\begin{align*} P(A \cup B \cup C \cup D) & = P(A) + P(B) + P(C) + P(D)\\ & - P(A \cap B) - P(A \cap C) - P(A \cap D) \\ & - P(B \cap C) - P(B \cap D) - P(C \cap D)\\ & + P(A \cap B \cap C) + P(A \cap B \cap D) \\ & + P(A \cap C \cap D) + P(B \cap C \cap D)\\ & - P(A \cap B \cap C \cap D) \end{align*}