Paraconsistent logic consistency, contradiction and negation

1.2 the situation could be described by the pictorialequation:contradictorinessþ consistency ¼ trivialityThe Logics of Formal Inconsistency, from now on LFIs, introduced in [2] andadditi

Logic, Epistemology, and the Unity of Science 40

Walter Carnielli

Marcelo Esteban Coniglio

Paraconsistent Logic:

Consistency,

Contradiction and Negation

Volume 40

Series editors

Shahid Rahman, University of Lille III, France

John Symons, University of Texas at El Paso, USA

Editorial Board

Jean Paul van Bendegem, Free University of Brussels, Belgium

Johan van Benthem, University of Amsterdam, The Netherlands

Jacques Dubucs, CNRS/Paris IV, France

Anne Fagot-Largeault, Collège de France, France

Göran Sundholm, Universiteit Leiden, The Netherlands

Bas van Fraassen, Princeton University, USA

Dov Gabbay, King’s College London, UK

Jaakko Hintikka, Boston University, USA

Karel Lambert, University of California, Irvine, USA

Graham Priest, University of Melbourne, Australia

Gabriel Sandu, University of Helsinki, Finland

Heinrich Wansing, Ruhr-University Bochum, Germany

Timothy Williamson, Oxford University, UK

of the unity of science in light of recent developments in logic At present, no singlelogical, semantical or methodological framework dominates the philosophy ofscience However, the editors of this series believe that formal techniques like, forexample, independence friendly logic, dialogical logics, multimodal logics, gametheoretic semantics and linear logics, have the potential to cast new light on basicissues in the discussion of the unity of science.

This series provides a venue where philosophers and logicians can apply specifictechnical insights to fundamental philosophical problems While the series is open

to a wide variety of perspectives, including the study and analysis of argumentationand the critical discussion of the relationship between logic and the philosophy ofscience, the aim is to provide an integrated picture of the scientific enterprise in allits diversity

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Department of Philosophy and Centre

for Logic, Epistemology and the History

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Campinas, São Paulo

Brazil

Department of Philosophy and Centrefor Logic, Epistemology and the History

of Science (CLE)University of Campinas (UNICAMP)Campinas, São Paulo

Brazil

ISSN 2214-9775 ISSN 2214-9783 (electronic)

Logic, Epistemology, and the Unity of Science

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memoriam To our children: B á, Juju, Matheus, Paolo, Gabriela, Vittorio … and to the kids we would have had, and to Juli and Tati Sine qua non.

Campinas, February 29, 2016

I protest against the use of in finite magnitude as something completed, which in matics is never permissible In finity is merely a façon de parler, the real meaning being a limit which certain ratios approach inde finitely near, while others are permitted to increase without restriction.

mathe-C.F Gauss, Brief an Schumacher (1831); Werke 8, 216 (1831)

In a letter to the astronomer H.C Schumacher in 1831, Gauss was rebukingmathematicians for their use of the infinite as a number, and even for their use of thesymbol for the infinite It would be difficult to sustain that kind of finitism,regardless of any epistemological considerations: a good part of mathematicssimply cannot survive with only the potential infinite

The reaction against the infinite, as well as against complex or imaginarynumbers, and against negative numbers before, are interesting examples of thedifficulties faced, even by great minds, in accepting certain abstractions Aristotle inChaps.4–8of Book III of Physics argued against the actual infinite, advocating forthe potential infinite His idea was that natural numbers could never be conceived as

a whole

Euclid in a certain sense never proved that there exist infinitely many primenumbers What was actually stated in Proposition 20 of Book IX, carefully avoidingthe term infinite, was that “prime numbers are more than any previously thought(total) number of primes”, which agrees with his tradition

It was only in the nineteenth century that G Cantor dispelled all those acceptedviews by showing that an infinite set can be treated as a totality, as a full-fledgedmathematical object with honorable properties, no less than the natural numbers.Imaginary numbers were introduced to mathematics in the sixteenth century(through Girolamo Cardano, though others had already used them in differentguises) These numbers caused an embarrassment among mathematicians for cen-turies, since they faced astonishing difficulties in accepting an extension of theconcept of number, especially in light of the problem of computing the square root

of −1 Only after the fundamental works of L Euler and Gauss did the complex

vii

numbers rid themselves of the label“imaginary” given them by Descartes in 1637,and even then not without difficulty.

The mathematics of the infinite and of complex numbers, and all they represent

in contemporary science, are triumphant cases of amplified concepts, but are not theonly ones A notable case of expansion of concepts, with deep implications for thedevelopment of contemporary logic, can be traced back to Frege and his famousarticle of 1891, Funktion und Begriff (see [1])

In this seminal paper, Frege recalls how the meaning of the term‘function’ haschanged in the history of mathematics, and how the mathematical operations used

to define functions have been extended by, as he says, ‘the progress of science’:basically, through passages (or transitions) to the limit, as in the process of defining

a new function y0¼ f0ðxÞ from a function y ¼ f ðxÞ (provided that the limitsinvolved in the calculus exist), and through accepting complex numbers in domainsand images of functions

Starting from this point, Frege goes further into adding expressions that now wecall predicates, such as‘=’, ‘<’ and ‘>’ Leaving aside his philosophical motivationsfor seeing arithmetic as a“further development of logic”, what Frege started was areal revolution, that made possible the development of quantifiers and anunprecedented unification of propositional and predicate logic into a far morepowerful system than any that preceded it

Not only could the truth-values, True and False, be taken as outputs of afunction, but any object whatsoever could be similarly treated To rephrase anexample from Frege himself, if we suppose‘the capital of x’ expresses a function,

of which‘the German Empire’ is the argument, Berlin is returned as the value of thefunction In this way, Frege’s system could represent non-mathematical thoughtsand predications, and founded the basis of the modern predicate calculus.Frege’s idea of defining an independent notion of ‘concept’ as a function whichmaps every argument to one of the truth-values True or False was instrumental inthe development of a strict understanding of the notions of‘proof’, ‘derivation’, and

‘semantics’ as parts of the same logic mechanism Regarding ‘concept’ as a wideand independent notion based on an amplification of the idea of function was anessential step for Frege’s fundamental break between the older Aristotelian traditionand the contemporary approach to logic

Paraconsistency is the study of logical systems in which the presence of acontradiction does not imply triviality, that is, logical systems with a non-explosivenegation: such that a pair of propositions A and :A does not (always) trivialize thesystem However, it is not only the syntactic and semantic properties of thesesystems that are worth studying Some questions arise that are perennial philo-sophical problems The question of the nature of the contradictions allowed byparaconsistent logics has been a focus of debate on the philosophical significance

of paraconsistency Although this book is primarily focused on thelogico-mathematical development of paraconsistency, the technical resultsemphasized here aim to help, and hopefully to guide, the study of some of thosephilosophical problems

Paraconsistent logics are able to deal with contradictory scenarios, avoidingtriviality by means of the rejection of the Principle of Explosion, in the sense thatthese theories do not trivialize in the presence of (at least some) contradictorysentences Different from traditional logic, in paraconsistent logics triviality is notnecessarily connected to contradictoriness; in intuitive terms (a more formalaccount in given in Sect 1.2) the situation could be described by the pictorialequation:

contradictorinessþ consistency ¼ trivialityThe Logics of Formal Inconsistency, from now on LFIs, introduced in [2] andadditionally developed in [3], are a family of paraconsistent logics that encom-passes a great number of paraconsistent systems, including the majority of systemsdeveloped within the Brazilian tradition An important characteristic of LFIs is thatthey are endowed with linguistic resources that permit to express the notion ofconsistency of sentences inside the object language by using a sentential unaryconnective referred to as ‘circle’: A meaning A is consistent Explosion in thepresence of contradictions does not hold in LFIs, as much as in any other para-consistent logic But LFIs are so designed that some contradictions will causedeductive explosion: consistent contradictions lead to triviality–intuitively, one canunderstand the notion of a‘consistent contradiction’ as a contradiction involvingwell-established facts, or involving propositions that have conclusive favorableevidence In this sense, LFIs are logics that permit one to separate the sentences forwhich explosion hold, from those for which explosion does not hold It is notdifficult to see that, in this way, reasoning under LFIs extend and expand thereasoning under classical logic: although LFIs are technically subsystems ofclassical logic, classical logic can be identified with that portion of LFIs that dealswith‘consistent contradictions’ Therefore LFIs subsume classical reasoning Thispoint will be explained in more detail in Sect.1.2

We may say that afirst step in paraconsistency is the distinction between iality and contradictoriness But there is a second step, namely, the distinctionbetween consistency and non-contradictoriness In LFIs the consistency connective

triv- is not only primitive, but it is also not necessarily equivalent to non-contradiction.This is the most distinguishing feature of the logics of formal inconsistency.Once we break up the equivalence betweenA and :ðA ^ :AÞ, some quite inter-esting developments become available Indeed, A may express notions of con-sistency independent from freedom from contradiction

The most important conceptual distinction between LFIs and traditional logic isthat LFIs start from the principle that assertions about the world can be divided intotwo categories: consistent sentences and non-consistent sentences Consistentpropositions are subjected to classical logic, and consequently a theory T thatcontains a pair of contradictory sentences A; :A explodes only if A is taken to be aconsistent sentence, linguistically marked as A (or :A) This is the only dis-tinction between LFIs and classical logic, albeit with far-reaching consequences:

classical logic in this form is augmented, in such a way that in most cases an LFIencodes classical logic.

