Skip to main content
SpringerLink
Log in
Book cover

New Directions in Paraconsistent Logic pp 3–38Cite as

Tutorial on Inconsistency-Adaptive Logics

  • Conference paper
  • First Online:

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

Abstract

This paper contains a concise introduction to a few central features of inconsistency-adaptive logics. The focus is on the aim of the program, on logics that may be useful with respect to applications, and on insights that are central for judging the importance of the research goals and the adequacy of results. Given the nature of adaptive logics, the paper may be read as a peculiar introduction to defeasible reasoning.

I am indebted to Mathieu Beirlaen for careful comments on a previous draft.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    See, for example, [17] for many other real-life examples of reasoning forms for which there is no positive test. The import of a positive test is discussed further in the text.

  2. 2.

    Uniform Substitution is rule of propositional logic. Predicative classical logic is traditionally axiomatized in terms of a finite set of rules and axiom schemata, rather than axioms. So no substitution rule is then required. Substitution rules in predicate logic have been studied [56] and the outcome is very instructive.

  3. 3.

    The \(\mathbf {L}\)-consequence set of \(\Gamma \) is defined as \(\textit{Cn}_{\mathbf {L}}(\Gamma ) =_{\textit{df}}\{ A \mid \Gamma \vdash _\mathbf {L} A \}\).

  4. 4.

    Just think about usual proofs. Every formula in the proof is a consequence of the premise set and every proof may be extended into a longer proof by applications of the rules.

  5. 5.

    A logic \(\mathbf {L}\) is paracomplete (with respect to a negation \(\lnot \)) iff some A may false together with its negation \(\lnot A\); syntactically: iff there are \(\Gamma \), A and B such that \(\Gamma , A \vdash _\mathbf {L} B\) and \(\Gamma , \lnot A \vdash _\mathbf {L} B\), but \(\Gamma \nvdash _\mathbf {L} B\).

  6. 6.

    In the context of \(\mathbf {CL}^+\), Excluded Middle together with Ex Falso Quodlibet define the classical negation.

  7. 7.

    So \(p\wedge \lnot q \vDash _{\mathbf {CL}^+} \lnot q\), \(\forall x\lnot Px \vDash _\mathbf {CL^+} \lnot Pa\), and \(a=b, Px \vDash _\mathbf {CL^+} Pb\), but \(a=b, \lnot Px \nvDash _\mathbf {CL^+} \lnot Pb\).

  8. 8.

    Names and notation may obviously be different and the model may be more complex.

  9. 9.

    Take conjunction as an example. The clause allowing for gluts: \(v_M(A\wedge B)=1\) iff (\(v_M(A)=1\) and \(v_M(B)=1\)) or \(v(A\wedge B)=1\); the one allowing for gaps: \(v_M(A\wedge B)=1\) iff (\(v_M(A)=1\) and \(v_M(B)=1\)) and \(v(A\wedge B)=1\); the one allowing for both: \(v_M(A\wedge B)=v(A\wedge B)\).

  10. 10.

    I heard the claim that restricting the formation rules of natural language so as to classify “this sentence is false” as non-grammatical is illegitimate because the sentence is ‘perfect English’. I also heard the claim that invalidating Disjunctive Syllogism is illegitimate because this reasoning form is ‘perfectly sound’.

  11. 11.

    This formula is \(\mathbf {CL}\)-equivalent to A but not \(\mathbf {CLuN}\)-equivalent to it.

  12. 12.

    As q is \(\mathbf {CLuN}\)-derivable from the premises, so is \(\lnot p\vee q\). However, relying on p to repeat the move described in the text delivers a formula that was already derivable, viz. q. The same story may be retold for every \(\mathbf {CLuN}\)-consequence of \(\Gamma _{1}\) and each time the move will be harmless because nothing new will come out of it.

  13. 13.

    Do not read the “not derived” as “not derivable”. Indeed, a formula may be derivable in several ways from the same premise set.

  14. 14.

    A more accurate wording requires that one adds: in a proof from \(\Gamma _{1}\) that extends the present stage 8. Indeed, the logic we are considering is non-monotonic. So extending the premise set may result in line 6 being marked.

  15. 15.

