Journal of Logic, Language and Information

, Volume 26, Issue 2, pp 109–141 | Cite as

In All But Finitely Many Possible Worlds: Model-Theoretic Investigations on ‘Overwhelming Majority’ Default Conditionals



Defeasible conditionals are statements of the form ‘if A then normally B’. One plausible interpretation introduced in nonmonotonic reasoning dictates that (\(A\Rightarrow B\)) is true iff B is true in ‘mostA-worlds. In this paper, we investigate defeasible conditionals constructed upon a notion of ‘overwhelming majority’, defined as ‘truth in a cofinite subset of \(\omega \)’, the first infinite ordinal. One approach employs the modal logic of the frame \((\omega , <)\), used in the temporal logic of discrete linear time. We introduce and investigate conditionals, defined modally over \((\omega , <)\); several modal definitions of the conditional connective are examined, with an emphasis on the nonmonotonic ones. An alternative interpretation of ‘majority’ as sets cofinal (in \(\omega \)) rather than cofinite (subsets of \(\omega \)) is examined. For these modal approaches over \((\omega , <)\), a decision procedure readily emerges, as the modal logic \({\mathbf {K4DLZ}}\) of this frame is well-known and a translation of the conditional sentences can be mechanically checked for validity; this allows also for a quick proof of \(\mathsf {NP}\)-completeness of the satisfiability problem for these logics. A second approach employs the conditional version of Scott-Montague semantics, in the form of \(\omega \)-many possible worlds, endowed with neighborhoods populated by collections of cofinite subsets of \(\omega \). This approach gives rise to weak conditional logics, as expected. The relative strength of the conditionals introduced is compared to (the conditional logic ‘equivalent’ of) KLM logics and other conditional logics in the literature.


Conditional Logics of Normality NonMonotonic Logic 



We wish to thank the two anonymous JoLLI referees for their useful suggestions, their constructive criticism and the insightful comments which helped us strengthen our results and improve the readability of the paper. The positive impact of their valuable feedback is gratefully acknowledged. This paper is a revised and extended version of the ECSQARU 2015 paper (Koutras and Rantsoudis 2005) under (almost) the same title. We are indebted to James Delgrande, Johannes Marti & Riccardo Pinossio for various comments in previous versions of this work and for drawing our attention to the ‘ordering semantics’ of D. Lewis.


