Skip to main content
SpringerLink
Log in

On the Transparency of Defeasible Logics: Equivalent Premise Sets, Equivalence of Their Extensions, and Maximality of the Lower Limit

  • Chapter
  • First Online:
Adaptive Logics for Defeasible Reasoning

Part of the book series: Trends in Logic ((TREN,volume 38))

  • 871 Accesses

Abstract

For Tarski logics, there are simple criteria that enable one to conclude that two premise sets are equivalent. We shall show that the very same criteria hold for adaptive logics, which is a major advantage in comparison to other approaches to defeasible reasoning forms. A related property of Tarski logics is that the extensions of equivalent premise sets with the same set of formulas are equivalent premise sets. This does not hold for adaptive logics. However a very similar criterion does. We also shall show that every monotonic logic weaker than an adaptive logic is weaker than the lower limit logic of the adaptive logic or identical to it. This highlights the role of the lower limit for settling the adaptive equivalence of extensions of equivalent premise sets.

A previous version of this chapter has been published under the name “On the Transparency of Defeasible Logics: Equivalent Premise Sets, Equivalence of Their Extensions, and Maximality of the Lower Limit” in Logique et Analyse [1]. The paper is co-authored by Diderik Batens and Peter Verdée. Any mistakes in the new material that has been added are alone my responsibility.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
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

Institutional subscriptions

Notes

  1. 1.

    With respect to the semantics, useful unifying work was done by Shoham and associates, for example [2–4].

  2. 2.

    These are older results, not in standard format, that soon will be improved upon.

  3. 3.

    In the text, we neglect some border cases, which are irrelevant to the present discussion, for example the case in which \({ Cn}_{{\mathbf{{L}}}}\left( \varGamma \right) \) is either empty or trivial.

  4. 4.

    One way to show this is as follows. We already argued in Sect. 2.7. that \(\varGamma \Vdash _{\mathbf{{AL^m}}} A\) and that \(A \notin Cn ^{{\mathcal {L}}^+}_{{\mathbf {AL^m}}}\left( \varGamma \right) \). Obviously, \({\mathcal {M}}_{\mathbf{{LLL}}}\bigl (\varGamma \bigr ) = {\mathcal {M}}_{\mathbf{{LLL}}}\bigl ( Cn ^{{\mathcal {L}}^+}_{{\mathbf {LLL}}}\left( \varGamma \right) \bigr )\) and hence both premise sets have the same minimally abnormal models. This implies, \( Cn ^{{\mathcal {L}}^+}_{{\mathbf {LLL}}}\left( \varGamma \right) \Vdash _{\mathbf{{AL^m}}} A\). Thus, by Theorem 2.7.1, \(A \in Cn ^{{\mathcal {L}}^+}_{{\mathbf {AL^m}}}\left( Cn ^{{\mathcal {L}}^+}_{{\mathbf {LLL}}}\left( \varGamma \right) \right) \).

  5. 5.

    Suitable axioms are \((A\supset \,\check{\lnot }A)\supset \,\check{\lnot }A\) and \(A\supset (\;\check{\lnot }A\supset B)\). The other classical symbols are stipulated to be identical to the corresponding standard symbols.

  6. 6.

    Except that, in order to define \(\varGamma \vDash _\mathbf{{T}} A\), a \(\mathbf{{T}}\)-model is defined as \(M = \langle W, w_0, R, v\rangle \) with \(w_0\in W\) and \(M\) is said to verify \(A\) iff \(v_M(A, w_0)=1\).

  7. 7.

    The property does not hold for all premise sets but is typical for premise sets \(\varGamma ^{\vartriangleright }\) with \(\varGamma \) a set of modal-free formulas.

  8. 8.

    The paraconsistent negation is there written as \(\sim \) (here as \(\lnot )\) and the classical negation as \(\lnot \) (here as \(\;\check{\lnot }\)).

  9. 9.

    It is for instance used in order to derive (4.8).

  10. 10.