    The reader might think that, as p is also a \(\mathbf {CLuN}\)-consequence of \(\Gamma _{1}\), \((p\wedge q)\wedge \lnot (p\wedge q)\) is also a \(\mathbf {CLuN}\)-consequence of \(\Gamma _{1}\). This however is mistaken. \(\lnot q \nvdash _\mathbf {CLuN} \lnot (p\wedge q)\).

  16. 16.

    Names like \(\mathbf {LLL}\), \(\mathbf {AL}\), \({\mathbf {AL}^\textit{r}}\), and \(\mathbf {ULL}\) are used as generic names to define the standard format and to study its features. The names refer to arbitrary logics that stand in a certain relation to each other.

  17. 17.

    Similarly for those models together with the trivial model—the model that verifies all formulas.

  18. 18.

    The notion played a rather central role in discussions on scientific heuristics. A very clear and argued position was for example proposed by Dudley Shapere [60].

  19. 19.

    This obviously does not mean that \(\mathbin {\hat{\vee }}\) is a symbol of the language. It is a conventional name to refer to a symbol of the language that has the meaning of classical disjunction. It may even refer ambiguously: if there are several classical disjunctions, \(\mathbin {\hat{\vee }}\) need not always refer to the same one.

  20. 20.

    Axioms are suppose to be closed formulas. So \(A\in \mathcal {W}_s\). The idea is that \(\mathbf {CLuN}\)-valid rules are fully retained in the extension. One of these rules is: from \(\vdash A(a)\supset B\) to derive \(\vdash \exists xA(x)\supset B\) provided a does not occur in B.

  21. 21.

    The axiom schema may be restricted to \(A \in \mathcal {W}_s^a\), but there is no need to do so.

  22. 22.

    The exception may be caused by the logic, which is then called a flip-flop, or by the premise set—for example if the premise set comprises the formulas verified by a \(\mathbf {LLL}\)-model.

  23. 23.

    Infinite stages can be extended by inserting lines in the sequence.

  24. 24.

    It is ironic that the study of the computational complexity of adaptive logics started with a paper arguing that they are too complex [41]. The philosophical complaints and misunderstandings in that paper were answered in [26]; a mistaken theorem was corrected in [68]. Extremely interesting and more detailed studies followed [54, 55].

  25. 25.

    Stepwise: the language \(\mathcal {L}_s\) of \(\mathbf {CLuN}\) is extended with the symbol \(\mathord {\sim }\) and \(\mathbf {CLuN}\) is extended with axioms or rules that give \(\mathord {\sim }\) its classical meaning—for example the schemas \(A\supset (\mathord {\sim }A\supset B)\) and \((A\supset \mathord {\sim }A)\supset \mathord {\sim }A\).

  26. 26.

    The classical symbols were actually superimposed on \(\mathcal {L}\): in the extended language, they never occur within the scope of the original logical symbols of \(\mathcal {L}\).

  27. 27.

    The distinction warrants that the reference to a finite proof stage in Definition 1.5 is all right.

  28. 28.

    The mistake is caused by a confusion between symbols and concepts. If \(\check{\vee }\) occurs in a premise, and so in \(\mathcal {L}\), then \(\check{\vee }\) is not a new symbol of the extended language. So one needs to extend the language with another symbol, say \(\tilde{\vee }\), and call that the checked disjunction.

  29. 29.

    All that is new in the restored version is the notion of an inferred \(\textit{Dab}\)-formula.

  30. 30.

    Adaptive versions of \(\mathbf {D2}\) and other Jaśkowski logics were extensively studied [48–50].

  31. 31.

    The low computational complexity of the consequence set is rather artificial. We suppose that at least one \(\varphi \cap \Delta = \emptyset \) is given, but precisely locating a \(\varphi \) may be a very complex task.

  32. 32.

    The logic-like entity has a rather limited application field. For some \(\Gamma \), \(\Phi (\Gamma )\) is not only infinite but also uncountable.

  33. 33.

    That is (i) \(\mathcal {W} \subseteq \mathcal {W}^+\) and (ii) if \(A,B \in \mathcal {W}^+\), then \((A\mathbin {\hat{\vee }}B) \in \mathcal {W}^+\).

  34. 34.

    Classical theories, which have \(\mathbf {CL}\) as underlying logic, fail to define such a theory. Their consequence relation is much less complex. If A is not a theorem of a classical theory, humans or Turing machines may never find this out. However, if A is a theorem of the classical theory, humans or Turing machines will find that out at a finite point in time. As this point may be two million years from here, the point is slightly theoretical.