  1. Arlo-Costa, H. (2014). The logic of conditionals. In E. N. Zalta, (eds.), The stanford encyclopedia of philosophy. Summer 2014 edition.Google Scholar
  2. Adams, E. (1975). The logic of conditionals. Dordrecht: D. Reidel Publishing Co.CrossRefGoogle Scholar
  3. Allen, J. F., Fikes, R., & Sandewall, E. (Eds.). (1991). Proceedings of the 2nd international conference on principles of knowledge representation and reasoning (KR’91), Cambridge, MA, USA, April 22–25, 1991. Morgan Kaufmann.Google Scholar
  4. Askounis, D., Koutras, C. D., & Zikos, Y. (2016). Knowledge means ’all’, belief means ’most’. Journal of Applied Non-Classical Logics, 26(3), 173–192.CrossRefGoogle Scholar
  5. Blackburn, P., de Rijke, M., & Venema, Y. (2001). Modal Logic. Number 53 in Cambridge Tracts in Theoretical Computer Science. Cambridge: Cambridge University Press.Google Scholar
  6. Bell, J. (1990). The logic of nonmonotonicity. Artificial Intelligence, 41(3), 365–374.CrossRefGoogle Scholar
  7. Besnard, Ph, & Hunter, A. (Eds.). (1998). Reasoning with Actual and Potential Contradictions, volume 2 of handbook of defeasible reasoning and uncertainty management systems. Dordrecht: Kluwer Academic Publishers.Google Scholar
  8. Bochman, A. (2001). A logical theory of nonmonotonic inference and belief change. Berlin: Springer.CrossRefGoogle Scholar
  9. Boutilier, C. (1992). Conditional logics for default reasoning and belief revision. PhD thesis, University of Toronto.Google Scholar
  10. Balbiani, Ph., Suzuki, N., Wolter, F., & Zakharyaschev, M. (Eds.). (2003). Advances in modal logic 4, papers from the fourth conference on “advances in modal logic”, held in Toulouse (France) in October 2002. King’s College Publications.Google Scholar
  11. Burgess, J. P. (1981). Quick completeness proofs for some logics of conditionals. Notre Dame Journal of Formal Logic, 22(1), 76–84.CrossRefGoogle Scholar
  12. Crocco, G., Fariñas del Cerro, L., & Herzig, A. (Eds.). (1996). Conditionals: From philosophy to computer science. Studies in logic and computation. Oxford: Oxford University Press.Google Scholar
  13. Chellas, B. (1975). Basic conditional logic. Journal of Philosophical Logic, 4, 133–153.CrossRefGoogle Scholar
  14. Chellas, B. (1980). Modal logic, an introduction. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  15. Chagrov, A. V., & Rybakov, M. N. (2003) How many variables does one need to prove PSPACE-hardness of modal logics. In Balbiani et al. (2003), pp. 71–82.Google Scholar
  16. Destercke, S., & Denoeux, Th. (Eds.). (2015). Proceedings of symbolic and quantitative approaches to reasoning with uncertainty—13th European conference, ECSQARU 2015, Compiègne, France, July 15-17, 2015, volume 9161 of lecture notes in computer science. Springer.Google Scholar
  17. Delgrande, J. P. (1987). A first-order conditional logic for prototypical properties. Artificial Intelligence, 33(1), 105–130.CrossRefGoogle Scholar
  18. Delgrande, J. P. (1988). An approach to default reasoning based on a first-order conditional logic: Revised report. Artificial Intelligence, 36(1), 63–90.CrossRefGoogle Scholar
  19. Delgrande, J. P. (1998). Conditional logics for defeasible reasoning, pp. 135–173, Volume 2 of Besnard and Hunter (1998) .Google Scholar
  20. Delgrande, J. P. (2003). Weak conditional logics of normality. In Gottlob and Walsh (2003), pp. 873–878.Google Scholar
  21. Delgrande, J. P. (2006). On a rule-based interpretation of default conditionals. Annals of Mathematics and Artificial Intelligence, 48(3–4), 135–167.Google Scholar
  22. D’Agostino, M., Gabbay, D., Haehnle, R., & Posegga, J. (Eds.). (1999). Handbook of tableau methods. Dordrecht: Kluwer Academic Publishers.Google Scholar
  23. Doyle, J., Sandewall, E., & Torasso, P. (Eds.). (1994). Proceedings of the 4th international conference on principles of knowledge representation and reasoning (KR’94), Bonn, Germany, May 24–27, 1994. Morgan Kaufmann.Google Scholar
  24. Eiter, Th, & Lukasiewicz, Th. (2000). Default reasoning from conditional knowledge bases: Complexity and tractable cases. Artificial Intelligence, 124(2), 169–241.CrossRefGoogle Scholar
