Paraconsistent logic consistency, contradiction and negation

Logic, Epistemology, and the Unity of Science 40

Walter Carnielli
Marcelo Esteban Coniglio

Paraconsistent
Logic:
Consistency,
Contradiction
and Negation


Logic, Epistemology, and the Unity of Science
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


Logic, Epistemology, and the Unity of Science aims to reconsider the question
of the unity of science in light of recent developments in logic. At present, no single
logical, semantical or methodological framework dominates the philosophy of
science. However, the editors of this series believe that formal techniques like, for
example, independence friendly logic, dialogical logics, multimodal logics, game
theoretic semantics and linear logics, have the potential to cast new light on basic
issues in the discussion of the unity of science.
This series provides a venue where philosophers and logicians can apply specific
technical insights to fundamental philosophical problems. While the series is open
to a wide variety of perspectives, including the study and analysis of argumentation
and the critical discussion of the relationship between logic and the philosophy of
science, the aim is to provide an integrated picture of the scientific enterprise in all
its diversity.

More information about this series at />

Walter Carnielli Marcelo Esteban Coniglio
•

Paraconsistent Logic:
Consistency, Contradiction
and Negation

123


Walter Carnielli

Department of Philosophy and Centre
for Logic, Epistemology and the History
of Science (CLE)
University of Campinas (UNICAMP)
Campinas, São Paulo
Brazil

Marcelo Esteban Coniglio
Department of Philosophy and Centre
for 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
ISBN 978-3-319-33203-1
ISBN 978-3-319-33205-5 (eBook)
DOI 10.1007/978-3-319-33205-5
Library of Congress Control Number: 2016936981
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To absent friends, and to Elias Alves, in
memoriam. To our children: Bá, 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


Preface

I protest against the use of infinite magnitude as something completed, which in mathematics is never permissible. Infinity is merely a façon de parler, the real meaning being a
limit which certain ratios approach indefinitely near, while others are permitted to increase
without restriction.
C.F. Gauss, Brief an Schumacher (1831); Werke 8, 216 (1831)

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

The reaction against the infinite, as well as against complex or imaginary
numbers, and against negative numbers before, are interesting examples of the
difficulties faced, even by great minds, in accepting certain abstractions. Aristotle in
Chaps. 4–8 of Book III of Physics argued against the actual infinite, advocating for
the 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 prime
numbers. What was actually stated in Proposition 20 of Book IX, carefully avoiding
the 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 accepted
views by showing that an infinite set can be treated as a totality, as a full-fledged
mathematical 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 different
guises). These numbers caused an embarrassment among mathematicians for centuries, since they faced astonishing difficulties in accepting an extension of the
concept 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


viii

Preface

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 the

only ones. A notable case of expansion of concepts, with deep implications for the
development of contemporary logic, can be traced back to Frege and his famous
article of 1891, Funktion und Begriff (see [1]).
In this seminal paper, Frege recalls how the meaning of the term ‘function’ has
changed 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 y0 ¼ f 0 ðxÞ from a function y ¼ f ðxÞ (provided that the limits
involved in the calculus exist), and through accepting complex numbers in domains
and images of functions.
Starting from this point, Frege goes further into adding expressions that now we
call predicates, such as ‘=’, ‘<’ and ‘>’. Leaving aside his philosophical motivations
for seeing arithmetic as a “further development of logic”, what Frege started was a
real revolution, that made possible the development of quantifiers and an
unprecedented unification of propositional and predicate logic into a far more
powerful system than any that preceded it.
Not only could the truth-values, True and False, be taken as outputs of a
function, but any object whatsoever could be similarly treated. To rephrase an
example 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 the
function. In this way, Frege’s system could represent non-mathematical thoughts
and predications, and founded the basis of the modern predicate calculus.
Frege’s idea of defining an independent notion of ‘concept’ as a function which
maps every argument to one of the truth-values True or False was instrumental in
the development of a strict understanding of the notions of ‘proof’, ‘derivation’, and
‘semantics’ as parts of the same logic mechanism. Regarding ‘concept’ as a wide
and independent notion based on an amplification of the idea of function was an
essential step for Frege’s fundamental break between the older Aristotelian tradition
and the contemporary approach to logic.
Paraconsistency is the study of logical systems in which the presence of a

contradiction does not imply triviality, that is, logical systems with a non-explosive
negation : such that a pair of propositions A and :A does not (always) 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 perennial philosophical problems. The question of the nature of the contradictions allowed by
paraconsistent logics has been a focus of debate on the philosophical significance
of paraconsistency. Although this book is primarily focused on the
logico-mathematical development of paraconsistency, the technical results
emphasized here aim to help, and hopefully to guide, the study of some of those
philosophical problems.


