Paraconsistent Logics

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  • B. H. SLATER


    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. Thus in his article 'The Logic of Paradox', Graham Priest composes

    a well known 'paraconsistent' logic. Its semantics involves evalua- tions which can be taken to be onto { 1 ,0 , - 1 } (Priest uses {t, p, f}, see Priest 1979, p. 227), such that V(-~A) = -V(A), and V(A A B) = min{V(A), V(B)}. Logical truth is then equated with 'for all evalua- tions, V(A) >>. 0'. In this logic all the classical tautologies hold, but it may be that V(A A -~A) = 0, and hence it looks like the logic allows contradictories to be both true.

    However, it is not the case that '-~A' is the contradictory of 'A' in this logic. Priest shows (Priest 1979, p. 238) that 'A is true' (V(A) >f 0) is equivalent to 'A', and that 'A is false' (V(A) ~< 0) is equivalent to '-~A'. But it is clear that 'A is true' and 'A is false' are not then contradictory, since what contradicts 'V(A)/> 0' is 'V(A) < 0'. So 'A is true' is merely subcontrary to 'A is false', making '-,' not a contra- diction forming functor, in this logic.

    The obvious danger that 'paraconsistent' logics should be dealing in subcontraries rather than contradictories has been noted by Priest else- where. Thus, in Paraconsistent Logics, Essays on the Inconsistent, the editors, Priest and Richard Routley, speaking of da Costa's 'positive plus' systems, say (Priest and Routley 1989, pp. 164/5):

    That an account of negation violates the law of non-contradiction provides prima facie evidence that account is wrong. This is [one] piece of evidence that da Costa negation is not negation.

    In fact we can make the claim more precise, Traditionally A and B are subcon- traries if A V B is a logical truth. A and/3 are contradictofies if A V/3 is a logical

    Journal of Philosophical Logic 24: 451-454, 1995. 9 1995 Kluwer Academic Publishers. Printed in the Netherlands.

  • 452 B. H. SLATER

    truth and A A B is logically false. It is the second condition which distinguishes con- tradictories from subcontraries. Now in da Costa's approach we have that A V ~A is a logical truth. But A A -~A is not logically false. Thus A and -~A are subcontraries, not contradictories. Consequently da Costa negation is not negation, since negation is a contradiction forming functor, not a subcontrary forming functor.

    This defect in da Costa's systems Priest nominally remedied in 'The Logic of Paradox', since there (Priest 1979, p. 228) not only is 'A V -~A' a 'logical truth', but also 'A A -~A' is 'logically false'. 1 However, the scare quotes are essential, because the remedy is only a face-lift. A formula 'X ' being 'logically false' only means that 'V(X)


    the well-known paradoxes with which he has been concerned. But that, for one thing, does not affect the above point, that there is no 'paraconsistent' logic. Moreover, Priest's ideas about the paradoxes, on which he might base a claim about the concept of contradiction not being in fact instantiated, themselves use a concept of truth which is surely not instantiated: the Tarskian concept of truth as a predicate of sentences. 6 Since Montague, we surely now know that syntactic treatments of modality must be replaced by operator formulations. 7 And not only must that hold with the identity modality 'It is true that', as a consequence, but there are clearly no paradoxes with the opera- tor notion, since it was proved many years ago, by Goodstein, to be consistent. 8

    As a result, while 'truth' and 'falsity' are only subcontrary in Priest's language, that does not show, in any way, that truth and falsity are only subcontraries. For no change of language can alter the facts, only the mode of expression of them, as we saw before. And one central fact is that contradictories cannot be true together - by definition. 9

    Now, in another place, Priest, with Chris Mortensen, has main- tained, in effect, that such a definition cannot be held to. Thus they say (Priest and Mortensen, p. 386):

    We might feel that we can resolve to make 'false' exclude 'true', but this is an illus- sion. There is no guarantee that we can keep control of the resolution once we allow 'false' and 'true' to have additonal logical properties.

    But, to keep to the definition of contradiction, in such a case, one sim- ply denies that falsity and truth (as defined) do have those additonal logical properties. Certainly other properties of falsity and truth, like being a predicate of sentences, may be incompatible with the definition of contradiction, but that in no way means we cannot hold onto that definition.


    1 Specifically, '-~(A A -~A)' is said to be 'logically valid', i.e. invariably V(-~(A A ~A))/> 0, so invariably V(A A -~A)

  • 454 B. H. SLATER

    p. 319.) But Meyer and Martin's discussion is somewhat confused, since they talk elsewhere (Meyer and Martin 1986, p. 309), in their own (Australian) case, of 'hold- ing that A' and 'not denying that A', in place of straight 'truth' and 'falsity'. 6 Priest 1979, pp. 222-3, see also Priest 1987, w167 4.2, 4.3. 7 Montague 1963, see also, for instance, Reinhardt 1980. 8 Goodstein, 1958, p. 419; see also Prior 1971, Ch 7. Prior shows (item (3) p. 105) that the paradoxical cases merely give rise to ambiguity. For further details see Slater 1986, 1991, and Sayward 1987. 9 For the record, this definition is given, for instance, in Sainsbury 1991, p. 16.


    Belnap, N.D. Jr. 1977, 'A Useful Four-Valued Logic', in Dunn, J.M. and Epstein, G. eds, Modern Uses of Multiple-Valued Logic, Reidel, Dordrecht, 8-37.

    Copeland, J. 1979, 'On when a Semantics is not a Semantics', Journal of Philosophi- cal Logic 8: 399-413.

    Copeland, J. 1986, 'What is a Semantics of Classical Negation?', Mind 95: 478-90. Goodstein, R.L. 1958, 'On the Formalisation of Indirect Discourse', Journal of Sym-

    bolic Logic 23: 417-9. Meyer, R.K. and Martin, E.P. 1986, 'Logic on the Australian Plan', Journal of Philo-

    sophical Logic 15: 305-32. Montague, R. 1963, 'Syntactic Treatments of Modality', Acta Philosophica Fennica

    6: 153-67. Priest, G.G. 1979, 'The Logic of Paradox', Journal of Philosophical Logic 8: 219-

    41. Priest, G.G. 1987, In Contradiction, Martinus Nijhoff, Dordrecht. Priest, G.G. and Mortensen, C. 1981, 'The Truth Teller Paradox', Logique et Analyse

    24: 381-88. Priest, G.G. and Routley, R. 1989, Paraconsistent Logics, Essays on the Inconsistent,

    Philosophia Verlag, Munich. Prior, A.N. 1971, Objects of Thought, O.U.E Oxford. Reinhardt, W.N. 1980, 'Necessity Predicates and Operators', Journal of Philosophical

    Logic 9: 437-50. Sainsbury, M. 1991, Logical Forms, Blackwell, Oxford. Sayward, C. 1987, 'Prior's Theory of Truth', Analysis 47: 83-7. Slater, B.H. 1986, 'prior's Analytic', Analysis 46: 76-81. Slater, B.H. 1991, 'Liar Syllogisms and Related Paradoxes', Analysis 51: 146-53.

    Department of Philosophy, University of Western Australia, Nedlands, W.A. 6009, Australia