Logic, Thought, and Action 20

Chapter 20 ON THE USEFULNESS OF PARACONSISTENT LOGIC Newton C.A. da Costa,1 Jean-Yves B´´eziau,2 and Ot´ a´vio Bueno3 1 University of S˜o ˜ Paulo 2 Institut de Logique, Universit´ ´ de Neuchˆ atel ˆ 3 University of South Carolina

Abstract

In this paper, we examine some intuitive motivations to develop a paraconsistent logic. These motivations are formally developed using semantic ideas, and we employ, in particular, bivaluations and truth-tables to characterise this logic. After discussing these ideas, we examine some applications of paraconsistent logic to various domains. With these motivations and applications in hand, the usefulness of paraconsistent logic becomes hard to deny.

If geometrical space were a framework imposed on each of our representations considered individually, it would be impossible to represent to ourselves an image without this framework, and we should be quite unable to change our geometry. But this is not the case; geometry is only the summary of the laws by which these images succeed each other. —Henri Poincare´ [1905], p. 64.

D. Vanderveken (ed.), Logic, Thought & Action, 465–478. 2005 Springer. Printed in The Netherlands.

c


466

1.

LOGIC, THOUGHT AND ACTION

Introduction

All of us, at some point, have heard questions about the usefulness of some branch of knowledge. What use is mathematics? Or topology? Or, for that matter, what use is logic? Of course, depending on the context in which such questions appear (for instance, a mathematician trying to understand a bit more of his or her own field, or a student upset with his or her final grades in mathematics), the particular features of the answer will change. What may not change, in a sense, is the nature of the answer. In most cases, it will indicate certain traits of the ‘pragmatics’ of the field under consideration, spelling out some of the connections between the theories formulated in such a field and their users, as well as the targets and constraints of the latter. In the course of such an investigation, some of the applications of such theories (either to their standard domain or to others) may be discussed and presented as reasons for their usefulness. Put in very general terms, these reasons can be understood in terms of the problem-solving resources disclosed by the theories (including their explicative power, the conceptual systematisation they supply, and the tools for the representation and analysis of the relevant phenomena). To a certain extent, the same holds for logic. Taken in a very strict sense, applied logic is concerned (among other issues) with the study of structures that can be employed to understand the formal features of our reasoning processes.1 At this level, just as with empirical theories, applied logic has its particular domain, being appropriate for representing certain kinds of phenomena, and hopeless for the examination of others (for instance, classical logic is by no means adequate for a constructive study of constructive mathematical thought, but can be seen as an idealised perspective on the representation of certain inferences usually found in classical mathematics). To the extent that the structures employed in a domain are appropriate to model the relevant features of it (thus ‘saving the phenomena’, as it were), we can claim that a particular applied logic has ‘explanatory power’; it indicates, after all, how such ‘phenomena’ can be understood in terms of the structures supplied by such a logic. Moreover, similarly to empirical theories, applied logic also offers a conceptual systematisation of inferences that are allowed in a certain domain, in particular spelling out the constraints imposed by them. Consider, for instance, the differences between constructive math-

1 For

a development of this theme, with special emphasis on paraconsistency, see da Costa and Bueno [2001], and da Costa and Bueno [1996].

On the Usefulness of Paraconsistent Logic

467

ematics (with the restrictions it brings to ‘classical’ mathematics)2 and paraconsistent mathematics (with the extensions and new structures it brings to ‘classical’ mathematics).3 Of course, the differences in the conceptual systematisation presented are due, in good part, to the different tools that each applied logic under consideration supply. In a sense, the very first step that would subsequently lead to such differences was taken by changing basic features of classical logic. Each logic, just as each geometry in Poincare’s ´ view (see the epigraph above), supplies a possible perspective for systematising our ways of representing certain phenomena (‘images’ in the case of geometry, and inferences in the case of applied logic). And if this is so, new perspectives can be offered by changing the logic (or, for that matter, the geometry). This straightforward fact already suggests a hint of the usefulness of nonclassical logics. In the present note, we wish to consider an instance of this general question, namely: what use is paraconsistent logic? In order to do so, we will first present such a logic from a perspective that can be easily understandable even by those who have little knowledge of the technical aspects of logic, namely, the semantic point of view. We will then consider, in connection with the preceding discussion, some straightforward applications of it, and this will convey at least a partial answer to our question. Finally, we shall briefly discuss some philosophical issues generated by such applications.

