Consider the following premises. If it is cloudy, then Robert is playing squash. If James cooks, then Robert is playing squash. Robert is not playing squash. If it is late, then Michael is not happy. It is cloudy or James is cooking. Can we infer the following from them? If we can, name the inference rule being used: It is not late.
 From the fact that if it is cloudy, then Robert is playing squash, that if James cooks, then Robert is playing squash, and that it is cloudy or James is cooking we can infer that Robert is playing squash via disjunction elimination. From the fact that Robert is playing squash we can infer that Robert is playing squash or Michael is happy via addition. From the fact that Robert is playing squash or Michael is happy, and that Robert is not playing squash we can infer that Michael is happy via disjunctive syllogism. Finally, from the fact that if it is late, then Michael is not happy, and that Michael is happy we can infer that it is not late via modus tollens. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.