" *f one accepts the plausible

view that statements concerning legal rights, obligations, and statuses, can be truth-

value apt, we seem to have a dialetheia" %f course, legal systems sometimes havemechanisms that can be used to remove such inconsistencies (e"g", by ordering

different kinds of laws in a hierarchy from customary laws, to established

#urisprudence, to ordinary legislation, to constitutional norms, etc"; or via the le)

posterior principle, giving priority to the more recent norm in case of conflict!" +ut

this is not always the case: the inconsistent laws may be of the same rank, enacted

at the same time, etc"

ach of the above arguments undoubtedly calls for further development, which

cannot be done here; but one can check riest, /EFC, for detailed discussions of all

of them"



'. &()ections to Dialetheism

9e now turn to arguments against dialetheism" &he only sustained defence of the

) in the history of philosophy is, as mentioned, that given by Aristotle in

)hapter 3 of Metaph'sics, " Diven the influence this chapter has had, thearguments are surprisingly poor" Aristotle's main argument, which takes up the first

half of chapter, is tangled and contorted" *t is not clear what it is, let alone that it

works" About the best one can say for it is that it depends on substantial and moot

principles of Aristotelian metaphysics, and, in any case, as a suasive argument,

begs the $uestion" &he si or seven arguments that Aristotle deploys in the second

half of the chapter are varied, swift, and fare little better" 5any of them seem also

to beg the $uestion" 9orse: many of them simply confuse dialetheism and

trivialism" (<or an analysis of Aristotle's arguments, see riest, /EEFb"!

'.1 The "r*#ment from +xplosion

A standard modern argument against dialetheism is to invoke the logical principle

of plosion, in virtue of which dialetheism would entail trivialism" Dranted that

trivialism is absurd (though why this is so is not as easy a $uestion as it might

appear: see riest, 2000a, riest, 200=, )h" , and Iabay, 20/0!, dialetheism must

be re#ected" *t is clear that this argument will fail against someone who subscribes

to a paraconsistent, non-eplosive logic, as (non-trivialist! dialetheists certainly

will"

*nterestingly enough, whereas Aristotle's defence of the ) cheerfully slides

between attacking dialetheism and trivialism (that is, between attacking the claim

that some contradictions are true, and the one that all contradictions are!,

Aristotelian syllogistic 4 the first formally articulated logic in 9estern philosophy

4 is not eplosive" Aristotle held that some syllogisms with inconsistent premises

are valid, whereas others are not (An" r" =3a /1!" Gust consider the inference:

(/! >ome logicians are intuitionists;

(2! o intuitionist is a logician;

()! &herefore, all logicians are logicians"

&his is not a valid syllogism, despite the fact that its premises are inconsistent" &he

principle of plosion had a certain tenure at places and times in 5edieval logic,

but it became well-established mainly with the <regean and post-<regean

development of classical logic, as it is nowadays called (rather inappropriately, as

we can see!"



'.2 The "r*#ment from +xcl#sion

Another argument against dialetheism that is sometimes deployed (it can be found,

for instance, in 5c&aggart, /E22, F; see also +erto, 200=, 20/2! is as follows" A

sentence is meaningful only if it rules something out" +ut if the ) fails, A doesnot rule out ¬ A, or, a #ortiori, anything else" @ence meaningful language

presupposes the )"

&here are many problems with this argument" %ne is, for instance, that even though

a dialetheia does not rule out its negation, it still may rule out several other things"

+ut the central trouble is that the first premise is simply false" )onsider again the

sentence 7verything is true8" &his entails everything, and so rules out nothing" Uet

it is meaningful" *t is something that everyone, ecept a trivialist, re#ects"

%ne might attempt a more sophisticated eplanation of the notion of rulin( out , for

instance in terms of information theory, or perhaps possible worlds" %ne may claim

that a statement 7rules out8 something insofar as there are situations, or worlds, at

which it fails" *n this sense, 7verything is true8 does rule something out" +ut now,

it is this account of propositional meaning which is wrong in general" *f

mathematical truths have a strictly necessary status (which may safely be assumed

here!, <ermat's ast &heorem rules out nothing: being a necessary truth, it holds at

all possible worlds" +ut it is perfectly meaningful; people have been wondering

whether it was true or false for centuries; and its proof by Andrew 9iles has been a

substantial discovery"

&he argument from eclusion has a more ad hoine twist, in which it is argued

(see arsons, /EE0, >hapiro, 2003, ittman and >immons, 2003! that the dialetheist

has trouble with ruling out things, or epressing disagreement with rival positions"

<or when the dialetheist utters 7¬ A8, 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 ¬ A"

>imilarly, 7 A is false8 and even 7 A is not true8 might not do the trick, since for the

dialetheist some A's being false, or untrue, does not rule out its being true"

&o this, the dialetheist has various replies" %ne is to epress eclusion via a

primitive notion of re5ection: to re#ect A is to positively refuse to believe that A"

