 In some (though not all) worlds semantics, logical validity and consequence are defined relative to just the normal worlds. The idea is that the normal worlds behave ‘appropriately’ with respect to the logic in question, whereas the non-normal worlds do not. For example, \(\Box(A \vee \neg A\) is valid in \(\mathbf{S2}\) and \(\mathbf{S3}\), but (by definition) it is false at the non-normal worlds. So we need to ignore the non-normal worlds when defining validity and consequence, which we do as follows: