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I understand that:

$True \implies True$, is true.

$True \implies False$, is False.

But why is it that

$False \implies True$, is True.

and

$False \implies False$, is True.

If $a$ is false I don't understand how we can say $a \implies b$ is true.

pancini
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Sean Hill
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    Have a look at these oldies: http://math.stackexchange.com/questions/70736/in-classical-logic-why-is-p-rightarrow-q-true-if-p-is-false-and-q-is-tr http://math.stackexchange.com/questions/48161/in-classical-logic-why-is-p-rightarrow-q-true-if-both-p-and-q-are-false – colormegone Jan 12 '16 at 04:23
  • If $a$ is false and $a$ is true, then we can prove anything, so $b$ is true. So if $a$ is false, then $a\implies\text{ anything}$ – Thomas Andrews Jan 12 '16 at 04:31
  • @ThomasAndrews not sure I understand. Why are either of those the case when $a$ is false? – Sean Hill Jan 12 '16 at 04:34
  • This is a vacuous truth. – Henricus V. Jan 12 '16 at 04:35
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    See http://math.stackexchange.com/questions/1551320/understanding-vacuously-true-truth-table/1551525#1551525 – Dan Christensen Jan 12 '16 at 04:35
  • There are several questions practically identical to this, which are generally considered duplicates of both http://math.stackexchange.com/questions/70736/in-classical-logic-why-is-p-rightarrow-q-true-if-p-is-false-and-q-is-tr and http://math.stackexchange.com/questions/48161/in-classical-logic-why-is-p-rightarrow-q-true-if-both-p-and-q-are-false (of course the "duplicate of" pointer can only point to one of those questions; but they are also cross-referenced to each other). – David K Jan 12 '16 at 04:57

2 Answers2

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I was taught to think of implications like contracts. For example, say I told you "if you wash my car, I'll pay you ten dollars." Then the only way I could end up lying - the only way this statement could be false - is if I break my contract to you, and I stiff you after you wash my car. If you don't wash my car, I never lied to you, whether or not I end up paying you, and the contract always holds if the first part of the conditional is false.

Sam Birns
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  • Makes sense, but I still struggle with the idea that just because I never wash the car, it means the promise was kept. It wasn't broken, sure, because I never washed the car. But since the deal was never enacted, how can we say anything about the status of whether or not the promise is kept one way or the other? – Sean Hill Jan 12 '16 at 04:48
  • I guess another way of thinking about it is to remember that things here are either true or false and nothing in between. So if I didn't break my promise to you, it must be the case that I actually kept my promise to you and my statement is true. – Sam Birns Jan 12 '16 at 05:01
  • In the structure A => B, A is the antecedent and B is the consequent. According to wikipedia https://en.wikipedia.org/wiki/Vacuous_truth, a material conditional is defined to be true when the antecedent is false (regardless of whether the conclusion is true). If you'd like, you can take it as a definition. – Vizuna Jan 12 '16 at 05:01
  • Just to clarify further, A => B is known as a proposition, and in logic, propositions are statements that are either true or false. In the case where A is false, we still need to assign a truth value to the proposition A => B, and logicians have defined it to be (vacuously) true. – Vizuna Jan 12 '16 at 05:05
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What do you think of the statement "If 638,544 is divisible by 14, then it is divisible by 7"?

You probably already agreed that this statement was true before realizing that it was of the form $F \Rightarrow F$.

David
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