&hat the notion is taken as primitive means, in particular, that it is not reducible to

the acceptance of negation: it is a sui (eneris act" &he linguistic counterpart of

re#ection is the speech act of denial" &hen the dialetheist can rule out that A is the

case by denying A; and this does not amount to the assertion of ¬ A (see riest,

200=, )h" =; of course, one can often epress denials by uttering ordinary language

negations: 7not8 is, in this sense, pra(aticall'ambiguous!" 9e will come back

below to how, and why, re#ection-denial may not be reducible to the acceptance-assertion of any negation" Another way in which the dialetheist may epress the



eclusion that A is the case is by uttering 7 A S ⊥8, where again ⊥ is or entails

7verything is true8" 6ecent considerations by @artry <ield, 200F, )h" 2C, and

+erto (forthcoming!, though, seem to show that 7arrow- #alsu8 cannot work as a

dialetheic eclusion-epressing device in all cases, due to side-effects of the

aforementioned )urry parado"

<inally, even if one persisted in subscribing to an account of propositional content

in terms of splitting situations, or worlds, into those where it holds and those where

it does not, this would not affect a dialetheic challenge to the )" <or a given Ato

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"

%f course, such an overlap re$uires dismissing the account of negation embodied

in (so-called! classical logic, and to this issue we now turn"

'.3 The "r*#ment from ,e*ation

&here are other arguments one might take into account in this contet, which are

focused on the concept of logical negation" &he main one goes as follows" &he

truth conditions for negation are: ¬ A is true iff A is not true" @ence, if A and

¬ A were true, A would be both true and not true, which is impossible"

&his argument has various troubles as well" <irst, the truth conditions for negation

employed here are contentious" An alternative view has it that ¬ A is true iff A is

false, and ¬ A is false iff A is true 4 and in the semantics of many paraconsistent

logics (for instance, the logic of <irst ?egree ntailment!, truth and falsity may

overlap" >uch an account preserves our intuition that negation is the operator which

(truth-functionally! switches truth and falsity" *t also preserves our intuition on

contradictoriness, in the form: A and - are contradictories iff, if A is true, - is

false, and if A is false, - is true" 9hat has to go is 7only8 the assumption that truth

and falsity are eclusive in all cases: there eist dialetheias, that is, sentences

falling simultaneously under both categories"

>econdly, and more importantly, the argument against dialetheism based on the

truth conditions for (classical! negation fails, since it begs the $uestion at its last

step: why should we assume that it is impossible for A to be both true and not trueL

9ell, because it is a contradiction" +ut we were supposed to be arguing for the

impossibility of any contradiction holding to begin with" *n fact, the dialetheist

may even accept a characterisation of the truth conditions for negation as: 7¬ A is

true iff A is not true8" <or if the 7metalanguage8 in which the characterisation is

epressed can be inconsistent in its turn, as a thoroughgoing dialetheist is likely to

allow, then there is no guarantee that the 7not8 in that clause behaves consistently"



5any other arguments for the ), whatever other failings they have, seem

ultimately to beg the $uestion in similar ways"

A variant on the anti-dialetheic argument from negation comes from a Vuinean

conception of logical vocabulary" *t goes as follows" ven granting that there is anoperator, say, W, which behaves as dialetheists claim (namely, such that in particular

in some cases A is true together with W A!, it is still perfectly possible to de#ine a

+oolean negation with all the properties of classical negation (in particular, the

property of being eplosive!" And since +oolean negation is the standard operator

in logic, it is not worth translating anything non-+oolean as 7not8: such a

translation may simply amount to calling 7negation8 something different" A change

in logical vocabulary is a change of sub#ect., as the Vuinean slogan goes"

%ne line of reply available to the dialetheist is that the ob#ection is confused between a logical theory and what the theory is a theory of" &here are many

different and well-worked out logical theories of negation (minimal negation,

intuitionistic negation, ?e 5organ negation, etc"!" *nsofar as each one of them

characteriJes its own theoretical ob#ect, there is no rivalry between logics" 6ivalry

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 o# something, and the theoretical ob#ect has to fit the real

ob#ect" ow, to assume beforehand that the classical, +oolean account of negation

is the correct one, in the sense that it captures how negation works in thevernacular, again begs the $uestion against the dialetheist (and, indeed, against

most non-classical logicians!: one cannot #ust assue that classical negation gets it

right" >omeone who proposes a treatment of negation alternative to the classical-

+oolean one is not thereby proposing to revise ne(ation, but an account of it, the

+oolean one, which she considers incorrect"

&here certainly are various other arguments against dialetheism in the

philosophical market" %ne worth mentioning is by Nalta, who argues that

preserving our pretheoretic understanding of what it is to eemplify or instantiate

a property. re$uires us to preserve the ) (2003, 32!" &his entry is not the place

to debate them all" +ut it is worth noting that, by forcing philosophers to struggle

to find arguments for what previously was an undisputed belief, namely the one in

the ), dialetheism may have rendered a valuable service to philosophy even if it

turned out that it is ultimately wrong"

