Abstract
Any attempt to characterize formally the set of sentences which a given person X knows to be true probably requires a certain amount of idealization. For the fact that he knows A to be true and knows B to be true can, one hopes, be determined by some kind of empirical investigation. If so, these facts are stubborn and irreducible, and X may know A but not B despite the fact that in some proposed formal system, A implies B. Still, it may be argued that X can be persuaded of B, given that he knows A, without informing him of any new contingent truths. For example, B may simply be a logical consequence of A, or it may follow, given certain other assumptions about knowledge which are held to be true of rational knowers.
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References
Hintikka is right, I think, to reject the second assumption, which is stronger, and thereby base his acceptance of the first on grounds other than introspection. See Jaakko Hintikka, Knowledge and Belief, Cornell University Press, Ithaca, N.Y., 1962, especially pp. 53–57 and 43–47. The notion of rational persuasion referred to in the preceding paragraph is the basis of Hintikka’s notion of defensibility, of which more below.
The proposal that certainty be taken to mean belief that one knows was made by G. E. Moore. See his article ‘Certainty’, in Philosophical Papers, George Allen and Unwin, London, 1959. It follows from this definition, in the combined system KTBJ, below, that knowledge does not imply certainty, although X cannot believe that he knows but is not certain. The suggestion that perfect certainty be interpreted as knowledge that one knows can be found in Peirce, see Collected Papers, Harvard University Press, Cambridge, Mass., 1933, 4.61. James and Dewey also construed certainty as belief that one knows, probably deriving the idea from Peirce. An extended discussion of whether certainty is belief that one knows (and of certain related matters) may be found in ‘Certainty and Criteria of Truth’ (unpublished Doctoral dissertation, Yale University, 1967) by Phillip Jacklin.
This is an epistemic version of the system KT which is discussed, for example, in E. J. Lemmon and Dana Scott, Tntensional Logic’ (unpublished Preliminary Draft, July, 1966). The addition of schema G i , below, gives us Lewis’ system S4. In what follows, we shall tolerate the ‘abuse of language’ of calling our meta-theorems theorems.
We may prove this as follows: Suppose first that there is a sequence A 1 , A 2 ,…, A n such that A n is A, and Г is a set of formulae of L, and that for every i, 1 ≤ i ≤ n, either (1) A i is in Г, (2) ├ L A i , or (3) there are j, k< i such that A k is A j ⊃A i . Let B 1, B 2,…, B m be all the members of this sequence which are in Г, and let ∆ be the set of those A i which are theorems of L. Then there is a proof of A in the propositional calculus from ∆U {B 1,B 2,…, B m }. m applications of the Deduction Theorem for propositional calculus then yields ├ l B 1 &B 2 &… &B m ⊃ A. Conversely, a proof in L of B 1&B 2&… &B m ⊃ A’ together with m applications of Modus Ponens will yield a sequence satisfying (1)-(3) above. Q.E.D. The conclusion thus established is important. For the rules do not enable us to derive ‘A ⊃ K i A’ as a theorem of KT unless ‘A’ is already a theorem. In particular, ‘p 2 ⊃K 1 p 2 is not a valid formula. In fact, we shall show that there is a formula ‘P’ which satisfies the conditions of the paradox such that P⊃p 2 ’ and ‘K 1(P⊃p 2)’ are both theorems. But ‘K 1 p 2 ’ is not deducible, and indeed contradicts ‘P’
In terms of our definition of ‘unforeseen’, we would have, in place of P3 and P4 (P3’) p 2 ⊃ ~ K 1 p 2 (P4’) p 4 ⊃ ~K 4 ps. However, (P3) and (P4) are derivable from (P1), (P2), (P3’), and (P4’) in KT, and conversely (P3’) and (P4’) obviously are derivable from (P3) and (P4). As a matter of convenience, we choose the simpler formulation.
The system of semantics used here is due to S. A.Kripke, ‘Semantical Analysis of Modal Logic I: Normal Modal Propositional Calculi’, Zeitschrift Math. Logik Grundlagen Math. 9 (1963), 67–96. The formulation we employ here is essentially that of Lemmon and Scott, op. cit., except for using two accessibility relations. Intuitively, a possible world is accessible from t if and only if, in some sense, s is a realizable alternative to t.
We may prove this as follows: Suppose that all the axioms of a logic L are valid in a set of modal structures 𝔄, and that every formula of a set Г of formulae of L are true in an interpretation (ℳ, t) where ℳ belongs to. If A is an axiom of L and is any modal structure in then, since is valid in, for every s such that tRs, A is true in (, s). Hence, all consequences of the axioms by rule RK i are valid. Moreover, by the above definition of truth, Modus Ponens preserves validity. So all theorems of L are valid in. Hence, by the result proved in note 4, above, if Г├ i B, then B is true in (ℳ, t). Therefore, Г is consistent in L.
