Consider the following premises. Patricia is playing a game if and only if John is writing a letter. If Patricia plays a game, then Mary is a lawyer. If Mary were a lawyer, then Mary is a janitor. Mary is not a janitor. Can we infer the following from them? John is writing a letter.
 From the fact that if Mary were a lawyer, then Mary is a janitor, and that Mary is not a janitor we can infer that Mary is not a lawyer. From the fact that Patricia is playing a game if and only if John is writing a letter we can infer that if John writes a letter, then Patricia is playing a game. From the fact that if Patricia plays a game, then Mary is a lawyer, and that Mary is not a lawyer we can infer that Patricia is not playing a game. Finally, from the fact that if John writes a letter, then Patricia is playing a game, and that Patricia is not playing a game we can infer that John is not writing a letter, which contradicts that John is writing a letter. Therefore, the answer is no.