 The two measures are required for technical reasons—in order to capture additivity on content independence and inductive independence respectively. Two sentences \(s\) and \(s'\) are content independent when they do not have any worlds in common. Two sentences \(s\) and \(s'\) are inductively independent when the conditional probability of each sentence given the other is identical to their initial unconditional probability. Additivity on inductive independence fails for cont, since it might be the case that \(\tcont(s\wedge s') \neq \tcont(s) + \tcont(s')\) on account of \(p(s)\) and \(p(s')\) having worlds in common—that is, on account of them not being content independent in spite of their being inductively independent. For additivity to hold on cont, it is content independence, as opposed to inductive independence that is required. By contrast, additivity on inductive independence does not fail for inf. Bar-Hillel and Carnap’s proof is non-trivial (found on their 1952: 244–5).