18
$\begingroup$

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?

  • 7
    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."2012-04-28
  • 0
    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 $a, \neg b$.2012-04-28
  • 0
    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.2012-04-28
  • 1
    The answers to this [prior question](http://math.stackexchange.com/q/30437/242) should prove enlightening.2012-04-28
  • 0
    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.2012-04-28
  • 0
    Have a look at [this](http://math.stackexchange.com/q/138078/24567).2012-05-01
  • 1
    See [xkcd](http://www.explainxkcd.com/wiki/index.php/704:_Principle_of_Explosion)2014-03-28
  • 1
    Also [this earlier question](http://math.stackexchange.com/q/48161/11619)2015-10-11

1 Answers 1