The Evolution of Reason: Logic as a Branch of Biology by William S. Cooper

Author:William S. Cooper
Language: eng
Format: mobi, epub
Publisher: Cambridge University Press
Published: 0101-01-01T00:00:00+00:00

Alice is in southern France logically implies Alice is in France.

Alice is in France becomes believed.

Therefore Alice is in southern France becomes more probable.

The inference pattern can be visualized as taking place in the event space shown in Figure 6.3. B includes A because it is a logical consequence of A. At first (Figure 6.3a), neither A nor B is believed at a high level of subjective probability. Then (in Figure 6.3b) B comes to be believed, practically filling the diagram as its area rises to a level of belief close to 1.0. As the area interior to B is stretched the area of A within it increases proportionately, so that A becomes more probable.

The stretching operation depicted in the diagram makes Polya’s inference pattern work as advertised. A subtle assumption is involved though. It is that the area inside proposition B gets stretched evenly when B is accepted. Technically what is postulated is that when an event’s probability rises as B's does, all probabilities conditional on the event remain the same, and similarly all probabilities conditional on the event’s complement remain the same. An orderly belief state change of this sort has been described as belief revision by probability kinematics (Jeffrey 1983).

The plausible inference pattern may therefore be declared valid for all situations in which the probability increase in B proceeds by probability kinematics. It is thought that many or most probability changes do in fact proceed by probability kinematics. It is a good default assumption for use in deliberations about probability dynamics. A kinematic change is a modification prompted by making an observation that tends to confirm or otherwise change the probability of an event while giving no additional information about things not implied by the event. The rationality of belief change by probability kinematics under appropriate conditions has been defended by Skyrms (1987; 1990) and others. It is a plausible conjecture open to exploration that such arguments could be reinterpreted in such a way as to support the evolutionary fitness of kinematic change.

Another of Polya’s patterns translates into belief language as follows. [Page 115]

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