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Manuscrito – Rev. Int. Fil. Campinas, v. 41, n. 2, pp. 111-122, abr.-jun. 2018.
BOOK REVIEW: CARNIELLI, W., CONIGLIO, M.
Paraconsistent Logic: Consistency, Contradiction and Negation.
Logic, Epistemology, and the Unity of Science Series. (New
York: Springer, 2016. ISSN: 2214-9775.)
Henrique Antunes
State University of Campinas
Department of Philosophy
Campinas, SP
Brazil
antunes.henrique@outlook.com
Vincenzo Ciccarelli
State University of Campinas
Department of Philosophy
Campinas, SP
Brazil
ciccarelli.vin@gmail.com
Article info
CDD: 160
Received: 01.12.2017; Accepted: 30.12.2017
DOI: http://dx.doi.org/10.1590/0100-6045.2018.V41N2.HV
Keywords:
Paraconsistent Logic
LFIs
ABSTRACT
Review of the book 'Paraconsistent Logic: Consistency, Contradiction and
Negation' (2016), by Walter Carnielli and Marcelo Coniglio
The principle of explosion (also known as ex contradictione
sequitur quodlibet) states that a pair of contradictory formulas
entails any formula whatsoever of the relevant language and,
accordingly, any theory regimented on the basis of a logic for
which this principle holds (such as classical and intuitionistic
logic) will turn out to be trivial if it contains a pair of
theorems of the form A and ¬A (where ¬ is a negation
operator). A logic is paraconsistent if it rejects the principle of
explosion, allowing thus for the possibility of contradictory
and yet non-trivial theories.
Among the several paraconsistent logics that have been
proposed in the literature, there is a particular family of
(propositional and quantified) systems known as Logics of
Formal Inconsistency (LFIs), developed and thoroughly studied
Henrique Antunes & Vincenzo Ciccarelli 112
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within the Brazilian tradition on paraconsistency. A
distinguishing feature of the LFIs is that although they reject
the general validity of the principle of explosion, as all other
paraconsistent logics do, they admit a a restrcited version of
it known as principle of gentle explosion. This principle asserts
that a contradiction that concerns a consistent formula
logically entails any other formula of the language. The
expression ‘consistent’ here is a generic term susceptible to
several alternative interpretations (not necessarily coinciding
with non-contradiction), depending on the particular LFI
under consideration. Another (related) feature that
distinguishes the LFIs from other paraconsistent logics is
that they internalize this unspecified notion of consistency
inside the object language by means of a unary sentential
operator ○ (called ‘consistency operator’ or simply ‘circle’).
When prefixed to a formula A, ○ expresses that A is
consistent or well behaved, however these expressions are to
be interpreted in each particular case.
Paraconsistent Logic: Consistency, Contradiction and Negation, by
Walter Carnielli and Marcelo Coniglio, is entirely devoted to
the Logics of Formal Inconsistency. The book covers the
main achievements in the field in the past 50 years or so,
presenting them in a systematic and (to a great extend) self-
contained way. Although the book is mostly concerned with
particular logical systems, the relations among them, and
their corresponding metatheoretical properties, it also sets
the basis of a new philosophical interpretation of
paraconsistent logics.
The book contains nine chapters, which altogether cover
several topics about the LFIs. In Chapter 1 the authors
explain the rationales behind paraconsistent logics in general
and the LFIs in particular, and discuss the philosophical
problems related to paraconsistency under the light of some
general issues in the philosophy of logic (such as the nature
of logic and the nature of contradictions). It is argued that
since there are some real life situations in which
contradictions do actually turn up, paraconsistent logics are
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justified, no matter how those contradictions are interpreted
– whether they are seen as concerning reality or knowledge.
The chapter also discusses the relation between
paracomplete and paraconsistent logics and analyzes some
key notions related to paraconsistency, such as consistency,
contradiction (and the principle of non-contradiction) and
negation.
