greater

I kindof don't like this rule; it seems to be one of those rules that takes you 80% of the way, and then all of a sudden lets you down. This rule implies that we can say halogens such as chlorine are always -1, which is always the case in Organic chemistry, but look at the I which is the central atom in Sample Exercise 8.10 on p. 270, and also look at the binary oxides on page 278. About two thirds of the way down the first colume Cl has an oxidation state of +7.
I'm going to speculate here that for any case where chlorine or another halogen can't take -1, that +7 is the only other alternative. Any other halogen oxide would probably need to do the +7 thing since oxygen is always -2.