    Since it is easy to observe that \({ C n}^{{\mathcal {L}}}_{\mathbf{{AL_\bullet }}}\) is monotonic, the reader may conjecture Theorem 4.6.1 equipped with the antecedent (†) may hold at least for ALs for which \({ C n}^{{\mathcal {L}}}_{\mathbf{{AL}}}\) is non-monotonic. But this is not true either. Suppose we add two logical constants to our language \({\mathcal {L}}\): \(\star \) and \(\circ \). \(\mathbf{{LLL}}\) and \(\mathbf{{L}}\) are defined as before. We alter our \(\mathbf{{AL}}\) by defining \(\varOmega \) by: \(\{\bullet A \check{\wedge }\; \check{\lnot }A \mid A\) is a propositional atom\(\} \cup \{\star \} \cup \{\star \mathop {\check{\vee }}\,\; \check{\lnot }\circ \}\). Note that \(\emptyset \vdash _{\mathbf{{AL}}} \circ \) (each minimally abnormal model has the abnormal part \(\emptyset \) and hence validates \(\,\,\check{\lnot }\star \) and \(\circ \)) but \(\{\star \} \not \vdash _{\mathbf{{AL}}} \circ \) (note that the minimally abnormal models all have the abnormal part \(\{\star , \star \mathop {\check{\vee }}\, \check{\lnot }\circ \}\) due to \(\star \) being a premise: some also validate \(\,\,\check{\lnot }\circ \)). This demonstrates that \(\mathbf{{AL}}\) is non-monotonic (relative to \({\mathcal {L}}\)). Furthermore, as before (4.14)–(4.16) hold.

References

  1. Batens, D., Straßer, C., Verdée, P.: On the transparency of defeasible logics: equivalent premise sets, equivalence of their extensions, and maximality of the lower limit. Logique et Analyse 52(207), 281–304 (2009)

    Google Scholar 

  2. Shoham, Y.: A semantical approach to nonmonotonic logics. In: Ginsberg, M.L. (ed.) Readings in Non-Monotonic Reasoning, pp. 227–249. Morgan Kaufmann, Los Altos (1987)

    Google Scholar 

  3. Shoham, A.L.B.Y. Jr.: New results on semantical non-monotonic reasoning. In: Reinfrank, M., de Kleer, J., Ginsberg, M.L., Sandewall, E. (eds.) NMR. Lecture Notes in Computer Science, vol. 346, pp. 19–26. Springer (1988)

    Google Scholar 

  4. Lin, F., Shoham, Y.: Epistemic semantics for fixed-points non-monotonic logics. Morgan Kaufmann Publishers Inc., Pacific Grove, CA (1990)

    Google Scholar 

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

    Google Scholar 

  6. Batens, D.: A strengthening of the Rescher-Manor consequence relations. Logique et Analyse 183–184, 289–313 (2003). Appeared 2005

    Google Scholar 

  7. 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  Google Scholar 

  8. Verhoeven, L.: Proof theories for some prioritized consequence relations. Logique et Analyse 183–184, 325–344 (2003). Appeared 2005

    Google Scholar 

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

    Article  Google Scholar 

  10. Benferhat, S., Dubois, D., Prade, H.: Some syntactic approaches to the handling of inconsistent knowledge bases: a comparative study. Part 1: the flat case. Stud. Logica 58, 17–45 (1997)

    Article  Google Scholar 

  11. Benferhat, S., Dubois, D., Prade, H.: Some syntactic approaches to the handling of inconsistent knowledge bases: a comparative study. Part 2: The prioritized case. In: Orłowska, E. (ed) Logic at Work. Essays Dedicated to the Memory of Helena Rasiowa, pp. 473–511. Physica Verlag (Springer), Heidelberg (1999)

    Google Scholar 

  12. Batens, D., Meheus, J., Provijn, D.: An adaptive characterization of signed systems for paraconsistent reasoning (Unpublished).

