## Paraconsistent logic! (A reply to Slater)

Por • 18 mar, 2020 • Sección: Opinion

Jean-Yves Béziau

Foot note 1_1

EDITOR’S NOTE

Je ne discute jamais du nom pourvu qu’on m’avertisse quel sens on lui donne.

Blaise Pascal, Les Provinciales

Contents

1. Paraconsistent logic?
2. Contradictories, subcontraries and contraries in the tradition
3. Contradictories, subcontraries and contraries in classical logic
4. Contradictories, subcontraries and contraries in paraconsistent logic
5. Da Costa’s logic C1
6. Priest’s logic LP
7. A general result about contradictories and paraconsistent logic
8. Conclusion
9. Bibliography
1. Paraconsistent logic

Paraconsistent logic is the study of logics in which there are some theories embodying contradictions but which are not trivial, in particular in a paraconsistent logic, the ex contradictione sequitur quod libet, which can be formalized as Cn(T, a,¬a)=F is not valid. Since nearly half a century various systems of paraconsistent logic have been proposed and studied. This field of research is classified under a special section (B53) in the Mathematical Reviews and watching this section, it is possible to see that the number of papers devoted to paraconsistent logic is each time greater and has recently increased due in particular to its applications to computer sciences (see e.g. Blair and Subrahmanian, 1989).

However in a recent paper entitled «Paraconsistent logics?», a philosopher from Perth, B.H.Slater, pretends to show in less than ten lines that paraconsistent logic doesn’t exist. Here is his laconic argument:

If we called what is now «red», «blue», and vice versa, would that show that pillar boxes are blue, and the sea is red? Surely the facts wouldn’t change, only the mode of expression of them. Likewise, if we called «subcontraries», «contradictories», would that show that «it’s not red» and «it’s not blue» were contradictories? Surely the same point holds. And that point shows that there is no «paraconsistent logic». (Slater 1995, p.451)

Are these few lines, the death sentence of paraconsistent logic?

Slater’ argumentation is based on the traditional notions of «contradictories» and «subcontraries». Unfortunately the Perthian doesn’t give precise definitions of them. After giving such definitions and proving a general result about them, we will show that Slater’s argument is not valid or, in the best case, is tautological.

1. Contradictories, subcontraries and contraries in the tradition

Such notions as «subcontraries» and «contradictories» belong to traditional logic, i.e. logic in the tradition of Aristotle. The first point is to precise what is their meaning in this tradition and the second point is to see how they can be understood in the light of modern mathematical logic.

One of the sad defect of Slater’s argument is that both of these points are eluded and that therefore his argument is viciated by fuzziness. The farther precision Slater is getting at is when he says that contradictories cannot be true together – by definition» (Slater 1995, p.453). Even this precision is quite ambiguous because, due to the fact the Perthian doesn’t give any definition of contradictories, one may imagine that the definition of contradictories is that two sentences are contradictories iff they cannot be true together, which is not the correct definition according to the tradition as we shall see very soon.

Of course one can imagine that it is not necessary to precise what is the exact meaning of notions such as contradictories and subcontraries, that everybody knows what their meaning is, and that this meaning is clear. But it is not so obvious, due to the fact that these notions belong to traditional logic, and that most concepts of traditional logic appear as confuse in the light of modern logic, and that at least their interpretations is not straightforward.

We will not enter into philological details to explain what is the meaning of «contradictories», and «subcontraries». The following excerpt from p.56 of (Kneale and Kneale 1962) will provide all the necessary information for our discussion including the standard definitions of contradictories, subcontraries and contraries (the concept of subalterns is not relevant for us here):

… the square of opposition, is also not to be found in Aristotle’s text, but it provides a useful summary of his doctrine. According to his explanations, statements are opposed as contradictories when they cannot both be true and cannot both be false, but as contraries only when they cannot both be true but may both be false [De Interpretatione 7 (17b 16-25)] … Although he does not use these expressions subaltern and sub-contrary), Aristotle (…) assumes that subcontraries cannot be false though they may both be true. This is shown by his description of them as contradictories of contraries.

For more details about the square of opposition, the reader may consult e.g. (Parsons 1997).

1. Contradictories, subcontraries and contraries in classical logic

Let F be the set of propositional formulas built with the connectives ¬, ∧, ∨, →. Formulas will be denoted by ab, etc., sets of formulas by TU, etc. The set C of classical valuations is defined as usual: it is a set of functions from F to {0,1} and its members obey the standard conditions, in particular we have: for any v in C and for any a in Fv(a)=1 iff va)=0.

With this framework we are now able to define precisely the discussed notions in the context of the semantics of classical logic.

Given two formulas a and b, we say that they are:

- contradictories iff for any v in Cv(a)=0 iff v(b)=1;

- contraries iff for any v in Cv(a)=0 or v(b)=0 and there exists v in Cv(a)=0 and v(b)=0;

- subcontraries iff for any v in Cv(a)=1 or v(b)=1, and there exists v in Cv(a)=1 and v(b)=1;

Let us note that if we remove the second part of the definition of subcontraries

«there exists v in Cv(a)=1 and v(b)=1», which translates «may both be true», then all contradictories are subcontraries. In this case confusing subcontraries with contradictories would not be the same as switching red with blue, or cats with dogs, but rather would amount of confusing dogs with canines. Let us call global confusion this kind of error by contrast to the first one that we can call switching confusion. As Slater claims through his red and blue example that paraconsistent logicians are making a switching confusion rather than a global one, it seems implicit that he doesn’t consider that all contradictories are subcontraries, neither do we here.

It is clear that for any formula aa and ¬a are contradictories. The connective ¬ is said to be a contradictory forming relation.

Which examples of subcontraries can we find? For any two atomic formulas a and ba and ¬ab are subcontaries, as the reader can easily check. This can be illustrated by «Plato is a cat» and «Plato is not a cat or snow is blue», which cannot both be false but can both be true.

Can we define the relation which associates to any formula a the set of formulas {¬abbF} as a subcontrary forming relation? That sounds reasonable but we must be aware that in this case this relation includes pairs of formulas like a and ¬a∨(a∧¬a) which are contradictories.

It is clear that inside classical logic, there are a lot of subcontrary forming relations; however the question is: are paraconsistent negations part of these subcontrary forming relations? And the answer is: no. Because these negations are not definable in classical logic.

For example da Costa’s paraconsistent negation of the logic C1 is not definable in classical logic because it is not self-extensional (i.e. the replacement theorem does not hold for it).

A paraconsistent negation is not in general a subcontrary forming relation inside classical logic, maybe be it is a subcontrary forming relation from another point of view – this question will be examined later on – but anyway we must remember that in general paraconsistent negations are not definable in classical logic and that for example the logic C1 of da Costa is strictly stronger than classical logic in the sense that classical logic is definable in C1 but not the converse. The same happens with intuitionistic logic, and that is why from this point of view, intuitionistic negation is not a contrary forming relation, erroneous conclusion that someone may reach applying an argument similar to Slater’s one.

Thus paraconsistent logic is not merely the result of changing the names of concepts of classical logic already existing, but the appearance of a new phenomenon. This is a first point against Slater.

Even if someone thinks that notions such as negation and contradictory cannot be used in another way that the way they are used in classical logic, he must admit that there are notions of non classical logic that cannot be defined in classical logic (and that therefore, however they are named, these notions cannot be named by names naming some notions definable in classical logic).

As I have pointed out in my review of Slater’s paper for Mathematical Reviews (96e03035), paraconsistent logic is not a result of a verbal confusion similar to the one according to which in Euclidean geometry «point» will be exchanged with «line», but rather the shift of meaning of «negation» in paraconsistent logic is comparable to the shift of meaning of «line» in non-Euclidean geometry.

1. Conclusion

In view of the above result, to say that a negation is not a negation because it is not a contradictory forming relation, is just to say that a negation is not a negation because it is not a classical negation, because only classical negation is a contradictory forming relation.

To state, without argumentation, that only classical negation is a negation and to claim that paraconsistent negations are therefore not negations, is just to make a tautological affirmation without any philosophical value.

But the real discussion does not reduce to such a trivial point. The question is to know what are the properties of classical negation which are compatible with the rejection of the ex contradictione sequitur quodlibet, rejection which is the basis of paraconsistent negation (on this topic see Béziau 2000).

Paraconsistent logic has shown in fact that a paraconsistent «negation» can have some strong properties, that for example it does not reduce to a mere modal operator and that it can make sense to use the word «negation» in the context of paraconsistency, in a similar way that it can make sense to speak of «intuitionistic negation» or of «Johansson’s negation».

Moreover, obviously the meaning of the word «negation» in natural language does not reduce to the meaning of classical negation of classical logic and nobody has yet tried to prohibit the use of this word in natural language.

Finally, a possible way to consider that a paraconsistent negation (or another non classical negation) is a contradictory forming relation, despite of our negative result of SECTION 4, is to change the definition of contradictory forming relation and to say that two formulas a and b are contradictories iff one is the «negation» of the other.

