 But the right-to-left direction will have to go, if \(w\) is one of Rescher and Brandom’s impossible worlds. These worlds are nonadjunctive: they allow two sentences to be true even though their conjunction is not true. (Adjunction is the principle that the truth of a conjunction follows from the truth of its conjuncts.) Rescher and Brandom’s worlds can also be nonprime: a disjunction may hold at them even though neither disjunct does. They may also make some \(A\) and its negation \(\neg A\) both true. But the corresponding conjunction, \(A \wedge \neg A\), doesn’t follow. These impossible worlds retain a certain amount of logical structure. They are closed under any classically valid, essentially single-premised inference (such as Disjunction Introduction); but they are not closed under essentially multiple-premised inferences (such as Adjunction).