Consider the following premises. w <-> s_4. w or w_2. w_2 <-> s_4. Can we infer ~s_4 from them?
 From w <-> s_4 we can infer w -> s_4. From w_2 <-> s_4 we can infer w_2 -> s_4. Finally, from w -> s_4, w_2 -> s_4, w or w_2 we can infer s_4, which contradicts ~s_4. Therefore, the answer is no.