In Chapter 2 the concept of LFI is precisely defined, as
well as other basic technical notions employed throughout
the book. A minimal propositional LFI, called mbC, is
introduced by means of an axiomatic system. mbC results
from positive classical propositional logic by the inclusion of two
additional axioms: the principles of excluded middle and
gentle explosion – A ∨ A and ○A → (A → (¬A → B),
respectively. mbC is then provided with a valuation semantics
with respect to which it is proved to be sound and complete.
The relations between mbC and classical propositional logic
are carefully analyzed. The analysis reveals that mbC can be
viewed both as a sublogic and as an extension of classical logic,
when these terms are suitably qualified.
Chapter 3 presents several extensions of mbC and
analyzes the relations between the notions of
consistency/inconsistency and contradictoriness/non-contradictoriness –
formally expressed by the formulas ○A/¬○A and A ˄
¬A/¬(A ˄ ¬A), respectively. As it turns out, although
consistency and non-contradictoriness (and inconsistency
and contradictoriness) are partially independent in mbC,
they may or may not coincide in some of its extensions. In
addition, the notion of a C-system is introduced. Despite the
complexity of the relevant definition, a C-system simply
amounts to an LFI within which the consistency operator is
definable in terms of the other connectives of the language.
Da Costa’s hierarchy of paraconsistent logics – a family of paradigm
examples of C-systems – is briefly presented and explained.
The chapter also deals with the important notions of
propagation and retro-propagation of the consistency
operator.
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The first part of Chapter 4 is devoted to the problem of
the algebraizability of some LFIs, and the second part
discusses some many-valued LFI-systems. In Section 4.1
some preliminary concepts concerning logical matrices are
introduced. Section 4.2 contains a Dugundji-style proof of
the uncharacterizability by finite matrices of the LFIs
presented so far. Section 4.3 contains a proof of the
algebraizability of some extension of mbC in the broader
sense of Block and Pigozzi. The remaining sections deal
separately with different many-valued LFIs, most of which
were proposed several decades before the emergence of the
concept of Logic of Formal Inconsistency.
Chapter 5 represents a partial detour from the main
exposition, for the systems presented therein are not
extensions of positive classical propositional logic. The first
case considered by the authors is that of intuitionistic logic:
more specifically, it is shown how a consistency operator ○
can be defined within Nelson’s logic N4 in terms of a strong
negation ~ operator (i.e., ○A ≡ ~(A ˄ ¬A)). Another
interesting case covered by the chapter is that of modal logic,
where the consistency operator is shown to be interpretable
as having a sort of “modal flavor”. In particular, the
definition ○A ≡ A → □A can be introduced in normal non-
degenerate modal logics. Some systems of fuzzy logic are
also analyzed in the chapter. In all of the aforementioned
logics, the strategy pursued by the authors consists in
defining a consistency operator within the system in question
and then showing that it satisfies the general definition of an
LFI.
Chapter 6 is devoted to the problem of defining non-
deterministic semantics for non-algebraizable systems (even
in the broader sense of Block and Pigozzi). It presents three
main formal semantics – based, respectively, on F-structures,
non-deterministic logical matrices, and possible translations. Of
particular interest, especially from a more philosophical
point of view, is the so-called possible translation semantics,
whose main idea is to translate a given logic into logics whose
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semantics are well known and deterministic. The relevant
notion of translation is that of a mapping preserving logical
consequences and the rationale for this approach is the
interpretation of a logic as a combination of “possible world
views”.
Chapter 7 concerns first-order LFIs. The chapter is
mainly devoted to two systems: QmbC, the first-order
extension of mbC, and QLFI1. Due to the non-
deterministic nature of mbC, a non-standard semantics is
defined for its first-order extension: the authors introduce
the notion of a Tarskian paraconsistent structure, defined as an
ordered pair composed of a Tarskian structure (in the
classical sense) together with a non-deterministic valuation.
