 Given exclusive disjunction, equivalence, symmetry, and asymmetry, Royce proposed that such relations could be expressed as a single ordering relation, the obverse or \(O\)-relation.[7]  Represented ‘\(O(A \: B)\)’, the \(O\)-relation marks two possible courses or modes of action, \(A\) and \(B\), and indicates that these courses of action are incompatible: that is, \(O(A\: B)\) can be read as the assertion that action A and action B cannot both be taken or that if \(A\) is taken \(B\) cannot be taken and vice versa. \(O\)-relations simultaneously represent a state of affairs such that two possible courses of action are available, one of which can be actualized and the other excluded. \(O\)-relations are also such that, in the course of things, one of the disjuncts must be taken. If \(O(A\: B)\), then it is also the case that “neither” is not an option in just the way that “neither” is not an option for \(O(A \: \neg A)\). The \(O\)-relation can include as many terms as are, when taken together, incompatible and, regardless of the number of terms, they may still be reduced to the dyad \(O(a \:\neg a)\). Taken as a set of incompatible actions, \(O(A \: B)\) is an \(O\)-collection (Royce 1905, 390), as are \(O(A\: B\: C)\) and \(O(A\: B\: C\: \ldots \: n)\). Now if \(A\) and \(B\) are incompatible (and so then are \(\neg A\) and \(\neg B\)), it is also the case that \(\neg A\) and \(B\) are compatible as are \(\neg B\) and \(A\). These terms are then related by the compatibility relation, equivalence, or the \(E\)-relation (Royce 1905, 390–391) such that \(E(\neg A\: B)\) and \(E(A \:\neg B)\).