Consider the following premises. For all x, if x listens to a song, then x is writing a letter. There is at least one x for which x will not write a letter. For all x, if x writes a letter, then x is listening to a song. Can we infer the following from them? If we can, name the inference rule being used: For all x, x will listen to a song.
 From the fact that for all x, if x writes a letter, then x is listening to a song, and that for all x, if x listens to a song, then x is writing a letter we can infer that for all x, x will write a letter if and only if x will listen to a song via universal biconditional introduction. From the fact that for all x, x will write a letter if and only if x will listen to a song we can infer that for all x, if x listens to a song, then x is writing a letter via universal biconditional elimination. Finally, from the fact that for all x, if x listens to a song, then x is writing a letter, and that there is at least one x for which x will not write a letter we can infer that there is at least one x for which x will not listen to a song via existential modus tollens, which contradicts that For all x, x will listen to a song. Therefore, the answer is no.