Preface

ix

Paraconsistent logics are able to deal with contradictory scenarios, avoiding
triviality by means of the rejection of the Principle of Explosion, in the sense that
these theories do not trivialize in the presence of (at least some) contradictory
sentences. Different from traditional logic, in paraconsistent logics triviality is not
necessarily connected to contradictoriness; in intuitive terms (a more formal
account in given in Sect. 1.2) the situation could be described by the pictorial
equation:
contradictoriness þ consistency ¼ triviality
The Logics of Formal Inconsistency, from now on LFIs, introduced in [2] and
additionally developed in [3], are a family of paraconsistent logics that encompasses a great number of paraconsistent systems, including the majority of systems
developed within the Brazilian tradition. An important characteristic of LFIs is that
they are endowed with linguistic resources that permit to express the notion of
consistency of sentences inside the object language by using a sentential unary
connective referred to as ‘circle’: A meaning A is consistent. Explosion in the
presence of contradictions does not hold in LFIs, as much as in any other paraconsistent logic. But LFIs are so designed that some contradictions will cause

deductive explosion: consistent contradictions lead to triviality–intuitively, one can
understand the notion of a ‘consistent contradiction’ as a contradiction involving
well-established facts, or involving propositions that have conclusive favorable
evidence. In this sense, LFIs are logics that permit one to separate the sentences for
which explosion hold, from those for which explosion does not hold. It is not
difficult to see that, in this way, reasoning under LFIs extend and expand the
reasoning under classical logic: although LFIs are technically subsystems of
classical logic, classical logic can be identified with that portion of LFIs that deals
with ‘consistent contradictions’. Therefore LFIs subsume classical reasoning. This
point will be explained in more detail in Sect. 1.2.
We may say that a first step in paraconsistency is the distinction between triviality and contradictoriness. But there is a second step, namely, the distinction
between consistency and non-contradictoriness. In LFIs the consistency connective
 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 between A and :ðA ^ :AÞ, some quite interesting developments become available. Indeed, A may express notions of consistency independent from freedom from contradiction.
The most important conceptual distinction between LFIs and traditional logic is
that LFIs start from the principle that assertions about the world can be divided into
two categories: consistent sentences and non-consistent sentences. Consistent
propositions are subjected to classical logic, and consequently a theory T that
contains a pair of contradictory sentences A; :A explodes only if A is taken to be a
consistent sentence, linguistically marked as A (or :A). This is the only distinction between LFIs and classical logic, albeit with far-reaching consequences:


x

Preface

classical logic in this form is augmented, in such a way that in most cases an LFI
encodes classical logic.
The concept of LFIs generalizes and extends the famous hierarchy of C-systems

introduced in [4] and popularized by hundreds of papers. At the same time, LFIs
expand the classical logical stance, and consequently the majority of the traditional
concepts and methods of classical logic, propositional or quantified (and even
higher-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 of
classical logic with paraconsistent logic. Because LFIs extend the classical stance,
the analogy with transfinite ordinal numbers and with complex numbers is compelling: 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 by
Sextus Empiricus1 against the sophists, is trumpeted against the need of paraconsistent 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 indeed
favorable to the kind of paraconsistency represented by LFIs. The notion of consistency—symbolized as  when applied to propositions—actually increases our
wisdom: it does not stop one to jump into the chasm, but rather marks out the
dangerous roads and, precisely, helps avoid such roads because of the chasm!
The idea that consistency can be taken as a primitive, independent notion, and be
axiomatized for the good profit of logic is a new idea, which permits one to separate
not only the notion of contradiction from the notion of deductive triviality, which is

true 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 unanticipated as the possibilities that imaginary numbers, completed infinite, and Frege’s
idealization of a ‘concept’ as a function mapping arguments to one of the
truth-values represented in mathematics, logic and philosophy. The rest of the book
will speak for itself.
1

Sextus Empiricus, Outlines of Pyrrhonism, LCL 273: 318–319. />view/sextus_empiricus-outlines_pyrrhonism/1933/pb_LCL273.3.xml


