Logic, Thought, and Action 20

Chapter 20

ON THE USEFULNESS OFPARACONSISTENT LOGIC

Newton C.A. da Costa,1 Jean-Yves Beziau,´ 2 and Otavio Bueno´ 3

1University of S˜o Paulo˜

2Institut de Logique, Universit´ de Neuchˆ´ atelˆ

3University of South Carolina

Abstract In this paper, we examine some intuitive motivations to develop a para-consistent logic. These motivations are formally developed using seman-tic ideas, and we employ, in particular, bivaluations and truth-tables tocharacterise this logic. After discussing these ideas, we examine someapplications of paraconsistent logic to various domains. With these mo-tivations and applications in hand, the usefulness of paraconsistent logicbecomes hard to deny.

If geometrical space were a framework imposed on each of our representationsconsidered individually, it would be impossible to represent to ourselves an imagewithout 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 whichthese images succeed each other.

—Henri Poincare [1905], p. 64.´

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

466 LOGIC, THOUGHT AND ACTION

1. IntroductionAll 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 thecontext in which such questions appear (for instance, a mathematiciantrying to understand a bit more of his or her own field, or a studentupset with his or her final grades in mathematics), the particular fea-tures of the answer will change. What may not change, in a sense, isthe nature of the answer. In most cases, it will indicate certain traitsof the ‘pragmatics’ of the field under consideration, spelling out some ofthe connections between the theories formulated in such a field and theirusers, as well as the targets and constraints of the latter. In the courseof such an investigation, some of the applications of such theories (eitherto their standard domain or to others) may be discussed and presentedas reasons for their usefulness. Put in very general terms, these reasonscan be understood in terms of the problem-solving resources disclosed bythe theories (including their explicative power, the conceptual systema-tisation they supply, and the tools for the representation and analysis ofthe relevant phenomena).

To a certain extent, the same holds for logic. Taken in a very strictsense, applied logic is concerned (among other issues) with the study ofstructures that can be employed to understand the formal features ofour reasoning processes.1 At this level, just as with empirical theories,applied logic has its particular domain, being appropriate for represent-ing certain kinds of phenomena, and hopeless for the examination ofothers (for instance, classical logic is by no means adequate for a con-structive study of constructive mathematical thought, but can be seen asan idealised perspective on the representation of certain inferences usu-ally found in classical mathematics). To the extent that the structuresemployed in a domain are appropriate to model the relevant features ofit (thus ‘saving the phenomena’, as it were), we can claim that a par-ticular applied logic has ‘explanatory power’; it indicates, after all, howsuch ‘phenomena’ can be understood in terms of the structures suppliedby such a logic. Moreover, similarly to empirical theories, applied logicalso offers a conceptual systematisation of inferences that are allowed ina certain domain, in particular spelling out the constraints imposed bythem. Consider, for instance, the differences between constructive math-

1For a development of this theme, with special emphasis on paraconsistency, see da Costaand 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 andparaconsistent mathematics (with the extensions and new structures itbrings to ‘classical’ mathematics).3 Of course, the differences in the con-ceptual systematisation presented are due, in good part, to the differenttools that each applied logic under consideration supply.

In a sense, the very first step that would subsequently lead to suchdifferences was taken by changing basic features of classical logic. Eachlogic, just as each geometry in Poincare’s view (see the epigraph above),´supplies a possible perspective for systematising our ways of representingcertain phenomena (‘images’ in the case of geometry, and inferences inthe case of applied logic). And if this is so, new perspectives can beoffered by changing the logic (or, for that matter, the geometry). Thisstraightforward fact already suggests a hint of the usefulness of non-classical logics.

In the present note, we wish to consider an instance of this generalquestion, namely: what use is paraconsistent logic? In order to do so,we will first present such a logic from a perspective that can be easilyunderstandable even by those who have little knowledge of the techni-cal aspects of logic, namely, the semantic point of view. We will thenconsider, in connection with the preceding discussion, some straightfor-ward applications of it, and this will convey at least a partial answer toour question. Finally, we shall briefly discuss some philosophical issuesgenerated by such applications.

2. Remarks on languageWe consider the usual language of propositional logic P with the con-

nectives ¬, ∧, ∨, →. Before we construct a semantics for this language,these four connectives are nothing but: a unary connective (the firstone) and three binary connectives (the remaining ones). There is, a pri-ori, no justification to call them negation, conjunction, disjunction andimplication.

In paraconsistent logic, we will denote by the symbol ¬, and call itnegation, a connective that is not the same as classical negation. Thereare 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 naturallanguage or mathematics). In the literature, the word negation and the

2See, for instance, Bishop [1967], Heyting [1971], and Dummett [1977].3Cf., for example, da Costa [1989], da Costa [2000], Mortensen [1995], and, for a discussionof the latter, da Costa and Bueno [1997].


corresponding symbols have long been used to denote different concepts,such as classical negation, intuitionistic negation, Johansson’s minimalnegation, Curry’s negation etc.

Of course, a unitary operator must have some basic properties to becalled a ‘negation’. For instance, no one will call the necessity operatora negation. Nevertheless, until now, no common agreement on what thebasic properties are that a unitary operator should obey to be called a‘negation’ has been achieved. We will not claim that the paraconsistentnegation presented here should absolutely be called as such. But we willtry to convince the reader that it has enough interesting properties todeserve this name.

3. Remarks on 0-1 semanticsAs 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 forconvenience by 0 and 1 (see da Costa and Beziau [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 (seeSuszko [1975]). This apparent paradox is solved by the distinction be-tween truth-functional semantics and non truth-functional semantics.Suszko’s 0-1 semantics is not truth-functional, neither will be the 0-1semantics that we present now.4

Given the standard propositional language P, a 0-1 semantics for P isa set B of functions from P to {0, 1}, called bivaluations. A 0-1 semanticsinduces a logic in the following way: given a set B of bivaluations, wesay that an object a of P (called a formula) is a consequence of a setT of objects of P (called a theory) iff for every bivaluation β ∈ B, if βgives the value 1 to every element of T, it also gives the value 1 to theformula a.

In other words, a 0-1 semantics defines a binary relation on the Carte-sian 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 conse-quence of T. In our view, a logic consists basically in presenting a set offormulas and a semantic consequence relation for that set.

It is easy to note that if a set of bivaluations B1 contains a set ofbivaluations B2, the corresponding logic L1 and L2 are ‘inversely pro-portional’, i.e. the consequence relation generated by B2 contains theconsequence relation generated by B1. This has a direct consequence

4For 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].´


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 logicthat is included in classical propositional logic LC and in which there aretheories that are non-trivial. (A theory T is called non-trivial if there isat least one formula of the language that is not a consequence of T.)

4. Definition of the set of paraconsistentbivaluations6

We will consider the following set BP of bivaluations. A function βfrom 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 isa conjunction or a disjunction.

For simplicity, we will call here LP the logic induced by BP. This logichas also been called C+

1 elsewhere, and is an improvement, due to Beziau´(see Beziau [1990]), of the logic C´ 1 of da Costa (see da Costa [1963]).

It is easy to see that BC is included in BP. Thus LP is included inLC. Note that generally BC is represented in terms of attributions oftruth-values to atomic formulas. This can be done because, in classicallogic, the set of bivaluations is freely generated by the set of bivaluationsrestricted to atomic formulas. But this is not the case with BP. (Thisproperty is connected with truth-functionality.)

The conditions for conjunction, disjunction and implication mutatismutandis are the standard ones. The condition [EM] can be interpretedas a semantic version of the principle of the excluded middle.

What is the intuitive explanation of the other conditions for paracon-sistent negation? The idea is as follows. We will say that a formulaobeys the principle of contradiction for a given bivaluation iff it cannothave the value 1 simultaneously with its negation. That is to say, as inthe classical case, it is true iff its negation is false.

6For a different presentation of paraconsistent logic, see da Costa, Beziau and Bueno [1995´ a].A historical perspective on paraconsistent logic can be found in Arruda [1980], Arruda [1989],D’Ottaviano [1990], and da Costa, Beziau and Bueno [1995´ b].


Now, the condition [SN], interpreted in this way, states that for anyformula a, a ∧ ¬a obeys the principle of contradiction. This conditionallows us to define a compound connective, ¬∗a = ¬a∧¬(a∧¬a), whichhas all the properties of classical negation; in particular, a is true iff¬∗a is false. Consequently, this allows us to ‘translate’ classical logicinto the paraconsistent logic LP. The translation consists in replacingthe ‘weak’ negation ¬ by the strong negation ¬∗. We therefore have asituation 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 toobey the principle of contradiction is that one of its direct subformu-las obeys this principle. This intuitive preservation principle will giveto paraconsistent negations interesting properties, such as parts of DeMorgan’s laws.

5. Truth-tablesIt 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 introducein the table of a formula a not only its subformulas, but also somenegations of its proper subformulas. To simplify, we will put together itssubformulas and all the negations of its proper subformulas; this set willbe called the sphere of a. Then a table for a is a set of functions from itssphere to {0, 1}, such that: (a) every such a function is the restrictionof an element of BP to this sphere, and (b) every restriction of BP tothis sphere appears. It is not difficult to construct these kinds of tablesfollowing the conditions that define BP (for details, see da Costa andAlves [1977]).

