Posts com a Tag ‘Consistência’
Contradictions, from Consistency to Inconsistency – CARNIELLI; MALINOWSKI (M)
CARNIELLI, Walter; MALINOWSKI, Jacek. Contradictions, from Consistency to Inconsistency. Trends in Logic 47. Springer International Publishing, 2018. VI+322 pagesp. Resenha de: TESTA, Rafael R. Manuscrito, Campinas, v.42 n.1 Jan./Mar. 2019.
Trends in Logic is the conference series of the journal Studia Logica, covering contemporary formal logic and its relations to other disciplines. The works collected in this volume were initiated by the discussions that took place at the conference to commemorate the 40th Anniversary of the Centre for Logic, Epistemology and History of Science. The title of the event celebrates one of the three main areas of CLE – that has been called as the epicentre of a “Brazilian school of paraconsistency”. The reasons for that are the original works of da Costa, followed by his pupils and collaborators that are part of CLE’s history. Simply put their interest include the development of systems strong enough to encompass most of mathematics, while avoiding some well-known logical paradoxes.
Not surprisingly most of the works in this volume are developed around paraconsistent logics or, as it is explained by the editors in the introductory chapter, they are concerned about the subtle distinctions between consistency and non-contradiction, as well as among contradiction, inconsistency and triviality. There are many interesting problems discussed in the book, some of them are well-known among readers familiar with paraconsistency. Setting aside the introduction, the book itself does not intend to be a historical review on the main questions regarding the subject. Rather, the chapters help the reader to taste some information about where paraconsistency is now and where it is heading, as well as cast new lights on some old questions regarding the consistency of formal theories. In what follows I succinctly present the central elements of each chapter.
The introduction briefly presents some state-of-the-art discussion regarding the central questions that permeate the book. In the homonym chapter, Carnielli and Malinowski explain the title of the book and show the relevance of the subject in contemporary discussions in logic and philosophy of science – themes that are familiar to the authors. Walter Carnielli is full professor of Logic at the State University of Campinas (Unicamp) and served as the director of CLE, as well as editor and member of editorial boards of major journals. Some of his works encompass for instance combinations of logics, many-valued and paraconsistent logics – like the logics of formal inconsistency advanced by Carnielli and Marcos (2002) that systematises a large class of paraconsistent logics. Jacek Malinowski is the editor-in-chief of Studia Logica, Head of the Department of Logic and Cognitive Science at the Polish Academy of Science and Head of the Section of Logical Semiotics at Nicolaus Copernicus University in Torun, Poland. He has published works in several areas, for instance logical foundations of computer sciences, nonmonotonic and cognitive logic, just to name a few. The book reflects the multidisciplinary interest of the editors.
The second chapter (the first of 13 collaborative papers) brings Arenhart’s investigation on an overlooked argument advanced by da Costa (1997) to the effect that there may be true contradictions about the concrete world. The novelty of the chapter “The Price of True Contradictions About the World” is bridging da Costa’s argument to a well-known dialetheist understanding of paraconsistency. By advancing several objections to the argument, the daring conclusion drawn by the author is that the acceptance of true contradictions about the world comes with heavy prices to pay: for instance adopting an inconvenient conservative and pessimistic attitude towards change in science.
In “The Possibility and Fruitfulness of a Debate on the Principle of Non-contradiction”, Estrada-González and del Rosario Martínez-Ordaz go back to the Aristotelian arguments regarding the principle of non-contradiction (PNC) originally advanced in his Metaphysics. The aim is to show how they can be used for a better understanding of the different standpoints that are present in the contemporary debate. The authors advance five major stances regarding the debate on the PNC, namely: Detractors, Fierce supporters, Demonstrators, Methodologists and Calm supporters. They suggest how we can find elements of those instances in several authors in the literature, from Aristotle up to the present. Maybe the main claim of this chapter is that one can find all the elements of Calm supporters already in Aristotle’s works.
