Logical omniscience and classical logic

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.