-. Dialetheism and !ationality

-.1 Consistency and &ther +pistemic irt#es



>ome have felt that what is wrong with dialetheism is not so much violation of the

) itself, as that an acceptance of the ) is a precondition for rationality" <or

eample, it is often suggested that it could not possibly be rational to accept a

contradiction"

9hilst the $uestion of the conditions under which it is rational to accept something

is a moot one, it is commonly agreed that, as @ume put it, the wise person

proportions his beliefs to the evidence. (/E11, p" //F!" @ence, if a sufficient case

can be made out for a contradiction, it will be rational to believe it" And sometimes

this does seem possible" 9e have seen that a seemingly compelling argument can

be made in favour of the truth of the strengthened iar sentence, 7&his sentence is

not true8" 9hether or not one takes the argument in $uestion to be completely

persuasive, it suggests that there is nothing in principle impossible about the

eistence of good arguments for true contradictions" %f course, if there wereconclusive evidence for the ), then no case for a contradiction could be strong

enough" +ut 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 a view held" <or if a person accepts A then, when an

argument for ¬ A is put up, they could simply accept both A and ¬ A"

+ut this is too fast" &he fact that some contradictions are rationally acceptable does

not entail that all are" &here is certainly a case to be made for the claim that the

iar sentence is both true and false, but this in no way shows that a case can also

be made for +risbane being and not being in Australia" (%f course, if one

subscribes to the claim that entailment is eplosive, a case for one contradiction is

a case for all; but if entailment is paraconsistent, this argument is of no use"! As

orthodo 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 ade$uacy to the data; its

simplicity, cleanness and elegance; its unity and freedom from ad hoc hypotheses;

its eplanatory and predictive power; etc" ot only do these (and other! criteria

come in degrees, but they may also be orthogonal to each other" *n 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" And it

may well turn out that a theory lacking the virtue of consistency 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 hinted at above"

And conversely, of course, an inconsistent theory may well be trumped by a



consistent theory, all things considered" >o it may be rational to re#ect an

inconsistent position, even if it is logically possible that it is true"

-.2 "cceptin* and "ssertin* Dialetheias

Diven all this, it is natural to epect that a dialetheist will sometimes accept, or

believe in, contradictions, and assert them" riest (200=, p" /0E! adopts the

following 6ationality rinciple:

(6! *f you have good evidence for (the truth of! A, you ought to accept A"

+elief, acceptance, and assertion have a point : when we believe and assert, what

we aim at is believing and asserting what is the case or, e$uivalently, the truth"

&herefore, the dialetheist will accept and, sometimes, assert both A and ¬ A, if she

has evidence that A is a dialetheia 4 that both A and ¬ A are true, as it happens, for

instance, with the iar sentences"

otice that this need not entail that the dialetheist both accepts and re5ects A at the

same time at all" 9e now come back to the issue flagged in >ection 3"2, on the

irreducibility of re#ection to negation" &hat re#ecting 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 <rege and eter

Deach" +ut this fusion is a confusion from a dialetheic viewpoint (see +erto, 200F,

on this issue!" &he point can be made independently of the issue of dialetheism: itis apparent as soon as we get out of the standard, bivalent framework"

aracompletists maintain that some sentences (notably, the iars! are neither true

nor false" ow if A is a truth-value gap (therefore, in particular, not a truth!, one

may well want to deny A; but it would be unfair to take such a denial as e$uivalent

to the assertion of ¬ A" *f A is truth-valueless, ¬ A is normally considered as truth-

valueless, too, therefore, not a truth, and so it should not be asserted in its turn" A

dual position can hold for dialetheism: given that accepting ¬ A is different from

re#ecting A, a dialetheist can do the former and not the latter 4 eactly when she

thinks that Ais a dialetheia"

/. Themes for 0#rther !esearch$ Dialetheism !ealism and "ntirealism

%ne of the signs of maturity in a research program is shown by its beginning to

confront itself with some ma#or topics in traditional and mainstream philosophy"

Among such topics, a prominent one is the debate between realists and anti-

realists(for eample, idealists and constructivists! in metaphysics" Bery roughly, to

be a realist about entities of some kind is to maintain that such entities ob#ectively

eist apart from, and antecedently to, anyone's thought of them; and, therefore, thatour thoughts, beliefs and theories concerning such entities are made ob#ectively

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true or ob#ectively false by them, apart from what we think of them (more refined

definitions of realism and anti-realism are certainly available; but this

characteriJation will suffice for our purposes!"