Concerning QLFI1, the approach is twofold: on one hand,
it is shown how the language may be interpreted in a suitable
Tarskian paraconsistent structure; on the other hand, a
different semantics is proposed, given that the propositional
fragment of QLFI1 can be characterized by a three-valued
matrix. The semantics is represented by a partial structure,
defined in a similar way to a classical Tarskian structure,
except for the fact that all predicate symbols are interpreted
as partial relations. Both QmbC and QLFI1 are proved to be
sound and complete with respect to the corresponding
semantics. Compactness and Lowenhëim-Skolem theorems
are proved for QmbC.
Chapter 8 concerns one of the most straightforward
applications of paraconsistent logics: set theory.
Nevertheless, the authors’ approach to the subject is
substantially different from what has been traditionally done
in the field of paraconsistent set theory – namely, to
formulate a non-trivial naïve set theory countenancing the
unrestricted comprehension principle for sets. The systems
presented in the chapter include all of Zermelo-Fraenkel set
theory’s axioms (except for the axiom of foundation, which is
replaced by a weaker version of it) with an LFI as the
underlying logic. Another distinguishing feature of those
systems is that they include a consistency predicate for sets
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whose behavior is governed by a set of additional axioms.
Hence, whereas in a propositional LFI the property of
consistency applies only to formulas, in the corresponding
paraconsistent set theories it applies to both formulas and
sets. The main results of the chapter are the derivability
adjustment theorem (establishing that any derivation in ZF can
be recovered within its paraconsistent counterpart) and a
proof of the non-triviality of the strongest system presented
in the chapter.
Chapter 9 discusses the significance of contradictions for
science, describing some historical paradigm examples where
contradictions seem to have played an important role in the
development of scientific theories. It also proposes an
interpretation of paraconsistent logics according to which
they are better viewed as possessing an epistemological,
rather than an ontological, character; in a nutshell, this means
that they are not supposed to deal primarily with reality and
truth (as in the case of classical logic), but with the epistemic
notion of evidence. This interpretation is meant to be a more
palatable alternative to dialetheism (the thesis that there are
true contradictions), since it neither affirms the existence of
true contradiction nor rejects classical logic as incoherent –
adhering thus to logical pluralism.
One of the main virtues of Paraconsistent Logic: Consistency,
Contradiction and Negation is that it keenly highlights the
pervasiveness and generality of the notion of logic of formal
inconsistency. Firstly, because it shows through the
definition of an LFI how several systems of paraconsistent
logic proposed in the literature – which at first sight might
have appeared to be quite unrelated with one another – can
be framed under a single unifying concept. Secondly,
because it emphasizes that the definition of an LFI is
applicable to systems based on logics of various different
kinds, such as classical, intuitionistic, fuzzy, and modal logic.
The resulting multiplicity of systems allows for various
alternative semantic approaches, which are carefully
described in several chapters of the book (e.g., valuation
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semantics, deterministic and non-deterministic matrices, F-
structures, swap structures, possible translations semantics).
The book is mainly devoted to the taxonomy of LFI-
systems, leaving little room for a more detailed discussion of
the intrinsic properties of each particular system. This is
understandable, though, since it is not meant to be a
textbook. However, it is possible to use the book as an
introductory text on formal paraconsistency by skipping
some of the more technical chapters (e.g., a reader merely
interested in those LFIs based on positive classical
propositional logic may well skip chapters 5, 6 and possibly
8).
Concerning the more philosophical chapters of the
book (chapters 1 and 9), the reader might think that the
issues discussed therein would have deserved a more
extended and rigorous analysis, especially when compared to
the painstakingness of the other chapters. In particular, she
might find the epistemic interpretation of paraconsistent
logics wanting, despite its initial plausibility, this view in not
sufficiently argued for. Moreover, specific relations between
the epistemic interpretation and the particular features of the
LFIs are missing. Nevertheless, this apparent shallowness is
presumably due to the fact the purpose of those chapters is
not to thoroughly develop a philosophical theory about
paraconsistency, but merely to indicate some conceptual
possibilities. After all, Paraconsistent Logic is mainly a technical
piece of work.
