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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 $\overline{A \cap \overline B}$.)

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  • $\begingroup$ Why $\overline{A \cap \overline B}$ rather than $\overline A \cup B$ $\endgroup$ – Taemyr Jan 16 '15 at 10:07
  • $\begingroup$ You are right. But I was exercising in using the upper line. The other explanation is that I conceptualise the "set implication" as the complement of the difference. I don't know why. $\endgroup$ – zoli Jan 16 '15 at 15:38
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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).

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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.

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    $\begingroup$ Yes, yes. But I, too, would like to understand the reasons behind. (Only a bad habit, so to speak a misnomer? What might be the historical explanation?) $\endgroup$ – zoli Jan 15 '15 at 23:01
  • $\begingroup$ Too many f 's up there! A implies B (in logic) does not imply that A is contained in B (as sets). $\endgroup$ – zoli Jan 15 '15 at 23:11
  • $\begingroup$ If $A$ and $B$ are events, or generally if it makes sense to ask whether $A$ is contained in $B$, then I think it does. Am I wrong? $\endgroup$ – Kevin Carlson Jan 15 '15 at 23:19
  • $\begingroup$ @zoli you might switch your acceptance to Henning's answer. $\endgroup$ – Kevin Carlson Jan 15 '15 at 23:28
  • $\begingroup$ No, thank you. I appreciate that you paid attention to my question. I accepted the agreement part :) But up voted I the other answer. $\endgroup$ – zoli Jan 15 '15 at 23:40
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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 $B$.

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