 Notice that on the face of it  IMP  and  CON  are distinct.  IMP,  in and of itself, does not prohibit inconsistent or even contradictory beliefs, all it requires is that my beliefs be closed under logical consequence.  CON,  on the other hand, does not require that I believe the consequences of the propositions I believe, it merely demands that the set of propositions I believe be consistent. However, given certain assumptions,  IMP  does entail  CON.  Against the background of classical logic, the entailment obtains provided we allow the following two assumptions: (i) one cannot (and, via the principle that “ought” implies “can”, ought not) both believe and disbelieve one and the same proposition simultaneously; and (ii) that disbelieving a proposition is tantamount to believing its  negation.[14]  For let \(\{ A_{1}, \dots, A_{n}\}\) be \(S\)’s inconsistent belief set. By classical logic, we have \(A_{1}, \dots, A_{n-1}\models \neg A_{n}\). Since \(S\)’s beliefs are closed under logical consequence, \(S\) believes \(\neg A_{n}\) and hence, by (ii), disbelieves \(A_{n}\). So, \(S\) both believes and disbelieves \(A_{n}\).