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Some Reflections on Quantified Epistemic Logic

In Knowledge and Belief Jaakko Hintikka presents a fairly detailed system of epistemic logic and tries to show its philosophical importance by bringing it to bear upon such problems as what goes wrong in saying “It’s raining, but I don’t believe it.” In setting up the system, Hintikka presents a way of symbolizing certain locutions containing expressions like ‘knows that’, ‘know who’, and ‘believes that’; he introduces certain notions which are analogous to the standard logical concepts consistency, validity, entailment (or logical implication), and logical equivalence; and he sets up (in effect) certain criteria which determine when these analogues of consistency, validity, etc. apply to epistemic and doxastic sentences and schemata-i.e., to sentences and schemata which contain, in their symbolic renditions, one or more of the so-called epistemic operators: ‘K’, ‘B’, ‘P’, and ‘C’. (⌈Ka⌉ is “the formal counterpart” of the words ⌈a knows that⌉; ⌈Ba⌉ is the counterpart of ⌈a believes that⌉; ⌈Pa⌉ is the counterpart of ⌈it is possible, for all that a knows, that⌉; and ⌈Ca⌉ is the counterpart of ⌈it is compatible with everything a believes that⌉ [p. 10].) The analogue of logical consistency which Hintikka introduces is called “defensibility.” Intuitively, a sentence is defensible provided that it is true in at least one of the “most knowledgeable of possible worlds” (p. 36)-i.e., provided it is true in at least one of the possible worlds which have the feature that each inhabitant which is capable of knowing and/or believing things (a) knows all the logical consequences of every proposition it knows and (b) believes all the logical consequences of every proposition it believes. The remaining notions are defined by reference to defensibility in just the way validity, entailment, and logical equivalence are often defined by reference to consistency. Thus self-sustenance, the epistemic analogue of validity, is defined thus: