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Xn]r and of a hyperplane [Yl:"': Yn]' being defined by n (1) L YiXi=Ü i=l Alllinear subspaees of Qn -1 form the lattiee L, with the elements a, b, e, .... Assume now that an operation a' with the properties L71-L73 in Seetion 9 exists: L71 L 72 L73 (a')' = a a n d = @ and au d acb implies d=>b'. = D, They imply (ef. Section 9) L74 (anb)'=dub' and (aub),=dnb'. Observe, that the relation ac b' is symmetrie in a, b, owing to L73 and L71. 18 GARRETT BIRKHOFF AND JOHN VON NEUMANN 2 If a: [Xl: ... : xn]r is a point, then a' is an [Yl: ...

True, the ordinary calculus of probability demands too much, but the mean value postulate [of von Neumann] demands too little; the ordinary calculus of probability is stilI needed if all quantities concerned commute. It may seem paradoxical in view of all this that the mean value postulate suffices for deducing the correct formulae. In this connection it must be pointed out that in von Neumann's deduction essential use is made of an extension of the mean value postulate to 'quantities' that are represented by projectian operators; as shown by von Neumann, these projectian operators represent statemenls on the measured values of these quantities rather than the quantities themselves - (to the eigenvalues 1 and 0 of the projectian operators correspond the truth values 'true' and 'false', respectively); thus, the calculus of the projectian operators represents a kind of sentential calculus.

A probability statement is a general implication between sentenees statingclass membership of elements and henee written in the form (Rl) R p here, 0 and P are class variabies, x and y are individual variabies, 3- is the sign for the prob relation ('probability implication'), and p is the numerieal value of the probability. A short-hand designation for (Rl) is R (O-3-P) p or w (0, P) = p. (R2) The multiplication theorem (axiom IV) may then be written in the form R (R IV) THE LOGlC OF COMPLEMENTARITY 35 Here, a compound of P and Q occurs only on the right-hand side 01 tne implication, and this (together with the rule of inference referred to above) makes it possible to deduce from meaningful expressions an expression that may be meaningless; in other words: R IV has the property of transfering existence.

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A Historical Perspective on the Topology and Physics of Hyperspace by Durham, I T

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