The concept of LFIs generalizes and extends the famous hierarchy of C-systemsintroduced in [4] and popularized by hundreds of papers At the same time, LFIsexpand the classical logical stance, and consequently the majority of the traditionalconcepts and methods of classical logic, propositional or quantified (and evenhigher-order), can be adapted, with careful attention to detail

Since, as much as intuitionistic logic, LFIs are more of an epistemic nature,rather than of an ontological, there is no point in advocating the replacement ofclassical logic with paraconsistent logic Because LFIs extend the classical stance,the analogy with transfinite ordinal numbers and with complex numbers is com-pelling: in such cases, there is no rejection of what has come before, but a

refinement of it

It is not infrequent that an argument as of the skeptics, such as that given bySextus Empiricus1against the sophists, is trumpeted against the need of paracon-sistent logic, in science or reasoning in general:

[If an argument] leads to what is inadmissible, it is not we that ought to yield hasty assent to the absurdity because of its plausibility, but it is they that ought to abstain from the argument which constrains them to assent to absurdities, if they really choose to seek truth,

as they profess, rather than drivel like children Thus, suppose there were a road leading up

to a chasm, we do not push ourselves into the chasm just because there is a road leading to

it but we avoid the road because of the chasm; so, in the same way, if there should be an argument which leads us to a confessedly absurd conclusion, we shall not assent to the absurdity just because of the argument but avoid the argument because of the absurdity So whenever such an argument is propounded to us we shall suspend judgement regarding each premiss, and when finally the whole argument is propounded we shall draw what conclusions we approve.

This argument, however, if it is not against the use of any logic, is indeedfavorable to the kind of paraconsistency represented by LFIs The notion of con-sistency—symbolized as  when applied to propositions—actually increases ourwisdom: it does not stop one to jump into the chasm, but rather marks out thedangerous roads and, precisely, helps avoid such roads because of the chasm!The idea that consistency can be taken as a primitive, independent notion, and beaxiomatized for the good profit of logic is a new idea, which permits one to separatenot only the notion of contradiction from the notion of deductive triviality, which istrue of all paraconsistent logics, but also the notion of inconsistency from the notion

of contradiction—as well as consistency from non-contradiction This refined idea

of consistency has great potential, as we shall see in detail in this book, as ticipated as the possibilities that imaginary numbers, completed infinite, and Frege’sidealization of a ‘concept’ as a function mapping arguments to one of thetruth-values represented in mathematics, logic and philosophy The rest of the bookwill speak for itself

unan-1 Sextus Empiricus, Outlines of Pyrrhonism, LCL 273: 318 –319 http://www.loebclassics.com/ view/sextus_empiricus-outlines_pyrrhonism/1933/pb_LCL273.3.xml

Book ’s Content: A Road Map

Section1.3will briefly retrace the motivations for the forerunners of LFIs andparaconsistency in general No discussion of paraconsistency can avoid touching

on, if only summarily, questions of the nature of logic, and Sect.1.4does this Nextchallenges to be faced are questions about the nature of contradictions Section1.5takes up this thorny philosophical topic from the times of the ancient Greece,cursorily discussing some remarks from Aristotle concerning three alleged versions

of the Principle of Non-Contradiction that correspond to the three traditionalaspects of logic, namely, ontological, epistemological, and linguistic

This stance helps to give a justification for the rational acceptance of dictory sentences, and to better appreciate the distinctions among contradiction,consistency, and negation, as characterized in Sect.1.6 It will also help to makepalatable the rationale behind the semantics of LFIs to be developed in all math-ematical details in Chaps.2and3, as well as to give support to alternative semanticsfor LFIs developed in Chap.6

contra-There is a wide variety of reasons for repudiating (or at least to be cautionedagainst) classical logic, and many of themfind an expression among paraconsistentlogics This chapter makes clear that LFIs are not coincidental with this spectrum ofphilosophical views, neither are they antagonistic, but can be combined with, andcan complement, some of them A summary of the main varieties of paraconsis-tency is given in Sect 1.7, which attempts to clarify the position of LFIs withrespect to other paraconsistent logics in the hope that this will justify some claimsmade in next chapters

Chapter2

Chapter2 offers a careful survey of the basic logic of formal inconsistency, mbC:

it is basic in the sense that, starting with positive classical logic CPL+and adding anegation and a consistency operator, it is endowed with minimal properties in order

to satisfy the definition of LFIs The chapter also lays out the main notation,ongoing definitions and main ideas that will be used throughout the book Positiveclassical logic is assumed as a natural starting point from which the LFIs will be

defined, although in Chap.5some LFIs will be studied starting from other logicsthan CPL+ A non-truth-functional valuation semantics for mbC is defined in Sect.2.2, and its meaning and consequences explored in Sect 2.3

A remarkable feature of LFIs in general, and of mbC in particular, as mentionedabove, is that classical logic (CPL) can be codified, or recovered, inside suchlogics, as shown, for instance, in Sect.2.4.

One of the criteria proposed by da Costa in [5], p 498, is that a paraconsistentcalculus must contain as many of the schemata and rules of classical logic as can beendorsed without validating of the laws of explosion and non-contradiction Thisvague criterion can be formalized in the sense that some LFIs can be proved to bemaximal with respect to CPL, as in the case, for instance, of some three-valuedLFIs treated in Chap.4

Moreover, in addition to being a subsystem of CPL, mbC is also an extension ofCPL, obtained by adding to the latter a consistency operator and a paraconsistentnegation: (see Sect.2.5) In this sense, mbC can be viewed, both, as a subsystemand as a conservative extension of CPL A similar phenomenon holds for severalother LFIs

That section also sheds light on how CPL can be codified in mbC, showing thatthis can be achieved by way of a conservative translation, or by establishing aDerivability Adjustment Theorem (or DAT) between CPL and mbC Section 2.5also discusses an alternative formulation for mbC called mbC?, showing that bymeans of linguistic adaptations mbC can be directly introduced as an extension ofCPL

Chapter3

Chapter3deals with extensions of mbC, which by its turn is a minimal extension

of CPL+ with a consistency operator  and a paraconsistent negation : terizable as an LFI This chapter defines several extensions of mbC, strengthening

charac-or expanding different characteristics of this basic system

In mbC, however, negation and consistency are totally separated concepts Thefirst extension of mbC, called mbCciw, is defined as the minimal extensionguaranteeing that the truth-values ofα and :α completely determine the truth-value

ofα

Besides being a subsystem of classical logic, mbC is strong enough to containthe germ of classical negation, possessing a kind of hidden classical negation, asexplained in Sect 2.4of Chap 2 Section 3.2 of this chapter shows that here isanother hidden operator in mbC: an alternative consistency operator β, one foreach formula β This operator establishes an important distinction, from a con-ceptual point of view, between mbC and mbC? as clarified in Sect.3.4

When he introduced his famous hierarchy Cn (1 n\ωÞ of paraconsistentsystems, da Costa defined, for each system Cn, a kind of “well-behavedness”operator (later identified with consistency) in terms of the paraconsistent negationand conjunction (see Sect 3.7) A special type of LFIs called dC-systems, char-acterized by the fact that the consistency operator can be defined in terms of theothers, has been defined in [2] The systems Cnof da Costa turn out to be examples

of dC-systems Section 3.3 of this chapter analyzes the formal notion ofdC-systems, and investigates how to expand mbC in order to define the consistency

and/or the inconsistency operator in terms of the other connectives of the givensignature.

In general terms, LFIs are concerned with the notion of consistency, expressed

by the operator The notion of inconsistency of α is usually defined via the newoperator :α, expressing the (formal) inconsistency of α Section3.5studies thebalance (or better, unbalance) between the formal concepts of consistency andinconsistency, defining a new LFI (mbC, which, in fact, is a dC-system) whereinconsistency is a primitive notion and consistency is a defined one

A natural requirement when characterizing consistency, as much as negation, ishow consistency can be propagated through the remaining connectives Sections3.6 and 3.8 analyze extensions of mbC enjoying propagation of consistency indifferent forms, in the spirit of the historical systems of da Costa

Chapter4

Chapter 4 deals with matrices and algebraizability, and their consequences Inparticular, the question of characterizability by finite matrices, as well as thealgebraizability of (extensions of) mbC is tackled Some negative results, in thestyle of the famous Dugundji’s theorem for modal logics, are shown for severalextensions of mbC This results in new, compact proofs of previously establishedresults, to the effect that a wide variety of LFIs extending mbC cannot besemantically characterized by finite matrices Despite these general results, somethree-valued extensions of LFIs can be characterized byfinite matrices, and most

of them are algebraizable in the well-known sense of Blok and Pigozzi This issurprising, considering that several extensions of mbC, including the systems Cnof

da Costa, cannot be algebraizable in Blok and Pigozzi’s sense (and consequently,not in Lindenbaum and Tarski’s sense)

On the topic of LFIs that can be defined matricially, the chapter also coversHalldén’s logic of nonsense as well as Segerberg’s variation, da Costa and

D’Ottaviano’s, logic J3, also known in its variants LFI1 and MPT, Sette’s logicP1, Priest’s logic LP, the system Ciore, and several other related systems.Chapter5

Chapter5is devoted to giving an account of LFIs based on other logics, distinctfrom what was done in previous chapters, in which LFIs based exclusively onpositive classical logic CPL+ were studied Although several extensions of thebasic system mbC have been proposed, including several three-valued logics (some

of them even algebraizable in the sense of Blok and Pigozzi, which is not possible

in the case of mbC) the underlying basis was always CPL+ This chapter, instead,analyzes LFIs defined over other logical basis, to wit: positive intuitionistic logic,the four-valued Belnap and Dunn’s logic BD, some families of fuzzy logics, andsome positive modal logics