Friend and del Rosario Martínez-Ordaz explore a formal method to model the fact that sometimes mathematicians and scientists reason with inconsistent premises while denying that this is possible or makes any sense – a tooll called Chunck and Permeate (C&P) advanced by Bryson and Priest (2004). Roughly speaking, C&P divides a given proof with inconsistent premises into consistent subsets, called chunks, and allows only some information to permeate from one chunk to the next. In “Keeping Globally Inconsistent Scientific Theories Locally Consistent”, the authors extend C&P by adding a visual representation of chunks in the form of bundle diagrams. By extending it, they apply the method to analyse a case in physics and discuss the implications of inconsistency toleration in science, possibly opening up avenues for other discussions in the role of logic in science.
In “What is a Paraconsistent Logic?”, Barrio, Pailos and Szmuc recall some canonical definitions of paraconsistent logics (advanced for instance by Priest, Tanaka and Weber (2016); Carnielli and Coniglio (2016); and Ripley (2015)) in order to suggest a new one. By taking into account a meta-inferential notion of explosion, the authors bring into the light the fact that some logical systems might validate the Explosion Principle but invalidate a meta-inferential version of it. Relaying on some well-formulated logical and philosophical reasons, this chapter advances the novel thesis that a logic is paraconsistent if it invalidates either the inferential or the meta-inferential notion of Explosion. Being so, a number of systems in the literature turn out to be, in that sense, paraconsistent logics.
Gaytán, D’Ottaviano and Morado present a system motivated by the problems of modelling explanation from the point of view of Philosophy of Science. In the chapter “Provided You’re not Trivial: Adding Defaults and Paraconsistency to a Formal Model of Explanation”, the authors advance the so-called GMD framework. Within that formal system it is possible to make an analysis of the interaction between rules and a minimal conception of context – composed by a set of beliefs (a minimal idea of a theory) in interaction with an inferential engine (a logic). In order to illustrate this novel epistemic system, the authors adopt it to analyse the concept of explanation using Reiter’s default theories and a specific paraconsistent logic of da Costa.
In the chapter “Para-Disagreement Logics and Their Implementation Through Embedding in Coq and SMT”, Woltzenlogel Paleo advances a novel approach to para-disagreement logics. The basic language is the usual propositional one, extended with box and diamond operators from modal logics and the @ operator from hybrid logics. The semantics are very similar to possible worlds for modal logics with small differences regarding the representation of world reachability. This framework allows a fine-tuned approach regarding information source, so that conflicting information from different sources can be consistently combined. By suggesting some possible semantical embeddings in Coq and SMT, the author advocates the implementation of automated reasoning tools for these logics.
Džamonja and Panza, in “Asymptotic Quasi-completeness and ZFC”, put forward a thesis that the axioms ZFC of first order set theory is actually very powerful at some infinite cardinal, contrary to what it could be stated. Since ZFC axioms are subject to Gödel’s Incompleteness Theorems (cf. Gödel (1931)), if they are assumed to be consistent then they are necessarily incomplete – a fact that can be supported by various concrete statements, including the celebrated Continuum Hypothesis. In order to illustrate their thesis, it is explained that by looking at limits of uncountable cardinals, such as אω, and working with singular cardinals (which are necessarily limits, cf. Kojman (2011)), at such cardinals there is a very serious limit to independence. Furthermore, many statements which are known to be independent on regular cardinals become provable or refutable by ZFC at singulars. The thesis then follows by the fact that the behaviour of the set-theoretic universe is asymptotically determined at singular cardinals by the behaviour that the universe assumes at the smaller regular cardinals. Being so, ZFC foundationally provides an asymptotically univocal image of the universe of sets around the singular cardinals.