Preface

xi

Book’s Content: A Road Map
Chapter 1
Chapter 1 purports to clarify the whole project behind LFIs, making sense of its
idealization and basic tenets. The paraconsistent viewpoint—materialized by means
of LFIs—objects to classical logic, but only on the grounds that contradiction and
triviality cease to coincide, and that contradiction ceases to coincide with inconsistency. But this requires no opposition to the classical stance, just the awareness
that ‘classical’ logic involves some hidden assumptions, as discussed above in this
chapter. In the light of this, Sect. 1.2 makes explicit some of the philosophical
underpinning implicit in LFIs.
Section 1.3 will briefly retrace the motivations for the forerunners of LFIs and
paraconsistency in general. No discussion of paraconsistency can avoid touching
on, if only summarily, questions of the nature of logic, and Sect. 1.4 does this. Next
challenges to be faced are questions about the nature of contradictions. Section 1.5
takes 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 traditional
aspects of logic, namely, ontological, epistemological, and linguistic.
This stance helps to give a justification for the rational acceptance of contradictory sentences, and to better appreciate the distinctions among contradiction,
consistency, and negation, as characterized in Sect. 1.6. It will also help to make
palatable the rationale behind the semantics of LFIs to be developed in all mathematical details in Chaps. 2 and 3, as well as to give support to alternative semantics
for LFIs developed in Chap. 6.
There is a wide variety of reasons for repudiating (or at least to be cautioned
against) classical logic, and many of them find an expression among paraconsistent
logics. This chapter makes clear that LFIs are not coincidental with this spectrum of
philosophical views, neither are they antagonistic, but can be combined with, and
can complement, some of them. A summary of the main varieties of paraconsistency is given in Sect. 1.7, which attempts to clarify the position of LFIs with
respect to other paraconsistent logics in the hope that this will justify some claims
made in next chapters.
Chapter 2
Chapter 2 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 a
negation 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. Positive
classical logic is assumed as a natural starting point from which the LFIs will be
defined, although in Chap. 5 some LFIs will be studied starting from other logics
than 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.


xii

Preface

A remarkable feature of LFIs in general, and of mbC in particular, as mentioned

above, is that classical logic (CPL) can be codified, or recovered, inside such
logics, as shown, for instance, in Sect. 2.4.
One of the criteria proposed by da Costa in [5], p. 498, is that a paraconsistent
calculus must contain as many of the schemata and rules of classical logic as can be
endorsed without validating of the laws of explosion and non-contradiction. This
vague criterion can be formalized in the sense that some LFIs can be proved to be
maximal with respect to CPL, as in the case, for instance, of some three-valued
LFIs treated in Chap. 4.
Moreover, in addition to being a subsystem of CPL, mbC is also an extension of
CPL, obtained by adding to the latter a consistency operator  and a paraconsistent
negation : (see Sect. 2.5). In this sense, mbC can be viewed, both, as a subsystem
and as a conservative extension of CPL. A similar phenomenon holds for several
other LFIs.
That section also sheds light on how CPL can be codified in mbC, showing that
this can be achieved by way of a conservative translation, or by establishing a
Derivability Adjustment Theorem (or DAT) between CPL and mbC. Section 2.5
also discusses an alternative formulation for mbC called mbC? , showing that by
means of linguistic adaptations mbC can be directly introduced as an extension of
CPL.
Chapter 3
Chapter 3 deals with extensions of mbC, which by its turn is a minimal extension
of CPL+ with a consistency operator  and a paraconsistent negation : characterizable as an LFI. This chapter defines several extensions of mbC, strengthening
or expanding different characteristics of this basic system.
In mbC, however, negation and consistency are totally separated concepts. The
first extension of mbC, called mbCciw, is defined as the minimal extension
guaranteeing that the truth-values of α and :α completely determine the truth-value
of α.
Besides being a subsystem of classical logic, mbC is strong enough to contain
the germ of classical negation, possessing a kind of hidden classical negation, as
explained in Sect. 2.4 of Chap. 2. Section 3.2 of this chapter shows that here is

another hidden operator in mbC: an alternative consistency operator β , one for
each formula β. This operator establishes an important distinction, from a conceptual point of view, between mbC and mbC ? as clarified in Sect. 3.4.
When he introduced his famous hierarchy Cn (1  n\ωÞ of paraconsistent
systems, da Costa defined, for each system Cn , a kind of “well-behavedness”
operator (later identified with consistency) in terms of the paraconsistent negation
and conjunction (see Sect. 3.7). A special type of LFIs called dC-systems, characterized by the fact that the consistency operator can be defined in terms of the
others, has been defined in [2]. The systems Cn of da Costa turn out to be examples
of dC-systems. Section 3.3 of this chapter analyzes the formal notion of
dC-systems, and investigates how to expand mbC in order to define the consistency


