If we are to have equations \(E\) behave well in a logic setting, each \(P_i\) should have the same valuation no matter where we see it—there should be no distinction between what we have been calling left and right sides; \(\bP_i\) and \(\bP'_i\) should be the same. In other words, we would like to have partial relations \(\bP_1\), …, \(\bP_k\) to interpret \(P_1\), …, \(P_k\) so that \([E](\langle\bP_1, \ldots, \bP_k\rangle) = \langle\bP_1, \ldots, \bP_k\rangle\)—‘before’ and ‘after’ values agree. This is called a fixed point of \([E]\). So, we need to know that \([E]\) has a fixed point, and if it has more than one then there is a plausible candidate we can choose as the best one.