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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?


marked as duplicate by quid, Jyrki Lahtonen Oct 12 '15 at 8:08

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    $\begingroup$ In informal speech, "if $A$ then $B$" and "$A$ implies $B$" are mainly used when it is believed there is a causal connection between $A$ and $B$. The truth-functional connective $\rightarrow$ does not capture this feature of "implies." $\endgroup$ – André Nicolas Apr 28 '12 at 3:24
  • $\begingroup$ I tend to think of it this way: when you draw out a truth table, a statement is considered false statements if and only if it is incompatible with the truth values of $a$ and $b$. For example, $a \to b$ is compatible with $\neg a, (\neg) b$ and $a, b$ but not $a, \neg b$. $\endgroup$ – Brett Frankel Apr 28 '12 at 3:26
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    $\begingroup$ The answers to this prior question should prove enlightening. $\endgroup$ – Bill Dubuque Apr 28 '12 at 3:58
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    $\begingroup$ See xkcd $\endgroup$ – Tim S. Mar 28 '14 at 17:24
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    $\begingroup$ Also this earlier question $\endgroup$ – Jyrki Lahtonen Oct 11 '15 at 15:21

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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