Example 1. The first example of a truth-table shows that the for-mula

((a ∧ ¬a) ∧ (a→ b))→ (¬a→ ¬b) (20.1)

is not a tautology of LP.

a ¬a b ¬b a ∧ ¬a a → b (a ∧ ¬a) ∧ (a → b) ¬a → ¬b (1.1)

0 1 0 1 0 1 0 1 10 1 1 0 0 1 0 0 10 1 1 1 0 1 0 1 11 0 0 1 0 0 0 1 11 0 1 0 0 1 0 1 11 0 1 1 0 1 0 1 11 1 0 1 1 0 0 1 11 1 1 0 1 1 1 0 01 1 1 1 1 1 1 1 1


Example 2. The following table shows that

¬(a ∧ ¬b)→ (a→ b) (20.2)

is a tautology of LP.

a ¬a b ¬b a ∧ ¬b ¬(a ∧ ¬b) (a → b) (1.2)

0 1 0 1 0 1 1 10 1 1 0 0 1 1 10 1 1 1 0 1 1 11 0 0 1 1 0 0 11 0 1 0 0 1 1 11 0 1 1 1 0 1 11 1 0 1 1 0 0 11 1 1 0 0 1 1 11 1 1 1 1 0 1 11 1 1 1 1 1 1 1

6. A change of paradigmThe main feature of paraconsistent logic is that, as opposed to clas-

sical logic, inconsistency and triviality cease to coincide. In LP, thereare some inconsistent theories (theories in which a formula and its nega-tion are both consequences) that are not trivial (not every formula is aconsequence). Such theories are called paraconsistent theories.

For example, as the method of tables shows, the atomic formula b isnot a consequence of the inconsistent theory constituted by the atomicformula a and its negation ¬a. Such a theory, therefore, is not trivial.

It is clear that the concept of triviality is more fundamental thanthe one of inconsistency. Moreover, it is more abstract in the sensethat its definition does not depend upon the particular connectives (inparticular, the negation).

7. Inconsistency and reasoningIn 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 dealtwith. In the mechanical treatment of information, contradictions alsooften appear. In both cases, classical logic, because it merges inconsis-tency with triviality, is a useless tool. Let us see now how paraconsistentlogic can be useful when classical logic fails.

Imagine that we are to construct an expert system. In order to doso, we start collecting the opinion of several hundreds of experts in aparticular subject. The information we get comes from reliable sources,and there is no way to tell ‘good’ information from ‘bad’ one. After


interviewing all these experts, there is no way to avoid the incompati-bilities found, for they in fact express opposite opinions.7 Among suchbits 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 statementwhatsoever. For example, we cannot derive from this contradiction thefollowing 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 reasoningthat 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 ofchocolate 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: paraconsistencyAs these examples show, despite the inconsistency, paraconsistent

logic allows us to draw interesting conclusions in a context where, werewe to cling exclusively to the classical logic paradigm, we would getstuck, inevitably deriving anything! Thus this supplies part of our an-swer to the question about the usefulness of paraconsistency: it opens

7In certain cases, due to the huge amount of data to be taken into account, we may even notnotice the existence of inconsistencies.


up an altogether different perspective to examine issues in which incon-sistencies are fundamentally involved.

In a sense, faced with a contradiction, the classical paradigm will notoffer 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 therelevant premises may in some way be entangled in our conceptual sys-tem, having such important connections with other statements of thesystem, that their rejection will lead to dramatic conceptual losses (seeda Costa and French [1989], p. 441). And even if this were not thecase, in contrast with the classical paradigm, with the employment ofthe tools supplied by paraconsistent logic, it is possible to take inconsis-tencies at face value, exploring thus the consequences that can be drawnfrom 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 sec-ond alternative within the classical paradigm. If the rejection of certainpremises in some cases cannot be recommended, people working withinthe classical paradigm can perhaps reject the validity of (some of) theinference rules used in order to obtain the contradiction under consid-eration, in such a way that the latter cannot be drawn any longer. Thetrouble with such a move, for the classicist, is that it means changing theunderlying logic (just as Poincare’s remark quoted above suggested with´regard to geometry), and moving to another paradigm. In order to dealwith this kind of inconsistency problem, this is exactly the suggestion wepresent (although, within paraconsistent logic, the change in the infer-ence rules is not meant to avoid the derivation of certain contradictions,but to formulate a system in which such contradictions do not lead to atrivial system).

The paraconsistent paradigm also advances new perspectives here. Infact, in several cases, and in stark contrast with the classical paradigm,given an inconsistency, we do not need to elaborate more or less ad hocstrategies to reject it: we can simply accept the premises and the infer-ences that led to the contradiction in question (provided such inferencesare among the ones to be found in a paraconsistent system, and thatwe have changed our logic to a paraconsistent one). In such a perspec-tive, we claim, we can learn more, having a truly pluralist account ofknowledge.

