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Paraconsistency in classical logic

Abstract

Classical propositional logic can be characterized, indirectly, by means of a complementary formal system whose theorems are exactly those formulas that are not classical tautologies, i.e., contradictions and truth-functional contingencies. Since a formula is contingent if and only if its negation is also contingent, the system in question is paraconsistent. Hence classical propositional logic itself admits of a paraconsistent characterization, albeit “in the negative”. More generally, any decidable logic with a syntactically incomplete proof theory allows for a paraconsistent characterization of its set of theorems. This, we note, has important bearing on the very nature of paraconsistency as standardly characterized.

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Notes

  1. 1.

     Some systems of paraconsistent logic have more than one negation operator, and \(\mathsf {ECQ}\) need not fail for all. For instance, da Costa’s C-systems (da Costa 1974) have a “strong negation” that satisfies \(\mathsf {ECQ}\). Of course, the question of what counts as a bona fide negation operator is by itself a delicate matter in this context (see e.g. Lenzen 1998; Béziau 2002b); we shall come back to this below.

  2. 2.

     We take a system \({\mathsf {S}}\) to be syntactically incomplete if and only if there is a formula \(\alpha \) in the language of \({\mathsf {S}}\) and a negation operator \(\lnot \) such that neither \(\vdash _{\mathsf {S}}\alpha \) nor \(\vdash _{\mathsf {S}}\lnot \alpha \).

  3. 3.

     It is also possible to consider a stronger notion of complementarity, whereby \({\mathsf {S}}\) and \(\overline{{\mathsf {S}}}\) are complementary just in case \(\Gamma \nvdash _{\mathsf {S}}\alpha \) if and only if (or, more generally, \(\Gamma \nvdash _{\mathsf {S}}\Delta \) if and only if , where \(\Gamma \) and \(\Delta \) are sets of formulas). In that case, is just the complement of \(\vdash _{{\mathsf {S}}}\). For our purposes, however, it will suffice to restrict our attention to the notion defined in \(\mathsf {COMP}\), which is generally weaker unless both \({\mathsf {S}}\) and \(\overline{{\mathsf {S}}}\) satisfy the deduction theorem (i.e., unless \(\Gamma \cup \{ \alpha \} \vdash _{\mathsf {S}}\beta \) implies \(\Gamma \vdash _{\mathsf {S}} \alpha \rightarrow \beta \) for all \(\Gamma , \alpha , \beta \), and similarly for ).

  4. 4.

     This simple system builds on Łukasiewicz’s refutation method (Łukasiewicz 1939; revisited in Łukasiewicz 1951, §27). For a survey of the method and of related theories, see Skura (2011).

  5. 5.

     Derivation in \(\overline{\mathsf {CL}}\) is defined as usual: if and only if there is a finite sequence \(\alpha _1, \dots , \alpha _m\) such that \(\alpha _m = \alpha \) and every \(\alpha _k\) is either a member of \(\Gamma \cup \{\bot \}\) or else follows from some \(\alpha _i\), \(i < k\), by one of the rules. When \(\Gamma = \varnothing \), \(\alpha \) counts as a \(\overline{\mathsf {CL}}\)-theorem and the sequence \(\alpha _1, \dots , \alpha _m\) as a \(\overline{\mathsf {CL}}\)-proof of \(\alpha \).

  6. 6.

     Gentzen-style complementary systems are typically defined in terms of the stronger notion of complementarity mentioned in note 3. They differ, therefore, from the version of \(\overline{\mathsf {CL}}\) defined above, which does not satisfy the deduction theorem. (We have, for instance, but not .)

  7. 7.

     An axiomatization of the classical contingencies can also be obtained from the version of \(\overline{\mathsf {CL}}\) given above by taking as axiom an arbitrary sentence variable, \(p_i\), and counting as equivalent the following pairs of formulas for \(\alpha \in \{\bot , \lnot \bot , p_i, \lnot p_i\}\): \(\lnot \lnot \alpha , \alpha \) | \(\alpha \vee \bot , \alpha \) | \(\bot \vee \alpha , \alpha \) | \(\alpha \vee \lnot \bot , \lnot \bot \) | \(\lnot \bot \vee \alpha , \lnot \bot \). See (Varzi 1990, §5).

