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I'm teaching my little sister propositional logic per her request.

I was trying to explain to her why $A \implies B$ holds whenever $A$ is false, and I didn't succeed with that.

I referred her to the definition: $A \implies B$ is true if whenever $A$ true, $B$ is true. If $A$ is false, then it is "never" true. If we want to test whether the implication is true, we need to check if $B$ follows when $A$ is true, but $A$ is never true, so we need not check anything, hence the implication is true.

Is my reasoning even correct? I forgot how to do these stuff anyway.

Please help me explain this idea. It would be nice if you can give an intuitive way to think about it as well.

Barely mathematical
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    An example never hurts:"If it's raining then it's cloudy" – user222031 May 17 '15 at 23:29
  • I know this has been discussed before, but I like to think in terms of an example. Suppose I tell you, "If I go to the store, then I'll give you a candy bar." I've really only lied to you (false implication) if I go to the store but don't give you a candy bar, right? So with a false premise, I get off on a technicality :) – pjs36 May 17 '15 at 23:30
  • If I said, "If a bird is a raven, then it is black", then, surely, seeing a bird which was not a raven would not make you doubt my statement, regardless of whether it were black. – Milo Brandt May 17 '15 at 23:30
  • That question asks about when both of these are false, it doesn't handle the case where the latter is true. – Barely mathematical May 17 '15 at 23:33

6 Answers6

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Yes, your reasoning is correct. It may help to present a situation close to the student's range of interests. Ask her to say "if I steal, then I don't get a cookie". Then ask her if she stole anything, to which, presumably, she will answer "no, I did not steal". Now give her a cookie and accuse her of lying in the first place when she said "if I steal, then I don't get a cookie" and have her defend her honour. Repeat the process, but this time do not give her a cookie and accuse her again of lying. If she gets convinced that the claim "if I steal, then I don't get a cookie" is not a lie in the situation where she did not steal, then she must accept the statement is true (regardless of whether or not she has a cookie). This is precisely the situation of $A\implies B$ where $A$ is "I steal" and $B$ is "I don't get a cookie", which is true if $A$ is false.

Ittay Weiss
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Assuming your kitchen is entirely within your house you can make the true statement "When you are in the kitchen you are in your house" - clearly it is still true even if you are not in the kitchen, the statement isn't saying anything about where you are, it's all about the fact that the kitchen is in your house.

This could lead to a discussion around Venn diagrams and amusing flights of fancy about what it would be like to live in a house where the kitchen was partly inside and partly outside or even entirely outside your house ( which probably is actually the case for a lot of people.)

WW1
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  • +1 I've always tended to use reasoning along the lines of "All the instances where $p$ is true are contained within the instances where $q$ is true." – Akay May 25 '17 at 04:19
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You can think of $A \Rightarrow B$ as a promise; if you do A, then you'll get B. BUT if you don't do A, I'm not breaking my promise.

For example, the gravity promises you that if you drop and object, then it will fail.

But what if I dont drop it? Well, the gravity isn't breaking his promise. The only way that gravity will break his promise is if you drop and object and it doesn't fall.

HeMan
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When you write $A \implies B$, you mean "$A$ implies $B$". And you want to say that $A$ does not imply $B$ only if $A$ is true and $B$ isn't. Meaning that "$A \implies B$" is false only if $A$ is true and $B$ is false. If $A$ is false, we're not in this situation, so automatically $A \implies B$ is true.

Ivo Terek
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In a not very rigourous, but perhaps expressive way, I would say $A\Rightarrow B$ is true when $A$ is false because you cannot produce a counter-example, i.e. a situation in which $A$ is true and $B$ false.

Bernard
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You could try making some statements involving something that is obviously false. For example, pick up a spoon made entirely of metal, and say,

"If this spoon is made of wood, then I'm a monkey's uncle."

Or say,

"If this spoon is made of wood, then I'll eat my hat."

People say things like this sometimes specifically to emphasize that the part before "then" is false. That is, while both of these statements use the language of logical implication, $A \implies B$, what the speaker of such a sentence is really trying to communicate is that $A$ is false. After all, if I thought it were possible that the metal spoon was actually made of wood, I would not want to say the first sentence, because I do not want people to think I'm a monkey's uncle; and likewise I would not utter the second sentence, because I certainly do not want to risk having to eat my hat!

And while the mathematical logic of implication does not always work the same way as colloquial speech, in this case it does. My knowledge that the spoon is not made of wood allows me to safely say that anything at all is implied if the spoon is wooden; I know I can never be held to account for having said the $B$ part of $A \implies B$, because the $A$ part will never be true.

By the way, another example of the "monkey's uncle" argument appears in this answer to a related question.

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David K
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