Logical omniscience and classical logic

  • Reinhard Muskens
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 633)


In all respectable logics a form of Leibniz's Law holds which says that logically equivalent expressions can be interchanged salva veritate. On the other hand, in ordinary language syntactically different expressions in general are not intersubstitutable in the scope of verbs of propositional attitude. It thus seems that logics of knowledge and belief should not be subject to Leibniz's Law. In the literature (e.g. Rantala [1982a, 1982b], Wansing [1990]) we indeed find attempts to define epistemic logics in which the interchangeability principle fails. These systems are based on the notions of possible and impossible worlds: in possible worlds all logical connectives get their standard interpretation, but in impossible worlds the interpretation of the connectives is completely free. It is easily seen, however, that the resulting systems are no logics if we apply standard criteria of logicality. There is a simple way out that saves the idea: once it is accepted that the English words ‘not’, ‘and’, ‘or’, ‘if’, and the like are not to be treated as logical operations, we might as well be open about it and overtly treat them as non-logical constants. This allows us to retain classical logic. Possible worlds can be defined as those worlds in which ‘not’, ‘and’, ‘or’, ‘if’ etc. get the standard logical interpretation. On the basis of this idea a small fragment of English is provided with a very fine-grained semantics: no two syntactically different expressions get the same meaning. But on sentences that do not contain a propositional attitude verb there is a classical relation of logical consequence.


Logical Omniscience Propositional Attitudes Impossible Worlds Epistemic Logic 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Andrews, P.B.: 1986, An Introduction to Mathematical Logic and Type Theory: to Truth through Proof, Academic Press, Orlando, Florida.Google Scholar
  2. Barwise, J.: 1974, Axioms for Abstract Model Theory, Annals of Mathematical Logic 7, 221–265.CrossRefGoogle Scholar
  3. Benthem, J.F.A.K. Van, and Doets, K.: 1983, Higher-Order Logic, in Gabbay & Guenthner [1983] Vol I, 275–329.Google Scholar
  4. Church, A.: 1940, A Formulation of the Simple Theory of Types, The Journal of Symbolic Logic 5, 56–68.Google Scholar
  5. Cresswell, M.J.: 1972, Intensional Logics and Logical Truth, Journal of Philosophical Logic 1, 2–15.CrossRefGoogle Scholar
  6. Fagin, R. and Halpern J.Y.: 1988, Belief, Awareness and Limited Reasoning, Artificial Intelligence 34, 39–76.CrossRefGoogle Scholar
  7. Gabbay, D. and Guenthner, F. (eds.): 1983, Handbook of Philosophical Logic, Reidel, Dordrecht.Google Scholar
  8. Gallin, D.: 1975, Intensional and Higher-Order Modal Logic, North-Holland, Amsterdam.Google Scholar
  9. Henkin, L.: 1950, Completeness in the Theory of Types, The Journal of Symbolic Logic 15, 81–91.Google Scholar
  10. Henkin, L.: 1963, A Theory of Propositional Types, Fundamenta Mathematicae 52, 323–344.Google Scholar
  11. Hintikka, J.: 1975, Impossible Possible Worlds Vindicated, Journal of Philosophical Logic 4, 475–484.CrossRefGoogle Scholar
  12. Hoek, W. Van der, and Meyer, J.-J.: 1988, Possible Logics for Belief, Rapport IR-170, Vrije Universiteit, Amsterdam.Google Scholar
  13. Levesque, H.J.: 1984, A Logic of Implicit and Explicit Belief, Proceedings AAAI-84, Austin, Texas, 198–202.Google Scholar
  14. Lewis, D.: 1974, 'Tensions, in Munitz, M.K. and Unger, P.K. (eds.), Semantics and Philosophy, New York University Press, New York.Google Scholar
  15. Mates, B.: 1950, Synonymity, reprinted in Linsky (ed.), Semantics and the Philosophy of Language, The University of Illinois Press, Urbana, 1952, 111–136.Google Scholar
  16. Montague, R.: 1970, Universal Grammar, reprinted in Montague [1974], 222–246.Google Scholar
  17. Montague, R.: 1973, The Proper Treatment of Quantification in Ordinary English, reprinted in Montague [1974], 247–270.Google Scholar
  18. Montague, R.: 1974, Formal Philosophy, Yale University Press, New Haven.Google Scholar
  19. Moore, R.C.: Propositional Attitudes and Russellian Propositions, in R. Bartsch, J.F.A.K. van Benthem and P. van Emde Boas (eds.), Semantics and Contextual Expression, Proceedings of the Sixth Amsterdam Colloquium, Foris, Dordrecht, 147-174.Google Scholar
  20. Muskens, R.A.: 1989a, A Relational Formulation of the Theory of Types, Linguistics and Philosophy 12, 325–346.CrossRefGoogle Scholar
  21. Muskens, R.A.: 1989b, Going Partial in Montague Grammar, in R. Bartsch, J.F.A.K. van Benthem and P. van Emde Boas (eds.), Semantics and Contextual Expression, Proceedings of the Sixth Amsterdam Colloquium, Foris, Dordrecht, 175–220.Google Scholar
  22. Muskens, R.A.: 1989c, Meaning and Partiality, Dissertation, University of Amsterdam.Google Scholar
  23. Putnam, H.: 1954, Synonymity and the Analysis of Belief Sentences, Analysis 14, 114–122.Google Scholar
  24. Quine, W.V.O.: 1966, Quantifiers and Propositional Attitudes, in The Ways of Paradox, New York.Google Scholar
  25. Rantala, V.: 1982a, Impossible Worlds Semantics and Logical Omniscience, in I. Niiniluoto and E. Saarinen (eds.), Intensional Logic: Theory and Applications, Helsinki.Google Scholar
  26. Rantala, V.: 1982b, Quantified Modal Logic: Non-normal Worlds and Propositional Attitudes, Studia Logica 41, 41–65.CrossRefGoogle Scholar
  27. Russell, B.: 1908, Mathematical Logic as Based on the Theory of Types, American Journal of Mathematics 30, 222–262.MathSciNetGoogle Scholar
  28. Thijsse, E.: 1992, Partial Logic and Knowledge Representation, Dissertation, Tilburg University.Google Scholar
  29. Vardi, M.Y.: 1986, On Epistemic Logic and Logical Omniscience, in J.Y. Halpern (ed.), Theoretical Aspects of Reasoning about Knowledge: Proceedings of the 1986 Conference, Morgan Kaufmann, Los Altos, 293–305.Google Scholar
  30. Wansing, H.: 1990, A General Possible Worlds Framework for Reasoning about Knowledge and Belief, Studia Logica 49, 523–539.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Reinhard Muskens
    • 1
  1. 1.Department of LinguisticsTilburg UniversityLE Tilburg

Personalised recommendations