Preface

xiii

and/or the inconsistency operator in terms of the other connectives of the given
signature.
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 new
operator :α, expressing the (formal) inconsistency of α. Section 3.5 studies the
balance (or better, unbalance) between the formal concepts of consistency and
inconsistency, defining a new LFI (mbC, which, in fact, is a dC-system) where
inconsistency is a primitive notion and consistency is a defined one.
A natural requirement when characterizing consistency, as much as negation, is
how consistency can be propagated through the remaining connectives. Sections
3.6 and 3.8 analyze extensions of mbC enjoying propagation of consistency in
different forms, in the spirit of the historical systems of da Costa.
Chapter 4
Chapter 4 deals with matrices and algebraizability, and their consequences. In
particular, the question of characterizability by finite matrices, as well as the

algebraizability of (extensions of) mbC is tackled. Some negative results, in the
style of the famous Dugundji’s theorem for modal logics, are shown for several
extensions of mbC. This results in new, compact proofs of previously established
results, to the effect that a wide variety of LFIs extending mbC cannot be
semantically characterized by finite matrices. Despite these general results, some
three-valued extensions of LFIs can be characterized by finite matrices, and most
of them are algebraizable in the well-known sense of Blok and Pigozzi. This is
surprising, considering that several extensions of mbC, including the systems Cn of
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 covers
Halldén’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 logic
P1, Priest’s logic LP, the system Ciore, and several other related systems.
Chapter 5
Chapter 5 is devoted to giving an account of LFIs based on other logics, distinct
from what was done in previous chapters, in which LFIs based exclusively on
positive classical logic CPL+ were studied. Although several extensions of the
basic 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, and
some positive modal logics.
Section 5.1 starts by defining LFIs based on positive intuitionistic logic, instead
of CPL+, beginning with paraconsistent logics based on IPL+ (taking as a basis
Johansson’s minimal logic and Nelson’s logic). A weaker version of mbC called


xiv

Preface

imbC obtained from the former by changing the positive basis CPL+ to IPL+ is
also investigated,
Section 5.2 is dedicated to the task of combining two paradigms of uncertainty:
fuzziness and paraconsistency, with exciting possibilities. Taking as a basis the
monoidal t-norm based logic MTL introduced in [6] as a generalization of the
famous basic fuzzy logic BL due to P. Hájek (which, in turn, simultaneously
generalizes three chief fuzzy logics, namely Łukasiewicz, Gödel-Dummet and
Product logics) several new LFIs had been recently developed (see [7]).
Justified by the fact that MTL is the most general residuated fuzzy logic whose
semantics is based on t-norms, the LFIs defined over MTL give a finely controlled
combination 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 and
inconsistency.
Section 5.3 investigates a four-valued modal LFI based on N. Belnap and J.M.
Dunn’s logic BD, a logic (based on their famous bilattice logic F OUR) suitable for
representing lack of information (a sentence is neither true nor false) or excess of
information (a sentence is both true and false). The logic BD was defined from the
notion of proposition surrogates introduced by J.M. Dunn about five decades ago
as a set-theoretic tool for representing De Morgan Lattices. The logic M4m , a matrix
logic 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 logic
preserving degrees of truth of the variety of bounded distributive lattices. The logic
M4m is 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 LFIs
and their unfoldings.
Chapter 6

Chapter 6 studies alternative semantics for the LFIs presented in Chaps. 2 and 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 relational
semantics, topological semantics, and neighborhood semantics, among others) to
better clarify their meaning, LFIs also naturally require a plurality of semantics. But
unlike 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 for
LFIs 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 of
twist structures (studied in Chap. 5), and also discusses the close relationship
between the concept of Fidel structures, swap structures and non-deterministic
matrices (or Nmatrices).
Section 6.8 surveys the possible-translations semantics (PTSs), a broad
semantical concept introduced in 1990 that gives new philosophical interpretations


Preface

xv

for some non-classical logics, and especially for paraconsistent logics. It happens
that PTSs is a very general semantical notion, to the point that virtually any logic
may have a PTS interpretation, under certain conditions. It also happens that
several other semantical notions can be seen as particular cases of PTSs; those
points are carefully explained in that section.
Chapter 7
Chapter 7 gives a full account of LFIs for first-order languages. The quantified

