Why is there this strange contradiction between the language of logic and that of set theory? 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 $\overline{A \cap \overline B}$.)
 A: Yes, it is inconsistent and confusing.
In logic's defense, the $\supset$ notation for logical implication is rare nowadays; it is more often notated $\to$ (Hilbert, 1922) or $\Rightarrow$ (Bourbaki, 1954) -- possibly in recognition of the potential for confusion with the subset/superset relation.
The "$\supset$" symbol for implication was originally a backwards "C" and dates back to Peano (1895). He wrote "$pCq$" for "p is a Consequence of q", and also reversed it to "$q\supset p$ for "q has p as a consequence".
According to some sources, writing "$\subset$" (first used by Schröder, 1890) for "is a subset of" replaced earlier use of "$<$" when authors felt set operations ought to be distinguished from the arithmetic notation that was first used by analogy.
Others claim that "$\subset$" dates back to J. Gergonne (1817) who used "C" for "Contained in".
(Most of the above information is from Earliest Uses of Various Mathematical Symbols by Jeff Miller).
A: 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 $B$.
A: It's just the notation, not the language itself: it's correct, though confusing, to say in this context $A\subset B$ iff $A\supset B$. I don't know the history of how $\supset$ came to be used for "implies" in logic, and have been bothered by this myself.