2.

Remarks on language

We consider the usual language of propositional logic P with the connectives ¬, ∧, ∨, →. Before we construct a semantics for this language, these four connectives are nothing but: a unary connective (the first one) and three binary connectives (the remaining ones). There is, a priori, no justification to call them negation, conjunction, disjunction and implication. In paraconsistent logic, we will denote by the symbol ¬, and call it negation, a connective that is not the same as classical negation. There are those who criticise such an abuse of language. Let us note, however, that it is difficult to claim that there is only one negation, let us say, classical negation (which would exactly model the negation of natural language or mathematics). In the literature, the word negation and the

2 See,

for instance, Bishop [1967], Heyting [1971], and Dummett [1977]. for example, da Costa [1989], da Costa [2000], Mortensen [1995], and, for a discussion of the latter, da Costa and Bueno [1997].

3 Cf.,

468

LOGIC, THOUGHT AND ACTION

corresponding symbols have long been used to denote different concepts, such as classical negation, intuitionistic negation, Johansson’s minimal negation, Curry’s negation etc. Of course, a unitary operator must have some basic properties to be called a ‘negation’. For instance, no one will call the necessity operator a negation. Nevertheless, until now, no common agreement on what the basic properties are that a unitary operator should obey to be called a ‘negation’ has been achieved. We will not claim that the paraconsistent negation presented here should absolutely be called as such. But we will try to convince the reader that it has enough interesting properties to deserve this name.

3.

Remarks on 0-1 semantics

As is known, it is possible to construct a wide range of logics taking as basic notions two truth-values, the false and the true, designated for convenience by 0 and 1 (see da Costa and B´ ´eziau [1994]). Even the so-called ‘many-valued’ logics can be treated in this way. For instance, Suszko has given a 0-1 semantics for Lukasiewicz three-valued logic (see Suszko [1975]). This apparent paradox is solved by the distinction between truth-functional semantics and non truth-functional semantics. Suszko’s 0-1 semantics is not truth-functional, neither will be the 0-1 semantics that we present now.4 Given the standard propositional language P, a 0-1 semantics for P is a set B of functions from P to {0, 1}, called bivaluations. A 0-1 semantics induces a logic in the following way: given a set B of bivaluations, we say that an object a of P (called a formula) is a consequence of a set T of objects of P (called a theory) iff for every bivaluation β ∈ B, if β gives the value 1 to every element of T, it also gives the value 1 to the formula a. In other words, a 0-1 semantics defines a binary relation on the Cartesian product of the power set of P and P, that we shall denote by |=. And we will write T |= a to say that T, a∈|=, i.e. that a is a consequence of T. In our view, a logic consists basically in presenting a set of formulas and a semantic consequence relation for that set. It is easy to note that if a set of bivaluations B1 contains a set of bivaluations B2, the corresponding logic L1 and L2 are ‘inversely proportional’, i.e. the consequence relation generated by B2 contains the consequence relation generated by B1. This has a direct consequence

4 For

more details on this subject, see da Costa, Beziau ´ and Bueno [1996]; for a related discussion of the concept of semantics, see da Costa, Bueno and Beziau ´ [1995].

On the Usefulness of Paraconsistent Logic

469

that we will use below. Let BC be the set of classical bivaluations. Then any set of bivaluations B that contains BC will generate a logic that is included in classical propositional logic LC and in which there are theories that are non-trivial. (A theory T is called non-trivial if there is at least one formula of the language that is not a consequence of T.)

4.

