 David Lewis (1981) proved that the truth conditions of premise semantics and ordering semantics fundamentally correspond to each other. One can start with a premise model and define a strict partial ordering for an o-model that gives the same results: define \(u \prec_w v\) iff the set of all premises in \(\Gamma(w)\) true at world \(u\) is a proper superset of the set of premises true at world \(v\). And conversely, one can start with an o-model based on a strict partial ordering and define a premise model that gives the same results: define \(\Gamma(w)\) as the set of all propositions of the form