So much for the general considerations. There are two
specific points that we think would deserve a more detailed
discussion. The first one concerns the cumbersome notation
employed in the characterization of the semantics of first-
order LFIs (Chapter 7): the strategy adopted by the authors
in that chapter consists in extending the (non-deterministic)
propositional valuations to the first-order case, combining
these with a (classical) Tarskian structure – characterized, as
usual, by a non-empty domain together with an
interpretation function. The resulting first-order valuations
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apply thus only to sentences and the notion of truth, as in
the propositional case, is not defined in terms of
assignments, sequences, or any other technical device usually
employed in order to interpreted the variables. The absence
of any of these devices leads the authors to locally indicate
all the relevant substitutions of individual constants for the
free variables of a given formula. In the case of QmbC, for
example, the semantic value of a quantified formula ∀xA
(under a structure 𝔄 and a valuation v) is defined by means
of the following clause:
v(∀xA) = 1 iff v(A[x / ā]) = 1, for every a in the domain of 𝔄
where A[x / ā] denotes the result of substituting the constant
ā for all free occurrences of x in A, and where the language
is supposed to have at least one individual constant ā for
each elements a of the domain of 𝔄 (that is, the language is
supposed to be diagrammatic). At first sight, the use of the
notation [x / ā] (and its generalization [x1,…, xn / ā1,…, ān]
to multiple simultaneous substitutions) does not seem to
compromise readability at all – in fact, they are usually
employed in the definition of substitutional semantics for
first-order logic. However, matters become much more
complicated when it comes to the additional clauses
introduced in the definition of v(A) in order to guarantee that
the substitution lemma holds for Tarskian paraconsistent
structures. One of these clauses, which concerns the
negation operator, is formulated as follows:
(sNeg) For every contexts (x
; z) and (x
; y), for every sequence
(a ; b
) in the domain of 𝔄 interpreting (x
; y
), for every A ∊
L(𝔄)x
; z and every t ∊ T(𝔄)x
; y
such that t is free for z in A, if
A[z/t] ∊ L(𝔄)x
; y
and c = (t[x
; y
/ a ; b
])𝔄 `then:
If v((A[z/t])[x
; y
/ a ; b
]) = v(A[x
; z / a ; c]) then
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v((¬A[z/t])[x
; y
/ a ; b
]) = v(¬A[x
; z / a ; c])
Without attempting to individually explain every piece of
notation above, (sNeg) merely expresses that if the
substitution lemma holds for a formula A, then it holds for
its negation as well (the introduction of this clause, absent in
the definition of classical first-order structures, is necessary
given the non-deterministic behavior of the negation
operator in mbC). Now, it is quite clear that the reader
would probably take several minutes to read and understand
(sNeg). Moreover, this situation is not restricted to (sNeg),
but it also happens with the similar clause concerning the
consistency operator and the formulation and proof of
various semantic theorems enunciated in Chapter 7. The
notational cumbersomeness of the chapter is further
worsened by the introduction of the notion of extended
valuation, which assigns a truth value to an arbitrary formula
A (not necessarily a sentence) by indicating a sequence of
individual constants with respect to which A is to be
evaluated. More precisely, if the free variables in A are
among x1,…, xn (abbreviated by x ) then the truth value of
A under the extended valuation vx
a is simply v(A[x1,…, xn
/ ā1,…, ān]). This notion represents a simile of the notion of
satisfaction and is necessary in order to provide an
interpretation for the open formulas.
