P(A U B U C) = P(A U B) + P(C) - P((A U B)^C)
                   = P(A) + P(B) - P(A^B) + P(C) - P((A^C) U (B^C))
                   = P(A) + P(B) - P(A^B) + P(C) - [P(A^C) + P(B^C) - P((A^C)^(B^C))]
                   = P(A) + P(B) + P(C) - P(A^B) - P(A^C) - P(B^C) + P(A^B^C)

You may have noticed that you find the probability by adding the probabilities of the individual events, then taking away the probabilities of each combination of two events, and finally adding the probability for all three to happen. This pattern is called the inclusion-exclusion principle, and it applies to the union of any number of probabilities.

So P(A U B U C U D) =

P(A) + P(B) + P(C) + P(D) - P(A^B) - P(A^C) - P(A^D) - P(B^C) - P(B^D) - P(C^D) + P(A^B^C) + P(A^B^D) + P(A^C^D) + P(B^C^D) - P(A^B^C^D)

From a Venn Diagram?
I don't think there's an algebraic way to do it.

Thanks DC92, much appreciated.