Consider the following premises. Jessica is a cashier if and only if William is an astronaut. If it is sunny, then William is an astronaut. Jessica is not a cashier. If William were an astronaut, then it is sunny. Can we infer the following from them? It is sunny.
 From the fact that Jessica is a cashier if and only if William is an astronaut, and that Jessica is not a cashier we can infer that William is not an astronaut. From the fact that if William were an astronaut, then it is sunny, and that if it is sunny, then William is an astronaut we can infer that William is an astronaut if and only if it is sunny. Finally, from the fact that William is an astronaut if and only if it is sunny, and that William is not an astronaut we can infer that it is not sunny, which contradicts that It is sunny. Therefore, the answer is no.