TITLE: Why is there this strange contradiction between the language of logic and that of set theory?
QUESTION [18 upvotes]: In standard probability theory events are represented by sets consisting of elementary events. Consider two events for which (as sets) $A \subset B$. If an elementary event $x \in A$ takes places then we say that $B$ takes place as well. Since $A \subset B$, $x \in A$ implies that $B$ takes place too.
It seems that when for sets (representing events) $$A \subset B$$ $$then \ A \ implies \ B .$$
On the other hand, if we use the language of logic for the events then $$A \supset B$$ $$ \ means\  that \ A\  implies\ B .$$
Why is this strange virtual contradiction between the language of sets and the language of logic?
(In order to avoid down votes and unplesant comments I reveal that I happen to know that the true translation of the sentence $A \supset B$ of logic to the language of sets (representing events) is $\overline{A \cap \overline B}$.)

REPLY [2 votes]: I do not know whether this has anything to do with the origin of the use of $\supset$ as $\implies$, but the statements that can be inferred from $A$ when $A \implies B$ include those that can be inferred from $B$.