AbstractThis paper makes two contributions. First, we give a semantics for sets of clauses of the syntactic form L0 ⇍ L1 &⋯& Ln where each Li is a literal. We call such clauses generally Horn clauses. Any such endeavour has to give a coherent, formal treatment of inconsistency (in the sense of two-valued logic). Thus, as a second contribution, we give a robust semantics for generally Horn programs that allows us to “make sense” of sets of generally Horn clauses that are inconsistent (in the two-valued logic sense). This applies to the design of very large knowledge bases where inconsistent information is often present.