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In propositional logic, the statement "If pigs can fly, then elephants can lay eggs." validates to true because the antecedent validates to false.

In other words, given $a \rightarrow b$, if a is false, the entire statement is true. Why?

Just because the antecendent is false doesn't mean that another fact depends on it, right?

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    Also [this earlier question](http://math.stackexchange.com/q/48161/11619)2015-10-11

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There are some plausible arguments for having "if $a$ then $b$" true when $a$ is false (like suggested ex falso quodlibet). But the fact is $\rightarrow$ doesn't even try to capture the if-then relation between propositions. $a \rightarrow b$ is defined as $\neg a \vee b$, and it's obvious why that's true when $a$ is false.

The actual if-then relation can be more appropriately captured by, for example, $a \Rightarrow b$. This is not propositional logic statement (rather metalogical), it says "it's impossible for $a$ to be true when $b$ is false".

Or better yet, use modal logics with modalities of necessity (physical, metaphysical, logical etc.): $\square (a \rightarrow b)$. This is much closer to capturing if-then relation of everyday use. Interpretation is "it's (physically/metaphysically/logically/...) impossible that $a$ is, but $b$ isn't". In fact trying to formalize if-then was perhaps the main reason why alethic modal logic was invented in the first place.

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    You can read more about the modal version of implication Luka mentions here: http://en.wikipedia.org/wiki/Strict_conditional2012-06-29