Whilst a straightforward condition given a classical bivalent system with values true and false, bringing truth value gluts into the picture raises some novel matters.

In a classical framework, ‘true’ is equivalent to ‘not false’, so the condition could simply be rephrased as: if S knows that p then p is not false.

But when the possibility of propositions which are both true and false is introduced, there are three plausible knowledge operators (K):

• Kp is true if and only if p is true
• Kp is true if and only if p is not false
• Kp takes the value of p, so it could be both true and false

The main question is, if one knows that p, does this require that p is true or that p is not false?

Title: The Logic of Knowledge and the Flow of Information

Abstract: In this talk I cover some work still in development which concerns the notions of information and knowledge as exemplified in Fred Dretske’s ‘Knowledge and The Flow of Information’. In particular, I cover (1) some work on the logic of information flow and (2) the issue of developing an epistemic logic which captures Dretske’s notion of knowledge as a semi-penetrating operator.

Interestingly, this is strongly reminiscent of a subsequent measure found in Shannon information theory. In philosophy literature, this formula can be found in Dretske’s formulation of information transmission derived from Shannon’s work as well as a measure of transmitted information given by Hintikka.

It is very easy to give examples of true beliefs that are not knowledge. There is the man who looks at a clock which is not going, though he thinks it is, and who happens to look at it at the moment when it is right; this man acquires a true belief as to the time of day, but cannot be said to have knowledge.

This passage is found in Section ‘D. Knowledge’, at the end of the Chapter ‘Fact, Belief, Truth, and Knowledge’.

Most would agree that this is not a case of knowledge. But how far can we carry this? Take the following example: A famous historical figure (X) dies and a medical staff member in attendance records X’s exact time of death using a clock hanging on the wall. This clock is also broken but happens to be stuck on the actual time, say, 6pm. Now, although the staff member records a fact (a true proposition), they do not actually know that X died at 6pm. Furthermore, by standard Dretskean information-theoretic epistemology, neither are they informed by the clock that X died at 6pm.

If the staff member does not have knowledge of nor are they informed of the time of death, is what they record a piece of information? Can a recorded fact be information or knowledge if the source of that record neither was informed of nor knew the proposition in question? If their record is used in a biographical book on X, can someone who reads this book 100 years later come to know that X died at 6pm?

Rohit Parikh proposed a criterion for relevance in belief change, which AGM is not stringent enough to meet. An example given is as follows:

Let p,q be two distinct elementary letters, and put K = Cn(p, q). Then there is an AGM maxichoice contraction that puts [the contraction of K by p ] to be Cn(), thus eliminating not only p but also q from K.

Subsequent expansion of Cn() with ~p or ~q would then result in K = Cn(~p, ~q).

Whilst it is clear to see how some sense of relevance is violated here, I think that such results can be legitimate. Take a Grove sphere modelling of the above example. The possible world where both p and q are true is in the centre. The closest outer shell contains the possible world where both p and q are false. The second outer shell contains the two possible worlds where one is true and one is false. So the ordering is:

This ordering can be motivated in a very reasonable and natural way. Say p stands for ‘Tom went to the party’ and q stands for ‘Harry went to the party’.

A third person, Dick, believes that both Tom and Harry went to the party. He also believes that it is almost certain that Tom and Harry only go to parties if both of them are going. So his second most plausible possible world is the one in which neither go to the party. Therefore, if he is told that Tom did not go to the party, as well as contracting p, q will also be contracted.