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

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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." – André Nicolas Apr 28 '12 at 3:24
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 $a, \neg b$. – Brett Frankel Apr 28 '12 at 3:26
The answers to this prior question should prove enlightening. – Bill Dubuque Apr 28 '12 at 3:58
See xkcd – Tim S. Mar 28 '14 at 17:24
Also this earlier question – Jyrki Lahtonen Oct 11 at 15:21

1 Answer 1

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: – Qiaochu Yuan Jun 29 '12 at 22:28
+1; This was a fun read. – Domagoj Pandža Jun 29 '12 at 23:18

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