References

  1. Avron, A.: On an implication connective of RM. Notre Dame J. Formal Logic 27, 201–209 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  2. Avron, A.: Natural 3-valued logics–characterization and proof theory. J. Symbolic Logic 56, 276–294 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  3. Avron, A.: Non-deterministic matrices and modular semantics of rules. In: J.-Y. Béziau (ed.) Logica Universalis, pp. 149–167. Birkhäuser Verlag, Basel–Boston–Berlin (2005)

    Google Scholar 

  4. Avron, A., Konikowska, B.: Multi-valued calculi for logics based on non-determinism. J. Interest Group Pure Appl. Logic 10, 365–387 (2005)

    MathSciNet  Google Scholar 

  5. Avron, A., Lev, I.: Non-deterministic multiple-valued structures. J. Logic Comput. 15, 241–261 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Avron, A., Zamansky, A.: Non-deterministic semantics for logical systems. In: Gabbay, D., Guenthner, F. (ed.) Handbook of Philosophical Logic, vol. 16, pp. 227–304. Springer (2011)

    Google Scholar 

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

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

    MathSciNet  Google Scholar 

  9. Batens, D.: Dialectical dynamics within formal logics. Logique et Analyse 114, 161–173 (1986)

    MathSciNet  Google Scholar 

  10. Batens, D.: Dynamic dialectical logics. In: Priest, G., Routley, R., Norman, J. (eds.) Paraconsistent Logic. Essays on the Inconsistent, pp. 187–217. Philosophia Verlag, München (1989)

    Google Scholar 

  11. Batens, D.: Blocks. The clue to dynamic aspects of logic. Logique et Analyse, 150–152, pp. 285–328 (1995). Appeared 1997

    Google Scholar 

  12. 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, New York (1999)

    Google Scholar 

  13. Batens, D.: Minimally abnormal models in some adaptive logics. Synthese 125, 5–18 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  15. Batens, D.: A general characterization of adaptive logics. Logique et Analyse, 173–175, 45–68 (2001). Appeared 2003

    Google Scholar 

  16. Batens, D.: On a partial decision method for dynamic proofs. In: Decker, H., Villadsen, J., Waragai, T. (eds.) PCL 2002. Paraconsistent Computational Logic, pp. 91–108. (=Datalogiske Skrifter vol. 95) (2002). Also available as cs.LO/0207090 at http://arxiv.org/archive/cs/intro.html

  17. Batens, D.: The need for adaptive logics in epistemology. In: Gabbay, D., Rahman, S., Symons, J., Van Bendegem, J.P. (eds.) Logic. Epistemology and the Unity of Science, pp. 459–485. Kluwer Academic Publishers, Dordrecht (2004)

    Google Scholar 

  18. Batens, D.: A procedural criterion for final derivability in inconsistency-adaptive logics. J. Appl. Logic 3, 221–250 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Batens, D.: Narrowing down suspicion in inconsistent premise sets. In Malinowski and Pietruszczak [42], pp. 185–209

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Batens, D.: Towards a dialogic interpretation of dynamic proofs. In Cédric D., Laurent K., Helge R. (eds.) Dialogues, Logics and Other Strange Things. Essays in Honour of Shahid Rahman, pp. 27–51. College Publications, London (2009). 558 pp

    Google Scholar 

  22. Batens, D.: New arguments for adaptive logics as unifying frame for the defeasible handling of inconsistency. In: Tanaka, K., Berto, F., Mares, E., Paoli, F. (eds.) Paraconsistency: Logic and Applications. Logic, Epistemology, and the Unity of Science, vol. 26, pp. 101–122. Springer, Dordrecht (2013)

    Chapter  Google Scholar 

  23. 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, volume 24 (sometimes 22) of Tributes, pp. 11–59. College Publications, London (2014)

    Google Scholar 

  24. Batens, D.: Adaptive Logics and Dynamic Proofs. Mastering the Dynamics of Reasoning. 201x. Forthcoming

    Google Scholar 

  25. Batens, D., De Clercq, K.: A rich paraconsistent extension of full positive logic. Logique et Analyse. 185–188, 227–257 (2004). Appeared 2005

    Google Scholar 

  26. Batens, D., De Clercq, K., Verdée, P., Meheus, J.: Yes fellows, most human reasoning is complex. Synthese 166, 113–131 (2009)