See Lemmon and Scott, op. cit., for a proof that every theorem of KT is valid in the set of modal structures for which R is reflexive. (For our system R 1 and R 3 must both be reflexive.) It is clear that axiom N is satisfied if R 3 ⊆R 1 . Intuitively, we may interpret this to mean that with time, as new information comes to light, the range of epistemic alternatives to the actual world becomes smaller. In other words, although X generally does not know which of several different possible worlds is the actual world, he is able, as time passes, to narrow down the possibilities. It turns out, however, that the consistency of S can be proven in an interpretation in which R 3 = R 1 . This is not surprising, since as we have shown, ‘p2’ follows from S, so thai if ‘S’ is true, there is only one day on which the examination can, in fact, be held, and there are no alternatives to narrow down. In the model of S to be given, Xknows that there is to be an unforeseen examination on one of the two possible days, but he is not absolutely certain that this is so.
Note that there are only finitely many formulae A such that ├ tr K i A. For, let Г be
the set consisting of all formulae of the form K i A, where A is a logically true subformula of P⊃p 2 ’, together with all formulae of the form ‘K i (A ⊃ B) ⊃ (K i A ⊃ K i B) or A ⊃ K i A’ where A and B are subformulae of ‘P⊃p 2 ’. Let Гcontain n formulae, and let T’r be the logic which has T i and N as axioms (in addition to those of Propositional Calculus), and Modus Ponens as its only rule. Then TV is equivalent to T’, Г (i.e., the result of adding to T’ r all the formulae of Г as axioms). Now, if A is any formula, then not — ├ T’ r K i A. For, if we give the usual truth-value assignments to the truth-functional connectives, but make KiA always take the value ‘false’, then all the theorems of T’ r will come out true, whereas K i A will never do so. Thus, all theorems of T r of the form K i A must be consequences of Г in T’ r . But of these there are finitely many. For since Г contains n formulae, at most n — 1 of them are conditionals from which an immediate consequence can be derived by Modus Ponens. Thus, Г has at most in(n +1) consequences. I would suggest that a person ignorant of a logical truth, the proof of which he is quite capable of understanding once he is shown it, is ignorant not because he does not know the logical rules of inference or the axioms, but because he has not considered the appropriate substitution instances.
When we let i = j, we shall refer to the resulting schema B i P ⊃ B i B i P as H i . Similarly, for the schema J ij , below, by J i we shall mean B i P ⊃ ~ B i ~ B i P.
This schema is a generalization of one considered by Binkley. For the purpose of proving consistent in BD, we chose to give B 1 and B 3 the same interpretation, so that J ij reduces to J i , But J i is equivalent to B i ~ B i P ⊃ ~ B i P’, so it too is a weaker version of ‘B i P⊃ P’ The latter, as remarked above, is true in any interpretation in which the accessibility relations R 1 = R 3 are reflexive. Hence, is consistent in BJ as well as in BD.
Here, and hereafter, we shall only be concerned with the schema Ji and not with J ij (see note 10, above).
However, J i is not a theorem of BD. The theorems of BD are valid in all modal structures for which the R i are serial, i.e., (x) (∃y) xR i y. See Lemmon and Scott, op. cit. However, B i p i ⊃ ~B i ~B i p i is false at a in the following modal structure: U= {a, b, c}; R 1 = R 3 = {(a, b), (b, c), (c, a)}; P 1 = {b}, where, as before, B i P’ is true at t if and only if for every s in U such that tR i S, ‘P’ is true at. All theorems of BD are true in this structure. This example thus gives an instance of the denial of J i which is consistent in BD. Hence, Ji itself is not derivable in BD, and a fortiori, neither is J ij . (BD is of course consistent, since its axioms are valid in all modal structures with serial R.)
We may show this as follows: First, as remarked in note 11, above, the axioms of BJ are valid in any modal structure in which the R i are reflexive. Hence, the interpretation which proved consistent in KT will also be one in which H i is false, if the ‘K i are construed as B i throughout. For example, B 1(~p 2 ⊃p 4)’ is true (this is just (V2)), but ‘B 1 B 1(~p 2 ⊃p 4)’ is false, since we have aRb and bRc, but ‘(~p2 ⊃p4)’ is false at c. As for ‘B i B i P ⊃ B i P’ consider the following interpretation: U={a, b, c}; R 1 = {(a, a), (b, b), (c, c), (a, c), (a, b)};P 1 = {c}. Then ‘B i B i p 1 is true at a, but ‘Bip 1 is not.
Charles Pailthorp, ‘Hintikka and Knowing that One Knows’, The Journal of Philosophy 64, No. 16 (August 24, 1967), 488.
Hintikka, op. cit., p. 123.
Indeed I would go further. Let us call a formula P fully modalized with respect to i, if every atomic formula in P is within the scope of a B i ’ and there are no occurrences of ‘B j ’ in P for j ≠ i. If belief is conscious judgment, then it is reasonable to suppose that X is aware of what judgments he has or has not made. So we assume (Mt) B4P⊃P For all P fully modalized with respect to i. The system BHM, however, is equivalent to adopting as axioms D i and H i , together with (Mo) B i (B i P ∨ B i Q) ⊃ (B i P ∨B i Q). See, for example, Lemmon and Scott, op. cit.
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© 1969 D. Reidel Publishing Company, Dordrecht, Holland
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Harrison, C. (1969). The Unanticipated Examination in View of Kripke’s Semantics for Modal Logic. In: Davis, J.W., Hockney, D.J., Wilson, W.K. (eds) Philosophical Logic. Synthese Library, vol 20. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-9614-0_5
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