Definition of the set of paraconsistent bivaluations6

We will consider the following set BP of bivaluations. A function β from P to {0, 1} is in BP iff it obeys the following conditions: [C] β[a ∧ b] = 1 iff β[a] = 1 and β[b] = 1; [D] β[a ∨ b] = 0 iff β[a] = 0 and β[b] = 0; [I] β[a → b] = 0 iff β[a] = 1 and β[b] = 0; [EM] if β[a] = 0, then β[¬a] = 1; [SN] if β[a ∧ ¬a] = 1, then β[¬(a ∧ ¬a)] = 0; [PN/N] if β[a] = 0, then β[¬¬a] = 0; [PN/I] if β[a → b] = 1 and β[a] = 0 or β[¬a] = 0 or β[b] = 0 or β[¬b] = 0, then β[¬(a → b)] = 0; [PN/C], [PN/D] are conditions similar to [PN/I], when the formula is a conjunction or a disjunction. For simplicity, we will call here LP the logic induced by BP. This logic ´eziau has also been called C+ 1 elsewhere, and is an improvement, due to B´ (see B´´eziau [1990]), of the logic C1 of da Costa (see da Costa [1963]). It is easy to see that BC is included in BP. Thus LP is included in LC. Note that generally BC is represented in terms of attributions of truth-values to atomic formulas. This can be done because, in classical logic, the set of bivaluations is freely generated by the set of bivaluations restricted to atomic formulas. But this is not the case with BP. (This property is connected with truth-functionality.) The conditions for conjunction, disjunction and implication mutatis mutandis are the standard ones. The condition [EM] can be interpreted as a semantic version of the principle of the excluded middle. What is the intuitive explanation of the other conditions for paraconsistent negation? The idea is as follows. We will say that a formula obeys the principle of contradiction for a given bivaluation iff it cannot have the value 1 simultaneously with its negation. That is to say, as in the classical case, it is true iff its negation is false. a different presentation of paraconsistent logic, see da Costa, Beziau ´ and Bueno [1995a]. A historical perspective on paraconsistent logic can be found in Arruda [1980], Arruda [1989], D’Ottaviano [1990], and da Costa, B´ ´eziau and Bueno [1995b].

6 For

470

LOGIC, THOUGHT AND ACTION

Now, the condition [SN], interpreted in this way, states that for any formula a, a ∧ ¬a obeys the principle of contradiction. This condition allows us to define a compound connective, ¬∗ a = ¬a ∧ ¬(a ∧ ¬a), which has all the properties of classical negation; in particular, a is true iff ¬∗ a is false. Consequently, this allows us to ‘translate’ classical logic into the paraconsistent logic LP. The translation consists in replacing the ‘weak’ negation ¬ by the strong negation ¬∗ . We therefore have a situation that is similar to the case of intuitionistic logic. In one sense, LP is weaker than LC; in another sense, LP is stronger than LC. The conditions [PN] state that a sufficient condition for a formula to obey the principle of contradiction is that one of its direct subformulas obeys this principle. This intuitive preservation principle will give to paraconsistent negations interesting properties, such as parts of De Morgan’s laws.

5.

Truth-tables

It is possible to adapt the classical method of truth-tables for the case of LP. The basic difference is that, in the case of LP, we must introduce in the table of a formula a not only its subformulas, but also some negations of its proper subformulas. To simplify, we will put together its subformulas and all the negations of its proper subformulas; this set will be called the sphere of a. Then a table for a is a set of functions from its sphere to {0, 1}, such that: (a) every such a function is the restriction of an element of BP to this sphere, and (b) every restriction of BP to this sphere appears. It is not difficult to construct these kinds of tables following the conditions that define BP (for details, see da Costa and Alves [1977]). Example 1. The first example of a truth-table shows that the formula ((a ∧ ¬a) ∧ (a → b)) → (¬a → ¬b)

(20.1)

is not a tautology of LP. a 0 0 0 1 1 1 1 1 1

¬a 1 1 1 0 0 0 1 1 1

b 0 1 1 0 1 1 0 1 1

¬b 1 0 1 1 0 1 1 0 1

a ∧ ¬a 0 0 0 0 0 0 1 1 1

a→b 1 1 1 0 1 1 0 1 1

(a ∧ ¬a) ∧ (a → b) 0 0 0 0 0 0 0 1 1

¬a → ¬b 1 0 1 1 1 1 1 0 1

(1.1) 1 1 1 1 1 1 1 0 1

471

On the Usefulness of Paraconsistent Logic

Example 2. The following table shows that ¬(a ∧ ¬b) → (a → b)

(20.2)

is a tautology of LP. a 0 0 0 1 1 1 1 1 1 1

6.