    Article  MATH  Google Scholar 

  27. Batens, D., Meheus, H.: A tableau method for inconsistency-adaptive logics. In Roy D. (ed.) Automated Reasoning with Analytic Tableaux and Related Methods, Lecture Notes in Artificial Intelligence, vol. 1847, pp. 127–142. Springer (2000)

    Google Scholar 

  28. Batens, D., Meheus, Joke: Shortcuts and dynamic marking in the tableau method for adaptive logics. Stud. Logica 69, 221–248 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  29. Batens, D., Vermeir, T.: Direct dynamic proofs for the Rescher-Manor consequence relations: the flat case. J. Appl. Non-Class. Logics 12, 63–84 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  30. Brown, B.: How to be realistic about inconsistency in science. Stud. Hist. Philos. Sci. 21, 281–294 (1990)

    Article  MathSciNet  Google Scholar 

  31. Carnap, R.: Logical Foundations of Probability. University of Chicago Press, Chicago (1950)

    MATH  Google Scholar 

  32. 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 

  33. Carnielli, W.A., Marcos, J., de Amo, S.: Formal inconsistency and evolutionary databases. Logic Logical Philos. 8, 115–152 (2001). Appeared 2002

    Google Scholar 

  34. De Clercq, K.: Logica in communicatie. Cahiers du Centre de Logique, vol. 14. Academia-Bruylant, Louvain-la-Neuve (2005)

    Google Scholar 

  35. D’Ottaviano, I.M.L.: Sobre uma Teoria de Modelos Trivalente (in Portuguese). PhD thesis, State University of Campinas (Brazil), 1982

    Google Scholar 

  36. D’Ottaviano, I.M.L.: The completeness and compactness of a three-valued first-order logic. In: Proceedings of the 5th Latin American Symposium on Mathematical Logic, pp. 77–94. Revista Colombiana de Matemáticas, 1–2, 1985

    Google Scholar 

  37. D’Ottaviano, I.M.L.: The model extension theorems for J \(_{3}\)-theories. In: Di Prisco C.A. (ed.) Methods in Mathematical Logic: Proceedings of the 6th Latin American Symposium on Mathematical Logic, Lecture Notes in Mathematics, vol. 1130, pp. 157–173. Springer (1985)

    Google Scholar 

  38. D’Ottaviano, I.M.L.: Definability and quantifier elimination for J\(_3\)-theories. Stud. Logica 46(1), 37–54 (1987)

    Google Scholar 

  39. D’Ottaviano, I.M.L., Epstein, R.L.: A paraconsistent many-valued propositional logic: J\(_3\). Rep. Math. Logic 22, 89–103 (1988)

    Google Scholar 

  40. Esser, O.: A strong model of paraconsistent logic. Notre Dame J. Formal Logic 44, 149–156 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  41. Horsten, L., Welch, P.: The undecidability of propositional adaptive logic. Synthese 158, 41–60 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  42. Jacek, M., Andrzej, P. (eds.): Essays in Logic and Ontology. Poznań Studies in the Philosophy of the Sciences and the Humanities, vol. 91. Rodopi, Amsterdam/New York (2006)

    Google Scholar 

  43. Meheus, J.: Adaptive logic in scientific discovery: the case of Clausius. Logique et Analyse 143–144, 359–389 (1993). Appeared 1996

    Google Scholar 

  44. Meheus, J.: Clausius’ discovery of the first two laws of thermodynamics. A paradigm of reasoning from inconsistencies. Philosophica 63, 89–117 (1999). Appeared 2001

    Google Scholar 

  45. Meheus, J.: Adaptive logics for question evocation. Logique et Analyse 173–175, 135–164 (2001). Appeared 2003

    Google Scholar 

  46. Meheus, J.: An adaptive logic for pragmatic truth. In: Carnielli, W.A., Coniglio, M.E., D’Ottaviano, I.M.L. (eds.) Paraconsistency. The Logical Way to the Inconsistent, pp. 167–185. Marcel Dekker, New York (2002)

    Google Scholar 

  47. Meheus, J.: Inconsistencies in scientific discovery. Clausius’s remarkable derivation of Carnot’s theorem. In: Krach, H., Vanpaemel, G., Marage, P. (eds.) History of Modern Physics. Acta of the XXth International Congress of History of Science, pp. 143–154. Brepols, Turnhout (Belgium) (2002)

