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Some Adaptive Contributions to Logics of Formal Inconsistency

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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 152)


Some insights were gained from the study of inconsistency-adaptive logics. The aim of the present paper is to put some of these insights to work for the study of logics of formal inconsistency. The focus of attention is application contexts of the aforementioned logics and their theoretical properties in as far as they are relevant for applications. As the questions discussed are difficult but important, a serious attempt was made to make the paper concise but transparent.


  • Paraconsistent logic
  • Logics of formal inconsistency
  • Inconsistency-adaptive logic

Mathematics Subject Classification (2000)

  • 03B53
  • 03B60
  • 03A05

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  1. 1.

    Being merely an abbreviation in \(\mathbf {C}_{n}\) logics, the consistency operator adds nothing to the expressive power of the language. That the definiens, for example \(\lnot (A\wedge \lnot A)\) in \(\mathbf {C}_{1}\), expresses the consistency of A is somewhat awkward in the context of the \(\mathbf {C}_{n}\) logics.

  2. 2.

    Some paraconsistent logicians defend a specific negation as the correct one. Priest [17], for example, seems to assign this role to the negation of \(\mathbf {LP}\). Other paraconsistent logicians, for example da Costa [12], consider a multiplicity of operators as paraconsistent negations, but sometimes impose certain conditions. Often a more general outlook is taken, as for example by Béziau [9].

  3. 3.

    The insight was Suszko’s [18]. The resulting semantics may be ugly but is obviously adequate.

  4. 4.

    The reference to \(\Gamma \) may be dropped for Tarski logics (reflexive, transitive, and monotonic logics).

  5. 5.

    The syntactic justification refers to the complementing character of the non-paracomplete paraconsistent negation. \(A, \mathord {\sim }A \vdash _\mathbf {L} \lnot A\) by explosion for the classical \(\mathord {\sim }\) and \(\lnot A, \mathord {\sim }A \vdash _\mathbf {L} \lnot A\) by reflexivity. Both together entail \(\mathord {\sim }A \vdash _\mathbf {L} \lnot A\) in view of the is complementing character of \(\lnot \).

  6. 6.

    Here too the reference to \(\Gamma \) may be dropped for Tarski logics.

  7. 7.

    This is the reason why the converses of the inferences mentioned in Fact 15.11 do not hold for all consistency connectives.

  8. 8.

    The situation may have been influenced by the somewhat odd behaviour of the (defined) consistency operator \(A^{(n)}\) in da Costa’s \(\mathbf {C}_{n}\) systems with \(n>1\). Another relevant consideration might have been that the consistent behaviour of a formula A on a premise set \(\Gamma \), viz. that the logic does not require \(\Gamma \) to entail A as well as \(\lnot A\), should not cause \(\mathord {\circ }A\) to be derivable from \(\Gamma \). However, this danger is nonexistent even in case \(\mathord {\circ }A\) is the suitable truth-function of A and \(\lnot A\).

  9. 9.

    A non-monotonic logic may assign to \(\Gamma \) a selection of models that verify all members of \(\Gamma \). The lemma contains references to all \(\mathbf {L}\)-models.

  10. 10.

    Adding or removing the trivial model—the model verifying all closed formulas—to the set of models defined by a semantics may require that the semantic clauses are adjusted. In view of the definition of the semantic consequence relation, it is obvious that such addition or removal does not affect the consequence relation.

  11. 11.

    There is no reason to handle Modus Ponens and Modus Tollens on a par. The first states a property of the implication. The justification of Modus Tollens requires a reference to negation: if A is true, then B is true (by Modus Ponens); but \(\lnot B\) is true; so if inconsistencies are not true, then neither is A.

  12. 12.

    This regularity requirement is stronger than the requirement for being a negation in the sense of Lemma 15.14. If S contains all \(\mathbf {CL}\)-models as well as the trivial model, \(\lnot \) is a negation but the regularity requirement is not fulfilled.

  13. 13.

    Do not confuse the question whether a logic is regular with the question whether a specific semantics of this logics is deterministic. See Sect. 2 of another paper [7] in this volume for a method to turn an indeterministic semantics of a certain type into a deterministic one.

  14. 14.

    There is no need to add “with respect to \(\mathbf {L}\)” as \( Dab \)-consequences of \(\Gamma \) will always be considered for a specific logic.

  15. 15.