“Interpretation and Truth in Set Theory” also presents an inquiry on some fundamental questions of set theory. In this chapter, Freire grasps concrete axiom systems in terms of a double-layer schema: respectively containing the conceptual and the deductive components of the system. The conceptual component is identified with a criterion given by directive principles, supposable bounding the subject matter of the system. After advancing two lists of directive principles for the set theory, the set-theoretic truth and the fixation of truth-values in each double-layer picture that emerged from these lists are then analysed. It is worth noticing that the general approach that is forwarded in this chapter can be applied to other mathematical theories with interesting results.
In the short but sturdy chapter “Coherence of the Product Law for Independent Continuous Events”, Mundici demonstrates a formal result regarding probability theory: the product law for logically independent events (for Boolean as well as for continuous MV-algebraic events) follows from de Finetti’s fundamental notion of a coherent set of betting odds, in the same sense that it was originally demonstrated for the additivity law by de Finetti’s 1932 Dutch Book theorem.
In the chapter “A Local-Global Principle for the Real Continuum”, Magossi and Rioul present a logical flow of proofs in the most influential undergraduate and graduate textbooks on Real Analysis in the U.S.A., France and Brazil in order to start a discussion regarding the local-global principle (LG) as a new efficient and enjoyable tool for proving the basic theorems of real analysis. Both, LG (any local and additive property is global ) and the related principle of global-limit (GL: any global and subtractive property has a limit point) could be used as basis for a new presentation of the integral, just as Cousin’s lemma was used to build the Kurzweil-Henstock integral – what the authors intend to advance in future works.
The chapter “Quantitative Logic Reasoning” by Finger brings an unifying approach on some logical systems, namely propositional Probabilistic Logic (classical propositional logic enhanced with probability assignments over formulas); first-order logic with counting quantifiers over a fragment containing unary and limited binary predicates; and propositional Łukasiewicz Infinitely-valued Probabilistic Logic (a multi-valued logic for which there exists a well-founded probability theory). From the viewpoint of Quantitative Logic Reasoning, the author shows that analogous properties hold throughout that class of systems, and presents for each one a language, semantics and decision problem, followed by normal form presentation and satisfiability characterization. Furthermore, complexity results and decision algorithms are also advanced.
Carnielli, Mariano and Matulovic advance an algebraic method based on the polynomial representation of first-order sentences in order to introduce algebraic semantics for first-order logic, departing from modern forms of “algebraizing a Logic” tradition like presented by Blok and Pigozzi (1989). In “Reconciling First-Order Logic to Algebra” the authors employ the notion of M-rings, rings equipped with infinitary operations that can be naturally associated to the first-order structures and each first-order theories. It is shown that infinitary versions of the Boolean sums and products are able to express algebraically first-order logic from a new perspective. This chapter also suggests an unifying algebraical approach to logic by opening-up avenues for possible generalizations of the method to n-valued and other non-classical logics.
In the book’s last chapter, Marcelino, Caleiro and Rivieccio clarify the efficiency of some novel techniques in the study of Hilbert-style logics. In “Plug and Play Negations” the authors focus on the negation fragments of logics which result from different possible choices of well-known rules involving the connectives {→,⊥}, with a few variations – in fact negation is usually introduced as a derived connective by making ¬p:=p→⊥¬p:=p→⊥ (that is, using the material implication → and the falsum constant ⊥). In turn the degree of tolerance to inconsistencies of a logic (degree of paraconsistency) can be determined by the interaction among these three connectives. The techniques used allow for a modular analysis of the logics, providing complete semantics based on (non-deterministic) logical matrices and complexity upper bounds.
Albeit the great diversity of themes discussed in the chapters, all of them can be subsumed into a broadly understood study of consistency – the book perfectly demonstrates how issues surrounding that study go well beyond traditional inquiries on paraconsistent logics, taking novel perspectives that are not too far away from such inquiries. The selection helps the reader to perceive how those works intersect with core traditional mathematical and philosophical questions. Contradictions, from Consistency to Inconsistency nicely supplements the existing literature on the subject. This is a volume that it is well worth reading!
