1. This expository paper has been motivated by various discussions with L. A. Zadeh in Berkeley, and also with J. Marschak in Los Angeles. The manuscript was completed when I visited the Hebrew University, Jerusalem in the Summer of 1972.
2. This distinction between fuzziness and randomness nevertheless is consistent with probability statements on fuzzy events (for example, the probability of having a warm day tomorrow), whereas it is not clear what we mean by fuzzified probability statements. Fuzzified probability, however, does seem to make sense on a qualitative level of probability theory which in its subjective interpretation is the core of modern decision theory under uncertainty. Then the qualitative ordering of two events A ≤ B may be interpreted as ‘A is not much less probable than B’ This might give rise to interesting representation theorems of probability strictly compatible with a qualitative fuzzy ordering.
3. The reader will find no difficulty to picture the theorem for the simple case n = 1 and coordinate hyperplane H= {x | h(x)=}.
4. One might also think of a reverse process, namely to decompose a given term into more primitive terms with corresponding semantics. A very simple example is this: Assume that the term ‘green’ is decomposable into ‘green1’, ‘green2’… Then ‘greeni’, i=1,2,… might be taken as a primitive term and ‘green’ obviously corresponds to a second-order semantic.
5. If we assume that the domain of the performance function is a real convex, (finite- dimensional) vector space, then it is natural to view this function as quasi-concave.