¬a 1 1 1 0 0 0 1 1 1 1

b 0 1 1 0 1 1 0 1 1 1

¬b 1 0 1 1 0 1 1 0 1 1

a ∧ ¬b 0 0 0 1 0 1 1 0 1 1

¬(a ∧ ¬b) 1 1 1 0 1 0 0 1 0 1

(a → b) 1 1 1 0 1 1 0 1 1 1

(1.2) 1 1 1 1 1 1 1 1 1 1

A change of paradigm

The main feature of paraconsistent logic is that, as opposed to classical logic, inconsistency and triviality cease to coincide. In LP, there are some inconsistent theories (theories in which a formula and its negation are both consequences) that are not trivial (not every formula is a consequence). Such theories are called paraconsistent theories. For example, as the method of tables shows, the atomic formula b is not a consequence of the inconsistent theory constituted by the atomic formula a and its negation ¬a. Such a theory, therefore, is not trivial. It is clear that the concept of triviality is more fundamental than the one of inconsistency. Moreover, it is more abstract in the sense that its definition does not depend upon the particular connectives (in particular, the negation).

7.

Inconsistency and reasoning

In everyday life, it is quite common for one to face contradictions. Such contradictions may not be real contradictions, whatever this means, but in several cases they cannot be trivially eliminated and must be dealt with. In the mechanical treatment of information, contradictions also often appear. In both cases, classical logic, because it merges inconsistency with triviality, is a useless tool. Let us see now how paraconsistent logic can be useful when classical logic fails. Imagine that we are to construct an expert system. In order to do so, we start collecting the opinion of several hundreds of experts in a particular subject. The information we get comes from reliable sources, and there is no way to tell ‘good’ information from ‘bad’ one. After

472

LOGIC, THOUGHT AND ACTION

interviewing all these experts, there is no way to avoid the incompatibilities found, for they in fact express opposite opinions.7 Among such bits of information, let us suppose that a group of experts, called X1 , asserts that: The price of chocolate will raise

and that a second group of experts, X2 , states that: The price of chocolate will not raise.

We are therefore facing a contradiction. Firstly, let us note that, using paraconsistent logic LP, as opposed to the classical case, we cannot derive from this contradiction any statement whatsoever. For example, we cannot derive from this contradiction the following claim (which, in particular, is not a classical tautology): If someone eats lots of chocolate, he or she will grow enormously fat.

Secondly, in the presence of this contradiction, all the bits of reasoning that are not valid in classical logic are still not valid in LP. For instance, from such a contradiction and the following statement on which both X1 and X2 agree If the price of chocolate raises, people will buy less chocolate

we cannot infer that If the price of chocolate does not raise, people will not buy less chocolate

(see the first truth-table above). Now, let us see a positive reasoning that we can perform in LP. Both experts X1 and X2 agree that It is not the case that the price of chocolate will raise and the price of chocolate cookies will not raise.

As the second truth-table shows, it is implied by this that If the price of chocolate raises, the price of chocolate cookies will raise.

8.

A new perspective: paraconsistency

As these examples show, despite the inconsistency, paraconsistent logic allows us to draw interesting conclusions in a context where, were we to cling exclusively to the classical logic paradigm, we would get stuck, inevitably deriving anything! Thus this supplies part of our answer to the question about the usefulness of paraconsistency: it opens 7 In

certain cases, due to the huge amount of data to be taken into account, we may even not notice the existence of inconsistencies.

