Theory of multibin tests: Definition and existence of extraneous tests

Dirac's sets of commuting observables, in the guise of lists of mutually orthogonal projections which add to the unit matrix, are readily extended to a convex completion. But, furthermore, there exist lists of nonnegative Hermitian matrices which sum to the unit matrix which do not even belong to this convex completion. It is shown that these ``extraneous'' lists appear as tests in ordinary quantum‐mechanical experiments. This circumstance leads to simpler rules for injecting measurement theory into the social sciences than might otherwise be proposed. Various relationships between lists of orthogonal projections and more general tests are given. The problem of devising rules of inference by direct computation is very briefly engaged.