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S. R. So, Proof of (ii): For any pairwise distinct options x; y; z 2 A, we have xP y ` R;fyP zg xRz (= :zRx), and xRz`RxRz`xRz`R;fzP yg xP y, in each case properly and irreversibly. Proof of Remark 7 Assume classical relevance Constrained and conditional entailment coincide by Remark 4. This implies the ?rst bullet point. The second bullet point follows from the additional fact that, for any p; q 2 X; Z X, the following are equivalent (see Dokow and Holzman [15] for a parallel argument): (i) p irreversibly constrainedly (= conditionally) entails q in virtue of Z, i.e., p ` R;Z q while fqg[Z 6 ` p; (ii) there is an instance of pair-negatability, i.e., the set Y := fp; :qg[Z is inconsistent and becomes consistent if one negates p and/or :q. Finally, pathlinkedness implies proper pathlinkedness, because any path of conditional entailments from a proposition to its negation must contain at least one properly conditional entailment (as is well-known since Nehring and Puppe [37]), and because 'conditional'is equivalent to 'constrained'. Proof of Theorem 5. Let the assumptions hold. By Theorem 4, there is a dictator i. To show that i is a strong dictator, I consider any (J 1 ; :::; J n ) 2 J n , and show that J i = F (J 1 ; :::; J n ) It su¢ ces to show that J i F (J 1 ; :::; J n ) Suppose q 2 J i . By assumption, q is the disjunction of some set, J n ). So, as q is the disjunction of a set containing p, we have q 2 F (J 1 ; :::; J n ), pp.2-3