ow, it has been claimed (see riest, 2000b, and riest, 200=, )h" 2! thatdialetheism is not by itself committed to a specific conception of truth (deflationist,

semantic, correspondentist, coherentist, constructivist, etc"!" evertheless, if we

accept even a mild form of realism, the truth of some contradictions entails the

eistence of inconsistent ob#ects andMor states of affairs: those that make the

contradictions true (see +erto, 200Cb!" %ne may claim that it makes no sense to

talk of inconsistent ob#ects, situations, or states of affairs" &he world is all there, all

together: how could some pieces of it contradict some other piecesL )onsistency

and inconsistency might be taken as properties of sentences, or theories (sets of

sentences closed under logical conse$uence!, or propositions (what sentencesepress!, or maybe thoughts, or (sets of! beliefs, etc" )ontradiction (6iderspruch,

the atin contradictio! has to do with discourse (diction, sprechen, dicere!" &he

world, with its non-mental and non-linguistic inhabitants 4 armchairs, trees,

people 4 is not the right +ind of thing that can be consistent or inconsistent, and

ascribing such properties to (a part of! the world is, to use Dilbert 6yle's

terminology, a category mistake"

&hese considerations might drive dialetheism towards an anti-realist interpretation

of the claim that there are dialetheias, true contradictions; and anti-realist dialetheictheories of truth have, in fact, been proposed (see e"g" G) +eall's 7constructive

methodological deflationism8, in +eall, 2003!" +ut other options are available to a

dialetheist who wants to embrace some form of metaphysically robust realism

about truth" <or 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 #ust is to say that some true purely descriptive sentences

about the world have true negations" )onse$uently, and not accidentally, it is $uite

common in the current literature both for and against dialetheism to

straightforwardly speak of inconsistent ob#ects, states of affairs, and entireinconsistent worlds" A dialetheic correspondence theory of truth might be

committed, in particular, to negative facts (re$uiring the simultaneous eistence 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" riest, 200=, pp" 1/K!"

&here may also be room for a further intermediate position, that is to say, a

7semantic dialetheism8 which accepts true contradictions without inconsistent

ob#ects or states of affairs as their truth-makers" &his position has been eplored in

the literature, and one may consider Iroon, 2003, and 5ares, 2003, as early and

interesting efforts in this direction" G) +eall's most recent position, epressed in his



transparent truth theory in +eall 200E, may also be seen as a form of semantic

dialetheism" &ransparency can be naturally paired with a deflationary view of truth"

<or suppose the truth predicate is a merely semantic device, coined, as Vuine

famously stressed, for epressive, 7dis$uotational8 purposes" &hen dialetheias such

as the iar(s! may well be semantic side-effects (7spandrels8, in +eall'sterminology! of the introduction of such device, not involving any metaphysically

committing contradiction in a language- and mind- independent world"

9oodbridge and Armour-Darb (forthcoming! recently argued 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 paradoes"

%f course, such debates on realism and anti-realism $uickly spill over into

$uestions concerning the nature of reality in general, that is, into metaphysicalissues: if reality is dialetheic, how should the ontology of a dialetheic world be

spelt outL *t is likely that this is another ma#or direction for future dialetheic

research" *f 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" &his was, after all, Aristotle's view, too: he decided to speak on behalf of

the unconditional validity of the ), not in his r(anon (his writings on the

sub#ect of logic!, but in the Metaph'sics, for this was for him an issue to be

addressed ontologically, not (only! via formal logical tools"

. Concl#sion

9e think it fair to say that, since Aristotle's defence of the ), consistency has

been something of a shibboleth in 9estern philosophy" &he thought that

consistency is a sine ua non for central notions such as validity, truth,

meaningfulness, rationality, is deeply ingrained into its psyche" %ne thing that has

come out of the modern investigations into dialetheism appears to be how

superficial such a thought is" *f consistency is, indeed, a necessary condition for

any of these notions, it would seem to be for reasons much deeper than anyone has

yet succeeded in articulating" And if it is not, then the way is open for the

eploration of all kinds of avenues and $uestions in philosophy and the sciences

that have traditionally been closed off"

Bi(lio*raphy

9e break up the references into sections corresponding to those of the tet" 9here

a reference is not eplicitly referred to in the tet, we add a sentence concerning its

relevance"

Some Basic Concepts





• +eall, G) and D" 6estall, 200=, !o(ical luralis, %ford: %ford

Hniversity ress"

• +erto, <", 200C, 7o" to ell a &ontradiction The !o(ic and Metaph'sics o#

.nconsistenc', ondon: )ollege ublications"

• Iabay, ", 20/0, n the lenitude o# Truth: A De#ense o# Trivialis,

>aarbrXcken: ambert Academic ublishing"

• riest, D", G) +eall, and +" Armour-Darb (eds"!, 2003, The !a" o# $on%

&ontradiction $e" hilosophical ssa's, %ford: %ford Hniversity ress"

• riest, D", 6" 6outley, and G" orman (eds"!, /EFE, araconsistent !o(ic:

ssa's on the .nconsistent , 5Xnchen: hilosophia Berlag"

• 9ittgenstein, ", /E1=, *ear+s on the Foundations o# Matheatics,

%ford: +asil +lackwell, rd edition, /ECF"

• 9oods, G", 200, arado) and araconsistenc', )ambridge: )ambridge

Hniversity ress" (*ncludes an ample discussion of the dialetheic approach to

paraconsistency and its wider outcomes within abstract sciences"!