versions of LFIs are essential for certain mathematical applications, such as set
theory, and also for concrete applications in computer science, such as databases
and logic programming. The combination of the consistency operator  with
quantifiers 8 and 9 demands a careful treatment: now, the propagation of consistency through quantifiers has to be duly balanced, generalizing from the propagation of consistency for conjunction and disjunction. The intuitive idea, of course, is
to regard the existential quantifiers as arbitrary conjunctions and disjunctions, but
this has to be done taking a certain technical care.
The chapter is structured around a complete treatment of the system QmbC, a
quantified exension of the system mbC, the basic LFI studied in Chap. 2. Other
extensions of QmbC, such as QCi and QmbC  (the latter including an equality
predicate), are also treated, keeping QmbC at the horizon. From the point of view
of semantics, Tarskian first-order structures are now endowed with a paraconsistent
bivaluation, and what results is a wide generalization of familiar model theory. An
alternative approach to three-valued first-order LFIs is developed in detail in Sect.
7.9, based on the theory of quasi-truth. This treatment, of course, can be extended to
other many-valued paraconsistent logics.
The paradigm of quasi-truth, which provides a way of accommodating the
conceptual incompleteness inherent in scientific theories as studied in [8], views
scientific theories from the perspective of paraconsistent logic. This paradigm offers
a rational account for the dynamics of theory change, allowing for theories
involving contradictions without triviality, with deep implications for the foundations 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], by
means of a three-valued model theory for an LFI called LPT1, which in turn
coincides (setting aside some details of language) with the quantified version of the
three-valued paraconsistent logic LFI1 introduced in Definition 4.4.41. An additional discusion on quasi-truth can be found in Sect. 9.3 of Chap. 9.
One of the aims of this chapter is to endorse the claim that basically the same
results of classical model theory hold for QmbC, and for first-order LFIs in general, with certain provisos. Well-established results in traditional model theory such
as the Completeness, Compactness and Lowenhëim–Skolem Theorems can be
proved for first-order LFIs along the same lines as the classical case. In this way,
the chapter makes clear that first-order LFIs expand traditional logic, and allows for
a revision of the uses of logic in mathematics and computer science from the

vantage point of richer logics.


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Preface

Chapter 8
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. The
Principle of Comprehension (also referred to as the Principle of Naïve
Comprehension, or Abstraction) was proposed in the nineteenth century, fruit of the
somewhat romantic ideas of Dedekind, Cantor, and Frege, and states that for every
property, expressed as a predicate, there exists a set consisting of exactly those
objects that satisfy the predicate. This principle lurks behind certain tough paradoxes, such as Russell’s paradox, and the history of contemporary set theory has
much to do with efforts to rescue Cantor’s naïve set theory from triviality, an
inevitable consequence, in traditional logic, of the contradictions entailed by those
paradoxes. Paraconsistent set theory has been an endeavor to save set theory from
certain (it not all) paradoxes for at least three decades. Chapter 8 aims to offer a new
approach to this question by means of employing LFIs and their powerful consistency operator. By assuming that not only sentences, but sets themselves can be
classified as consistent or inconsistent objects, the basis for new paraconsistent
set-theories that can resist certain paradoxes without falling into trivialism is
established. One of the main motivations of this chapter, as stated in Sect. 8.1, is to
rescue, together with Cantor’s naïve set theory, the proper Cantor’s intuition
towards ‘inconsistent sets’. Indeed, the chapter attempts to show that Cantor’s
treatment of inconsistent collections can be related to the one provided by means of
LFIs.
Section 8.2 defines ZFmbC, a basic system of paraconsistent set theory whose
underlying logic is QmbC  , and which contains two non-logical predicates (besides the equality predicate ): the binary predicate “2” (for membership), and the
unary predicate C (for consistency of sets). Section 8.3 proposes some extensions of

ZFmbC by means of employing stronger LFIs as underlying logics and setting
appropriate axioms for the consistency operator C for sets. Section 8.4 discusses the
relationship between the notions of ‘to be a consistent object in set theory’ (as
formalized 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. In
the same spirit, proper classes can be regarded as inconsistent objects. Such
affinities between consistent objects in set theory and sets, and between proper
classes and inconsistent objects, though it cannot be strengthened into equivalence,
testify to the richness of this approach.
Section 8.5, the last in the chapter, starts the discussion of models of paraconsistent set theory. If the construction of models for standard set theory is a fraught
task, the analogue for paraconsistent set theory is adventurous, to say the least. One
might consider standard models of paraconsistent set theory, where the ε relation of
that model corresponds exactly to the membership relation 2 of the universe of
ZFmbC and its extensions, and the same for the consistency operator , but it is
also reasonable to make room for non-standard models. Only in this way could one
venture into deeper waters, such as extending forcing machinery to paraconsistent


Preface

xvii

set theory. Although this is not done in this book, and it may be an ambitious
project, it is not unrealistic.
Chapter 9
Chapter 9 attempts to clarify the close connections between paraconsistency and
philosophy of science: in a nutshell, there are so many cases of contradictions, even
if temporary, arising between scientific theories, as well as between facts and
theories, that a paraconsistent approach to the foundations of science seem to be
almost inevitable. Section 9.1 advocates an epistemological understanding of

