Explanation of $A \to B = \neg A \lor B$

Asked Nov 13 '15 at 21:21

Active Nov 13 '15 at 21:27

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Can anybody explain intuitively why the logical implication $A \to B$ is equivalent to $$ \neg A \lor B$$

If e.g. $A$ is "It rains." and $B$ is "The street is wet.", then obviously $A$ implies $B$. Why is this the same as "It doesn't rain or the street is wet."?

edited Nov 13 '15 at 21:27

hmakholm left over Monica

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asked Nov 13 '15 at 21:21

Breaking M

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Also: In classical logic, why is $p\Rightarrow q$ true if $p$ is false and $q$ is true? – hmakholm left over Monica Nov 13 '15 at 21:26
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It's a bit easier if you see it as $\lnot(A\land \lnot B)$. In that case, it means "It is not (never) the case that it rains and the street is not wet". – Arthur Nov 13 '15 at 21:26
It either doesn't rain or it does. If it rains, the street is wet. Therefore, it either doesn't rain or the street is wet. – John Douma Nov 13 '15 at 21:51

Explanation of $A \to B = \neg A \lor B$

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