References
ARISTOTLE: Metaphysics. In The Complete Works of Aristotle, ed. J Barnes (vol. 2). Princeton N.J.: Princeton University Press, 1984. [ Links ]
BLOK, W.J. and D. PIGOZZI. “Algebraizable Logics”. Memoirs of the AMS. 396, American Mathematical Society, Providence, USA, 1989. [ Links ]
BROWN, B. and PRIEST, G. “Chunk and permeate, a paraconsistent inference strategy. part i: the infinitesimal calculus”. Journal of Philosophical Logic 33 (4): 379-388. August, 2004. [ Links ]
CARNIELLI, W. and CONIGLIO, M. Paraconsistent Logic: Consistency, Contradiction and Negation. Dordrecht: Springer, 2016. [ Links ]
_____ and MARCOS, J. “A taxonomy of C-systems”. Paraconsistency: The Logical Way to the Inconsistent. Lecture Notes in Pure and Applied Mathematics 228, 1-93, 2002. [ Links ]
DA COSTA, N.C.A. Logiques Classiques et Non Classiques. Essai sur les fondements de la logique. Paris: Masson, 1997. [ Links ]
DE FINETTI, B. Theory of Probability, A Critical Introductory Treatment, Translated by Antonio Machí and Adrian Smith. Chichester, UK: Wiley, 2017. [ Links ]
GÖDEL, K. “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I”. Monatshefte für Mathematik und Physik 38: 173-198, 1931. [ Links ]
KOJMAN, M. “Singular Cardinals: from Hausdorff’s gaps to Shelah’s pcf theory”. In Sets and Extensions in the Twentieth Century ed. by Dov M. Gabbay, Akihiro Kanamori, and John Woods, vol. 6 of Handbook of the History of Logic, pp. 509-558. Elsevier, 2011. [ Links ]
PRIEST, G., TANAKA, K. and WEBER, Z. “Paraconsistent logic”. In The Stanford Encyclopedia of Philosophy, Winter 2016 ed, ed. Edward Zalta. Stanford University, 2016. http://plato.stanford.edu/archives/win2016/entries/logic-paraconsistent/. [ Links ]
RIPLEY, D. “Paraconsistent logic”. Journal of Philosophical Logic 44 (6): 771-780, 2015. [ Links ]
Rafafel R. Testa – University of Campinas. Center for Logic and Epistemology. Campinas, SP. Brazil. E-mail: [email protected]
Paraconsistent Logic: Consistency, Contradiction and Negation – CARNIELLI; CONIGLIO (M)
CARNIELLI, W.; CONIGLIO, M.. Paraconsistent Logic: Consistency, Contradiction and Negation. Logic, Epistemology, and the Unity of Science Series. New York: Springer, 2016. Resenha de: ANTUNES, Henrique; CICCARELLI, Vicenzo. Manuscrito, Campinas, v.41 n.2 Apr./June 2018.
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 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 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.
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 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 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 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 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 ? 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 ?
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 ? (that is, the language is supposed to be diagrammatic). At first sight, the use of the notation [x / ā] (and its generalization [x 1,…, x n / ā 1,…, ā n] 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 ? interpreting (x→ ; y→ ), for every A ∈ L(?) x→ ; z and every t ∈ T(?) x→ ; y→ such that t is free for z in A, if A[z/t] ∈ L(?) x→ ; y→ and c = (t[x→ ; y→ / a→ ; b→ ]) ?“ then:
If v((A[z/t])[x→ ; y→ / a→ ; b→ ]) = v(A[x→ ; z / a→ ; c]) then
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 x 1,…, x n (abbreviated by x → ) then the truth value of A under the extended valuation v x→ a→ is simply v(A[x 1,…, x n / ā 1,…, ā n]). 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:
(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 ? 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 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 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. [ Links ]
Henrique Antunes – State University of Campinas, Department of Philosophy, Campinas, SP, Brazil, antunes. E-mail: [email protected]
Vincenzo Ciccarelli – State University of Campinas, Department of Philosophy, Campinas, SP, Brazil. E-mail: [email protected]