Translate the following inference to logic notation: There is at least one x for which x_6 is excited or x_6 will write a letter. For all x, if x_6 writes a letter, then x_6 is an astronaut. For all x, x_6 is not excited or x_6 is a scientist. For all x, x_6 is a scientist if and only if x_6 is an astronaut. Therefore there is at least one x for which x_6 is an astronaut.
exists x: P(x) or Q(x). forall x: Q(x) -> R(x). forall x: ~P(x) or S(x). forall x: S(x) <-> R(x). Therefore exists x: R(x).