Consider the following premises. For all x_10, x_10 will play a game if and only if x_10 is a cashier. If Joseph were a cashier, then Joseph is playing a game. Joseph is not playing a game. Can we infer the following from them? Joseph is a cashier.
 From the fact that for all x_10, x_10 will play a game if and only if x_10 is a cashier we can infer that for all x_10, if x_10 plays a game, then x_10 is a cashier. From the fact that for all x_10, if x_10 plays a game, then x_10 is a cashier we can infer that if Joseph plays a game, then Joseph is a cashier. From the fact that if Joseph plays a game, then Joseph is a cashier, and that if Joseph were a cashier, then Joseph is playing a game we can infer that Joseph is playing a game if and only if Joseph is a cashier. Finally, from the fact that Joseph is playing a game if and only if Joseph is a cashier, and that Joseph is not playing a game we can infer that Joseph is not a cashier, which contradicts that Joseph is a cashier. Therefore, the answer is no.