Of course this can lead to nonsense if we are dealing with something which has nothing to do with negation. But if we reasonably change the meaning of «negation», it makes sense to accordingly change the meaning of «contradictories».

It seems that this is the option Priest has now taken after we present to him our present criticisms to his paper with Routley.

It is worth emphasized that from this point of view Priest’s negation LP does not present any superiority to da Costa’s negation C1 or other paraconsistent negations.

Postface

This paper was originally written in 1996, just after I wrote the review of Slater’s paper, «Paraconsistent logics?» for Mathematical Reviews; a Romanian translation of it was published in 2004 in I.Lucica et al. (eds), Ex falso qodlibet, Tehnica, Bucarest. In particular this paper was written before the publication of Greg Restall’s paper, «Paraconsistent logics!», Bulletin of the Section of Logic 26/3 (1997), with a title which is quite the same. However the contents of the papers are completely different. After writing this paper I wrote several papers which are a continuation of it:

• J.-Y.Béziau, «Paraconsistent logic from a modal viewpoint», Journal of Applied Logic, 3 (2005), pp.7-14. [<http://www.unine.ch/unilog/jyb/jyb-wopalo-elsevier.pdf>]
• J.-Y. Béziau, «New light on the square of oppositions and its nameless corner», Logical Investigations, 10, (2003), pp.218-232. [<http://www.unine.ch/unilog/jyb/sep.pdf>]
• J.-Y.Béziau, «Are paraconsistent negations negations?», in Paraconsistency: the logical way to the inconsistent, W.Carnielli et al. (eds), Marcel Dekker, New-York, 2002, pp.465-486.
1. Bibliography
• [Bez94] J.-Y.Béziau, 1994, «Théorie législative de la négation pure», Logique et Analyse147-148, pp.209-225.
• [Bez96] J.-Y.Béziau, 1996, review of [Sla], Mathematical Reviews, 96e03035.
• [Bez97] J.-Y.Béziau, 1997, «What is many-valued logic?», in Proceedings of the 27th International Symposium on Multiple-Valued logic, IEEE Computer Society, Los Alamitos, pp.117-121.
• [Bez00] J.-Y.Béziau, 2000, «What is paraconsistent logic?», in Frontiers in paraconsistent logic, D.Batens et al. (eds), Research Studies Press, Baldock, pp.95-111.
• [BlS] H.A.Blair and V.S.Subrahmanian, 1989, «Paraconsistent logic programming», Theoretical Computer Science68, pp.135-153.
• [Cos] N.C.A.da Costa, 1963, «Calculs propositionnels pour les systèmes formels inconsistants», Comptes Rendus de l’Académie des Sciences de Paris257, pp.3790-3793.
• [CoA] N.C.A.da Costa and E.H.Alves, 1976, «Une sémantique pour le calcul C1», Comptes Rendus de l’Académie des Sciences de Paris238A, pp.729-731.
• [CoB] N.C.A.da Costa and J.-Y.Béziau, 1994, «Théorie de la valuation», Logique et Analyse146, pp.95-117.
• [KnK] W.Kneale and M.Kneale, 1962, The development of logic, Clarendon Press, Oxford.
• [LoC] A.Loparic and N.C.A.da Costa, 1984, «Paraconsistency, paracompleteness and valuations», Logique et Analyse106, pp.119-131.
• [Mal] G.Malinowski, 1993, Many-valued logics, Clarendon Press, Oxford.
• [Mali] G.Malinowski, 1994, «Inferential many-valuedness», in Philosophical logic in Poland, J.Wolenski (ed), Kluwer, Dordrecht, pp.75-84.
• [Par] T.Parsons, 1997, «The traditional square of opposition», Stanford Encyclopedia of Philosophyhttp://plato.stanford.edu.
• [Pri] G.Priest, 1979, «Logic of paradox», Journal of Philosophical Logic8, pp.219-241.
• [PrR] G.Priest and R.Routley, 1989, «Systems of paraconsistent logic», in Paraconsistent logic: Essays on the inconsistent, Philosophia, Munich, pp.151-186.
• [Sla] B.H.Slater, 1995, «Paraconsistent logics?», Journal of Philosophical logic24, pp.451-454.

Acknowledgments

I would like to thank N.C.A. da Costa, G.Priest and B.H.Slater for discussions and comments.

Jean-Yves Béziau Institute of Logic and Semiological Research Center University of Neuchâtel, Espace Louis Agassiz 1 CH – 2000 Neuchâtel, Switzerland <jean-yves.beziau [at] unine.ch>

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