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Knowledge and its Limits$

Timothy Williamson

Print publication date: 2002

Print ISBN-13: 9780199256563

Published to Oxford Scholarship Online: November 2003

DOI: 10.1093/019925656X.001.0001

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(p.316) Appendix 5 A Non‐Symmetric Epistemic Model

(p.316) Appendix 5 A Non‐Symmetric Epistemic Model

Source:
Knowledge and its Limits
Publisher:
Oxford University Press

A creature stores information in sentential form. Its language L has two atomic sentences H (‘It is hot’) and C (‘It is cold’), the logical constants ∼ and ∧ with their usual interpretations, and the unary sentence functor K (‘I know that . . . ’). Let be the proposition expressed by the sentence A on this interpretation, and W = {w 1, w 2, x}. In order to specify which sentences are stored in which worlds, recursively define an auxiliary function φ from L to W:

Let the creature be so connected to its environment that for all y∈W and A∈L:
  • # It is disposed in y to store A if and only if y∈ φKA.

For example, since φ∼ C = {w 1, x}, φK∼ C = {w 1, x}, so it is disposed to store ∼ C in w 1 and x but not in w 2. Since φKC = {}, it is not disposed to store C in any world. Thus # agrees with the example in section
10.6 on the storage of information about whether it is cold; likewise for information about whether it is hot.

We can argue plausibly that φA is the set of worlds in which is true. The argument is by induction on the complexity of A. The only non‐routine case is the induction step for K. The induction hypothesis is that φA is the set of worlds in which is true. Since the clause for φKA implies that φKA φA, # implies that A is stored only in worlds in φA. Thus, by the induction hypothesis, the creature is disposed to store A only when is true. We can therefore reasonably suppose that if the creature is disposed to store A then it knows . Conversely, if it is not disposed to store A, then it does not know . But is true if and only if it knows . Thus is true if and only if the creature is disposed to store A. It follows by # that φKA is the set of worlds in which is true. This completes the induction step. ▪

Given that φA is the set of worlds in which is true, the definition of φ recursively specifies the truth‐conditions of sentences of L. One can easily check that its results coincide with those of a semantics in possible worlds style, using (p.317) the accessibility relation in the diagram in section 10.6. Since the accessibility relation is reflexive and transitive, every theorem of the modal system S4 is true in every world, when □ is replaced by K and propositional variables by arbitrary sentences of L. Consequently, the creature knows every logical consequence of what it knows; moreover, whenever it knows p, it knows that it knows p. But sometimes, when it does not know p, it does not know that it does not know p. That is because it cannot survey the totality of its knowledge. It is a failure of self‐knowledge, not of rationality in any ordinary sense.