Section5.1starts by defining LFIs based on positive intuitionistic logic, instead

of CPL+, beginning with paraconsistent logics based on IPL+ (taking as a basisJohansson’s minimal logic and Nelson’s logic) A weaker version of mbC called

imbC obtained from the former by changing the positive basis CPL+ to IPL+isalso investigated,

Section5.2is dedicated to the task of combining two paradigms of uncertainty:fuzziness and paraconsistency, with exciting possibilities Taking as a basis themonoidal t-norm based logic MTL introduced in [6] as a generalization of thefamous basic fuzzy logic BL due to P Hájek (which, in turn, simultaneouslygeneralizes three chief fuzzy logics, namely Łukasiewicz, Gödel-Dummet andProduct logics) several new LFIs had been recently developed (see [7])

Justified by the fact that MTL is the most general residuated fuzzy logic whosesemantics is based on t-norms, the LFIs defined over MTL give a finely controlledcombination of fuzzy and consistency (as well inconsistency) operators, giving rise

to mathematical models for the novel notion of fuzzy (in)consistency operators,which formalizes the nice and natural idea of degrees of consistency andinconsistency

Section5.3investigates a four-valued modal LFI based on N Belnap and J.M.Dunn’s logic BD, a logic (based on their famous bilattice logic FOUR) suitable forrepresenting lack of information (a sentence is neither true nor false) or excess ofinformation (a sentence is both true and false) The logicBD was defined from thenotion of proposition surrogates introduced by J.M Dunn aboutfive decades ago

as a set-theoretic tool for representing De Morgan Lattices The logic M4m, a matrixlogic expanding Belnap and Dunn’s logic BD by adding a modal operator, is then

defined and proved to be an LFI Moreover, it is a dC-system based on the logicpreserving degrees of truth of the variety of bounded distributive lattices The logic

M4mis based on the previous work by A Monteiro on tetravalent modal algebras.The chapter closes, in Sect.5.4, with an overview of the notion of modal LFIsand their unfoldings

Chapter6

Chapter6studies alternative semantics for the LFIs presented in Chaps 2and 3,concentrating on the novel notion of swap structures As much as modal logics,LFIs are in general non-truth-functional, and (as much as modal logics) have access

to different kinds of semantics (like algebraic semantics, Kripke or relationalsemantics, topological semantics, and neighborhood semantics, among others) tobetter clarify their meaning, LFIs also naturally require a plurality of semantics Butunlike modal logics, LFIs in general do not have non-trivial logical congruences,and the question of defining other semantics for LFIs becomes more sensible.Standard tools, like categorial or algebraic semantics, will not work so easily forLFIs and the development of alternative semantical techniques for certain LFIs is

an ongoing and relevant task

The chapter clarifies the heritage of swap structures from M Fidel’s notion oftwist structures (studied in Chap 5), and also discusses the close relationshipbetween the concept of Fidel structures, swap structures and non-deterministicmatrices (or Nmatrices)

Section 6.8 surveys the possible-translations semantics (PTSs), a broadsemantical concept introduced in 1990 that gives new philosophical interpretations

for some non-classical logics, and especially for paraconsistent logics It happensthat PTSs is a very general semantical notion, to the point that virtually any logicmay have a PTS interpretation, under certain conditions It also happens thatseveral other semantical notions can be seen as particular cases of PTSs; thosepoints are carefully explained in that section.

Chapter7

Chapter 7 gives a full account of LFIs for first-order languages The quantifiedversions of LFIs are essential for certain mathematical applications, such as settheory, and also for concrete applications in computer science, such as databasesand logic programming The combination of the consistency operator  withquantifiers 8 and 9 demands a careful treatment: now, the propagation of consis-tency through quantifiers has to be duly balanced, generalizing from the propaga-tion of consistency for conjunction and disjunction The intuitive idea, of course, is

to regard the existential quantifiers as arbitrary conjunctions and disjunctions, butthis has to be done taking a certain technical care

The chapter is structured around a complete treatment of the system QmbC, aquantified exension of the system mbC, the basic LFI studied in Chap 2 Otherextensions of QmbC, such as QCi and QmbC (the latter including an equalitypredicate), are also treated, keeping QmbC at the horizon From the point of view

of semantics, Tarskianfirst-order structures are now endowed with a paraconsistentbivaluation, and what results is a wide generalization of familiar model theory Analternative approach to three-valuedfirst-order LFIs is developed in detail in Sect.7.9, based on the theory of quasi-truth This treatment, of course, can be extended toother many-valued paraconsistent logics

The paradigm of quasi-truth, which provides a way of accommodating theconceptual incompleteness inherent in scientific theories as studied in [8], viewsscientific theories from the perspective of paraconsistent logic This paradigm offers

a rational account for the dynamics of theory change, allowing for theoriesinvolving contradictions without triviality, with deep implications for the founda-tions of science and for the understanding of the scientific method A generalization

of the logical aspects of the theory of quasi-truth has been undertaken in [9], bymeans of a three-valued model theory for an LFI called LPT1, which in turncoincides (setting aside some details of language) with the quantified version of thethree-valued paraconsistent logic LFI1 introduced in Definition 4.4.41 An addi-tional discusion on quasi-truth can be found in Sect.9.3of Chap 9

One of the aims of this chapter is to endorse the claim that basically the sameresults of classical model theory hold for QmbC, and forfirst-order LFIs in gen-eral, with certain provisos Well-established results in traditional model theory such

as the Completeness, Compactness and Lowenhëim–Skolem Theorems can beproved forfirst-order LFIs along the same lines as the classical case In this way,the chapter makes clear thatfirst-order LFIs expand traditional logic, and allows for

a revision of the uses of logic in mathematics and computer science from thevantage point of richer logics

The confusion between the concept of set on the one hand, and of class, or species,

on the other hand, has plagued the foundations of set theory since its birth ThePrinciple of Comprehension (also referred to as the Principle of NạveComprehension, or Abstraction) was proposed in the nineteenth century, fruit of thesomewhat romantic ideas of Dedekind, Cantor, and Frege, and states that for everyproperty, expressed as a predicate, there exists a set consisting of exactly thoseobjects that satisfy the predicate This principle lurks behind certain tough para-doxes, such as Russell’s paradox, and the history of contemporary set theory hasmuch to do with efforts to rescue Cantor’s nạve set theory from triviality, aninevitable consequence, in traditional logic, of the contradictions entailed by thoseparadoxes Paraconsistent set theory has been an endeavor to save set theory fromcertain (it not all) paradoxes for at least three decades Chapter8aims to offer a newapproach to this question by means of employing LFIs and their powerful con-sistency operator By assuming that not only sentences, but sets themselves can beclassified as consistent or inconsistent objects, the basis for new paraconsistentset-theories that can resist certain paradoxes without falling into trivialism isestablished One of the main motivations of this chapter, as stated in Sect.8.1, is torescue, together with Cantor’s nạve set theory, the proper Cantor’s intuitiontowards ‘inconsistent sets’ Indeed, the chapter attempts to show that Cantor’streatment of inconsistent collections can be related to the one provided by means ofLFIs

Section8.2defines ZFmbC, a basic system of paraconsistent set theory whoseunderlying logic is QmbC, and which contains two non-logical predicates (be-sides the equality predicate): the binary predicate “2” (for membership), and theunary predicate C (for consistency of sets) Section8.3proposes some extensions ofZFmbC by means of employing stronger LFIs as underlying logics and settingappropriate axioms for the consistency operator C for sets Section8.4discusses therelationship between the notions of ‘to be a consistent object in set theory’ (asformalized in the chapter) and‘to be a set’ It shows that consistent objects can be(without risk of trivialism) regarded as sets, by means of an appropriate axiom Inthe same spirit, proper classes can be regarded as inconsistent objects Such

affinities between consistent objects in set theory and sets, and between properclasses and inconsistent objects, though it cannot be strengthened into equivalence,testify to the richness of this approach

Section8.5, the last in the chapter, starts the discussion of models of sistent set theory If the construction of models for standard set theory is a fraughttask, the analogue for paraconsistent set theory is adventurous, to say the least Onemight consider standard models of paraconsistent set theory, where theε relation ofthat model corresponds exactly to the membership relation 2 of the universe ofZFmbCand its extensions, and the same for the consistency operator, but it isalso reasonable to make room for non-standard models Only in this way could oneventure into deeper waters, such as extending forcing machinery to paraconsistent

paracon-set theory Although this is not done in this book, and it may be an ambitiousproject, it is not unrealistic.

Section 9.3 reviews—from a more philosophical perspective—the concept ofpragmatic truth, also referred to as quasi-truth, or partial truth, already analyzedfrom the formal point of view in Chap.7 Quasi-truth, developed as part of efforts toexpand the bounds of the traditional Tarskian account of formalized truth, proposes

a partial (or pragmatic) notion of truth, intending to capture the meaning of wider,moreflexible, theories of truth held by anti-realist thinkers in philosophy of science.Section9.4emphasizes the evidence-based approach to paraconsistency, in thesense of understanding a pair of contradictory sentences as representing, andallowing us to reason about, conflicting evidence, defending this view as particu-larly promising for philosophical interpretations of paraconsistent logics

The last section, Sect 9.5, succinctly wraps up one of the chief points behindLFIs: they are concerned with truth, since classical logic can be fully recoveredinside most of the LFIs, but they are also concerned with the notion of evidence, anotion weaker than truth that allows for an intuitive and plausible understanding

of the acceptance of contradictions in some reasoning contexts In this regard, bothintuitionistic and paraconsistent logics may be conceived as normative theories oflogical consequence endowed with an epistemic character This view not only

stresses the brotherhood between the intuitionistic and the paraconsistent digms, but explains the adequacy of LFIs for wider accounts in the philosophy ofscience, and also their applicability in thefields of linguistics, theoretical computerscience, inferential probability, and confirmation theory.