On the Usefulness of Paraconsistent Logic

473

up an altogether different perspective to examine issues in which inconsistencies are fundamentally involved. In a sense, faced with a contradiction, the classical paradigm will not offer any alternative but, in order to avoid trivialisation, that of rejecting (some of) the premises in terms of which the contradiction was reached. Unfortunately, this alternative may not always be open to us, since the relevant premises may in some way be entangled in our conceptual system, having such important connections with other statements of the system, that their rejection will lead to dramatic conceptual losses (see da Costa and French [1989], p. 441). And even if this were not the case, in contrast with the classical paradigm, with the employment of the tools supplied by paraconsistent logic, it is possible to take inconsistencies at face value, exploring thus the consequences that can be drawn from the system that includes them (as is clear from the examples above; see additional examples below). Nonetheless, one can claim that, to a certain extent, there is also a second alternative within the classical paradigm. If the rejection of certain premises in some cases cannot be recommended, people working within the classical paradigm can perhaps reject the validity of (some of) the inference rules used in order to obtain the contradiction under consideration, in such a way that the latter cannot be drawn any longer. The trouble with such a move, for the classicist, is that it means changing the underlying logic (just as Poincar´ ´e’s remark quoted above suggested with regard to geometry), and moving to another paradigm. In order to deal with this kind of inconsistency problem, this is exactly the suggestion we present (although, within paraconsistent logic, the change in the inference rules is not meant to avoid the derivation of certain contradictions, but to formulate a system in which such contradictions do not lead to a trivial system). The paraconsistent paradigm also advances new perspectives here. In fact, in several cases, and in stark contrast with the classical paradigm, given an inconsistency, we do not need to elaborate more or less ad hoc strategies to reject it: we can simply accept the premises and the inferences that led to the contradiction in question (provided such inferences are among the ones to be found in a paraconsistent system, and that we have changed our logic to a paraconsistent one). In such a perspective, we claim, we can learn more, having a truly pluralist account of knowledge. For someone who is classically minded, the last assertion might seem bizarre. How can a proponent of the paraconsistent view truly learn anything? After all, in a sense, part of our learning process depends upon our way of changing our beliefs, given contrary evidence. If this

474

LOGIC, THOUGHT AND ACTION

proponent, faced with such contrary evidence, simply adds it to the stock of his or her beliefs, claiming that ‘No problem, it won’t lead to trivialisation’, how could he or she ever change his or her mind? How could he or she ever come to the ‘saturation point’, from which everything will follow? Such questions seem to be still more pressing given the classical accounts of belief change that apparently underlie them. Put in very abstract and rough terms, such accounts will run like this. We can just keep adding any beliefs we wish to our belief system, provided we meet some consistency-preserving rule. If we fail to do that, and introduce inconsistencies into our system, it will simply be trivialised, becoming useless for any systematisation and cognitive purposes. However, if consistency is not a necessary constraint, as is the case in the domain of paraconsistency, a different perspective on the nature of belief change will emerge. Instead of consistency preservation, the ultimate constraint now will shift to the avoidance of trivialisation. After all, within the paraconsistency paradigm, we can deal with inconsistencies, whereas triviality clearly represents cognitive bankruptcy. Indeed, while an inconsistent theory may have several interesting features, at least from a heuristic point of view (Bohr’s atomic model and naive set theory are obvious examples), and we have learnt a lot from them (trying, although not exclusively, to devise consistent successors to them for instance), trivial theories are useless for any cognitive purposes. So, when paraconsistent logic, as against the classical one, clearly demarcates inconsistency from triviality, we can trace this demarcation to an epistemic distinction: between theories that, despite being inconsistent, can lead to (even inconsistent) fruitful successors, and those that are altogether hopeless for explanation, cognitive systematisation etc. To a certain extent, part of the usefulness of paraconsistency derives from such an epistemic distinction. Inconsistent theories may be rich, interesting, full of fruitful consequences, whereas trivial theories are simply useless. With paraconsistency, the whole new domain of the inconsistent, left in complete darkness by the classical approach, is thus open to investigation. And this domain has in fact received detailed consideration since the inception of paraconsistent logic. Let us conclude this note briefly mentioning three applications of paraconsistent logic that show this trend. They are respectively concerned with three distinct fields: mathematics, artificial intelligence, and philosophy. (1) Cantor’s naive set theory is characterised chiefly by two basic principles: the postulate of extensionality (if two sets have the same elements, then they are equal) and the postulate of comprehension (every

