5.1241 “p. q” is one of the propositions that “p” asserts and also one of the propositions that “q” asserts.
Two propositions are opposed to one another if asserting them both makes no meaningful proposition [wenn es keinen sinnvollen Satz gibt].
Every proposition that contradicts another denies it.
Black (p. 242) says of the last sentence that “It is hard to see the point of this remark.” Presumably the point has to do with sentences referring to or dealing with each other. A red light and a green light just as such do not deny each other, but when used as traffic signals they do. Incompatible sentences are not merely incompatible, so that you cannot have both, as it were. They actually rule each other out. The reason why you cannot have both is internal to the sentences.