3 Carnielli, Walter A., Marcelo E Coniglio, and Jo ão Marcos 2007 Logics of Formal Inconsistency In Handbook of Philosophical Logic (2nd edn), ed Dov M Gabbay and Franz Guenthner, vol 14, 1 –93 Springer, doi: 10.1007/978-1-4020-6324-4_1

4 da Costa, Newton C.A 1963 Sistemas formais inconsistentes (Inconsistent formal systems, in Portuguese) Habilitation thesis, Universidade Federal do Paran á, Curitiba, Brazil, Republished

by Editora UFPR, Curitiba, Brazil, 1993

5 da Costa, Newton C.A 1974 On the theory of inconsistent formal systems (Lecture delivered

at the First Latin-American Colloquium on Mathematical Logic, held at Santiago, Chile, July 1970) Notre Dame Journal of Formal Logic 15(4): 497 –510.

6 Esteva, Francesc and Llu ís Godo 2001 Monoidal t-norm based logic: Towards a logic for left-continuous t-norms Fuzzy Sets and Systems 124(3): 271 –288.

7 Coniglio, Marcelo E., Francesc Esteva, and Llu ís Godo 2014 Logics of formal inconsistency arising from systems of fuzzy logic Logic Journal of the IGPL 22(6): 880 –904, doi: 10.1093/ jigpal/jzu016

8 Bueno, Ot ávio and Newton C A da Costa 2007 Quasi-truth, paraconsistency, and the foundations of science Synthese 154(3): 383 –399.

9 Coniglio, Marcelo E., and Luiz H Silvestrini 2014 An alternative approach for quasi-truth Logic Journal of the IGPL 22(2): 387 –410, doi: 10.1093/ljigpal/jzt026

We would like to express our gratitude to the many organizations and people whoread, wrote, and offered criticisms and comments, allowed us to quote their papers,assisted in editing and proofreading, and provided support of all kinds—monetary,philosophical, personal, and emotional We would like to acknowledge supportfrom FAPESP (Thematic Project LogCons 2010/51038-0, Brazil) and from indi-vidual research grants from The National Council for Scientific and TechnologicalDevelopment (CNPq), Brazil The intellectual environment of the Centre for Logic,Epistemology and the History of Science (CLE) of the State University ofCampinas—UNICAMP deserves a special mention: we thank the colleagues andthe officers of CLE for having provided all necessary facilities, from libraryfacilities, to secretarial work, computers, and good coffee.

Personal thanks go to Abílio Rodrigues (Belo Horizonte), Henrique AntunesAlmeida (Campinas), Peter Verdée (Campinas and Brussels), Raymundo Morado(Campinas and Mexico City), Itala D’Ottaviano (Campinas), Giorgio Venturi(Campinas), David Gilbert (Campinas and Urbana), Gabriele Pulcini (Campinas),Rodolfo Ertola (Campinas), Francesc Esteva (Barcelona), Lluís Godo (Barcelona),Josep Maria Font (Barcelona), Ramón Jansana (Barcelona), Tommaso Flaminio(Varese), Carles Noguera (Prague), João Marcos (Natal), Juliana Bueno-Soler(Limeira), Newton Peron (Chapecó), Rafael Testa (Campinas), Marcio Ribeiro(Guarulhos), Erin O’Connor (Sorocaba), and Gareth J Young (Glasgow)

Thanks also to Christi Lue and to the Springer team (Dordrecht) for theircontinuous support over the many years this book took to complete We begforgiveness from all those we might have unintentionally failed to mention: wehave made every effort to leave that set consistently empty

xix

1 Contradiction and (in)Consistency 1

1.1 Introduction 1

1.2 On the Philosophy of the Logics of Formal Inconsistency 4

1.3 A Historical Sketch: The Forerunners of the Logics of Formal Inconsistency 10

1.4 Paraconsistency and the Nature of Logic 11

1.5 Paraconsistency and the Nature of Contradictions 15

1.6 Contradiction, Consistency and Negation 18

1.6.1 On Contradiction 19

1.6.2 On Consistency 20

1.6.3 On Negation 21

1.7 Varieties of Paraconsistency Involvement 24

References 26

2 A Basic Logic of Formal Inconsistency: mbC 29

2.1 Introducing mbC 29

2.2 A Valuation Semantics for mbC 35

2.3 Applications of mbC-Valuations 39

2.4 Recovering Classical Logic Inside mbC 43

2.5 Reintroducing mbC as an Expansion of CPL 50

2.5.1 The New Presentation mbC? of mbC 51

2.5.2 Valuation Semantics for mbC 53

2.5.3 Equivalence Between mbC and mbC? . 55

References 61

3 Some Extensions of mbC 63

3.1 A Wider Form of Truth-Functionality for Consistency 63

3.2 A Hidden Consistency Operator 68

3.3 Consistency and Inconsistency as Derived Connectives 71

3.4 Some Conceptual Differences Between mbC andmbC? . 91

3.5 Inconsistency Operators and Double-Negations 95

xxi

3.6 Propagating Consistency 104

3.7 da Costa’s Hierarchy and Consistency Propagation 111

3.8 A Stronger Consistency Propagation 115

References 119

4 Matrices and Algebraizability 121

4.1 Logical Matrices 121

4.2 Uncharacterizability by Finite Matrices 122

4.3 The Problem of Algebraizability of LFIs 129

4.4 Some 3-Valued LFIs 136

4.4.1 Halldén’s Logic of Nonsense (1949) 136

4.4.2 Segerberg’s Logic of Nonsense (1965) 138

4.4.3 da Costa and D’Ottaviano’s Logic J3 (1970) 140

4.4.4 Sette’s Logic P1 (1973) 144

4.4.5 Asenjo-Priest’s Logic LP (1966–1979) 149

4.4.6 Ciore and Other Related Systems 151

4.4.7 LFI1, MPT and J3 158

References 168

5 LFIs Based on Other Logics 171

5.1 LFIs Based on Positive Intuitionistic Logic 171

5.1.1 Basic Features of Positive Intuitionistic Logic 171

5.1.2 Johansson’s Minimal Logic 175

5.1.3 Nelson’s Paraconsistent Logic N4 179

5.1.4 An Intuitionistic Version of mbC 187

5.2 LFIs Based on Fuzzy Logics 191

5.2.1 Preliminaries on MFL 191

5.2.2 Fuzzy Logics with a Consistency Operator 197

5.2.3 Propagation of Consistency and DAT 205

5.2.4 Fuzzy Logics with an Inconsistency Operator 208

5.3 A Modal LFI Based on Belnap and Dunn’s Logic BD 211

5.3.1 The Logic M4m of Tetravalent Modal Algebras 213

5.3.2 M4mas an LFI 218

5.3.3 M4mas a dC-System 222

5.3.4 The Contrapositive Implication 224

5.3.5 A Hilbert-Style Axiomatization of M4mc 226

5.4 Paraconsistent Modalities, Consistency and Determinedness 229

References 233

6 Semantics of Non-deterministic Character for LFIs 237

6.1 Fidel Structures for mbC 238

6.2 Fidel Structures for Some Extensions of mbC 242

6.3 Non-deterministic Matrices 251

6.4 Swap Structures for mbC 253

6.5 Swap Structures for Some Extensions of mbC 260

6.6 Axiom (cl) and Uncharacterizability by Finite Nmatrices 272

6.7 Some Remarks on Fidel Structures and Swap Structures 2796.8 The Possible-Translations Semantics 2806.8.1 Possible-Translations Semantics for Some LFIs 2806.8.2 A 3-Valued Possible-Translations Semantics for Cila 2826.8.3 Some Remarks on Possible-Translations Semantics 288References 290

7 First-Order LFIs 2937.1 The Logic QmbC 2947.2 Basic Properties of QmbC 2967.3 Tarskian Paraconsistent Structures 3027.4 Soundness Theorem for QmbC 3087.5 Completeness Theorem for QmbC 3107.5.1 Henkin Theories 3117.5.2 Canonical Interpretations 3127.6 Compactness and Lowenhëim-Skolem Theorems for QmbC 3167.7 QmbCwith Equality 3187.8 First-Order Characterization of Other Quantified LFIs 3227.9 First-Order LFI1 and the Logic of Quasi-truth 3247.9.1 Semantics of Partial Structures 3247.9.2 The LogicQLFI1 3327.10 First-Order P1 and Partial Structures 335References 343

8 Paraconsistent Set Theory 3458.1 Antinomic Sets and Paraconsistency 3468.2 LFIs Predicating on Consistency 3498.3 Some Extensions of ZFmbC 3548.4 Inconsistent Sets and Proper Classes 3608.5 On Models 363References 365

9 Paraconsistency and Philosophy of Science: Foundations

and Perspectives 3699.1 An Epistemological Understanding of Paraconsistency,

and Its Significance for Science 3699.2 Consistency and Contradiction in Scientific Theories 3719.2.1 The Heritage of Kant 3719.2.2 Some Historical Examples 3739.2.3 The Beginning of Quantum Theory and

