Name of these lemmas in set theory

Question

Lemma 1.2 If $S$ is countable and $S'\subset S$, then $S'$ is also countable

Lemma 1.3 If $S'\subset S$ and $S'$ is uncountable, then so is $S$.

I was wondering if there was a name for the logic/proof that these two lemmas act upon. That if a component lacks a quality, it disqualifies the greater set containing that component from having that quality. And the other, that should a component have a quality (in this case, uncountability), then so does its greater set containing it.

Is there a name for this reasoning?

My question is derived from Plato's Cratylus, where Socrates essentially makes the argument that:

1. There are true and false propositions.
2. A proposition being true is contingent on the parts (by Socrates, called names, which loosely translates to words) of the proposition also being true
3. That the part of a true proposition must also be true, and of a false proposition false.

I thought his rational was similar to the lemmas used for sets, how for the set to be countable (true), the parts (elements) must also be true.

Answer 1

I'd say 1.3 follows from 1.2 using uncountable = not countable and excluded middle:

• given $S' \subset S$, $S'$ uncountable.
• Suppose $S$ were countable.
• 1.2 says that $S'$ is countable.
• This contradicts $S'$ not countable from the given data.
• so the assumption $S$ is countable was wrong:
• ergo $S$ is uncountable.

1.2 is the basic fact: countable sets are "downwards closed", 1.3 is a derived fact and needs no new principle.