Can Importance Be Different?

Importance is an uncountable noun, so one may shorten this sentence

We have external information and internal information. The importance of the external information and that of the internal information are different.

as follows,

We have both external and internal information. Their importance is different.

or as suggested (Is "they" valid for "information?"),

We have both external and internal information. They differ in importance.

Question 1: Can "both" (first sentence) be used for the singular object "information?"

Question 2: Can "their" (second sentence) be used for the singular object "information?"

Question 3: Can "importance" (second sentence) be different though it is singular?

Solution 1:

1. In your example, "both" can be used because you're not using "information" as a singular object, but rather "external information" and "internal information" are two separate entities.

2. As @YosefBaskin stated in his reply, "you do not have two owners of one importance" thus "their" is not appropriate. "Their importance" implies one importance belonging to a singular group. Your second example using "they" does work, however, because importance is not being treated as a singular object but as a property that "they differ" in. You could say: "Their importances are different" although that would sound awkward since importance isn't generally pluralized. To explain the difference, if you said "their car" it implies one car with multiple owners whereas "their cars" implies multiple cars with an owner each.

3. Importance is a subjective property and, as you mentioned in the question, an uncountable noun. The amount of something uncountable can be different in different contexts (here, the contexts of "external information" and "internal information"). Taking into account the answer for (2), though, you need to make it clear that each has an importance that differs rather than treating importance as a singular entity shared by both.