User:IssaRice/Strength of a mathematical statement

Strong vs subset

A puzzle: why do we say P is stronger than Q if P is a subset of Q, but we also say that a theorem is stronger if it is more general (so bigger)?

One reply/intuition uses something like possible world semantics, e.g. see Wei Dai's post on Aumann's agreement theorem. There is just one possible world (a single $ω \in Ω$ ), but our information state is the set of all possible worlds that we cannot distinguish, so the less we know, the more possible worlds we think we could be in.
One visualization is to use a Venn diagram. The stronger the statement, the more our movement is restricted, as we are forced to be in more and more sets.
When we say a strong statement like $\forall x P (x)$ , we are saying $P (x_{1}) \land P (x_{2}) \land \dots \land P (x_{n})$ . When we say a weak statement like $\exists x P (x)$ , we are saying $P (x_{1}) \lor P (x_{2}) \lor \dots \lor P (x_{n})$ . It seems like in both cases we are accumulating more and more things.