One way to classify adjectives is into 2 categories: intersective and non-intersective, with 3 subcategories of non-intersective adjective (subsective, plain and privative).
An example of an intersective adjective is carnivorous in ‘carnivorous mammal’. This describes something that is both a mammal and carnivorous.
The set of carnivorous mammals is the intersection of 2 sets: the set of mammals and the set of carnivorous things.
‘Jane is a good lawyer’ does not say that Jane is both a good person and a lawyer. It says she is a lawyer and good as a lawyer.
The set of good lawyers is a subset of the set of lawyers.
Plain adjectives are non-intersective and they are not subsective. In ‘Jane is a former senator’, Jane is not being described as a senator (now) and ‘Jane is former’ would be a meaningless statement.
The set of former senators is not a subset of the set of people who are (now) senators. And there is no set of people who are ‘former’.
For another example using ‘former’, please see A British former player – Language Miscellany
The 3rd sub-category of non-intersective adjectives is privative adjectives. They negate the noun they modify. A ‘fake pistol’ is not a pistol at all.
The set of fake pistols is not a subset of the set of pistols. Said differently, the intersection of (a) the set of pistols and (b) the set of things that are fake—is a set that is empty.
Adverbial and adjectival modification, Artemis Alexiadou in The Cambridge Handbook of Generative Syntax, edited by Marcel den Dikken (2013)