• 9oods, G", 2001, ?ialectical )onsiderations on the ogic of )ontradiction:

art *., !o(ic ournal o# the ./!, /: 2/K=0" (A discussion on the methodology

needed to conduct non-$uestion-begging disputes on the aw of on-

)ontradiction"!

Dialetheism in the History of Philosophy

• Aristotle, The &oplete 6or+s (ed" by G" +arnes!, rinceton: rinceton

Hniversity ress"

• ?eguchi U", G"" Darfield and D" riest, 200F, &he 9ay of the ?ialetheist:)ontradictions in +uddhism., hilosoph' ast and 6est 1F: E1K302" (An

eamination of the dialetheic aspects of +uddhism"!

• @egel, D"9"<", /F0, n<'+lop=die der der philosophischen 6issenscha#ten

in /rundrisse, in 6er+e in <"an<i( -=nde, hrg" von " 5oldenhauer und I"5"

5ichel, +Ynde FK/0, >uhrkamp, /EC0; page references are to the nglish

translation, The nc'clopaedia !o(ic >"ith the us=t<e?, *ndianapolis: @ackett,

/EE/"



• @egel, D"9"<", /F/, 6issenscha#t der !o(i+ , /F/, vols" // and /2

of /esaelte 6er+e, in Berbindung mit der ?eutschen <orschungsgemeinschaft,

hrg" von der 6heinisch-9estfYlischen Akademie der 9issenschaften, 5einer,

/E=Fff; page references are to the nglish translation, 7e(els cience o# !o(ic,

ew Uork: @umanity +ooks, /E=E"

• )usanus, icholas, /330, # !earned .(norance, D" @eron (trans"!, ondon:

6outledge and Iegan aul, /E13"

• Iant, *", /CF/, @riti+ der reinen ernun#t , /CF/, vols" and 3

of /esaelte chri#ten, de Druyter O )o", /E=E; page references are to the

nglish translation, &ritiue o# ure *eason, ew Uork: algrave 5acmillan,

200"

• riest, D", /EE0, ?ialectic and ?ialetheic., cience and ociet', 1: FFK

3/1"

• riest, D", /EE/, 9as 5ar a ?ialetheistL., cience and ociet', 13: 3=FK

C1"

• riest, D", /EE1, -e'ond the !iits o# Thou(ht , )ambridge: )ambridge

Hniversity ress, 2nd epanded edition, %ford: %ford Hniversity ress, 2002"

• riest, D", and 6" 6outley, /EFEa, &he @istory of araconsistent ogic.,

)hapter / of riest, 6outley and orman, /EFE (above!" (An account of

dialetheism and paraconsistency in the history of philosophy"!

• 6obinson, &"5", /EFC, 7eraclitus: Fra(ents, &oronto: Hniversity of

&oronto ress"

• 6outley, 6", /EF0, )plorin( Mein(on(s un(le and -e'ond , )anberra:

Australian ational Hniversity"

• >mart, ", /E=3, Doctrine and Ar(uent in .ndian hilosoph', ondon:

Allen and Hnwin"

• >uJuki, ?"&", /E=E, The en Doctrine o# $o Mind , ondon: 6ider and )o"

• NhuangJi, 6anderin( on the 6a': arl' Taoist Tales and arables o#

&huan( T<u, B" @" 5air (trans"!, ew Uork: +antam +ooks, /EE3"

Motivations for Dialetheism



• +eall, G), 200E, pandrels o# Truth, %ford: %ford Hniversity ress"

• +eall, G), forthcoming, <inding &olerance 9ithout Dluts., Mind "

•

+eall, G) and G" 5urJi, forthcoming, &wo <lavors of )urry'sarado., ournal o# hilosoph'"

• +eall, G) and ?" 6ipley, forthcoming, on-)lassical &heories of &ruth., in

5" DlanJberg (ed"!, The )#ord 7andboo+ o# Truth, %ford: %ford Hniversity

ress"

• +rady, 6", /EFE, &he on-&riviality of ?ialectical >et &heory., in riest,

6outley and orman (above!, pp" 3CKC/"

• )olyvan, 5", 200E, Bagueness and &ruth., in @" ?yke (ed"!, Fro Truth to

*ealit': $e" ssa's in !o(ic and Metaph'sics, %ford: 6outledge, 200E, pp" 2EK

30"

• <ield, @", 200F, avin( Truth #ro arado), %ford: %ford Hniversity

ress"

