Exercise 7.67

Suppose \(𝑠\) is a person alive throughout the interval \(𝑈\). Apply the above definition to the example \(𝑝(𝑠,𝑡)\) = “person \(𝑠\) is worried about news \(𝑡\)” from Example 7.65. Here, \(𝑇(𝑉)\) is the set of items that are in the news throughout the interval \(𝑉\).

1. What open subset of \(𝑈\) is \(∀(𝑡: 𝑇). 𝑝(𝑠,𝑡)\) for a person \(𝑠\)?

2. Does it have the semantic meaning you’d expect, given the less formal description in Section 7.4.4?

Author’s solution

1. The formula says that \(∀(𝑡: 𝑇). 𝑝(𝑠,𝑡)\) “returns the largest open set \(V ⊆ U\) for which \(p(s|_V,t) = V\) holds for all \(t ∈ T(V)\).” Note that \(𝑇(𝑉)\) is the set of items that are in the news throughout the interval \(𝑉\). Substituting, this becomes “the largest interval of time \(𝑉 ⊆ 𝑈\) over which person \(𝑠\) is worried about news \(𝑡\) for every item \(𝑡\) that is in the news throughout \(𝑉\).” In other words, for \(𝑉\) to be nonempty, the person \(𝑠\) would have to be worried about every single item of news throughout \(𝑉\). My guess is that there’s a festival happening or a happy kitten somewhere that person \(𝑠\) is not worried about, but maybe I’m assuming that person \(𝑠\) is sufficiently mentally “normal.” There may be people who are sometimes worried about literally everything in the news; we ask you to please be kind to them.

2. Yes, it is exactly the same description.

Alternative answer

It returns the largest open set \(V ⊆ U\) for which \(p(s|_V,t) = V\) holds for all \(t ∈ T(V)\). That is, it returns the largest open set where restricting the person \(s\) to the interval \(V\) is still the whole interval \(V\) for all active news items. As in Example 7.65, this corresponds to when the person is worried about everything in the news (rather than just something i.e. at least one thing).