fortune’, ‘In this case sacrifice’}. Since J’(1)
J’(0) this is a compact belief.
Remark: If s is the case, and one doesn’t believe J(s), so that J(s) = 0, then one believes some alternative J’(s). Someone unfamiliar with matches would have the unsound (perhaps only basic) information ‘this is just a piece of wood’. More complex situations need thorough analysis. E.g. someone may know the text of a theorem and benefit from that, but may not know its proof.
Lemma III: If there is sound information (J(1), J(0)) on a controlable dichotomous state s, then:
(i) if the information is not compact then there are 8 states of the system, with 4 states implying a hidden cause,
(ii) if the information is compact, these numbers are halved.
Proof:
We tabulate the possible states of the system (J(1), J(0), s) in Table 16.
In cases (rows) (3), (4), (6) and (7), the agent doesn’t possess sound information and believes some J(s) (e.g. ‘the world is as it is’), but he chances at s nevertheless. This implies that there is a hidden cause. (For example, the state of the system was inherited, and the agent wishes to keep things as they are. In that case (J’(1), J’(0), s) has causality within a more complex model, describing in more detail how people act on their beliefs.)
If the information is compact, we only consider states (1) to (4). Q.E.D.
Discussion: To understand the proof, look for example at row 6: There is a true model for sequential states {1, 1} and {1, 0}, or to maintain 1 or change to 0. But nothing is truly known about maintaining 0 or changing back from 0 to 1 (though beliefs can exist). Observed is s = 0. Perhaps it once was a conscious choice to go from 1 to 0, and perhaps one uses the implied control {chance(0, 1)}