    The set of minimal \( Dab \)-consequences obviously depends on the logic. For some paraconsistent logics, like \(\mathbf {CLuN}\) mentioned in a subsequent section, \((p\wedge \lnot p)\vee (q\wedge \lnot q)\) is the only \( Dab \)-consequence of \(\Gamma _{1}\). Other paraconsistent logics assign infinitely many \( Dab \)-consequences to \(\Gamma _{1}\). Still, I cannot picture any formal paraconsistent logic for which \(\mathord {\circ }t\) has an effect on the minimal \( Dab \)-consequences of \(\Gamma _{1}\). This is weaker than what is claimed in the text, but I shall buy you a beer if you show my imagination lacking at this point and that is stronger than what is said in the text.

  16. 16.

    To be more precise, this is the case for some (not necessarily all) sets \(\{A_1, \ldots , A_n\}\) such that \((A_1\wedge \lnot A_1)\vee \ldots \vee (A_n\wedge \lnot A_n)\) is a minimal \( Dab \)-consequence of the non-logical axioms of the theory.

  17. 17.

    The predicative logic \(\mathbf {CLuN}\), first introduced in [2], is the predicative extension of the propositional \(\mathbf {PI}\) from [1]. The latter extended with a suitable axiom for a consistency operator is the LFI \(\mathbf {mbC}\)—see Definition 42 of [10].

  18. 18.

    Suitable are a classical conjunction or a gappy one. Glutty conjunctions have to be considered contextually because they allow for models that verify a conjunction and falsify one of the conjuncts. While such models are clearly abnormal with respect to \(\mathbf {CL}\) and many other logics, it depends on further properties whether a consistency operator should handle this. See for example [8] on gluts and gaps of all kinds.

  19. 19.

    For present purposes, this may be identified with a compact Tarski logic.

  20. 20.

    The set \(\Omega \) may comprise formulas of the form \(\exists (A\wedge \lnot A)\). If A is any formula, the form is unrestricted; if A is required to be an atomic formula, the form is restricted.

  21. 21.

    The upper limit logic \(\mathbf {ULL}\) is obtained by extending \(\mathbf {LLL}\) with a rule that causes all abnormalities to entail triviality.

  22. 22.

    If one of those symbols would be glutty or gappy, the abnormalities would need to contain members that describe the gluts or gaps in the existential quantifier and the conjunction in order to handle the situation in an adequate way. See [8] for more information.

  23. 23.

    The absence of the restriction may cause the adaptive logic to be a flip-flop, which means that the adaptive consequence set reduces to the lower limit consequence set whenever the premise set is abnormal.

  24. 24.

    This \(\Omega \) will also be adequate for some combinations of non-classical quantifiers, but that need not concern us in the present paper.

  25. 25.

    If \(\Gamma \) has no \( Dab \)-consequences, \(\Phi (\Gamma )=\{\emptyset \}\); if \(\Gamma \) has no \(\mathbf {LLL}\)-models, \(\Phi (\Gamma )=\{\Omega \}\); \(\Phi (\Gamma )\ne \emptyset \) always holds.

  26. 26.

    Where a logic \(\mathbf {L}\) is defined over \(\mathcal {L}\), a set \(\Gamma \) is \(\mathbf {L}\)-trivial iff \( Cn _{\mathbf {L}}(\Gamma ) = \mathcal {W}\).

  27. 27.

    If consistency decisions do not trivialise the theory, consistency reclaims do not either; if consistency decisions trivialise the theory, consistency reclaims cannot make that situation worse.

  28. 28.

    Actually for all formulas not in the set \(\{p,q\}\), but never mind.

  29. 29.

    The reasons for the \(\mathcal {X}\) is as in (15.9); it would be tiresome to make this more precise here.

  30. 30.

    If you frown here, realize that \(\lnot p\) is not a \(\mathbf {mbC}\)-consequence of \(\lnot (p\vee q)\).