• @yde, ?", /EEC, <rom @eaps and Daps to @eaps of Dluts., Mind , /0=:

=30K=0"

• Iirkham, 6"", /EE2, Theories o# Truth A &ritical .ntroduction, )ambridge,

5ass: 5*& ress"

• Iripke, >", /EC1,%utline of a &heory of &ruth., ournal o# hilosoph', C2:

=E0KC/=" 6eprinted in 6"5" 5artin (ed"!, *ecent ssa's on Truth and the !iar

arado), %ford: %ford Hniversity ress, /EF3, pp" 1KF/"

• 5artin, 6"5", /E=C, &owards a >olution to the iar

arado., hilosophical *evie", C=: 2CEK//"

• 5ortensen, )", /EE1, .nconsistent Matheatics, ?ordrecht: Iluwer

Academic ublishers" (&he introduction contains a discussion of dialetheism"!

• riest, D", /EFC, .n &ontradiction, ?ordrecht: 5artinus i#hoff" 2nd

epanded edition, %ford: %ford Hniversity ress, 200="

• riest, D", 20/0, *nclosures, Bagueness, and >elf-6eference., $otre Dae

ournal o# Foral !o(ic, 1/: =EKF3"



• riest, D", and 6" 6outley, /EFEb, Applications of araconsistent ogic.,

)hapter / of riest, 6outley and orman, /EFE (above!" ()ontains some

discussion of most of the motivations for dialetheism"!

•riest, D", and 6" 6outley, /EFEc, &he hilosophical >ignificance and

*nevitability of araconsistency., )hapter /F of riest, 6outley and orman, /EFE

(above!" ()ontains some discussion of most of the motivations for dialetheism"!

• 6estall, D", 2000, An .ntroduction to ubstructural !o(ics, ondon-ew

Uork: 6outledge"

• 6ipley, ?", 20/2a, >orting %ut the >orites., in I" &anaka, <" +erto, "

5ares and <" aoli (eds"!, araconsistenc': !o(ic and Applications, ?ordrecht:

>pringer, 20/2, pp" 2CK31"

• 6ipley, ?", 20/2b, aradoes and <ailures of )ut., Australasian ournal o#

hilosoph', E/: /EK=3" (A nonstandard dialetheic treatment of the iar which

avoids trivialism by restricting the transitivity of entailment"!

• 6outley, 6", /ECE, ?ialectical ogic, >emantics and

5etamathematics., r+enntnis, /3: 0/K/" (A defence of a dialetheic account of

the paradoes of self-reference"!

• 6outley, 6", and 6"I" 5eyer, /EC=, ?ialectical ogic, )lassical ogic, and

the )onsistency of the 9orld., tudies in oviet Thou(ht , /=: /K21" (A classical

defence of a dialetheic approach to paraconsistency"!

• 6ussell, +", /E0, rinciples o# Matheatics, )ambridge: )ambridge

Hniversity ress"

• van <raassen, +", /E=F, resuppositions, *mplication and >elf-

6eference., ournal o# hilosoph', =1: /=K1/"

• BarJi, A", /EEC, *nconsistency without )ontradiction., $otre Dae

ournal o# Foral !o(ic, F: =2/KE"

• 9eber, N", 20/0a, A araconsistent 5odel of Bagueness., Mind , //E:

/02=K31"

• 9eber, N", 20/0b, tensionality and 6estriction in aive >et

&heory., tudia !o(ica, E3: FCK/03"



• 9eber, N", 20/2, otes on *nconsistent >et &heory., in I" &anaka, <" +erto,

" 5ares and <" aoli (eds"!, araconsistenc': !o(ic and Applications (above!, pp"

/K21"

&()ections to Dialetheism

• Armour-Darb, +" and G" 9oodbridge, 200=, ?ialetheism, >emantic

athology, and the %pen air., Australasian ournal o# hilosoph', F3: E1K3/="

(An ob#ection to dialetheism based on the notion of pathological sentence"!

• +eall, G) and D" riest, 200C, ot so deep inconsistency: a reply to

klund., Australasian ournal o# !o(ic, 1: C3KF3" (A reply to klund 2002!"

• +erto, <", 200=, 5eaning, 5etaphysics, and )ontradiction., Aerican

hilosophical Buarterl', 3: 2FKEC"

• +erto, <", 20/2, @ow to 6ule %ut &hings with 9ords., in D" 6estall and D"

6ussell (eds"!, $e" 6aves in hilosophical !o(ic, ew Uork: algrave 5acmillan,

20/2, pp" /=EKFE"

• +erto, <", forthcoming, Absolute )ontradiction, ?ialetheism, and

6evenge., *evie" o# 'bolic !o(ic"

• )arrara, 5", >" Daio and " 5artino, 20//, )an riest's ?ialetheism Avoid

&rivialismL., The !o(ica Cearboo+ 2010, ondon: )ollege ublications, pp" 1K

=3" (An argument to the effect that dialetheism may entail trivialism in a

cumbersome way"!

