What semantic notions underlie ex/in-tēnsiō with the logical meanings of ex/in-tension?

Question

1. How did 'ex/in-tēnsiō' semantically specialize to mean the logical meanings below?

2. 'ex/in-tēnsiō' obviously share the same root, and differ merely in prefixes. Does the difference in prefix explain their meanings?

3. 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?

4. To wit, why doesn't EXtension signify 'qualities or attributes that the term connotes'?

5. 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.

Etymonline on 'intension (n.)' :

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.

Answer 1

I will keep this answer about Latin only. For developments in other languages, please ask on another language site. The philosophy or linguistics sites might also be options for related questions.

1. How did 'ex/in-tēnsiō' semantically specialize to mean the logical meanings below?

This happened after Latin. For example, in physics we have intensive and extensive properties coming from those same Latin words. The meaning is quite different from the one you quote, and your quoted text mentions that the terms work differently in logic and grammar. Therefore it is safe to say that in scientific terminology words derived from intendere and extendere do not have any fixed meanings but depend greatly on context and convention.

Both the physical terms and the ones you quote were inspired by the Latin words, but not taken from their then current meaning. When a new concept is created, often an old word is taken and given a new purpose.

2. 'ex/in-tēnsiō' obviously share the same root, and differ merely in prefixes. Does the difference in prefix explain their meanings?

No. The same prefix can modify the meaning of a verb (like tendere) in many different ways. You cannot deduce the meaning of a prefixed verb from the simplex verb and the prefix. You can have a decent guess, but at least nuances will be a little off. This is the same as with English phrasal verbs; the preposition can change the meaning wildly, in a way that is not immediately related to the meaning of the preposition.

3. 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?

Whether they are opposites is a matter of taste. I am more familiar with physics, so let me comment on that example. There are essentially two ways a thermodynamic property can behave when objects are combined. One of them is called intensive, the other extensive. This makes it reasonable to treat them as opposites, although they are more accurately the two alternatives.

The pair in-/ex- is commonly recognized as an opposite pair or as a pair of alternatives. The pair of "in" and "out" makes a clear pair for either kind of use. Therefore it makes sense to define the opposite or alternative of ex-something as in-something. In the physical case, I could argue for the choice as it has been made or switching the prefixes. I am not familiar with the original argument for the choice; perhaps one term was introduced earlier and the other one later by analogy.

I assume the logical pair of terms is similar.

4. To wit, why doesn't EXtension signify 'qualities or attributes that the term connotes'?

5. Why doesn't INtension signify 'members of the class that the term denotes'?

This sounds like an arbitrary choice of nomenclature to me. And in my experience it is often most useful to take definitions as given and the go and work with them. Of course the rationale behind the choice is interesting, but it is secondary to the subject matter itself. It is important to get this order of precedence right; otherwise you fall victim to the etymological fallacy.

Why the terms ended up having this meaning is a question about the history of logic — or any other scholarly field where they appear — not a question of Latin. The semantic reasons you are looking for lie outside Latin.