 This limitation can be overcome (at least for technical purposes) if there is a pairing function on the relevant domain, that is, if there is a function \(\pi\) such that \(\pi(u, v) = \pi(u', v')\) just in case \(u = u'\) and \(v = v'\). For then quantification over dyadic relations can be represented by plural quantification over ordered pairs. Moreover, by iterated applications of the ordered pair function we can represent \(n\)-tuples and thus also quantification over \(n\)-adic relations. The question is how this pairing function is to be understood. One option is to proceed as in mathematics and simply postulate the existence of a pairing function as an abstract mathematical object. But this option has the obvious disadvantage of stepping outside of what most people are willing to call “pure logic”. A cleverer option, explored in the Appendix to Lewis 1991 and in Hazen 1997 and 2000, is to simulate talk about ordered pairs using only resources that arguably are purely logical. It turns out that talk about ordered pairs can be simulated in monadic third-order logic, given some plausible extra assumptions. Monadic third-order logic can in turn be interpreted either in a theory which combines plural quantification with mereology (Lewis 1991: ch. 3; Burgess and Rosen 1997: II.C.1) or in terms of higher-level plural quantification  (Section 2.4).