Paraconsistency 3749.2.4 Mercury’s Orbit and a Non-existent Planet 3749.2.5 Contradictions in Phlogiston, the Imponderable 3769.2.6 The Special Theory of Relativity 3779.2.7 Mathematics, and the Meaning of Objects

that Mean Nothing 379

9.3 Quasi-truth and the Reconciliation of Science

and Rationality 3829.4 An Evidence-Based Approach to Paraconsistency 3849.5 Summing Up 385References 387

Index 391

Index of Logic Systems 397

Contradiction and (in)Consistency

1.1 Introduction

The target audience of this book is mainly the philosopher, the logician interested inthe philosophical aspects of paraconsistency, and the computer scientist looking fornew logics for applications But the intended audience also includes the mathemati-cian intrigued by the possibility of working in a logic that allows contradictions (aparaconsistent logic), the linguist worried about the acceptance of contradictions inthe ordinary speech, and the scientist interested in the significance of contradictions

in the history of science.1

The reader of this book is invited, first of all, to take into account that contradictionsare pervasive in scientific theories, in philosophical argumentation, in several areas

of computer science such as abduction, automated reasoning, logic programming,belief revision and the semantic web People negotiating a contract, as buyers andsellers, many times encounter contradictions, and strive to overcome them in order

to strike a deal Paradoxes in formal semantics, as the famous liar paradox, areseen as dangerous to the standard theories of truth, and paradoxes in naive (albeitintuitively acceptable) set theory are seen as threats to the foundations of science andmathematics

However, contradictory information is not only frequent, and more so as systemsincrease in complexity, but can have a positive role in human thought, in some casesbeing desirable Finding contradictions in juridical testimonies, in statements fromsuspects of a crime or in suspects of tax fraud can be an efficient strategy Contra-dictions can be very informative: we will never know if people being questionedcoherently lie or not, unless they contradict each other!

1 This chapter corresponds in part with the tutorial on Logics of Formal Inconsistency presented in

the 5t h World Congress on Paraconsistency (Kolkata, India, February 2014), see [1 ] Parts of that material have already appeared in [ 2 ].

© Springer International Publishing Switzerland 2016

W Carnielli and M.E Coniglio, Paraconsistent Logic: Consistency, Contradiction

and Negation, Logic, Epistemology, and the Unity of Science 40,

DOI 10.1007/978-3-319-33205-5_1

1

The current orthodoxy is that all contradictions are equally virulent, in view of

the principle of Ex Contradictione Sequitur Quodlibet (ECSQ), or The Principle of

Explosion (PE), the principle that holds that from a contradiction, anything logicallyfollows But how can standard logic, which endorses ECSQ, impose a principle that isnot followed by common reasoning? Are all contradictions really equally hazardous?

half century ago, the clash between the notions of contradiction and semantic mation: the less probable a statement is, the more informative it is, and so contradic-tions carry the maximum amount of information, and in the light of standard logicare, as a famous quote by Bar-Hillel and Carnap has it, “too informative to be true”.This is a difficult philosophical problem for standard logic, which is forced to equatetriviality and contradiction, and to regard all contradictions as equivalent, as the fol-lowing example illustrates If two auto technicians tell me that the battery of mycar is flat, and its electrical system out of order, and add all the (potentially infinite)statements about car electrics, I have an excessive amount of information, including ahuge amount of irrelevant information Classically, this trivial amount of information

infor-is exactly the same as the information conveyed by the car technicians telling me acontradiction, such as the battery of my car is flat and that it is not flat However, ifone of the car technicians tells me (among his statements) that the battery is flat, andthe other that the battery is not flat, between them they are contradictory, but now Iknow where the problem is! Skipping all technicalities in favor of a clear intuition(details are given elsewhere), the Bar-Hillel-Carnap observation is not paradoxical

for LFIs since, as will be clear in the following, LFIs do not treat all contradictions

equivalently, and do not equate contradiction with triviality

The idea that any contradiction inexorably leads to deductive explosion (by means

of ECSQ) seems to have entered logical orthodoxy towards the end of the 19th century

at the hands of G Frege, B Russell, D Hilbert and W Ackermann, pioneered by G.Boole As outlined in [4], the logic of antiquity did not endorse the validity of ECSQ,and the principle only became a topic of debate in the Middle Ages or Medieval era

It is a plausible, though debatable, conjecture that what is now known as ECSQ,

Quodlibet3might have been originated in the 14th century ideas of John of Cornwall(quite possibly the ‘Pseudo-Duns Scotus’ himself)

The incorporation of the principle ECSQ into contemporary logic had resounding

consistent if and only if it is not deducibly trivial

2 Example 14 (p 15) of [ 5] provides an example of a logic that respects the principle of Ex Falso Sequitur Quodlibet, but not the ECSQ, showing that those principles do not need to be identified,

contrary to what is commonly held in the literature.

3To the best of our knowledge, the exact expressions Ex Contradictione Sequitur Quodlibet and Ex Contradictione Quodlibet have been independently coined by, respectively Priest and Bobenrieth-

Miserda, see [ 6 ].

Glorious results of the 20th century such as K Gödel’s proof of the consistency ofCantor’s continuum hypothesis with the axioms of set theory, his celebrated theorem

on the impossibility of proving the consistency of arithmetic by finitary means, as well

as P Cohen’s forcing technique for proving consistency and independence results inset theory, among many others, all depend upon the acceptance of that definition ofconsistency

However, a logical system need not endorse ECSQ; paraconsistent logic avoids

said to be paraconsistent if it is not explosive, and yet preserves enough properties

to be accounted as a logic Relatedly, the notion of consistency need not be seen

from the aforementioned perspective: the spirit of the LFIs to be developed in full in

the next chapters shows that other mathematically well-founded and philosophicallypalatable approaches exist, and have exciting and relevant consequences

The LFIs are a family of propositional and quantified paraconsistent logics that

encode consistency (and inconsistency) as operators independent of negation in theirobject language Encoding consistency and inconsistency in this way has the effect

of permitting an explicit separation between, respectively, contradiction from sistency, inconsistency from triviality, consistency from non-contradiction, and non-

incon-triviality from consistency The rich language of LFIs make it possible to investigate

contradictory theories without assuming that they are necessarily trivial

The LFIs are proper fragments of classical logic (hence non trivial) yet rejecting

ECSQ in the presence of a contradiction, unless the contradictory sentence is taken to

be consistent The family of LFIs incorporate a great number of paraconsistent

sys-tems of various sorts, in a sense to be explained, including the well-known hierarchy

of logics introduced by Newton da Costa in Brazil in the sixties

The idealization behind LFIs is that assertions about the world should be divided

into two categories: consistent sentences and non-consistent sentences Consistentpropositions are subjected to classical logic, and consequently a theory that contains

consistent sentence, linguistically marked as◦α (or ◦¬α) This is the only distinction

between LFIs and classical logic, albeit with far-reaching consequences: classical logic in this way is expanded, in such a way that in most cases an LFI encodes

are actually true, the LFIs will be on their side If not, the LFIs will continue to

be of value, independently of this debate In this regard LFIs are a free theory of deduction Third, the LFIs are extensible to first-order logic (and

metaphysically-to higher-order logics as well), and are powerful enough metaphysically-to reproduce, under tle adaptations, all the main metamathematical results of traditional logic, such ascompleteness, compacteness, most model theoretical properties of quantified logic,decidability, and so on

sub-The history of expansion of concepts in mathematics has a long tradition, since thenegative numbers, zero and imaginary numbers have been called, variously, “sophis-tic”, “impossible” or “inexplicable” Science and engineering would be hardly pos-sible without such ‘fictions’ today, and the centuries following the introduction ofimaginary numbers (the name itself suggested a pejorative connotation) have wit-nessed many other ‘sophistic’ creatures, such as G Cantor’s cardinal arithmetic,

A Robinson’s non-standard numbers, the hyperreal numbers, fractal dimensions,and a number of other similar ideas Perhaps the idea of consistency as a primi-tive concept will be recognized as of a similar sort—the reader is invited to judgeher/himself

The following section discusses some primary philosophical issues related to

paraconsistency in general, and especially to the LFIs Our basic standpoint is that

there are two basic and philosophically legitimate approaches to paraconsistency thatdepend on whether the contradictions are understood ontologically or epistemologi-

cally LFIs are well suited to both options, but we shall emphasize the epistemological

interpretation of contradictions The main argument depends on the duality betweenparaconsistency and paracompleteness Briefly, the idea is as follows: just as excludedmiddle may be rejected by intuitionists for epistemological reasons, explosion mayalso be rejected by paraconsistentists for epistemological reasons as well

1.2 On the Philosophy of the Logics of Formal

Inconsistency

It is a fact that contradictions appear in a number of real-life contexts of reasoning.Databases very often contain not only incomplete information but also conflicting

logicians and philosophers, and, more recently, mathematicians as well Scientifictheories are another example of real situations in which contradictions seem to beunavoidable There are several scientific theories, however successful in the areaswith which they are primarily concerned, that yield contradictions, either by them-selves or in combination with other successful theories Contradictions are problem-atic when the Principle of Explosion, or ECSQ, holds:

In this case, since anything follows from a contradiction, one may conclude anythingwhatsoever In order to deal rationally with contradictions, explosion cannot be validwithout restrictions, since triviality (that is, a circumstance such that everythingholds) is obviously unacceptable Given that in classical logic explosion is a validprinciple of inference, the underlying logic of a contradictory context of reasoningcannot be classical.