  8. 8.

     Here it is important to keep in mind that \(\lnot \) admits of a perfectly standard semantic characterization as a negation connective. Certainly there are ways of interpreting this operator that would not justify the claim of paraconsistency. For instance, with \(\lnot \) understood as a modal connective for necessity \(\Box \), or possibility \(\Diamond \), any normal modal logic would be non-explosive in the present sense, but of course such connectives do not qualify as negations in any reasonable sense. See also below, note 11.

  9. 9.

     As with the users of the Cretan Manual envisaged in Massey (1978), or the characters of the dialogue in Casati and Varzi (2000). The standard notion of logical consequence as truth preserving, by contrast, goes back to Tarski (1936). Of course, the validity of \(\mathsf {ECQ}\) is not just a byproduct of the standard notion; it depends also on specific logical principles. For example, C. I. Lewis’s classic argument for \(\mathsf {ECQ}\) requires disjunction introduction (\(\alpha \vdash \alpha \vee \beta \)) and disjunctive syllogism (\(\alpha , \lnot \alpha \vee \beta \vdash \beta \)); see (Lewis and Langford 1932, p. 250).

  10. 10.

     As already pointed out, Gentzen-style complementary systems are typically defined in terms of the stronger notion of \(\mathsf {COMP}\) mentioned in note 3. Thus, in that case the contrast between Alice’s and Bob’s attitudes is more naturally put in terms of validity and invalidity rather than truth and falsity: “Is that sequent valid [resp. invalid]? Then so is this one.”

  11. 11.

     There are other ways of blending paraconsistency and classicality. For instance, Béziau (2002a) points out that in classical first-order logic one can define a non-explosive negation operator \({\sim }\) just by setting \({\sim } \alpha =_{df} \exists x \lnot \alpha \). Our point, here, is that there is a paraconsistent side already to the classical negation operator \(\lnot \).

  12. 12.

     Note that \(\mathsf {AT}\) and \(\mathsf {BT}\) are not only classically invalid; they have instances that are classical contradictions, e.g. when \(\alpha \) is \(p_i \wedge \lnot p_i\).

  13. 13.

     To our knowledge, all familiar paraconsistent systems are syntactically incomplete, hence the argument at the end of Sect. 2 applies mutatis mutandis: if \({\mathsf {S}}\) is such a system and \(\mathsf {\overline{{\mathsf {S}}}}\) satisfies \(\mathsf {COMP}\), then \(\mathsf {\overline{{\mathsf {S}}}}\) must be paraconsistent, too. For the incompleteness of \({\mathsf {S}}\) implies the existence of some and , and from this we can conclude that for any \({\mathsf {S}}\)-theorem \(\beta \).

  14. 14.

     This would not be true if complementarity were defined in the stronger sense mentioned in note 3, yielding what Béziau and Buchsbaum (2016) call “anti-classical consequence relation”. For this reason, the following observations do not apply to Gentzen-style complementaries of \(\mathsf {CL}\), though it should be noted that such calculi, too, preserve several important properties of their classical counterparts, including the subformula property and, as already noted, cut elimination (Carnielli and Pulcini 2017).

  15. 15.

    We are most thankful to Walter Carnielli, Graham Priest, and to the referees of Synthese for they helpful comments and suggestions on earlier versions of this paper. G.P. also thankfully acknowledges the support from the Portuguese Science Foundation, FCT, through the project “Hilbert’s 24th Problem” PTDC/MHC-FIL/2583/2014.

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Pulcini, G., Varzi, A.C. Paraconsistency in classical logic. Synthese 195, 5485–5496 (2018). https://doi.org/10.1007/s11229-017-1458-0

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Keywords

  • Paraconsistency
  • Classical logic
  • Complementary system
  • Consequence relation
  • Decidability
  • Unprovability