The main purpose of this paper is to provide a new proof of the impossibility of local theories of hidden variables based on simpler and more general assumptions than those used by Bell (1964, 1966), Wigner (1970), and Suppes and Zanotti (1976). In particular, no assumptions requiring specific quantum mechanical calculations are required. They are replaced by the principle of exchangeability and the principle of identical conditional distributions given the hidden variable.

The results may be most easily discussed in terms of a system of two spin-½ particles initially in the singlet state, but generalizations to other quantum mechanical systems of a similar nature are apparent. In essential terms, the analysis of Bell, stated in a particularly clear form by Wigner, depends upon the following assumptions, which we state in intuitive form. (Mathematically explicit axioms are formulated in Appendix A.)

1. Axial symmetry. For any direction of the measuring apparatus the expected spin is 0, where spin is measured by +1 and -1 for spin +½ and spin -½, respectively. Further, the expected product of the spin measurements is the same for different orientations of the measuring apparatuses, as long as the angle between the measuring apparatuses remains the same.

2. Opposite measurement for same orientation. The correlation between the spin measurements is – 1 if the two measuring apparatuses have the same orientation.

3. Independence of λ. The expectation of any function of λ is independent of the orientation of the measuring apparatus.