Dialetheism And Its Applications

186 Pages·2019·2.055 MB·English

by  Adam Rieger, Gareth Young

Trends in Logic 52
Adam Rieger
Gareth Young Editors
Dialetheism
and its
Applications
Trends in Logic
Volume 52
TRENDS IN LOGIC
Studia Logica Library
VOLUME 52
Editor-in-Chief
HeinrichWansing,DepartmentofPhilosophy,RuhrUniversityBochum,
Bochum,Germany
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ArnonAvron,DepartmentofComputerScience,UniversityofTelAviv,TelAviv,Israel
KatalinBimbó,DepartmentofPhilosophy,UniversityofAlberta,Edmonton,AB,Canada
GiovannaCorsi,DepartmentofPhilosophy,UniversityofBologna,Bologna,Italy
JanuszCzelakowski,InstituteofMathematicsandInformatics,UniversityofOpole,
Opole,Poland
RobertoGiuntini,DepartmentofPhilosophy,UniversityofCagliari,Cagliari,Italy
RajeevGoré,AustralianNationalUniversity,Canberra,ACT,Australia
AndreasHerzig,IRIT,UniversityofToulouse,Toulouse,France
WesleyHolliday,UCBerkeley,Lafayette,CA,USA
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DanieleMundici,MathematicsandComputerScience,UniversityofFlorence,Firenze,Italy
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PeterSchroeder-Heister,Wilhelm-Schickard-Institut,UniversitätTübingen,Tübingen,
Baden-Württemberg,Germany
YdeVenema,ILLC,UniversiteitvanAmsterdam,
Amsterdam,Noord-Holland,TheNetherlands
AndreasWeiermann,VakgroepZuivereWiskundeenComputeralgebra,UniversityofGhent,
Ghent,Belgium
FrankWolter,DepartmentofComputing,UniversityofLiverpool,Liverpool,UK
MingXu,DepartmentofPhilosophy,WuhanUniversity,Wuhan,China
JacekMalinowski,InstituteofPhilosophyandSociology,PolishAcademyofSciences,
Warszawa,Poland
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RyszardWojcicki,InstituteofPhilosophyandSociology,PolishAcademyofSciences,
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Adam Rieger Gareth Young
(cid:129)
Editors
Dialetheism and its
Applications
123
Editors
AdamRieger Gareth Young
University of Glasgow University of Glasgow
Glasgow, UK Glasgow, UK
ISSN 1572-6126 ISSN 2212-7313 (electronic)
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Introduction
Dialetheismistheviewthattherearetruecontradictions.Formostofthehistoryof
Westernphilosophyatleast,ithasbeenregardedasabsurd,thoughasmallnumber
offigures, such as Heraclitus and Hegel, have dissented. Logical developments in
the twentieth century, traceable back at least to a 1954 Argentinian PhD thesis,1
have made possible a more systematic investigation of the position, which has
slowly built into a significant level of activity in recent decades. Undoubtedly, the
person who deserves the largest share of the credit (or blame) for this is Graham
Priest; starting with the 1979 paper ‘The Logic of Paradox’,2 through the com-
prehensive In Contradiction3 and numerous other publications, Priest has inde-
fatigablynurturedthetopicfromthe‘lunaticfringe’intowhatisnowaflourishing
researcharea.ThecurrentvolumearosefromaconferenceheldattheUniversityof
Glasgow in December 2012 to celebrate 25 years since the publication of the first
edition of In Contradiction. Some of the papers are versions or descendants of
material presented there; others are invited contributions from scholars who were
unable to attend the conference.
Whilstearlyworkondialetheismtendedtofocusonwhetheritisevencoherent,
and its application to the semantic paradoxes, increasingly dialetheism is finding
applications in wider areas of philosophy. The potential here, indeed, seems very
broad: in a typical philosophical dispute, we may have an apparently good argu-
ment for P and another one for not-P. The usual approach isto try to locate where
one argument is at fault, perhaps at the cost of giving up something highly plau-
sible,butdialetheismholdsouttheprospectofmaintainingthatboththearguments
arenotjustapparently,butactually,sound.Ofcourse,itisnotaseasyasthis.There
is a cost to a dialetheist solution as well, and every case must be judged on its
merits. By no means every author in this collection is a supporter of dialetheism;
1Asenjo,F.G.(1954).Laideadeuncalculodeantinomias.SeminarioMathemático,Universidad
deLaPlata.
2JournalofPhilosophicalLogic,8(1),219–241.
3Dordrecht: Martinus Nijhoff, 1987. 2nd expanded edition, Oxford: Oxford University Press,
2006.
v
vi Introduction
buttheyhaveincommonthattheythinkitisworthconsideringandengagingwith;
where it is rejected, the rejection must be argued for.