Simply put, paraconsistency is the study of logical systems in which the presence

of a contradiction does not imply triviality, that is, logical systems with a

trivialize the system However, it is not only the syntactic and semantic properties

of these systems that are worth studying Some questions arise that are perennialphilosophical problems The question of the nature of the contradictions allowed

in paraconsistent logic has been a particular focus of debates on the philosophicalsignificance of paraconsistency

In philosophical terminology, we say that something is ontological when it has to

do with reality, the world in the widest sense, and that something is epistemologicalwhen it has to do with knowledge and the process of its acquisition A central questionfor paraconsistency is the following: are the contradictions that paraconsistent logicdeals with ontological or epistemological? Do contradictions have to do with realityproper? That is, is reality intrinsically contradictory, in the sense that we really needsome pairs of contradictory propositions in order to describe it correctly? Or docontradictions have to do with knowledge and thought? Contradictions of the latterkind would have their origin in our cognitive apparatus, in the failure of measuringinstruments, in the interactions of these instruments with phenomena, in operations

of thought, or even in simple mistakes that in principle could be corrected later on.Note that in all of these cases the contradiction does not belong to reality properlyspeaking

The question of nature of contradictions, in its turn, is related to another centralissue in philosophy of logic, namely, the nature of logic itself As a theory of logicalconsequence, the task of logic is to formulate principles and methods for establishing

the principles of logic about? Are they about language, thought, or reality? That logic

is normative is controversial, but if logic is anyhow normative for thought, its mative character may be combined both with an ontological and an epistemologicalapproach

nor-The epistemological side of logic is present in the widespread (but not unanimous)characterization of logic as the study of laws of thought This concept of logic, whichacknowledges an inherent relationship between logic and human rationality, has beenput aside since classical logic has acquired the status of the standard account of logicalconsequence—for example, the work of Frege, Russell, Tarski, Quine and a manyother influential logicians

Classical logic is a very good account of the notion of truth preservation, but itdoes not give a sustained account of rationality This point shall not be developed

in detail here, but it is well known that some classically valid inferences are not

that anything impliesα; from α, to conclude the disjunction of α and anything; from

a contradiction, to conclude anything The latter is the principle of explosion, and

contradictory propositions Nevertheless, from the point of view of preservation oftruth, given the classical meaning of sentential connectives, all the inferences aboveare irreproachable

We assume here a concept of logic according to which logic is not restricted tothe idea of truth preservation Logical consequence is indeed the central notion oflogic, but the task of logic is to tell us which conclusions can be drawn from a givenset of premises, under certain conditions, in concrete situations of reasoning Weshall see that sometimes it may be the case that it is not only truth that is at stake.6

Among the contexts of reasoning in which classical logic is not the most suitabletool, two are especially important: contexts with ‘excess of information’ and ‘lack

of information’ The logics suited to such contexts are, respectively, paraconsistentand paracomplete—in the former, explosion fails, in the latter excluded middle fails.There are two basic approaches to paraconsistency If some contradictions aretrue, since it is not the case that everything holds, we need an account of logicalconsequence that does not collapse in the face of a contradiction On the other hand,

if contradictions are epistemological, we shall argue that the rejection of explosiongoes hand in hand with the rejection of excluded middle by intuitionistic logic Inthe latter case, the formal system has an epistemological character and combines adescriptive with a normative approach

In the next sections, some basic concepts will be introduced in order to guish triviality from inconsistency In addition, we shall make a first presentation

distin-of LFIs, distinguishing paraconsistency and paracompleteness from the classical

and the forerunners of the Logics of Formal Inconsistency (henceforward referred to

para-consistency and the issue of the nature of logic We will argue that, like the rejection

of excluded middle by intuitionistic logic, the rejection of explosion may be

of the issue of the nature of contradictions, and considers whether they should beunderstood ontologically or epistemologically We shall argue that both positions arephilosophically legitimate Finally, we will be ready to show how the simultaneous

attribution of the value 0 (or false) to a pair of sentences α and ¬α may be interpreted

as conflicting evidence, not as truth and falsity ofα.

We have seen that paraconsistent logics are able to deal with contradictory narios, avoiding triviality by means of the rejection of the principle of explosion.Let us put these ideas more precisely A theory is a set of sentences closed under

6 This idea has some consequences for Harman’s arguments [ 9 ] against non-classical logics, a point that we intend to develop elsewhere.

 and the underlying logic is L Suppose the language of T has a negation ∼ We

say that T is:

Contradictory if and only if there is a propositionα in the language of T such

Trivial if and only if for any propositionα in the language of T , T proves α;

Explosive if and only if T trivializes when exposed to any pair of contradictory

formulas—i.e.:

In books of logic we find two different but classically equivalent notions of

i S is consistent if and only if there is a formula β such that  S β;

ii S is consistent if and only if there is no formula α such that  S α and  S ∼α What (i) says is that S is non-trivial; and (ii) says that S is non-contradictory In

classical logic both are provably equivalent

So, a theory whose underlying logic is classical is contradictory if and only if it

is trivial But this is the case precisely because such a theory is explosive, since the

principle of explosion holds in classical logic It is clear, then, that contradictoriness

is not necessarily coincident with explosiveness The obvious move in order to deal

with contradictions is, thus, to reject the unrestricted validity of the principle ofexplosion This is a necessary condition if we want a contradictory but not-trivialtheory

The first formalization of paraconsistent logic to appear in the literature is to

contradictory but non-trivial logic must attend:

1 It must be non-explosive;

2 It should be “rich enough to enable practical inference”;

3 It should have “an intuitive justification”

Condition (1), as we have seen, is a necessary condition for any paraconsistentsystem We want to call attention to conditions (2) and (3) Indeed, the biggestchallenge for a paraconsistentist is to devise a logical system compatible with what

we intuitively think should follow (or not follow) from what This is the idea expressed

by the criteria (2) and (3) presented by Ja´skowski An intuitive and applicable notion

of logical consequence should be appropriate for describing and reconstructing theactual reasoning going on in real-life contexts An intuitive account of the meaning

of the logical connectives—more precisely, of paraconsistent negation—should be

an integral part of such an account of logical consequence It follows that an intuitiveinterpretation of a paraconsistent notion of logical consequence depends essentially

on an intuitive interpretation of negation

1 α ∧ ∼α 

According to (1), there is no model M such that α ∧ ∼α holds in M (2) says that

We say that a negation is paracomplete if it disobeys (2), and that a negation is

paraconsistent if it disobeys (1) From the point of view of rules of inference, the

duality is not between non-contradiction and excluded middle, but rather betweenexplosion and excluded middle Notice that the notion of logical consequence haspriority over the notion of logical truth: the latter must be defined in terms of theformer, not the contrary The principle of non-contradiction is usually taken as theclaim that reality is not contradictory But we may well understand the principle of

hold together, otherwise we get triviality From the above considerations it is clearthat in order to give a counterexample to the principle of explosion we need a weaker

M Dually, a paracomplete logic must have a model M such that both α and ¬α does

A central feature of classical negation∼ (but not of all negations, as we shall see)

is that it is a contradictory forming operator This is due to its semantic clause,

M (∼α) = 1 iff M(α) = 0

in the sense that they cannot simultaneously receive the value 0, nor simultaneouslythe value 1 In classical logic the truth-values 0 and 1 are understood respectively asfalse and true, but in non-classical logics this does not need to be the case It is not

nor that a paraconsistent logic takes them as both true

Obviously, neither a paracomplete nor a paraconsistent negation is a contradictory

that such negations are really negations? Our answer is yes

It should not be surprising that the meaning of a classical connective splits up intosome alternative meanings when its use in natural language and real-life arguments isanalyzed Indeed, different meanings are sometimes attached to conditional, disjunc-tion, and conjunction, and the connectives so obtained are still called the conditional,disjunction, and conjunction, of course with some qualifications What would be thereason by which the same cannot occur with negation? In fact, both paracompleteand paraconsistent negations do occur in real life An obvious example of the for-mer is intuitionistic negation: it may be the case that we do have a classical proof

7 For a more detailed account on the duality between paracompleteness and paraconsistency, see e.g [ 11 ].

proofs ofα and ∼α, from the constructive point of view, we have neither α nor ∼α.

On the other hand, sometimes it happens that we have to deal simultaneously with

α and ∼α, but we do not need to say that both are true Finally, the above

consid-erations show that a paraconsistent negation is a negation to the same extent that aparacomplete (including intuitionistic) negation is a negation Nevertheless, what is

of major importance is that the question of whether or not a paraconsistent negationmay have an intuitive meaning has a positive answer

In this section we shall present the basic ideas of LFIs without going into the technical details, which will be left for the following chapters As remarked, LFIs

have resources to express the notion of consistency inside the object language by

As in any other paraconsistent logic, explosion does not hold in LFIs But it is handled

in a way that allows distinguishing between contradictions that can be accepted fromthose that cannot The point of this distinction is that no matter the nature of thecontradictions a paraconsistentist is willing to accept, there are contradictions that

cannot be accepted In LFIs, negation is explosive only with respect to consistent

formulas (that is, formulas that are taken to be consistent):

α, ¬α  L F I β, while ◦α, α, ¬α  L F I β.

An LFI is thus a logic that separates the sentences for which explosion holds from

are called gently explosive.

