Proof of the impossibility of probabilistic induction

In what follows I restate and simplify the proof of the impossibility of probabilistic induction given in [1,2]. Other proofs are possible (cf. Given two statements x, y we write x y if x entails y. For example, if x "All men are mortal" y "Socrates is a man" z "Socrates is mortal" we can write (x y) z. Entailment is a "truth broadcasting" operation that allows the truth of the premises to flow to the conclusion. We also have the following simple theorem.