The first two papers in the volume are concerned with dialetheism’s most
familiarapplicationsofar,insolvinglogicalparadoxes.Curryparadoxes—suchas
‘If this sentence is true then the moon is made of green cheese’—are awkward for
the dialetheist. The classic set-theoretic and semantic paradoxes arise from appar-
ently sound reasoning; dialetheism has the attractive feature of endorsing this
reasoning, whilst avoiding trivialism by adopting a paraconsistent logic, which
abandonsexcontradictionequodlibet,aprincipletheintuitivevalidityofwhichis,
in any case, debatable. In contrast, the Curry paradox trivialises directly, without
passing through a contradiction (we can clearly replace the consequent of a Curry
conditional by any sentence we choose). The paradox relies only on simple and
apparently compelling principles of reasoning involving conditionals, and the
dialetheist has to pay the price of giving at least one of these up.
In their paper ‘Liars with Curry: dialetheism and the prospects of a uniform
solution’, Ben Burgis and Otávio Bueno consider in particular the implications of
Curry paradoxes for another claimed advantage of dialetheism, its ability (unlike
conventional solutions of the paradoxes) to obey the ‘Principle of Uniform
Solution’—the idea that paradoxes of similar structure should be given similar
solutions. Burgis and Bueno, however, argue that the Curry paradox casts severe
doubt on whether dialetheism really has this claimed advantage; whichever of a
number of options the dialetheist chooses, the cost of preserving the principle will
be prohibitively high.
A further advantage that has been claimed on behalf of a dialetheic solution to
theparadoxesisimmunitytoso-calledrevengeproblems.Theseconcernacommon
phenomenon, whereby an attempt to classify a liar sentence into some semantic
category merely results in a renewed paradox involving that category. In the sim-
plest case, attempting to classify ‘This sentence is false’ as, say, defective, where
defective sentences are neither true nor false, runs into problems with ‘This sen-
tence is either false or defective’. Some vital semantic notion must remain inex-
pressible in the object language or inconsistency results.
Byfacingcontradictionhead-on,dialetheismhasbeenclaimedtoavoidrevenge
problems and allow for genuinely semantically closed languages. But Gareth
Young, in his paper ‘A revenge problem for dialetheism’, argues that this claim is
ungrounded. Taking the dialetheism of Priest as the most promising revenge-free
solution, Young argues that dialetheists of this sort must accept as contradictory,
notonlytruth,butvalidity:inotherwords,someinferenceprinciplesarebothvalid
andinvalid.Hearguesthat,dependingonthedetailsofthetheory,all,oralmostall,
inference principles can be shown to be dialetheically invalid. This, he claims,
makes the notion of invalidity inexpressible for the dialetheist and deprives them
oftheabilitytoexpresscrucialsemanticclaimsabouttheirtheory.Hence,theytoo
fall foul of a revenge problem.
Thetwofollowingpapersdiscusstheprospectsforapplyingdialetheisminnew
areas,somewhatdistantfromitstraditional huntingground.Inhispaper‘Itwasso
revoltingIcouldn’ttakemyeyesoffit’,GrahamPriestintroduceswhathearguesis
Introduction vii
a new form of dialetheia. Before doing so, he briefly surveys the main areas in
which dialetheias have previously been thought to occur. Because the view is so
closelyassociated with theparadoxes, it isinteresting that Priest writes that he has
‘neverthoughtoftheparadoxesasthemosttransparentexample’.Rather,hethinks
cases of conflicting laws provide this, say when one law stipulates of a person,
Emily, that she is permitted to vote, and another law that she is forbidden.
Priest’s new examples concern psychological dialetheias, in particular cases
whereoneissimultaneouslyattractedandrepelledbysomething.Priestarguesthat
insuchcasesoneisbothattractedandnotattracted;this,heclaims,isnotsimplya
matter of having contradictory attitudes (as in both wanting and wanting not to do
something),butagenuinedialetheia.Hedefendsthisclaimagainstvariousattempts
to defuse the contradiction.
Sebastiano Moruzzi and Annalisa Coliva’s paper ‘Dialetheism in action: a new
strategyforsolvingtheequalvalidityparadox?’explorestheprospectsforgivinga
dialetheist account of cases of ‘faultless disagreement’, such as when one speaker
asserts‘gingeristasty’andanother‘gingerisnottasty’.Standardexamplessuchas
this concern judgments of taste, though, depending on one’s views, the phe-
nomenonmayoccurmorewidely,forexample,inaestheticsorethics.Thepuzzleis
that, whilst intuitively such cases seem possible (or so many have thought), it is
hard to analyse precisely what is going on in them whilst simultaneously vindi-
cating both that they are cases of genuine disagreement and that the parties are
faultless.
After exploring various alternative solutions to their version of the paradox,
MoruzziandColivainvestigatethepossibilityofadialetheicsolution,inwhichthe
law of non-contradiction is abandoned. The dialetheist, they argue, has some dif-
ficulty in explaining why these cases involve genuine disagreement, since in a
dialetheic framework the two parties could, it seems, each admitsthe truth of their
opponent’sstatementaswellastheirown.Afterconsideringdialetheicresponsesto
this problem, they investigate the relationship between the dialetheic solution and
trivialism(theviewthateverysentenceisbothtrueandfalse).Thisisathreatinthis
context, since, in contrast to the usual semantic and set-theoretic paradoxes, the
dialetheiasmaybeuniversal,atleastinthosedomainswherefaultlessdisagreement
can arise.