• )arrara, 5", B" 5orato and " 5artino, 20/2, %n ?ialetheic

ntailment., The !o(ica Cearboo+ 2011, ondon: )ollege ublications, pp" CK3F"

(A criti$ue of dialetheism based on entailment in paraconsistent logic"!

• ?enyer, ", /EFE, ?ialetheism and &rivialisation., Mind , EF: 21EK=" (Acriti$ue of a dialetheic account of the paradoes of self-reference"!

• klund, 2002, ?eep *nconsistency., Australasian ournal o# hilosoph',

F0: 2/K/" (Another criti$ue of a dialetheic account of the paradoes of self-

reference"!

• *rvine, A"?", /EE2, Daps, Dluts and arado., &anadian ournal o#

hilosoph', /F (>upplementary Bolume!: 2CKEE" (A criti$ue of a dialetheic

account of the paradoes of self-reference"!



• ittman, D", and I" >immons, 2003, A )riti$ue of ?ialetheism., in riest,

+eall and Armour-Darb (above!, pp" /3K1"

• 5c&aggart, G"5"", /E22, tudies in the 7e(elian Dialectic, 2nd edition,

)ambridge: )ambridge Hniversity ress"

• arsons, &", /EE0, &rue )ontradictions., &anadian ournal o# hilosoph',

20: 1K1" (A criti$ue of a dialetheic account of the paradoes of self-reference"!

• riest, D", /EFE, ?enyer's Z ot +acked by >terling Arguments., Mind , EF:

2=1KF" (A reply to ?enyer, /EFE"!

• riest, D", /EE1, Daps and Dluts: 6eply to arsons., &anadian ournal o#

hilosoph', 21: 1CK==" (A reply to arsons, /EE0"!

• riest, D", /EEFa, 9hat's >o +ad About )ontradictionsL., ournal o#

hilosoph', E1: 3/0K2=" 6eprinted in riest, +eall and Armour-Darb 2003, ch" /"

(A detailed discussion of some modern ob#ections to dialetheism"!

• riest, D", /EEFb, &o +e and ot to +e: &hat *s the Answer" %n Aristotle on

the aw of on-)ontradiction., hilosophie(eschichte und !o(ische Anal'se, /:

E/K/0" 6eprinted as )hapter / of riest 200="

• riest, D", 200, *nconsistent Arithmetic: *ssues &echnical and

hilosophical., in B" <" @endricks and G" 5alinowski (eds"!,Trends in !o(ic,

?ordrecht: Iluwer Academic ublishers, pp" 2CKEE" 6eprinted as )hapter /C of

the 2nd edition of riest /EFC" (A discussion of inconsistent arithmetics, including

a reply to >hapiro, 2002"!

• riest, D", and &" >miley, /EE,)an )ontradictions be &rueL., roceedin(s

o# the Aristotelian ociet', =F (>upplement!: /CK13" (A debate on the issue of

dialetheism"!

• 6estall, D", /EE, ?eviant ogic and the aradoes of >elf-

6eference., hilosophical tudies, C0: 2CEK0" (*ncludes a discussion of Vuinean

ob#ections to non-classical accounts of negation"!

• >hapiro, >", 2002, *ncompleteness and *nconsistency., Mind , ///: F/CK2"

(A criti$ue of the possibility of inconsistent arithmetic"!

• >hapiro, >", 2003, >imple &ruth, )ontradiction and )onsistency., in riest,

+eall and Armour-Darb (above!, pp" =K13"



• Nalta, ", 2003, *n ?efense of the aw of on-)ontradiction., in riest,

+eall and Armour-Darb (above!, pp" 3/=K="

Dialetheism and !ationality

• +eall, G) and 5" )olyvan, 200/, ooking for

)ontradictions., Australasian ournal o# hilosoph', CE: 1=3KE" (%n the spread of

dialetheias in the empirical world!"

• +erto, <", 200F, Ad'naton and 5aterial clusion., Australasian ournal

o# hilosoph', F=: /=1KE0"

• +remer, 5", 200F, 9hy and @ow to +e a ?ialetheist., tudia hilosophica

stonica, /: 20FK2C (A discussion of the conditions on the rational believability ofdialetheism!"

• ?utilh ovaes, )", 200F, )ontradiction: the 6eal )hallenge for

araconsistent ogic., in *n G"U" +[Jiau, 9" )arnielli, and ?" Dabbay

(eds"!, 7andboo+ o# araconsistenc', ondon: )ollege ublications" (A

specification of the conditions for a non-$uestion-begging debate between

dialetheists and supporters of the )"!