On the Usefulness of Paraconsistent Logic

475

property determines a set). As is well known, this postulate, in the standard language of set theory, is the following scheme of formulas: ∃y∀x(x ∈ y ↔ ϕ(x)) If we replace the formula ϕ(x), in the separation postulate, by x ∈ / x, Russell’s paradox is immediately derived. In other words, this postulate is inconsistent. Therefore, if it is added to first-order logic, viewed as the logic of set theoretic language, we obtain a trivial theory. Classical set theories are then constructed by imposing restrictions on the separation postulate, so that the paradoxes can be avoided. (Further axioms are then introduced in order that the resulting theory does not become too weak.) For instance, in Zermelo-Fraenkel set theory (ZF), comprehension is formulated as follows: ∃y∀x(x ∈ y ↔ (ϕ(x) ∧ x ∈ z)), where the variables are subject to obvious conditions. Hence, in ZF, ϕ(x) determines the subset of the elements of the set z that satisfy the formula ϕ(x). Using certain paraconsistent logics, it is possible to construct set theories in which the postulate of separation is subject either to restrictions weaker than those of the classical set theories or subject to no restrictions at all. Moreover, it is also possible to study, without trivialisation, the properties of ‘inconsistent’ objects, such as the Russell set, R = {x : x ∈ / x}. (Further details can be found, for instance, in da Costa [1986], da Costa and Bueno [2001], and da Costa, B´ ´eziau and Bueno [1998].) (2) In certain domains, such as in the construction of expert systems, the presence of inconsistencies is almost unavoidable. In order to construct these systems, enormous knowledge bases are elaborated, aggregating the opinion of several specialists in a particular field (let us say, medicine). As one can immediately imagine, such bases are inconsistent, and one of the problems consists in how to drawn inferences from them. Some paraconsistent logics have been especially devised to deal with this problem (see, for example, da Costa and Subrahmanian [1989]). (3) Surprisingly or not, inconsistent beliefs are frequently found, both in science and in everyday life. However, from such inconsistent belief sets, it is simply not the case that any statement whatsoever is derived. (So, apparently at least, we are not here concerned with ‘trivial’ systems.) In order to propose a formal framework to model some aspects of this phenomenon, certain paraconsistent doxastic logics have been constructed (see da Costa and French [1989]). In particular, the problem of

476

LOGIC, THOUGHT AND ACTION

self-deception and related problems that involve the holding of contradictory beliefs, can then receive a distinct approach (see da Costa and French [1990], and da Costa and French [1988]). Moreover, the relations between rationality and consistency can also be re-evaluated. After all, one of the main arguments to the effect that consistency is a minimum condition for rationality rests on the assumption that inconsistency leads to triviality; precisely the assumption challenged by the paraconsistent approach. (For details, see French [1990], da Costa and French [1995], and da Costa, Bueno and French [1998].) With the considerations advanced in this note, we hope to have indicated some aspects of the usefulness of paraconsistency. If we have not convinced you, gentle reader, of this point, we expect to have conveyed at least an idea of why paraconsistent logic is far from being useless.8

References Arruda A.I. (1980). “A Survey of Paraconsistent Logic”, in Arruda, Chuaqui, and da Costa (eds.), pp. 1–41. Arruda A.I. (1989). “Aspects of the Historical Development of Paraconsistent Logic”, in Priest, Routley and Norman (eds.), pp. 99–130. Arruda A., Chuaqui R., and da Costa N.C.A. (eds.) (1980). Mathematical Logic in Latin America. Amsterdam: North-Holland. Beziau ´ J.-Y. (1990). “Logiques construites suivant les m´´ethodes de da Costa”, Logique et Analyse 131–132, pp. 259–272. Bishop E. (1967). Foundations of Constructive Analysis. New York: McGraw-Hill. da Costa N.C.A. (1963). “Calculs propositionnels pour les syst`emes formels inconsistants”, Comptes-rendus de l’Acad´mie ´ des Sciences de Paris 257, 7 pp. 3790–3793. — (1986). “On Paraconsistent Set Theory”, Logique et Analyse 115, pp. 361–371. da Costa N.C.A. (1989). “Mathematics and Paraconsistency (in Portuguese)”, Monografias da Sociedade Paranaense de Matem´tica ´ 7. Curitiba: UFPR. — (2000). “Paraconsistent Mathematics”, in D. Batens, C. Mortensen, G. Priest and J.-P. Van Bendegem (eds.), Frontiers of Paraconsistency. Dordrecht: Kluwer Academic Publishers. da Costa N.C.A. and Alves E.H. (1977). “A Semantic Analysis of the Calculi Cn”, Notre Dame Journal of Formal Logic 16, pp. 621–630.