paraconsistency based upon the notion of evidence, and questions its significance
for science supported by some examples of real situations, examined in Sect. 9.2.
Consistency and contradiction in scientific theories can be understood by an epistemic approach to paraconsistency, we claim, inspired by some Kantian insights
about the limits of human reason. Some historical examples of cases where scientists have held contradictory positions, and where science as a whole has gained
from holding them, are reviewed in this section. The controversy surrounding the
movement of the luminiferous aether of the nineteenth century, the controversies in
the early development of quantum theory, the case of Mercury’s orbit and the
failure in hypothesizing Vulcan, a planet that only existed in the heads of certain
astronomers, and the contradictions arising from the ‘imponderable’ phlogiston in
the beginnings of the chemistry of the eighteenth century are illuminating cases.
The provisional contradictions faced by Einstein just before he formulated the
special theory of relativity in 1905 is another typical example of what we call
epistemic contradictions, which arise between two non-contradictory theories that,
when put together, yield contradictory results. The phenomenon is not restricted to
natural sciences: the imaginary numbers, which baffled mathematicians and
philosophers until the beginning of the twentieth century, is another piece of
epistemic contradiction.
Section 9.3 reviews—from a more philosophical perspective—the concept of
pragmatic truth, also referred to as quasi-truth, or partial truth, already analyzed
from the formal point of view in Chap. 7. Quasi-truth, developed as part of efforts to
expand 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,
more flexible, theories of truth held by anti-realist thinkers in philosophy of science.
Section 9.4 emphasizes the evidence-based approach to paraconsistency, in the
sense of understanding a pair of contradictory sentences as representing, and
allowing us to reason about, conflicting evidence, defending this view as particularly promising for philosophical interpretations of paraconsistent logics.
The last section, Sect. 9.5, succinctly wraps up one of the chief points behind
LFIs: they are concerned with truth, since classical logic can be fully recovered
inside most of the LFIs, but they are also concerned with the notion of evidence, a
notion weaker than truth that allows for an intuitive and plausible understanding

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


xviii

Preface

stresses the brotherhood between the intuitionistic and the paraconsistent paradigms, but explains the adequacy of LFIs for wider accounts in the philosophy of
science, and also their applicability in the fields of linguistics, theoretical computer
science, inferential probability, and confirmation theory.

References
1. Frege, Gottlob. 1891. Über Funktion und Begriff (in German). In Jenaischen Gesellschaft für
Medizin und Naturwissenschaft. Jena: Verlag Hermann Pohle. Translated by Peter Geach as
‘Function and Concept’. In Translations from the Philosophical Writings of Gottlob Frege, ed.
Peter Geach and Max Black, Blackwell, 1980.
2. Carnielli, Walter A., and João Marcos. 2002. A taxonomy of C-systems. In Paraconsistency: The
Logical Way to the Inconsistent. Proceedings of the 2nd World Congress on Paraconsistency
(WCP 2000), volume 228 of Lecture Notes in Pure and Applied Mathematics, eds. Carnielli,
Walter A., Marcelo E. Coniglio, and Itala M.L. D’Ottaviano, 1–94, New York: Marcel Dekker.
3. Carnielli, Walter A., Marcelo E. Coniglio, and João 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 Paraná, 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 Lluís 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 Lluís 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, Otávio 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.


Acknowledgments

We would like to express our gratitude to the many organizations and people who
read, 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 support
from FAPESP (Thematic Project LogCons 2010/51038-0, Brazil) and from individual research grants from The National Council for Scientific and Technological
Development (CNPq), Brazil. The intellectual environment of the Centre for Logic,
Epistemology and the History of Science (CLE) of the State University of
Campinas—UNICAMP deserves a special mention: we thank the colleagues and
the officers of CLE for having provided all necessary facilities, from library
facilities, to secretarial work, computers, and good coffee.
Personal thanks go to Abílio Rodrigues (Belo Horizonte), Henrique Antunes
Almeida (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 their
continuous support over the many years this book took to complete. We beg
forgiveness from all those we might have unintentionally failed to mention: we
have made every effort to leave that set consistently empty.

xix


Contents

1 Contradiction and (in)Consistency . . . . . . . . . . . . . . . . . . . .
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 On the Philosophy of the Logics of Formal Inconsistency.
1.3 A Historical Sketch: The Forerunners of the Logics
of Formal Inconsistency . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Paraconsistency and the Nature of Logic . . . . . . . . . . . .
1.5 Paraconsistency and the Nature of Contradictions . . . . . .
1.6 Contradiction, Consistency and Negation . . . . . . . . . . . .
1.6.1 On Contradiction . . . . . . . . . . . . . . . . . . . . . . .
1.6.2 On Consistency . . . . . . . . . . . . . . . . . . . . . . . .
1.6.3 On Negation . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7 Varieties of Paraconsistency Involvement . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 A Basic Logic of Formal Inconsistency: mbC . . . . .
2.1 Introducing mbC . . . . . . . . . . . . . . . . . . . . . .
2.2 A Valuation Semantics for mbC . . . . . . . . . . .