The idea of expressing a kind of logical ‘well-behavior’ in the object language

expressed byα◦, in such a way that:

α, ¬α  C1β, while α◦, α, ¬α  C1β

the concept of LFIs is quite significative: da Costa in [12] named his logical systems

“inconsistent formal systems”, while he was really referring to contradictory and

idea of consistency as a primitive notion represents a radical departure from stream paraconsistency, since it allows us to simultaneously capture a number ofparaconsistent systems (old and new) and to give a more comprehensible account ofparaconsistency generally

main-We may say, thus, that a first step in paraconsistency is the distinction betweentriviality and contradictoriness But there is a second step, namely, the distinction

8Actually, da Costa defined a hierarchy of systems, starting with the system C1 A full hierarchy

of calculi C n , for n natural, is defined and studied in [12] Each C n has its own definition of well-behavedness.

between consistency and non-contradictoriness In LFIs the consistency connective

◦ is not only primitive, it is also not always logically equivalent to non-contradiction.This is the most distinguishing feature of the logics of formal inconsistency Once we

necessarily related to freedom from contradiction

1.3 A Historical Sketch: The Forerunners

of the Logics of Formal Inconsistency

The advent of paraconsistency occurred more than a century ago In 1910 the Russianphilosopher and psychologist N.A Vasiliev proposed the idea of a non-Aristotelianlogic, free of the laws of excluded middle and non-contradiction By analogy with theimaginary geometry of Lobachevsky, Vasiliev called his logic ‘imaginary’, meaningthat it would hold in imaginary worlds Despite publishing, between 1912–1913,some conceptual papers on the subject, Vasiliev was not concerned with formalizinghis logic (see [13], pp 307ff.)

Ja´skowski in [10], trying to answer a question posed by Łukasiewicz, presented thefirst formal system for a paraconsistent logic, called ‘discussive logic’ This system

is connected to modalities, and later on came to be regarded as a particular member

of the family of the logics of formal inconsistency (see [5])

Intending to study logical paradoxes from a formal perspective, S Halldén

closely related to the nonsense logic introduced in 1938 by the Russian logicianD.A Bochvar Since its third truth-value is distinguished, Hállden’s logic is para-consistent, and it can also be reckoned as as one of the first paraconsistent formal

systems presented in the literature In fact, like Ja´skowski’s logic, it is also an LFI.

non-constructive features of intuitionistic negation By eliminating the principle ofexplosion from this system, [16] obtained a first-order paraconsistent logic, althoughparaconsistency was not his primary concern Indeed, the famous Nelson’s paracon-sistent logic was proposed some decades after, in [17]

Paraconsistency also has some early links to K Popper’s falsificationism In 1954

sub-mitted to the University of Oxford a thesis entitled ‘Alternative Systems of Logic’

in which he intended to develop a logic dual to intuitionistic logic In Cohen’slogic, the law of explosion is no longer valid, while the law of excluded middleholds as a theorem Cohen’s thesis, according to Kapsner et al., escaped scholarlyattention, having been only briefly mentioned in Popper’s famous ‘Conjectures and

9 The reader is warned that, along this book, the expression ‘strong negation’ is reserved for a negation with a Boolean character.

Refutations’ (see [19], footnote 8, p 321) It did, however, in some sense

paraconsistent)

In [20] we find a discussion of the status of contradiction in mathematics,

intro-ducing the Principle of Non-Trivialization, according to which non-triviality is more

important than non-contradiction The idea is that any mathematical theory is worthstudying, provided it is not trivial While we agree that non-trivial mathematical (andlogical) systems are worth studying, a little more is required for an account of logicalconsequence to be accepted as an account of reasoning

the broadest formal study of paraconsistency proposed up to that time It is worthmentioning here what has been said by da Costa, in private conversation: “As withthe discovery of America, many people are said to have discovered paraconsistentlogic before my work I can only say that, as with Columbus, nobody has discoveredparaconsistency after me, just as nobody discovered America after Columbus.”

three-valued logic as a formal framework for studying antinomies His logic is essentiallydefined by Kleene’s three-valued truth-tables for negation and conjunction, where a

third truth-value is distinguished Asenjo’s logic is structurally the same as the Logic

of Paradox LP studied by Priest in [22] more than a decade later

From the 1970s on, after the Peruvian philosopher F M Quesada, at da Costa’srequest, coined the name ‘paraconsistent logic’ to encompass all these creations,

1.4 Paraconsistency and the Nature of Logic

A central question in philosophy of logic concerns the nature of logical principles,and specifically whether these principles are about reality, thought, or language Wefind this issue brought forth, either implicitly or explicitly, in a number of places Inthis section we shall discuss the relationship between paraconsistent logic and theproblem of the nature of logic

Aristotle formulates three versions of the principle of non-contradiction, each onecorresponding to one of the aforementioned aspects of logic (more on this below)

view-point, relating the three approaches (ontological, epistemological and linguistic) toperiods in the history of philosophy—respectively ancient and medieval, modern, and

question is whether the principles of logic are:

(I.a) Laws of thought in the sense that they describe how we actually think; (I.b) Laws of thought in the sense that they are normative laws, i.e., laws that tell us

how we should think;

10 See ‘Carta de F.M Quesada a N.C.A da Costa, 29.IX.1975’ in [ 13 ], p 609.

(II) The most general laws of nature, i.e., laws that apply to any kind of object; (III) Laws of certain descriptive languages.

There are three basic options, which are not mutually exclusive: the laws of logichave (I) epistemological, (II) ontological, or (III) linguistic character With respect to(I), they may be (I.a) descriptive or (I.b) normative These aspects may be combined

In many accounts, logic is taken as having a normative character, no matter whether

it is conceived primarily as having to do with language, thought or reality The point

of asking this question is not really to find a definitive answer It is a perennial sophical question, which, nonetheless, helps us to clarify and understand importantaspects of paraconsistent logic

philo-According to widespread opinion, a linguistic conception of logic has prevailedduring the 20th century From this perspective, logic has to do above all with thestructure and functioning of certain languages We do not agree with this view For us,logic is primarily a theory about reality and thought.11The linguistic aspect appearsonly inasmuch as language is used in order to represent what is going on in realityand in thought Although the linguistic aspects of logic are related to epistemology(since language and thought cannot be completely separated) and to ontology (bymeans of semantics), we do not think that a linguistic conception of logic is going

to help much in clarifying a problem that is central for us here, that of whethercontradictions have to do with reality or thought

Aristotle, defending the principle of non-contradiction (PNC), makes it clear that

it is a principle about reality, language, and thought, but there is a consensus amongscholars that its main formulation is a claim about objects and properties: it cannot bethe case that the same property belongs and does not belong to the same object Put

in this way, PNC is ontological in character Like a general law of nature, space-time phenomena cannot disobey PNC, nor can mathematical objects.

The epistemological aspects of logic became clear in the modern period A very

illuminating passage can be found in the so-called Logic of Port-Royal ([25], p 23),where we read that logic has three purposes:

The first is to assure us that we are using reason well.

The second is to reveal and explain more easily the errors or defects that can occur in mental operations.

The third purpose is to make us better acquainted with the nature of the mind by reflecting

on its actions.

Notice how the passage above combines the normative character of logic with ananalysis of mind This view of logic does not fit very well with the account of logicalconsequence given by classical logic, but it has a lot to do with intuitionistic logic

Frege’s Begriffsschrift [26] had an important role in establishing classical logic

as the standard account of logical consequence Although there is no semantics in

Frege’s work, it is well known that we find in the Begriffsschrift a complete and

correct system of first-order classical logic At first sight, Frege’s approach is purely

11 A rejection of the linguistic conception of logic, and a defense of logic as a theory with ontological

and epistemological aspects, can be found in the Introduction to [24 ].

proof-theoretical, but one should not draw the conclusion that his system has noontological commitments We cannot loose sight of the fact that the idea of truthpreservation developed by Frege, although worked out syntactically, is constrained

by a realist notion of truth

Frege had a realist concept of logic, according to which logic is independent oflanguage and mind In fact, since he was a full-blooded platonist with respect tomathematics, and his logicist project was to prove that arithmetic is a development

of logic, he had to be a logical realist For Frege, the laws of logic are as objective as

of logic is very well suited to the idea of truth-preservation He indeed famouslyexplains the task of logic as being ‘to discern the laws of truth’ [28], or more precisely,the laws of preservation of truth Hence, it is not surprising that laws of logic cannot

be obtained from concrete reasoning practices In other words, logic cannot have adescriptive aspect, in the sense of (I.a) above.13It is worth noting that Frege proves

the principle of explosion as a theorem of his system; it is Proposition 36 of the

Begriffsschrift.

It is important to emphasize the contrast between Frege’s and Brouwer’s tions of logic This fact is especially relevant for our aims here because of the duality

the point of view of classical logic, the rejection of excluded middle by intuitionisticlogic is like a mirror image of the rejection of explosion

It is well known that for Brouwer mathematics is not a part of logic, contrary towhat Frege wanted to prove Quite the contrary, logic is abstracted from mathematicalreasoning Mathematics is a product of the human mind, and mathematical proofsare mental constructions that do not depend on language or logic The role of logic

in mathematics is only to describe methodically the constructions carried out by

an analysis of the functioning of mind in constructing mathematical proofs To theextent that intuitionistic logic intends to avoid improper uses of excluded middle,

it is normative, but it is descriptive precisely in the sense that, according to Frege,logic cannot be descriptive Intuitionistic logic thus combines a descriptive with anormative character

12 See [ 27 ], p 13: ‘they are boundary stones set in an eternal foundation, which our thought can overflow, but never displace’.

13 There is a sense in which for Frege laws of logic are descriptive: they describe reality, as well as laws of physics and mathematics But we say here that a logic is descriptive when it describes, in some way, actual reasoning.

14 Brouwer [ 29]: “Mathematics can deal with no other matter than that which it has itself constructed.

In the preceding pages it has been shown for the fundamental parts of mathematics how they can

be built up from units of perception […] The words of your mathematical demonstration merely

accompany a mathematical construction that is effected without words […] While thus mathematics

is independent of logic, logic does depend upon mathematics.” A more acessible presentation of the motivations for intuitionistic logic is to be found in [ 30 ].

