Consider the following premises. James is cooking. For all x_10, x_10 is an artist. For all x_10, if x_10 were an artist, then x_10 is a politician. Can we infer the following from them? If we can, name the inference rule being used: For all x_10, either x_10 will cook or x_10 is a politician are not true.
 From the fact that for all x_10, if x_10 were an artist, then x_10 is a politician, and that for all x_10, x_10 is an artist we can infer that for all x_10, x_10 is a politician via universal modus ponens. From the fact that James is cooking we can infer that there is at least one x_10 for which x_10 will cook via existential generalization. Finally, from the fact that there is at least one x_10 for which x_10 will cook, and that for all x_10, x_10 is a politician we can infer that there is at least one x_10 for which x_10 will cook and x_10 is a politician via existential conjunction, which contradicts that For all x_10, either x_10 will cook or x_10 is a politician are not true. Therefore, the answer is no.