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‘More specific information’, here, is informally referring to the value of the (positive) *likelihood ratio*: the ratio between how likely it is to get that information if a proposition is true, and how likely it is to get that information if the proposition is false. (I say ‘informally’ because, as you're aware, *specificity* is also a technical term in statistics, which is computed somewhat differently. The author here is using ‘specific’ in its general sense, not the technical sense.)

In this example, if the proposition is question is that both children are girls, the likelihood ratio corresponding to the information that the oldest child is a girl is $\frac{P(\text{oldest is girl} \mid \text{both are girls})}{P(\text{oldest is girl} \mid \text{at least one is not a girl})}$, or $\frac{1}{1/3} = 3$. The likelihood ratio corresponding to the information that at least one child is a girl is $\frac{P(\text{at least one is girl} \mid \text{both are girls})}{P(\text{at least one is girl} \mid \text{at least one is not a girl})}$, or $\frac{1}{2/3} = \frac32$. For information like a winter child is a girl, the likelihood ratio will be somewhere between these two values—in this case, $\frac{P(\text{at least one is winter girl} \mid \text{both are girls})}{P(\text{at least one is winter girl} \mid \text{at least one is not a girl})} = \frac{7/16}{1/6} = \frac{21}{8}$.

Note that the difference between ‘girl’ and ‘winter girl’ shows up on both sides of that ratio—the probability that at least one is a winter girl is less than the probability that at least one is a girl, both under the assumption that both are girls *and* under the assumption that at least one is not a girl. Any additional requirement placed on the girl will do this. But critically, the probability is hit harder in the case where at least one child is not a girl—because the both-girls case has two chances to meet the requirement, and the other case has one or zero. So additional requirements increase the likelihood ratio, and the stricter the requirement, the fewer girls there are who could meet it, and the stronger the effect. ‘The oldest child’ identifies exactly one girl, which is the strictest possible requirement short of impossibility, so we can't expect the likelihood ratio to increase beyond 3.

I'm guessing your book covers the value of likelihood ratios elsewhere; for the purposes of this example, it's enough to understand that a likelihood ratio completely captures the value of information in the context of a particular proposition. If you have an initial assumption for the probability of the proposition, the likelihood ratio is all you need to compute the probability of the proposition given the new information. The larger the likelihood ratio, the greater the conditional probability will be. So a likelihood ratio between two others will result in a conditional probability between the probabilities you'd get from those alternate pieces of information, and everything I just said about likelihood ratios applies to conditional probabilities.