  25. Friedman, N., & Halpern, J. Y. (1994). On the complexity of conditional logics. In Doyle et al. (1994), pp. 202–213.Google Scholar
  26. Giordano, L., Gliozzi, V., Olivetti, N., & Pozzato, G. L. (2009). Analytic tableaux calculi for KLM logics of nonmonotonic reasoning. ACM Transactions on Computational Logic, 10(3), 1–47.Google Scholar
  27. Ginsberg, M. L. (1986). Counterfactuals. Artificial Intelligence, 30(1), 35–79.CrossRefGoogle Scholar
  28. Goldblatt, R. (1992). Logics of time and computation. Number 7 in CSLI lecture notes (2nd ed.). Center for the Study of Language and Information, Stanford University.Google Scholar
  29. Goré, R. (1999). Tableau methods for modal and temporal logics, pp 297–396. In D’Agostino et al. (1999).Google Scholar
  30. Geffner, H., & Pearl, J. (1992). Conditional entailment: Bridging two approaches to default reasoning. Artificial Intelligence, 53(2–3), 209–244.CrossRefGoogle Scholar
  31. Goldszmidt, M., & Pearl, J. (1992). On the consistency of defeasible databases. Artificial Intelligence, 52(2), 121–149.CrossRefGoogle Scholar
  32. Gottlob, G., & Walsh, T. (Eds.). (2003). IJCAI-03, Proceedings of the eighteenth international joint conference on artificial intelligence, Acapulco, Mexico, August 9–15, 2003. Morgan Kaufmann.Google Scholar
  33. Halpern, J. Y. (1995). The effect of bounding the number of primitive propositions and the depth of nesting on the complexity of modal logic. Artificial Intelligence, 75(2), 361–372.CrossRefGoogle Scholar
  34. Jauregui, V. (2008). Modalities, conditionals and nonmonotonic reasoning. PhD thesis, Department of Computer Science and Engineering, University of New South Wales.Google Scholar
  35. Kraus, S., Lehmann, D. J., & Magidor, M. (1990). Nonmonotonic reasoning, preferential models and cumulative logics. Artificial Intelligence, 44(1–2), 167–207.CrossRefGoogle Scholar
  36. Koutras, C.D., & Rantsoudis, Ch. (2015). In all, but finitely many, possible worlds: Model-theoretic investigations on ’overwhelming majority’ default conditionals. In Destercke and Denoeux (2015), pp 117–126.Google Scholar
  37. Lamarre, Ph. (1991) S4 as the conditional logic of nonmonotonicity. In Allen et al. (1991), pp. 357–367.Google Scholar
  38. Lewis, D. (1973). Counterfactuals. Oxford: Blackwell.Google Scholar
  39. Lewis, D. (1981). Ordering semantics and premise semantics for counterfactuals. Journal of Philosophical Logic, 10(2), 217–234.CrossRefGoogle Scholar
  40. Lehmann, D. J., & Magidor, M. (1992). What does a conditional knowledge base entail? Artificial Intelligence, 55(1), 1–60.CrossRefGoogle Scholar
  41. Nute, D. (1980). Topics in Conditional Logic. Dordrecht: Kluwer.CrossRefGoogle Scholar
  42. Pacuit, E. (2007) Neighborhood semantics for modal logic: an introduction. Course Notes for ESSLLI 2007.Google Scholar
  43. Pearl, J. (1988). Probabilistic reasoning in intelligent systems: Networks of plausible inference. Burlington: Morgan Kaufmann.Google Scholar
  44. Pozzato, G. L. (2010). Conditional and preferential logics: Proof methods and theorem proving. Frontiers in artificial intelligence and applications. Amsterdam: IOS Press.Google Scholar
  45. Schlechta, K. (1995). Defaults as generalized quantifiers. Journal of Logic and Computation, 5(4), 473–494.CrossRefGoogle Scholar
  46. Schlechta, K. (1997). Filters and partial orders. Logic Journal of the IGPL, 5(5), 753–772.CrossRefGoogle Scholar
  47. Segerberg, K. (1970). Modal logics with linear alternative relations. Theoria, 36, 301–322.CrossRefGoogle Scholar
  48. Segerberg, K. (1971). An essay in classical modal logic. Uppsala: Filosofiska Studies.Google Scholar
  49. Veltman, F. (1985). Logics for conditionals. PhD thesis, University of Amsterdam.Google Scholar

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© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Department of Informatics and TelecommunicationsUniversity of PeloponneseTripolisGreece
  2. 2.Graduate Programme in Logic, Algorithms and Computation, Department of MathematicsUniversity of AthensAthensGreece

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