For someone who is classically minded, the last assertion might seembizarre. How can a proponent of the paraconsistent view truly learnanything? After all, in a sense, part of our learning process dependsupon our way of changing our beliefs, given contrary evidence. If this


proponent, faced with such contrary evidence, simply adds it to thestock of his or her beliefs, claiming that ‘No problem, it won’t leadto trivialisation’, how could he or she ever change his or her mind?How could he or she ever come to the ‘saturation point’, from whicheverything will follow?

Such questions seem to be still more pressing given the classical ac-counts of belief change that apparently underlie them. Put in very ab-stract and rough terms, such accounts will run like this. We can justkeep adding any beliefs we wish to our belief system, provided we meetsome consistency-preserving rule. If we fail to do that, and introduceinconsistencies into our system, it will simply be trivialised, becominguseless for any systematisation and cognitive purposes.

However, if consistency is not a necessary constraint, as is the case inthe domain of paraconsistency, a different perspective on the nature ofbelief change will emerge. Instead of consistency preservation, the ulti-mate constraint now will shift to the avoidance of trivialisation. Afterall, within the paraconsistency paradigm, we can deal with inconsisten-cies, whereas triviality clearly represents cognitive bankruptcy. Indeed,while an inconsistent theory may have several interesting features, atleast from a heuristic point of view (Bohr’s atomic model and naive settheory are obvious examples), and we have learnt a lot from them (try-ing, although not exclusively, to devise consistent successors to themfor instance), trivial theories are useless for any cognitive purposes. So,when paraconsistent logic, as against the classical one, clearly demar-cates inconsistency from triviality, we can trace this demarcation to anepistemic distinction: between theories that, despite being inconsistent,can lead to (even inconsistent) fruitful successors, and those that arealtogether hopeless for explanation, cognitive systematisation etc.

To a certain extent, part of the usefulness of paraconsistency derivesfrom such an epistemic distinction. Inconsistent theories may be rich, in-teresting, full of fruitful consequences, whereas trivial theories are simplyuseless. With paraconsistency, the whole new domain of the inconsistent,left in complete darkness by the classical approach, is thus open to in-vestigation. And this domain has in fact received detailed considerationsince the inception of paraconsistent logic.

Let us conclude this note briefly mentioning three applications of para-consistent logic that show this trend. They are respectively concernedwith three distinct fields: mathematics, artificial intelligence, and phi-losophy.

(1) Cantor’s naive set theory is characterised chiefly by two basicprinciples: the postulate of extensionality (if two sets have the sameelements, then they are equal) and the postulate of comprehension (every


property determines a set). As is well known, this postulate, in thestandard 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 postulateis inconsistent. Therefore, if it is added to first-order logic, viewed asthe logic of set theoretic language, we obtain a trivial theory.

Classical set theories are then constructed by imposing restrictions onthe separation postulate, so that the paradoxes can be avoided. (Furtheraxioms are then introduced in order that the resulting theory does notbecome 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 theformula ϕ(x).

Using certain paraconsistent logics, it is possible to construct set the-ories in which the postulate of separation is subject either to restric-tions weaker than those of the classical set theories or subject to norestrictions at all. Moreover, it is also possible to study, without triv-ialisation, the properties of ‘inconsistent’ objects, such as the Russellset, R = {x : x /∈// x}. (Further details can be found, for instance, inda Costa [1986], da Costa and Bueno [2001], and da Costa, Beziau and´Bueno [1998].)

(2) In certain domains, such as in the construction of expert sys-tems, the presence of inconsistencies is almost unavoidable. In orderto construct these systems, enormous knowledge bases are elaborated,aggregating the opinion of several specialists in a particular field (letus say, medicine). As one can immediately imagine, such bases are in-consistent, and one of the problems consists in how to drawn inferencesfrom them. Some paraconsistent logics have been especially devised todeal with this problem (see, for example, da Costa and Subrahmanian[1989]).

(3) Surprisingly or not, inconsistent beliefs are frequently found, bothin science and in everyday life. However, from such inconsistent beliefsets, it is simply not the case that any statement whatsoever is derived.(So, apparently at least, we are not here concerned with ‘trivial’ sys-tems.) In order to propose a formal framework to model some aspects ofthis phenomenon, certain paraconsistent doxastic logics have been con-structed (see da Costa and French [1989]). In particular, the problem of


self-deception and related problems that involve the holding of contra-dictory beliefs, can then receive a distinct approach (see da Costa andFrench [1990], and da Costa and French [1988]). Moreover, the relationsbetween rationality and consistency can also be re-evaluated. After all,one of the main arguments to the effect that consistency is a minimumcondition for rationality rests on the assumption that inconsistency leadsto triviality; precisely the assumption challenged by the paraconsistentapproach. (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 indi-cated some aspects of the usefulness of paraconsistency. If we have notconvinced you, gentle reader, of this point, we expect to have conveyedat least an idea of why paraconsistent logic is far from being useless.8

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Logic, Thought, and Action 20