    Google Scholar 

  48. Meheus, J.: Discussive adaptive logics: handling internal and external inconsistencies. In: Malinowski and Pietruszczak [42], pp. 211–223

    Google Scholar 

  49. 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 

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

    Google Scholar 

  51. Nersessian, N.: Inconsistency, generic modeling, and conceptual change in science. In: Meheus, J. (ed.) Inconsistency in Science, pp. 197–211. Kluwer, Dordrecht (2002)

    Google Scholar 

  52. Norton, J.: The logical inconsistency of the old quantum theory of black body radiation. Philos. Sci. 54, 327–350 (1987)

    Article  MathSciNet  Google Scholar 

  53. Norton, J.: A paradox in Newtonian gravitation theory. PSA 1992(2), 420–421 (1993)

    Google Scholar 

  54. Odintsov, S.P., Speranski, S.O.: On algorithmic properties of propositional inconsistency-adaptive logics. Logic Logical Philos. 21, 209–228 (2012)

    MathSciNet  MATH  Google Scholar 

  55. 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

    Article  MathSciNet  MATH  Google Scholar 

  56. Pogorzelski, W.A., Prucnal, T.: The substitution rule for predicate letters in the first-order predicate calculus. Rep. Math. Logic 5, 77–90 (1975)

    MathSciNet  MATH  Google Scholar 

  57. Priest, G.: In Contradiction. A Study of the Transconsistent. Nijhoff, Dordrecht (1987)

    Google Scholar 

  58. Rescher, N., Manor, R.: On inference from inconsistent premises. Theory Decis. 1, 179–217 (1970)

    Article  MATH  Google Scholar 

  59. Schütte, K.: Beweistheorie. Springer, Berlin (1960)

    MATH  Google Scholar 

  60. Shapere, D.: Logic and the philosophical interpretation of science. In: Weingartner, P. (ed.) Alternative Logics. Do sciences need them?, pp. 41–54. Springer, Berlin, Heidelberg (2004)

    Google Scholar 

  61. Smirnova, E.D.: An approach to the justification of semantics of paraconsistent logics. In: Batens, D., Mortensen, C., Priest, G., Van Bendegem, J.P. (eds.) Frontiers of Paraconsistent Logic, pp. 255–262. Research Studies Press, Baldock, UK (2000)

    Google Scholar 

  62. Smith, J.: Inconsistency and scientific reasoning. Stud. History Philos. Sci. 19, 429–445 (1988)

    Article  Google Scholar 

  63. Straßer, C.: Adaptive Logic and Defeasible Reasoning. Applications in Argumentation, Normative Reasoning and Default Reasoning, Trends in Logic, vol. 38. Springer (2014)

    Google Scholar 

  64. Van De Putte, F., Straßer, C.: Extending the standard format of adaptive logics to the prioritized case. Logique et Analyse 220, 601–641 (2012)

    Google Scholar 

  65. Van Kerckhove, B., Vanackere, G.: Vagueness-adaptive logic: a pragmatical approach to Sorites paradoxes. Stud. Logica 75, 383–411 (2003)

    Article  Google Scholar 

  66. Vanackere, G.: Ambiguity-adaptive logic. Logique et Analyse 159, 261–280 (1997). Appeared 1999

    Google Scholar 

  67. Vanackere, G.: The role of ambiguities in the construction of collective theories. Logique et Analyse, 173–174–175, 189–214 (2001). Appeared 2003

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre for Logic and Philosophy of Science, Ghent University, Blandijnberg 2, 9000, Gent, Belgium

    Diderik Batens

Authors
  1. Diderik Batens
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Diderik Batens .

Editor information

Editors and Affiliations

  1. Federal University of Rio de Janeiro, Rio de Janeiro, Brazil

    Jean-Yves Beziau

  2. Indian Statistical Institute Kolkat, Kolkata, India

    Mihir Chakraborty

  3. C.I.T Campus, Institute for Mathematical Sciences, Chennai, Tamil Nadu, India

    Soma Dutta

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer India

About this paper

Cite this paper

Batens, D. (2015). Tutorial on Inconsistency-Adaptive Logics. In: Beziau, JY., Chakraborty, M., Dutta, S. (eds) New Directions in Paraconsistent Logic. Springer Proceedings in Mathematics & Statistics, vol 152. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2719-9_1

Download citation

Publish with us

Policies and ethics

Search

Navigation