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

zoli
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    Why $\overline{A \cap \overline B}$ rather than $\overline A \cup B$ – Taemyr Jan 16 '15 at 10:07
  • 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. – zoli Jan 16 '15 at 15:38

3 Answers3

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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).

hmakholm left over Monica
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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 $B$.

Marko
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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.

Kevin Carlson
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    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?) – zoli Jan 15 '15 at 23:01
  • Too many f 's up there! A implies B (in logic) does not imply that A is contained in B (as sets). – zoli Jan 15 '15 at 23:11
  • 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? – Kevin Carlson Jan 15 '15 at 23:19
  • @zoli you might switch your acceptance to Henning's answer. – Kevin Carlson Jan 15 '15 at 23:28
  • No, thank you. I appreciate that you paid attention to my question. I accepted the agreement part :) But up voted I the other answer. – zoli Jan 15 '15 at 23:40
  • Regarding your remark: "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." If A and B are sets rep'ing events then an elementary event may not belong to their union, and yet the implication is TRUE. – zoli Jan 15 '15 at 23:50
  • I don't understand what you mean, but no worries. – Kevin Carlson Jan 16 '15 at 04:42
  • Think of the truth value of the set A as a predicate A(x) which isTRUE if x is in A and is false otherwise. – zoli Jan 16 '15 at 06:47