2.3 Applications of mbC-Valuations . . . . . . . . . . .
2.4 Recovering Classical Logic Inside mbC . . . . . .
2.5 Reintroducing mbC as an Expansion of CPL . .
2.5.1 The New Presentation mbC? of mbC .
2.5.2 Valuation Semantics for mbC . . . . . . .
2.5.3 Equivalence Between mbC and mbC? .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Some
3.1
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3.3
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Extensions of mbC . . . . . . . . . . . . . . . . . . . . . . . . . .
A Wider Form of Truth-Functionality for Consistency . .
A Hidden Consistency Operator . . . . . . . . . . . . . . . . .
Consistency and Inconsistency as Derived Connectives. .
Some Conceptual Differences Between mbC and mbC?
Inconsistency Operators and Double-Negations . . . . . . .

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Contents

3.6 Propagating Consistency . . . . . . . . . . . . . . . . . . .
3.7 da Costa’s Hierarchy and Consistency Propagation
3.8 A Stronger Consistency Propagation . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Matrices and Algebraizability. . . . . . . . . . . . . . . . . . . . .
4.1 Logical Matrices . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Uncharacterizability by Finite Matrices . . . . . . . . . . .
4.3 The Problem of Algebraizability of LFIs . . . . . . . . .
4.4 Some 3-Valued LFIs . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Halldén’s Logic of Nonsense (1949) . . . . . .
4.4.2 Segerberg’s Logic of Nonsense (1965) . . . . .
4.4.3 da Costa and D’Ottaviano’s Logic J3 (1970).
4.4.4 Sette’s Logic P1 (1973) . . . . . . . . . . . . . . .
4.4.5 Asenjo-Priest’s Logic LP (1966–1979) . . . . .
4.4.6 Ciore and Other Related Systems . . . . . . . .
4.4.7 LFI1, MPT and J3 . . . . . . . . . . . . . . . . . .

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 LFIs Based on Other Logics. . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 LFIs Based on Positive Intuitionistic Logic . . . . . . . . . . . .
5.1.1 Basic Features of Positive Intuitionistic Logic . . . .
5.1.2 Johansson’s Minimal Logic . . . . . . . . . . . . . . . . .
5.1.3 Nelson’s Paraconsistent Logic N4 . . . . . . . . . . . .
5.1.4 An Intuitionistic Version of mbC . . . . . . . . . . . .
5.2 LFIs Based on Fuzzy Logics. . . . . . . . . . . . . . . . . . . . . .
5.2.1 Preliminaries on MFL . . . . . . . . . . . . . . . . . . . .
5.2.2 Fuzzy Logics with a Consistency Operator . . . . . .
5.2.3 Propagation of Consistency and DAT. . . . . . . . . .
5.2.4 Fuzzy Logics with an Inconsistency Operator . . . .
5.3 A Modal LFI Based on Belnap and Dunn’s Logic BD . . . .
5.3.1 The Logic M4m of Tetravalent Modal Algebras . . .
5.3.2 M4m as an LFI . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.3 M4m as a dC-System . . . . . . . . . . . . . . . . . . . . .
5.3.4 The Contrapositive Implication . . . . . . . . . . . . . .
c
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5.4 Paraconsistent Modalities, Consistency and Determinedness
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Semantics of Non-deterministic Character for LFIs . . . . . . .
6.1 Fidel Structures for mbC . . . . . . . . . . . . . . . . . . . . . .
6.2 Fidel Structures for Some Extensions of mbC . . . . . . . .
6.3 Non-deterministic Matrices . . . . . . . . . . . . . . . . . . . . .
6.4 Swap Structures for mbC . . . . . . . . . . . . . . . . . . . . . .
6.5 Swap Structures for Some Extensions of mbC . . . . . . .
6.6 Axiom (cl) and Uncharacterizability by Finite Nmatrices

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6.7
6.8

Some Remarks on Fidel Structures and Swap Structures . . . .
The Possible-Translations Semantics . . . . . . . . . . . . . . . . .
6.8.1 Possible-Translations Semantics for Some LFIs. . . .
6.8.2 A 3-Valued Possible-Translations Semantics for Cila
6.8.3 Some Remarks on Possible-Translations Semantics .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 First-Order LFIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1 The Logic QmbC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Basic Properties of QmbC . . . . . . . . . . . . . . . . . . . . . . . .