  1. Batens, D.: Paraconsistent extensional propositional logics. Logique et Analyse 90–91, 195–234 (1980)

    Google Scholar 

  2. Batens, D.: Inconsistency-adaptive logics. In: Orłowska, E. (ed.) Logic at Work. Essays Dedicated to the Memory of Helena Rasiowa, pp. 445–472. Physica Verlag (Springer), Heidelberg (1999)

    Google Scholar 

  3. Batens, D.: Towards the unification of inconsistency handling mechanisms. Logic Log. Philos. 8, 5–31 (2000). Appeared 2002

    Google Scholar 

  4. Batens, D.: Narrowing down suspicion in inconsistent premise sets. In: Malinowski, J., Pietruszczak, A. (eds.) Essays in Logic and Ontology. Poznań Studies in the Philosophy of the Sciences and the Humanities, vol. 91, pp. 185–209. Rodopi, Amsterdam (2006)

    Google Scholar 

  5. Batens, D.: A universal logic approach to adaptive logics. Log. Univers. 1, 221–242 (2007)

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. Batens, D.: The consistency of Peano Arithmetic. A defeasible perspective. In: Allo, P., Van Kerkhove, B. (eds.) Modestly Radical or Radically Modest. Festschrift for Jean Paul van Bendegem on the Occasion of His 60th Birthday. Tributes, vols. 24, 22, pp. 11–59. College Publications, London (2014)

    Google Scholar 

  7. Batens, D.: Tutorial on inconsistency-adaptive logics. In: Béziau, J.-Y., Chakraborty,M., Dutta, S. (eds.) New Directions in Paraconsistent Logic. Springer (2015)

    Google Scholar 

  8. Batens, D.: Spoiled for choice? J. Logic Comput. (1913), in print. doi:10.1093/logcom/ext019

  9. Béziau, J.-Y.: S5 is a paraconsistent logic and so is first-order classical logic. Log. Invest. 9, 301–309 (2002)

    Google Scholar 

  10. Carnielli, W.A., Coniglio, M.E., Marcos, J.: Logics of formal inconsistency. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 14, pp. 1–93. Springer (2007)

    Google Scholar 

  11. da Costa, N.C.A.: Calculs propositionnels pour les systèmes formels inconsistants. Comptes rendus de l’Académie des sciences de Paris 259, 3790–3792 (1963)

    Google Scholar 

  12. da Costa, N.C.A.: On the theory of inconsistent formal systems. Notre Dame J. Form. Logic 15, 497–510 (1974)

    CrossRef  MATH  Google Scholar 

  13. Nasieniewski, M.: Logiki adaptujące sprzeczność (Inconsistency adapting logics). Ph.d. thesis, Chair of Logic, N. Copernicus University, Toruń, Poland (2002)

    Google Scholar 

  14. Nasieniewski, M.: An adaptive logic based on Jaśkowski’s logic \(\mathbf{D}_2\). Logique et Analyse 185–188, 287–304 (2004). Appeared 2005

    Google Scholar 

  15. Nasieniewski, M.: Wprowadzenie do logik adaptywnych. Wydawnictwo Naukowe, Universytetu Mikołaja Kopernika, Toruń (2008)

    Google Scholar 

  16. Odintsov, S.P., Speranski, S.O.: Computability issues for adaptive logics in multi-consequence standard format. Stud. Logica 101(6), 1237–1262 (2013). doi:10.1007/s11225-013-9531-2

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. Priest, G.: In Contradiction. A Study of the Transconsistent. Oxford University Press, Oxford (2006). Second expanded edition (first edition 1987)

    Google Scholar 

  18. Suszko, R.: The Fregean axiom and Polish mathematical logic in the 1920s. Stud. Logica 36, 377–380 (1977)

    CrossRef  MathSciNet  Google Scholar 

  19. Verdée, P.: Adaptive logics using the minimal abnormality strategy are \(\Pi ^1_1\)-complex. Synthese 167, 93–104 (2009)

    CrossRef  MathSciNet  MATH  Google Scholar 

  20. Verdée, P.: Strong, universal and provably non-trivial set theory by means of adaptive logic. Logic J. IGPL 21, 108–125 (2012)

    CrossRef  Google Scholar 

  21. Verdée, P.: Non-monotonic set theory as a pragmatic foundation of mathematics. Found. Sci. 18, 655–680 (2013)

    CrossRef  MathSciNet  Google Scholar 

  22. Verdée, P.: A proof procedure for adaptive logics. Logic J. IGPL 21, 743–766 (2013). doi:10.1093/jigpal/jzs046

    CrossRef  MATH  Google Scholar 

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Batens, D. (2015). Some Adaptive Contributions to Logics of Formal Inconsistency. In: Beziau, JY., Chakraborty, M., Dutta, S. (eds) New Directions in Paraconsistent Logic. Springer Proceedings in Mathematics & Statistics, vol 152. Springer, New Delhi.

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