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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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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
    
Mainly, because this is the definition that turns out to be useful; the other possibility, making $a\to b$ be false when $a$ is false and $b$ is true, leads to equivalence, which is much stronger. What we want is to capture things like "If it rains, then I'll run at the gym." This says what will happen if it rains, but it doesn't tell us what will happen if it doesn't rain; I may decide to run at the gym anyway, or not. That's the situation we are trying to capture. –  Arturo Magidin Apr 28 '12 at 3:48
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The answers to this prior question should prove enlightening. –  Bill Dubuque Apr 28 '12 at 3:58
    
Shouldn't the term be "evaluate" rather than "validate"? I believe the latter means confirmation, and you can't confirm something if it's false. –  anon Apr 28 '12 at 4:00

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

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