Translate the following inference to logic notation: For all x, if x_8 were fast, then x_8 is excited. For all x, x_8 is excited or x_8 will not write a letter. There is at least one x for which x_8 is a doctor. For all x, x_8 will write a letter or x_8 is fast. Therefore there is at least one x for which x_8 is a doctor.
forall x: P(x) -> Q(x). forall x: Q(x) or ~R(x). exists x: S(x). forall x: R(x) or P(x). Therefore exists x: S(x).