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Some Reflections on Quantified Epistemic Logic
Published online by Cambridge University Press: 01 January 2020
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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:
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1 Hintikka stresses that the primary vehicles of defensibility, etc. are statements, not sentences or schemata, where a statement is construed as “the act of uttering, writing, or otherwise expressing, a declarative sentence” (Knowledge and Belief [Ithaca, N.Y.: Cornell University Press, 1962], p. 6). However, most of the discussion is carried on in terms of sentences (or schemata if one wishes to so construe his notation (see ibid., pp. 11 f.]). Hintikka explains that a sentence will be defensible (derivatively) provided it is true that whenever the sentence is uttered (on one occasion, etc.) the resulting statement is defensible in so far as one can tell without knowing who the speaker is, when the [statement was] made, or any other facts about [it] except the form of words [it exemplifies]” (ibid., p. 8).
(Hereafter all pages references in parentheses refer to Knowledge and Belief.)
2 I use ‘→’ to mean: virtually implies; and ‘→’ to mean: does not virtually imply. Also, ‘p(c/b)’ means the same as ‘The result of substituting c for b in p’.
3 I refer here to the extended version of (C. =).
4 The formulation below is an adaptation of some of Quine’s remarks in “Reference and Modality,” in From a Logical Point of View (2nd rev.; New York: Harper and Row, 1961).
5 See J. Hintikka, “Individuals, Possible Worlds, and Epistemic Logic,” Nous, I (1967).
6 I owe this observation to Alvin Plantinga. It is true of course that if A and B share all the properties of this kind which either has, then A = B. But this tact in no way helps to elucidate what it is for A and B to be the same person since the condition in question presupposes that A and B both exist in both worlds.
7 I exclude again such properties as being fat in the world described by CN2.
8 See Williams, B. A. O. “Imagination and the Self,” in Studies in the Philosophy of Thought and Action, ed. by Strawson, P. (Oxford: Oxford University Press, 1968Google Scholar).
9 This point is made by R. Chisholm, “Identity Through Possible Worlds: Some Questions,” Nous, I (1967). I say ‘almost’ since categoreal boundaries cannot be transversed.
10 See Hinlikka. “Individuals, Possible Worlds, and Epistemic Logic,” op. cit., pp. 40f.
11 Ibid., p. 41.