    Google Scholar 

  13. Besnard, P., Schaub, T.: Signed systems for paraconsistent reasoning. J. Autom. Reason. 20, 191–213 (1998)

    Article  Google Scholar 

  14. De Clercq, K.: Two new strategies for inconsistency-adaptive logics. Logic Log. Philos. 8, 65–80 (2000). Appeared 2002

    Google Scholar 

  15. Batens, D.: Inconsistency-adaptive logics and the foundation of non-monotonic logics. Logique et Analyse 145, 57–94 (1994). Appeared 1996

    Google Scholar 

  16. Antoniou, G.: Nonmonotonic Reasoning. MIT Press, Cambridge (1996)

    Google Scholar 

  17. Brewka, G.: Nonmonotonic Reasoning: Logical Foundations of Commonsense. Cambridge University Press, Cambridge (1991)

    Google Scholar 

  18. Łukaszewicz, W.: Non-Monotonic Reasoning. Formalization of Commonsense Reasoning. Ellis Horwood, New York (1990)

    Google Scholar 

  19. Straßer, C.: An adaptive logic for rational closure. In: Carnielli, W., Coniglio, M.E., D’Ottaviano, I.M.L. (eds.) The Many Sides of Logic, pp. 47–67. College Publications (2009)

    Google Scholar 

  20. Lehmann, D.J., Magidor, M.: What does a conditional knowledge base entail? Artif. Intell. 55(1), 1–60 (1992)

    Article  Google Scholar 

  21. Straßer, C., Šešelja, D.: Towards the proof-theoretic unification of Dung’s argumentation framework: an adaptive logic approach. J. Logic Comput. 21, 133–156 (2010)

    Article  Google Scholar 

  22. Dung, P.M., Son, T.C.: Nonmonotonic inheritance, argumentation and logic programming. In: Marek, V.W., Nerode, A. (eds.) Logic Programming and Nonmonotonic Reasoning, Proceedings. Lecture Notes in Computer Science. Third International Conference, LPNMR’95, Lexington, USA, June 26–28, 1995, vol. 928, pp. 316–329. Springer (1995)

    Google Scholar 

  23. Dung, P.M.: On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artif. Intell. 77, 321–358 (1995)

    Google Scholar 

  24. Primiero, G., Meheus, J.: Majority merging by adaptive counting. Synthese 165, 203–223 (2008)

    Article  Google Scholar 

  25. Konieczny, S., Pino Pérez, R.: Merging information under constraints: a logical framework. J. Logic Comput. 12, 773–808 (2002)

    Google Scholar 

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

    Google Scholar 

  27. Meheus, J.: Erotetic arguments from inconsistent premises. Logique et Analyse 165–166, 49–80 (1999). Appeared 2002

    Google Scholar 

  28. Wiśniewski, A.: The Posing of Questions. Logical Foundations of Erotetic Inferences. Kluwer, Dordrecht (1995)

    Book  Google Scholar 

  29. Meheus, J., Verhoeven, L., Van Dyck, M., Provijn, D.: Ampliative adaptive logics and the foundation of logic-based approaches to abduction. In: Magnani, L., Nersessian, N.J., Pizzi, C., et al (eds.) Logical and Computational Aspects of Model-Based Reasoning, pp. 39–71. Kluwer, Dordrecht (2002)

    Google Scholar 

  30. Meheus, J., Provijn, D.: Abduction through semantic tableaux versus abduction through goal-directed proofs. Theoria 22/3, 295–304 (2007)

    Google Scholar 

  31. Meheus, J., Batens, D.: A formal logic for abductive reasoning. Logic J. IGPL 14, 221–236 (2006)

    Article  Google Scholar 

  32. Gauderis, T., Putte, F.V.D.: Abduction of generalizations. THEORIA Int. J. Theor. Hist. Found. Sci. 27(3), (2012)

    Google Scholar 

  33. Gauderis, T.: Modelling abduction in science by means of a modal adaptive logic. Found. Sci. (2013)

    Google Scholar 

  34. Aliseda, A.: Abductive Reasoning. Logical Investigations into Discovery and Explanation. Springer, Dordrecht (2006)

    Google Scholar 

  35. Provijn, D., Weber, E.: Adaptive logics for non-explanatory and explanatory diagnostic reasoning. In: Magnani, L., Nersessian, N.J., Pizzi, C., et al (eds.) Logical and Computational Aspects of Model-Based Reasoning, pp. 117–142. Kluwer, Dordrecht (2002)

    Google Scholar 

  36. Batens, D., Meheus, J., Provijn, D., Verhoeven, L.: Some adaptive logics for diagnosis. Logic Log. Philos. 11/12, 39–65 (2003)

    Google Scholar 

  37. Reiter, R.: A theory of diagnosis from first principles. Artif. Intell. 32, 57–95 (1987)

    Article  Google Scholar 

  38. Meheus, J.: Empirical progress and ampliative adaptive logics. In: Festa, R., Aliseda, A., Peijnenburg, J. (eds.) Confirmation, Empirical Progress, and Truth Approximation. Essays in Debate with Theo Kuipers. vol. 1, Poznan Studies in the Philosophy of the Sciences and the Humanities, vol. 83, pp. 193–217. Rodopi, Amsterdam/New York (2005)

    Google Scholar 

  39. Kuipers, T.A.F.: From Instrumentalism to Constructive Realism. On some Relations Between Confirmation, Empirical Progress, and Truth Approximation, Synthese Library, vol. 287. Kluwer, Dordrecht (2000)

    Google Scholar 

  40. Van De Putte, F., Verdée, P.: The dynamics of relevance: adaptive belief revision. Synthese 187(1), 1–42 (2012)

    Article  Google Scholar 

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

    Google Scholar 

  42. Mikenberg, I., da Costa, N.C.A., Chuaqui, R.: Pragmatic truth and approximation to truth. J. Symb. Logic 51, 201–221 (1986)

    Article  Google Scholar 

  43. da Costa, N.C., Bueno, O., French, S.: The logic of pragmatic truth. J. Philos. Logic 27, 603–620 (1998)

    Article  Google Scholar 

  44. Van Dyck, M.: Causal discovery using adaptive logics. Towards a more realistic heuristics for human causal learning. Logique et Analyse 185–188, 5–32 (2004). Appeared 2005

    Google Scholar 

  45. Leuridan, B.: Causal discovery and the problem of ignorance. An adaptive logic approach. J. Appl. Logic 7, 188–205 (2009)

    Article  Google Scholar 

  46. Pearl, J.: Causality. Models, Reasoning, and Inference. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  47. Straßer, C.: An adaptive logic framework for conditional obligations and deontic dilemmas. Logic Log. Philos. 19(1–2), 95–128 (2010)

    Google Scholar 

  48. Straßer, C., Meheus, J., Beirlaen, M.: Tolerating deontic conflicts by adaptively restricting inheritance. Logique Anal. 219, 477–506 (2012)

    Google Scholar 

  49. Goble, L.: A proposal for dealing with deontic dilemmas. In: Lomuscio, A., Nute, D. (eds.) DEON Lecture Notes in Computer Science, vol. 3065, pp. 74–113. Springer (2004)

    Google Scholar 

  50. Goble, L.: A logic for deontic dilemmas. J. Appl. Logic 3, 461–483 (2005)

    Article  Google Scholar 

  51. Verdée, P., van der Waart van Gulik, S.: A generic framework for adaptive vague logics. Studia Logica 90, 385–405 (2008)

    Google Scholar 

  52. Schotch, P.K., Jennings, R.E.: On detonating. In: Priest, G., Routley, R., Norman, J. (eds.) Paraconsistent Logic. Essays on the Inconsistent, pp. 306–327. Philosophia Verlag, München (1989)

    Google Scholar 

  53. Makinson, D.: Bridges from Classical to Nonmonotonic Logic, Texts in Computing, vol. 5. King’s College Publications, London (2005)

    Google Scholar 

  54. Hughes, G., Cresswell, M.: An Introduction to Modal Logic. Methuen, New York (1972). First published 1968

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sint-Pietersnieuwstraat 61, 9000, Ghent, Belgium

    Christian Straßer

Authors
  1. Christian Straßer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christian Straßer .

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Straßer, C. (2014). On the Transparency of Defeasible Logics: Equivalent Premise Sets, Equivalence of Their Extensions, and Maximality of the Lower Limit. In: Adaptive Logics for Defeasible Reasoning. Trends in Logic, vol 38. Springer, Cham. https://doi.org/10.1007/978-3-319-00792-2_4

Download citation

Publish with us

Policies and ethics

Search

Navigation