How did 'ex/in-tēnsiō' semantically specialize to mean the logical meanings below?
'ex/in-tēnsiō' obviously share the same root, and differ merely in prefixes. Does the difference in prefix explain their meanings?
Does the difference in prefix explain why 'ex/in-tēnsiō' didn't mean the opposite? Why didn't 'ex/in-tēnsiō' mean «in/ex-tēnsiō» respectively?
To wit, why doesn't EXtension signify 'qualities or attributes that the term connotes'?
Why doesn't INtension signify 'members of the class that the term denotes'?
Source: Hurley, P. A Concise Introduction to Logic (2014 12 ed, but ∃ 2017 13 ed). p. 92 Middle.
The previous section of this chapter explored the cognitive meaning of language in general. The cognitive meaning of terms comprises two kinds: intensional and extensional. The intensional meaning, or intension, consists of the qualities or attributes that the term connotes, and the extensional meaning, or extension, consists of the members of the class that the term denotes. For example, the intensional meaning of the term “cat” consists of the attributes of being furry, of having four legs, of moving in a certain way, of emitting certain sounds, and so on, while the extensional meaning consists of cats themselves—all the cats in the universe. The term connotes the attributes and denotes the cats.
The intensional meaning of a term is otherwise known as the connotation, and the extensional meaning is known as the denotation. Intension and extension are roughly equivalent to the more modern terms sense and reference, respectively. Also, note that logic uses the terms connotation and denotation differently from the way they are used in grammar. In grammar, connotation refers to the subtle nuances of a word, whereas denotation refers to the word’s direct and specific meaning.
c. 1600, "action of stretching; increase of degree or force,"
from Latin intensionem/intentionem (nominative intensio/intentio) "a stretching, straining," figuratively "exertion, effort," noun of action
from past participle stem of intendere in its literal sense of "stretch out, strain" (see intend, and compare intention, which has the figurative sense).
[ OED : ] Etymology: < Latin intensiōn-em stretching, straining, noun of action from intendĕre to stretch: see intend n., intense adj., and compare intention n., which is etymologically a doublet of this.