The notation of Chapter 7 could, however, be greatly
simplified in the following way: instead of importing the
notion of valuation from the corresponding propositional
LFI, the authors could well have defined a new notion of
valuation which assigns one of the truth values 0 or 1 to each
pair (s, A), where s is an assignment of objects of the domain
to first-order variables and A is an arbitrary formula (open
or closed). All definitions and theorems of the chapter could
then be easily adapted according to this strategy, yielding
much simpler formulations. In particular, clause (sNeg)
above would become:
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(sNeg’) Let A be a formula with at least one free variable z and let
t be a term free for z in A. Let s be an assignment in a structure 𝔄
and let s’ be the assignment which is just like s except that is assigns
the interpretation of t under s to the variable z. Then:
If v(s’, A) = v(s, A[z / t]) then v(s’, ¬A) = v(s, ¬A[z / t])
In addition to the evident simplicity of this new
formulation, it is worth mentioning that since the notion of
valuation above applies to any formula whatsoever of the
language (open or closed), it is unnecessary to introduce
extended valuations, resulting in a significant conceptual
simplification.
Our second criticism concerns the paraconsistent set
theories of Chapter 8. In general, the main motivation for a
paraconsistent set theory is to recover the intuitive notion of
set codified in the unrestricted principle of comprehension
– i.e., the idea that every property P determines a set of all
and only those objects having P. Of course, this can only be
achieved by renouncing to classical logic, since that principle
classically entails the existence of contradictory sets (e.g.,
Russell’s set, universal set, etc.). On the other hand, classical
set theories (such as ZF) maintain classical logic at the cost
of imposing what seems to be ad hoc restrictions to the
comprehension principle and countenancing additional
principles whose justification seems also ad hoc. Hence,
paraconsistent and classical set theories are symmetrically
opposed to one another: what the former tries to achieve
(i.e., preserve the intuitive notion of set) is given up by the
latter, and what the latter preserves (i.e., classical logic) the
former revises.
Nevertheless, the approach to paraconsistent set theory
adopted by the authors diverges significantly from these two
trends. Firstly, because the attempt to recover the intuitive
notion of set codified in the principle of comprehension is
explicitly given up once they opt for ZF-like axiomatizations
of their theories – ruling out well-known inconsistent
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collections from the outset. Secondly, given that those
theories are variations of ZF based on one or another LFI,
the revision of the underlying logical theory is achieved by
extending classical logic, rather than renouncing to it. In fact,
each of the set theories of Chapter 8 is equivalent to ZF
under the assumption that all sets enjoy the property of
consistency.
This particular take on paraconsistency may leave the
reader wondering what is the point of having a
paraconsistent set theory that does not explicitly
countenance contradictory collections (‘Why not just stick
with ZF?’, she might ask.). The book does not provide an
explicit answer to this question, though. However, it would
not be difficult to imagine a scenario in which the systems of
Chapter 8 would be vindicated: suppose that ZF is someday
shown to be inconsistent. Under this circumstance, any of
those systems could be used to preserve the strength of ZF
while avoiding its triviality. Even though a paraconsistent
set theory of this kind may turn out to be fruitful, its
fruitfulness turns on an unlikely possibility, though – namely,
that ZF could be inconsistent. In view of such a possible
application, we suggest that the approach to paraconsistent
set theory adopted by the authors is aimed at presenting
alternative versions of ZF that are more “cautious” in the
sense that they would be able to withstand contradictions,
should they ever arise within ZF. For this reason, we believe
that those theories should not be viewed as competitors to
classical set theories, but rather as interesting and possibly
useful variations of it, whose mathematical properties are
nonetheless worth investigating.
Paraconsistent Logic: Consistency, Contradiction and Negation is
a comprehensive text on the LFIs and fulfills an important
gap in the literature on paraconsistency. A huge amount of
significant results is presented for the first time in a single
text, providing the reader with an extensive survey of the
research in the area. Moreover, the content of the book is
not limited to the achievements of the so-called Brazilian
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school of logic, but also encompasses contributions coming
from other areas and research groups. As a result, it is highly
recommended for everyone interested in both the formal
and the philosophical aspects of paraconsistency, including
mathematicians, linguistics, computer scientists, and
philosophers of language, mathematics and science.
References
CARNIELLI, W., CONIGLIO, M. Paraconsistent Logic:
Consistency, Contradiction and Negation. Logic,
Epistemology, and the Unity of Science Series. New York:
Springer, 2016.
