 where “\(\rightarrow\)” is the standard material conditional. This doesn’t give the intuitive truth conditions of (32) on the reading that concerns us, i.e., that every donkey-owning farmer beats every donkey he owns. The third and final element we need to get the truth conditions to come out right is to say that a pair of assignments \(\langle h,h\rangle\) is in the interpretation of a conditional iff for all k such that \(\langle h,k\rangle\) satisfies the antecedent, there is a j such that \(\langle k,j\rangle\) satisfies the  consequent.[18]  This says, roughly, that for any output assignment k of a pair of assignments satisfying the antecedent of the conditional (assignment of a donkey-owning farmer to x and a donkey owned by x to y in the case of (32)/(32a)), k is the input of a pair \(\langle k,j\rangle\) that satisfies the consequent, for some j. Since the consequent of the conditional in (32)/(32a) is an atomic formula, j=k. So the account claims that any output of a pair of sequences that satisfies the existentially quantified antecedent also satisfies “x beats y”, and so also assigns to “x” and “y” to something that stands in the beating relation. That is, the truth of (32)/(32a) requires every donkey-owning farmer to beat every donkey he owns.