Thefollowingthreepapersallexplore,indifferentways,howdialetheismcanbe
helpful in reasoning about impossibilities, even if one is not convinced that these
occur in the actual world. Ira Kiourti’s paper ‘An excess of dialetheias: in defence
ofgenuineimpossibleworlds’examinesthequestionofwhetherimpossibleworlds
can be reconciled with a Lewis-style modal realism. There are some attractions to
admitting impossible worlds, for example, to give plausible semantics to counter-
factualswithimpossibleantecedents(theseallcomeouttriviallytrueonastandard
Lewis–Stalnaker theory). But Lewis himself thought impossible worlds should be
rejected. Kiourti proposes two ways of resisting Lewis’s conclusion. The more
moderate is to reject Lewis’s principle governing negation, that for A to be true at
world w is simply for A to fail to be true at w. But this principle, Kiourti argues,
begs the question against impossible worlds and can be rejected whilst retaining
viii Introduction
classical logic.Alternatively, andmore radically, onecanretain Lewis’ssemantics
for negation but abandon classical logic and embrace dialetheias. Kiourti defends
this approach against the charge that it trivialises, by making everything true at
every world.
Lewis’sscepticismaboutreasoningwithimpossibilityisalsoastartingpointfor
Frederick Kroon’s paper, ‘Game of Truth: truth, fictionalism, and semantic para-
dox’. Suppose the moral that one draws from the semantic paradoxes is that our
naive concept of truth is inconsistent, as has been a popular (though by no means
universal) reaction. One option is to accept the inconsistencies and become a
dialetheist.Anotheristoreplacethenaiveconceptbyanewone,salvagingasmuch
as possible from the naive conception whilst avoiding inconsistency. That might
seem to exhaust the possible responses, but Kroon’s paper is an exploration of a
third option: fictionalism about truth. The paradoxes show that there is really no
property of truth; nevertheless, argues Kroon, it is extremely useful to pretend that
there is one. Just as on Field’s view of mathematics, or Joyce’s view of morality,
there are no such things as numbers or the property of goodness, but it is a con-
venientpracticetobehaveasthoughthereare,Kroonthinkstheexistenceoftruthis
a useful fiction. Because of the paradoxes, the fiction, in this case, will be incon-
sistent—like Priest’s story of Sylvan’s box which is both empty and not-empty.4
The logic of the fiction will have to be paraconsistent, to avoid triviality. But the
contradictions will not, as Kroon puts it, ‘leach out into the class of ordinary
statements about reality’.
Chris Mortensen’s short paper, ‘Dialetheism and impossible figures’, has some
ideasincommonwithKroon’s.Themoralweshoulddrawfromimpossiblefigures
such as the Penrose triangle, he suggests, is that we can imagine contradictory
situations and reason about them; such reasoning is non-trivial, and so requires
paraconsistentlogic.(MortensenalsomentionsPriest’sstoryofSylvan’sboxinthis
context, as achieving a similar effect by literary, rather than visual, means.) Since
the objects depicted need not exist, the argument yields only weak rather than
strongparaconsistency—thatis,itdoesnotestablishthattherearedialetheiasinthe
world.
The finaltwopapers aremore technical innature.Amongst solutionstotheliar
paradox which preserve the naive T-rules in full but change the logic in order to
avoid catastrophe, a natural desideratum is classical recapture: the principle that
discourse not involving the problematic semantic notions such as truth should
continue to be governed by classical logic. Andrew Tedder and Stewart Shapiro’s
paper ‘Making truth safe for intuitionists’ considers the prospects of this kind of
recapture if we start with intuitionist logic. The usual paradoxical reasoning is
intuitionistically valid, and intuitionists accept ex contradictione quodlibet, so the
paradoxes are as potentially troubling for them as they are for a classicist. Tedder
andShapiroconsidertheprospectsfor‘intuitionistrecapture’foravarietyoflogics,
somewithadialetheicinterpretationandsomenot.Aswellasdemonstratingsome
4Priest,G.(1997).Sylvan’sBox:AShortStoryandTenMorals.NotreDameJournalofFormal
Logic,38(4),573–582.
Introduction ix
results, they sketch out a terrain which promises to be fertile ground for future
work.
Inconsistent mathematics looks likely to be a growth area in the coming years.
A natural idea is to preserve naive set theory, preventing triviality by adopting a
paraconsistent logic. But which logic? It must be weak enough to avoid paradox
(including Curry), but strong enough to be usable in proving mathematical theo-
rems. Guillermo Badia and Zach Weber’s paper ‘A substructural logic for incon-
sistent mathematics’ presents a logic designed for this purpose. The various
desiderataproveimpossibletosatisfywithastandardsetofconnectives;theirlogic,
therefore, has two conditionals and two negations. As an illustration of what the
logic can do, they argue that it justifies induction in naive set theory. As with the
Shapiro and Tedder paper, and indeed the collection as a whole, one is left
impressednotonlywithwhathasalreadybeenachieved,buthowmuchinteresting
work remains to be done.
Adam Rieger
Gareth Young