Abstract
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.
I would like to thank Ed Keenan and Heinrich Wansing for comments and criticisms.
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References
Andrews, P.B.: 1986, An Introduction to Mathematical Logic and Type Theory: to Truth through Proof, Academic Press, Orlando, Florida.
Barwise, J.: 1974, Axioms for Abstract Model Theory, Annals of Mathematical Logic 7, 221–265.
Benthem, J.F.A.K. Van, and Doets, K.: 1983, Higher-Order Logic, in Gabbay & Guenthner [1983] Vol I, 275–329.
Church, A.: 1940, A Formulation of the Simple Theory of Types, The Journal of Symbolic Logic 5, 56–68.
Cresswell, M.J.: 1972, Intensional Logics and Logical Truth, Journal of Philosophical Logic 1, 2–15.
Fagin, R. and Halpern J.Y.: 1988, Belief, Awareness and Limited Reasoning, Artificial Intelligence 34, 39–76.
Gabbay, D. and Guenthner, F. (eds.): 1983, Handbook of Philosophical Logic, Reidel, Dordrecht.
Gallin, D.: 1975, Intensional and Higher-Order Modal Logic, North-Holland, Amsterdam.
Henkin, L.: 1950, Completeness in the Theory of Types, The Journal of Symbolic Logic 15, 81–91.
Henkin, L.: 1963, A Theory of Propositional Types, Fundamenta Mathematicae 52, 323–344.
Hintikka, J.: 1975, Impossible Possible Worlds Vindicated, Journal of Philosophical Logic 4, 475–484.
Hoek, W. Van der, and Meyer, J.-J.: 1988, Possible Logics for Belief, Rapport IR-170, Vrije Universiteit, Amsterdam.
Levesque, H.J.: 1984, A Logic of Implicit and Explicit Belief, Proceedings AAAI-84, Austin, Texas, 198–202.
Lewis, D.: 1974, 'Tensions, in Munitz, M.K. and Unger, P.K. (eds.), Semantics and Philosophy, New York University Press, New York.
Mates, B.: 1950, Synonymity, reprinted in Linsky (ed.), Semantics and the Philosophy of Language, The University of Illinois Press, Urbana, 1952, 111–136.
Montague, R.: 1970, Universal Grammar, reprinted in Montague [1974], 222–246.
Montague, R.: 1973, The Proper Treatment of Quantification in Ordinary English, reprinted in Montague [1974], 247–270.
Montague, R.: 1974, Formal Philosophy, Yale University Press, New Haven.
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.
Muskens, R.A.: 1989a, A Relational Formulation of the Theory of Types, Linguistics and Philosophy 12, 325–346.
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.
Muskens, R.A.: 1989c, Meaning and Partiality, Dissertation, University of Amsterdam.
Putnam, H.: 1954, Synonymity and the Analysis of Belief Sentences, Analysis 14, 114–122.
Quine, W.V.O.: 1966, Quantifiers and Propositional Attitudes, in The Ways of Paradox, New York.
Rantala, V.: 1982a, Impossible Worlds Semantics and Logical Omniscience, in I. Niiniluoto and E. Saarinen (eds.), Intensional Logic: Theory and Applications, Helsinki.
Rantala, V.: 1982b, Quantified Modal Logic: Non-normal Worlds and Propositional Attitudes, Studia Logica 41, 41–65.
Russell, B.: 1908, Mathematical Logic as Based on the Theory of Types, American Journal of Mathematics 30, 222–262.
Thijsse, E.: 1992, Partial Logic and Knowledge Representation, Dissertation, Tilburg University.
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.
Wansing, H.: 1990, A General Possible Worlds Framework for Reasoning about Knowledge and Belief, Studia Logica 49, 523–539.
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Muskens, R. (1992). Logical omniscience and classical logic. In: Pearce, D., Wagner, G. (eds) Logics in AI. JELIA 1992. Lecture Notes in Computer Science, vol 633. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0023421
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DOI: https://doi.org/10.1007/BFb0023421
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