Science, Decision and Value: Proceedings of the Fifth by Patrick Suppes (auth.), James Leach, Robert Butts, Glenn

By Patrick Suppes (auth.), James Leach, Robert Butts, Glenn Pearce (eds.)

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Additional resources for Science, Decision and Value: Proceedings of the Fifth University of Western Ontario Philosophy Colloquium, 1969

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B. An B ~ AilB. AuB c;. But in general neither operation is idempotent. However there are many sets A for which A = A 0 A and A = A E9 A, these identities being logically equivalent: such sets will be called perfect. Every one-membered set is perfect; and if A, B are perfect, so also are A 0 B, A E9 B, A r. B. Thus every credal set for a game with nature is perfect, since it is constructible by these three operations from the one-membered fundamental credal sets. But a stronger theorem holds, namely that the credal set of every certainty is perfect, and hence every credal set is perfect, since it is constructible by the three operations from credal sets of certainties.

The desired positive addition to Rule A can now be given by reference to an action's optimality; and Rule AB, incorporating this addition can be stated in the form that, if Luke is certain that Matthew is playing in such a game and has such a preference system that Ms is inadmissible and M t is optimal in that game and for that preference system, then Luke must not allow for the possibility of Matthew's choosing Ms and must allow for the possibility of his choosing M t • Luke's following Rule AB instead of Rule A will not affect Luke's credal sets when only Matthew's monadic games are concerned.

But the argument requires that Luke should hold a creed of another sort about Matthew, and what this creed should be allows some freedom of choice to the philosophical analyst. It is prima facie plausible to take this creed as being a certainty or uncertainty of Luke's as to Matthew's certainty or uncertainty as to what action Lr (which row Lr) Luke will choose. But, on my view, to discuss the matter in this way will presuppose that, while Luke is playing a game with Matthew, 38 R. B. BRAITHWAITE he believes that Matthew is in fact playing a game with nature and treating Luke as an automaton.

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