8 We

wish to thank Steven French for his comments on an earlier version of this paper.

On the Usefulness of Paraconsistent Logic

477

da Costa N.C.A. and Beziau, ´ J.-Y. (1994). “Th´ ´eorie de la valuation”, Logique et Analyse 146, pp. 95–117. da Costa, N.C.A. Beziau ´ J.-Y. and Bueno O. (1995a). “Aspects of Paraconsistent Logic”, Bulletin of the Interest Group in Pure and Applied Logics 3, pp. 597–614. — (1995b). “Paraconsistent Logic in a Historical Perspective”, Logique et Analyse 150-151-152, pp. 111–125. — (1996). “Malinowski and Suszko on Many-Valuedness: On the Reduction of Many-Valuedness to Two-Valuedness”, Modern Logic 6, pp. 272–299. — (1998). Elements of Paraconsistent Set Theory (in Portuguese). Campinas: Cole¸c˜ao CLE. da Costa N.C.A. and Bueno O. (1996). “Consistency, Paraconsistency and Truth (Logic, the Whole Logic and Nothing but the Logic)”, Ideas y Valores 100, pp. 48–60. — (1997). “Review of Chris Mortensen (1995)”, Journal of Symbolic Logic 62, pp. 683–685. — (2001). “Paraconsistency: Towards a Tentative Interpretation”, Theoria 16, pp. 119–145. da Costa N.C.A., Bueno O. and Beziau ´ J.-Y. (1995). “What is Semantics? A Brief Note on a Huge Question”, Sorites - Electronic Quarterly of Analytical Philosophy 3, pp. 43–47. da Costa N.C.A., Bueno O. and French S. (1998). “Is There a Zande Logic?”, History and Philosophy of Logic 19, pp. 41–54. da Costa N.C.A. and French S. (1988). “Belief and Contradiction”, Cr´tica ´ XX, pp. 3–11. — (1989). “On the Logic of Belief”, Philosophy and Phenomenological Research XLIX, pp. 431–446. — (1990). “Belief, Contradiction and the Logic of Self-Deception”, American Philosophical Quarterly 27, 7 pp. 179–197. — (1995). “Partial Structures and the Logic of the Azande”, American Philosophical Quarterly 32, pp. 325–339. da Costa N.C.A. and Subrahmanian V.S. (1989). “Paraconsistent Logics as a Formalism for Reasoning About Inconsistent Knowledge Bases”, Artificial Intelligence in Medicine 1, pp. 167–174. D’Ottaviano I. (1990). “On the Development of Paraconsistent Logic 7, pp. 89–152. and da Costa’s Work”, Journal of Non-Classical Logic 7 Dummett M. (1977). Elements of Intuitionism. Oxford: Clarendon Press. French S. (1990). “Rationality, Consistency and Truth”, Journal of NonClassical Logic 7 7, pp. 51–71. Heyting A. (1971). Intuitionism: An Introduction. (3rd edition.) Amsterdam: North-Holland.

478

LOGIC, THOUGHT AND ACTION

Mortensen C. (1995). Inconsistent Mathematics. Dordrecht: Kluwer Academic Publishers. Poincare´ H. (1905). Science and Hypothesis. New York: Dover. Priest G., Routley R. and Norman J. (ed.) (1989). Paraconsistent Logic: Essays on the Inconsistent. Munich: Philosophia. Suszko R. (1975). “Remarks on Lukasiewicz’s Three-Valued Logic”, Bulletin of the Section of Logic 4, pp. 87–90.