Consider the following premises. Thomas is not listening to a song. If Thomas listens to a song, then John is writing a letter. If John writes a letter, then Thomas is listening to a song. If John writes a letter, then Thomas is listening to a song. Can we infer the following from them? If we can, name the inference rule being used: John is writing a letter.
 From the fact that if Thomas listens to a song, then John is writing a letter, and that if John writes a letter, then Thomas is listening to a song we can infer that Thomas is listening to a song if and only if John is writing a letter via biconditional introduction. From the fact that Thomas is listening to a song if and only if John is writing a letter we can infer that if Thomas listens to a song, then John is writing a letter via biconditional elimination. From the fact that if John writes a letter, then Thomas is listening to a song, and that if Thomas listens to a song, then John is writing a letter we can infer that John is writing a letter if and only if Thomas is listening to a song via biconditional introduction. Finally, from the fact that John is writing a letter if and only if Thomas is listening to a song, and that Thomas is not listening to a song we can infer that John is not writing a letter via biconditional elimination, which contradicts that John is writing a letter. Therefore, the answer is no.