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By Patrick Suppes

Paperback with none writing within the booklet. released 1979 north-holland publishing corporation Amsterdam

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19) (h) prob(PIG & prob(PIG) = r) = r. Note that if prob(PIG) f: r then prob(PIG & prob(PIG) counterlegal probability. = r) is a 4. THE NON-BOOLEAN THEORY OF PROPORTIONS The Boolean theory of proportions only concerns itself with the Boolean operations on sets. In this respect, it is analogous to the propositional calculus. , relations. There are a number of principles that ought to hold in that case but are not contained in the Boolean theory of proportions. The classical probability calculus takes no notice of relations, and to that extent it is seriously inadequate.

8), the concept of a proportion is still basically a measure-theoretic notion. 10) If X ~ Y then fly(X) = p(XIY). 11) If A ~ Yand B p(AIB) = ~ Yand p(BIY) -=I 0 then fly(A n B) . fly( B) This observation is helpful in mustering intuitions in support of certain principles regarding proportions. For example, suppose X ~ Y and there is a Z ~ Y such that fly(Z) = r, where r < ,Uy(X). 12) If X ~ Yand r < p(XIY) and (3Z)[Z ~ Y & p(ZIY) = r] then (3Z)[Z ~ X ~ Y & p(ZIY) = rJ. 12) as an axiom and call it The Denseness Principle.

4) If 03G then 0 ~ prob(FIG) ~ 1. 5) If 03G and (G ~ F) then prob(FIG) = 1. 6) If 03H and [H ~ - (F & G)] then 37 p p p prob(F v GIH) = prob(FIH) + prob(FIH). 7) If 03(F & H) then p prob(F & GIH) = prob(FIH)· prob(GIF & H). 3), according to which if - 03G then probw(FIG) is a weighted p average of the probiFIG) for the different nearest 03G-worlds x. 9) If (G ~ prob(FIG) ~ ~ 1. F) then prob(FIG) = 1. 7), the multiplicative axiom, is not so easily generalized. If 03H but - 03(F & H) then (H ~ - F) and [H ~ -(F & G)].

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