A majority social choice function of order fJ defines a relation" R between pairs of alternatives such that xRy if and only if the number of individuals who prefer x to y is at least fJ times the number who prefer y to x.A binary relation R is cyclic over a set S of m alternatives when we have (J1R(J2' (J2R(J3' ... , (Jm-1R(Jm and (JmR(J1 [(Ji E S]. R is "generally acyclic" over a set L of M alternatives when there is no subset S c: L over which R is cyclic.The connection between general acyclicity and transitivity is close, though the former may be viewed as a weaker property. A relation R is generally acyclic over a triple [x, y, z] only ifThis paper concerns general acyclicity and not transitivity because the suitably chosen majority function defines a binary relation that need not relate x and z even if xRy and yRz; we prove the negative theorem "not zRx", rather than the positive" x Rz ";We assume that any individual can have anyone of the M! logically possible, connected, strict preference orderings" of the set L. We shall prove that a sufficient condition for a majority social choice function of order fJ to define a generally acyclic binary relation over L is that fJ> M -1. We shall also show that fJ = M -1 is not sufficient. First we prove the following theorem, deducing the above result as a corollary. Theorem. A binary relation R will not be cyclic over a set S of m alternatives if R isdefined by a majority social choice function of order cP > m -1.Proof. Suppose that we have a majority social choice function of order > m -1. Let n(i,j) be the number of individuals who prefer a, to (JJ(Ji' (Jj E S], and let n' be the number of individuals who prefer (J1 to (J2' (J2 to (J3, ... , Suppose that n(l, 2)~.n(2, 1), n(2! 3)~.n(3.. 2), n(m-l:m)~.n (m, m-l).

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