7.3 Tarskian Paraconsistent Structures . . . . . . . . . . . . . . . . . . .
7.4 Soundness Theorem for QmbC . . . . . . . . . . . . . . . . . . . . .
7.5 Completeness Theorem for QmbC. . . . . . . . . . . . . . . . . . .
7.5.1 Henkin Theories . . . . . . . . . . . . . . . . . . . . . . . . .
7.5.2 Canonical Interpretations. . . . . . . . . . . . . . . . . . . .
7.6 Compactness and Lowenhëim-Skolem Theorems for QmbC .
7.7 QmbC with Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.8 First-Order Characterization of Other Quantified LFIs . . . . .
7.9 First-Order LFI1 and the Logic of Quasi-truth . . . . . . . . . .
7.9.1 Semantics of Partial Structures. . . . . . . . . . . . . . . .
7.9.2 The Logic QLFI1 . . . . . . . . . . . . . . . . . . . . . . .
7.10 First-Order P1 and Partial Structures . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8 Paraconsistent Set Theory . . . . . . . . . . .
8.1 Antinomic Sets and Paraconsistency.
8.2 LFIs Predicating on Consistency . . .
8.3 Some Extensions of ZFmbC. . . . . .
8.4 Inconsistent Sets and Proper Classes
8.5 On Models . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . .

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9 Paraconsistency and Philosophy of Science: Foundations
and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1 An Epistemological Understanding of Paraconsistency,
and Its Significance for Science . . . . . . . . . . . . . . . . .
9.2 Consistency and Contradiction in Scientific Theories . .
9.2.1 The Heritage of Kant . . . . . . . . . . . . . . . . . .
9.2.2 Some Historical Examples . . . . . . . . . . . . . .
9.2.3 The Beginning of Quantum Theory and
Paraconsistency . . . . . . . . . . . . . . . . . . . . . .
9.2.4 Mercury’s Orbit and a Non-existent Planet . . .
9.2.5 Contradictions in Phlogiston, the Imponderable
9.2.6 The Special Theory of Relativity . . . . . . . . . .
9.2.7 Mathematics, and the Meaning of Objects
that Mean Nothing . . . . . . . . . . . . . . . . . . . .

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xxiv

Contents

9.3

Quasi-truth and the Reconciliation of Science
and Rationality . . . . . . . . . . . . . . . . . . . . . . . .
9.4 An Evidence-Based Approach to Paraconsistency
9.5 Summing Up. . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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382
384
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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
Index of Logic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397


Chapter 1

Contradiction and (in)Consistency

1.1 Introduction
The target audience of this book is mainly the philosopher, the logician interested in
the philosophical aspects of paraconsistency, and the computer scientist looking for
new logics for applications. But the intended audience also includes the mathematician intrigued by the possibility of working in a logic that allows contradictions (a
paraconsistent logic), the linguist worried about the acceptance of contradictions in
the 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 contradictions
are 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 and
sellers, many times encounter contradictions, and strive to overcome them in order
to strike a deal. Paradoxes in formal semantics, as the famous liar paradox, are

seen as dangerous to the standard theories of truth, and paradoxes in naive (albeit
intuitively acceptable) set theory are seen as threats to the foundations of science and
mathematics.
However, contradictory information is not only frequent, and more so as systems
increase in complexity, but can have a positive role in human thought, in some cases
being desirable. Finding contradictions in juridical testimonies, in statements from
suspects of a crime or in suspects of tax fraud can be an efficient strategy. Contradictions can be very informative: we will never know if people being questioned
coherently 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 5th 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


2

1 Contradiction and (in)Consistency

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 logically
follows. But how can standard logic, which endorses ECSQ, impose a principle that is

not followed by common reasoning? Are all contradictions really equally hazardous?
The so called Bar-Hillel-Carnap paradox (see [3], p. 229) has already suggested,
half century ago, the clash between the notions of contradiction and semantic information: the less probable a statement is, the more informative it is, and so contradictions carry the maximum amount of information, and in the light of standard logic
are, 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 equate
triviality and contradiction, and to regard all contradictions as equivalent, as the following example illustrates. If two auto technicians tell me that the battery of my
car 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 a
huge amount of irrelevant information. Classically, this trivial amount of information
is exactly the same as the information conveyed by the car technicians telling me a
contradiction, such as the battery of my car is flat and that it is not flat. However, if
one of the car technicians tells me (among his statements) that the battery is flat, and
the other that the battery is not flat, between them they are contradictory, but now I
know 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,
sometimes confused2 with Ex Falso Sequitur Quodlibet, or Ex Impossibile Sequitur
Quodlibet 3 might 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
consequences: in standard logic, a theory is by definition consistent if no pairs
of contradictory sentences α, ¬α are deducible from . Consequently, in light of
ECSQ, the notion of consistency is inescapably dependent on negation, and is

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.
3 To the best of our knowledge, the exact expressions Ex Contradictione Sequitur Quodlibet and Ex
Contradictione Quodlibet have been independently coined by, respectively Priest and BobenriethMiserda, see [6].