• @ume, ?avid, /C3F, An .nuir' &oncernin( 7uan nderstandin( , )"9"

@endel (ed"!, *ndianapolis: +obbs-5erril )ompany *nc", /E11"

• riest, D", 2000a, )ould verything +e &rueL., Australasian ournal o#

hilosoph', CF: /FEKE1" 6eprinted as )hapter of riest 200="

• riest, D", 200=, Doubt Truth to -e a !iar , %ford: %ford Hniversity ress"

• &anaka, I", 2001, &he AD5 &heory and *nconsistent +elief

)hange., !o(iue et Anal'se, /FE: //K10" (A dialetheic approach to the logic of

belief revision"!

Themes for 0#rther !esearch$ Dialetheism !ealism and "ntirealism

• +eall, G), 2000, %n &ruthmakers for egative &ruths., Australasian

ournal o# hilosoph', CF: 2=3KF" (A discussion of the connections between

dialetheism, correspondence theory, and negative facts"!

• +eall, G), 2003, &rue and <alse K As *f., in riest, +eall and Armour-Darb

(eds"! 2003, /ECK2/="



• +erto, <", 200Cb, *s ?ialetheism an *dealismL., Dialectica, =/: 21K="

• Drim, ", 2003, 9hat is a contradiction., in riest, +eall and Armour-Darb

(eds"! 2003, 3EKC2"

• Iroon, <", 2003, 6ealism and ?ialetheism., in riest, +eall and Armour-

Darb (eds"! 2003, 231K="

• 5ares, ", 2003, >emantic ?ialetheism., in riest, +eall and Armour-Darb

(eds"! 2003, 2=3KC1"

• riest, D", 2000b, &ruth and )ontradiction., hilosophical Buarterl', 10:

01K/E" 6eprinted as )hapter 2 of riest 200="

• &ahko, &", 200E, &he aw of on-)ontradiction as a 5etaphysical

rinciple., Australasian ournal o# !o(ic, C,available online" (A defense of the

aw of on-)ontradiction as a metaphysical K as opposed to logical or semantic K

principle!"

• 9oodbridge, 6" and +" Armour-Darb, forthcoming, >emantic

?efectiveness and the iar., hilosophical tudies"

"cademic Tools

@ow to cite this entry"

review the ?< version of this entry at the <riends of the > >ociety"

ook up this entry topic at the *ndiana hilosophy %ntology ro#ect (*nh%!"

nhanced bibliography for this entry at hilapers, with links to its database"

&ther nternet !eso#rces

• ?ialetheism, entry in 9ikipedia"

!elated +ntries

Aristotle, Deneral &opics: metaphysics \ Aristotle, >pecial &opics: on non-

contradiction \ contradiction \ @egel, Deorg 9ilhelm <riedrich \ impossible

worlds \ liar parado \ logic: paraconsistent \ logic: substructural \ mathematics:

inconsistent \ paradoes: and contemporary logic \ 6ussell's parado \ self-

reference \ truth: correspondence theory of \ truth: deflationary theory of

"c4no5led*ments

http://philosophy.unimelb.edu.au/ajl/2009/2009_4.pdf

http://plato.stanford.edu/cgi-bin/encyclopedia/archinfo.cgi?entry=dialetheism

https://leibniz.stanford.edu/friends/preview/dialetheism/


https://leibniz.stanford.edu/friends/


https://inpho.cogs.indiana.edu/entity?sep=dialetheism&redirect=True


https://inpho.cogs.indiana.edu/


http://philpapers.org/sep/dialetheism/

http://philpapers.org/

http://en.wikipedia.org/wiki/Dialetheism


http://plato.stanford.edu/archives/fall2013/entries/aristotle-metaphysics/

http://plato.stanford.edu/archives/fall2013/entries/aristotle-noncontradiction/



http://plato.stanford.edu/archives/fall2013/entries/contradiction/

http://plato.stanford.edu/archives/fall2013/entries/hegel/








http://plato.stanford.edu/archives/fall2013/entries/paradoxes-contemporary-logic/








http://philosophy.unimelb.edu.au/ajl/2009/2009_4.pdf

http://plato.stanford.edu/cgi-bin/encyclopedia/archinfo.cgi?entry=dialetheism





http://philpapers.org/sep/dialetheism/

http://philpapers.org/


http://plato.stanford.edu/archives/fall2013/entries/aristotle-metaphysics/



http://plato.stanford.edu/archives/fall2013/entries/contradiction/

http://plato.stanford.edu/archives/fall2013/entries/hegel/
















&he authors would like to thank G) +eall, 5a )arrara, and two anonymous

referees, for providing helpful comments and suggestions"

Copyright © 2013 by

Graham PriestFrancesco Berto <[email protected] >

http://plato.stanford.edu/archives/fall2013/info.html#c

http://www.abdn.ac.uk/sdhp/philosophy/profiles/f.berto

mailto:[email protected]

http://plato.stanford.edu/archives/fall2013/info.html#c

http://www.abdn.ac.uk/sdhp/philosophy/profiles/f.berto

mailto:[email protected]

dialetheism

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