The view according to which intuitionistic logic has an epistemological characterthat contrasts with the ontological vein of classical logic is not new.15Note how theintuitionistic approach fits in well with the passage quoted above from logic of PortRoyal Furthermore, even if one wants to insist on an anti-realist notion of truth, thethesis that intuitionistic logic is not about truth properly speaking, but about mentalconstructions, is in line with the intuitionistic program as it was developed by Heyting

Now we may ask: does intuitionistic logic give an account of truth preservation?Our answer is negative, because intuitionistic logic is not only about truth; it isabout truth and something else We may say that it is about constructive truth in the

following sense: it is constrained by truth but it is not truth simpliciter; rather, it is truth

achieved in a constructive way Accordingly, not only the failure of excluded middle,but the whole enterprise of intuitionistic logic, may be seen from an epistemologicalperspective

It is worth noting that Brouwer’s and Heyting’s attempts to identify truth with anotion of proof have failed, as [32] shows, because the result is a concept of truth thatgoes against some basic intuitions about truth The notion of constructive provability

thatα is true, but the converse may not hold.

An analogous interpretation can be made with respect to contradictions in consistent logics While in intuitionistic logic (and paracomplete logics in general)the failure of excluded middle may be seen as a kind of lack of information (no proof

para-ofα, no proof of ¬α), the failure of explosion may be interpreted epistemologically

later) The acceptance of contradictory propositions in some circumstances need notmean that reality is contradictory: on the contrary, it may be accounted as a step inthe process of acquiring knowledge that, at least in principle, could be revised.Imagine a context of reasoning such that there are some propositions well estab-lished as true, or as false, and some others that have not been conclusively establishedyet Now, if among the latter there is a contradiction, one does not conclude that themoon is made of green cheese, but, rather, one takes a more careful stance with respect

to the specific contradictory proposition On the other hand, the inferences allowedwith respect to propositions already established as true are normally applied In fact,what does happen is that the principle of explosion is not unrestrictedly applied Thecontradictory propositions are still there, and it may happen that they are used insome inferences, but they are not taken as true propositions

15 See, for example, [ 31 ]: “two [logics] stand out as having a solid philosophical-mathematical justification On the one hand, classical logic with its ontological basis and on the other hand intuitionistic logic with its epistemic motivation”.

16 In [ 30 ], p 1, we read “You ought to consider what Brouwer’s program was […] It consisted

in the investigation of mental mathematical construction as such, without reference to questions regarding the nature of the constructed objects, such as whether these objects exist independently

of our knowledge of them”.

By means of a non-explosive negation and the consistency operator,◦, an LFI may

formally represent this scenario The discussion about this point will be later resumed

in more detail For now, we want to emphasize that the sketch of a paraconsistent logic

in which contradictions are epistemologically understood as conflicting evidence,and not as a pair of contradictory true sentences, is inspired by an analysis of realsituations of reasoning in which contradictions occur The notion of evidence isweaker than truth in the sense that, if we know thatα is true, then there must be some

evidence forα, but the fact that there is evidence for α does not imply that α is true.

A paraconsistent logic may thus be obtained analogously to the way intuitionisticlogic has been obtained

1.5 Paraconsistency and the Nature of Contradictions

We now turn to a discussion of paraconsistency from the perspective of the problem

of the nature of contradictions The latter is a very old philosophical topic that can

be traced back to the beginnings of philosophy in ancient Greece, and, as we havejust seen, is closely related to the issue of the nature of logic There is an extensive

discussion and defense of the principle of non-contradiction in Aristotle’s

Meta-physics, book  According to Aristotle, PNC is ‘the most certain of all principles’

(Metaphysics 1005b19) It is a proposition that ‘has no other propositions prior to it’ (Posterior Analytics 72a5), and, as such, cannot be demonstrated from more basic principles Although Aristotle claims that PNC is, strictly speaking, indemonstrable,

he presents arguments in defense of it This is not in fact a problem, since these

argu-ments may be thought of as elucidations, or informal explanations, of PNC, rather

PNC that correspond to the three aspects of logic mentioned above, ontological,

epis-temological and linguistic We refer to them here respectively as PNC-O, PNC-E, and PNC-L.

the most indisputable of all beliefs is that contradictory statements are not at thesame time true If it is impossible at the same time to affirm and deny a thing truly,

it is also impossible for contraries to apply to a thing at the same time

The point is that PCN-O is talking about objects and their properties, PCN-E about

ontological, psychological and semantic.17Łukasiewicz strongly attacks Aristotle’s

defense of PNC, and claims that the psychological (i.e epistemological) version is

simply false and that the ontological and the semantic (i.e., linguistic) versions havenot been proven at all He ends the paper by saying that Aristotle ‘might well havehimself felt the weaknesses of his argument, and so he announced his principle a

Aristotle’s arguments, nor Łukasiewicz’s criticisms in detail Rather, we are interested

in the following question: what should be the case in order to make true each one of

the formulations of PNC? We will see that the weaknesses of Aristotle’s arguments

have a lot to reveal about contradictions

The basic idea of PNC-O corresponds to a theorem of first-order logic:

∀x¬(Px ∧ ¬Px), i.e., the same property cannot both belong and not belong to

the same object An object may have different properties at different moments oftime, or from two different perspectives, but obviously these cases do not qualify as

counterexamples for PNC (see Metaphysics, 1009b1 and 1010b10) PNC-O depends

on an ontological categorization of reality in terms of objects and properties Thiscategorization has been central in philosophy and is present in logic since its begin-

of property It is enough to change ‘the object a has the property P’ to ‘the object a satisfies the predicate P’ In any case, we are speaking in the broadest sense, which

includes objects in space-time as well as mathematical objects

The linguistic formulation, here called PNC-L, although talking about language,

also has an ontological vein because of the link between reality and the notion oftruth If there is a claim that is to a large extent uncontentious about truth, it is that

if a proposition (or any other truth-bearer) is true, it is reality that makes it true;

or, in other words, truth is grounded in reality Understood in this way, PNC-O and

PNC-L collapse, the only difference being that the former depends on ontological

categorization in terms of objects and properties, while the latter depends on languageand an unqualified notion of truth Note that Aristotle seems to conflate both, since in

the passage III quoted above PNC-O is the conclusion of an argument whose premise

the former, a proof of PNC-O would be tantamount to showing that mathematics is

consistent But this cannot be proven, even with respect to arithmetic With respect

to the latter, there is an extensive literature about the occurrence of contradictions

in empirical theories (see, for example, Chap 5 of [35,36]) However, to date, there

is no indication whether these contradictions are due to the nature of reality, or

17 This tripartite approach is also found in [ 34 ], where these three versions are called, respectively, ontological, doxastic and semantic.

18 For example, the issue of particulars/universals, the Fregean distinction between object and tion, and even Quine’s attacks to the notion of property.

func-whether they belong to theories, which are nothing but attempts to give a model ofreality in order to predict its behavior In other words, there is no clear indication,far less a conclusive argument, that these contradictions are ontological and not onlyepistemological.

The linguistic version of PNC is exactly the opposite of the dialetheist thesis as

it is presented by [7]:

A dialetheia is a sentence,α, such that both it and its negation, ¬α, are true […] Dialetheism

is the view that there are dialetheias […] dialetheism amounts to the claim that there are true contradictions.

Thus, a proof of PNC-L would be tantamount to a disproof of dialetheism.

Although dialetheism has antecedents in the history of philosophy and is legitimatefrom the philosophical point of view, it is a thesis that is far from being conclusivelyestablished as true Further, if we accept that every sentence says something aboutsomething, a thesis that has not been rejected by logical analysis in terms of argu-

ments and functions, what makes PNC-O true would also make PNC-L true, and vice-versa Our conclusion is that neither PNC-O nor PNC-L has been conclusively

established as a true principle And this is not because Aristotle’s arguments, or anyother philosophical arguments in defense of the two principles are not good Rather,the point is that this issue outstrips what can be done a priori by philosophy itself Itseems to be useless for the philosopher to spend time trying to prove them

Now we turn to PNC-E As it stands, the principle says that the same person

cannot believe in two contradictory propositions Here, the point is not how it could

be proved, because it really seems that there are sufficient reasons to suppose that

it has already been disproved It is a fact that in various circumstances people havecontradictory beliefs Even in the history of philosophy, as [33], p 492 remarks, “con-tradictions have been asserted at the same time with full awareness” Indeed, sincethere are philosophers, like Hegel and the contemporary dialetheists, that defend theexistence of contradictions in reality, this should be an adequate counterexample

to PNC-E Furthermore, if we take a look at some contexts of reasoning, we will

find out that there are a number of situations in which one is justified in believingbothα and ¬α Sometimes we have simultaneous evidence for α and ¬α, which

does not mean that we have to take both as true, but we may have to deal taneously with both propositions Nevertheless, the problem we have at hand may

simul-be put more precisely: PNC-E is somewhat naive and does not go to the core of

the problem The relevant question is whether the contradictions we find in real uations of reasoning—databases, paradoxes, scientific theories—belong to realityproperly speaking, or have their origin in thought and/or in the process of acquiringknowledge

sit-Let us see which lessons may be taken from all of this It is a fact that tions appear in several contexts of reasoning Any philosophical attempt to give aconclusive answer to the question of whether there are contradictions that correctlydescribe reality, is likely to be doomed to failure However, the lack of such a conclu-sive answer does not imply that it is not legitimate to devise a formal system in whichcontradictions are interpreted as true If there are some ontological contradictions