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In algebra, one lesson we took was about logic. We learned that it is a true statement or logical expression to say that if a Beijing was the capital of the US then the moon existed last night, as this is convenient with a false statement implying a true one being a true statement. Agree?

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As an example of why the convention 'false implies true is true' is useful, consider the sentence "if a given number is smaller than $10$ then it is also smaller than $100$". This is clearly a true statement. Therefore, if we specialize the statement by replacing the words 'any number' by a number, we should still consider it to be true. So let's look at some of these specialized cases.

Using the number $5$ gives the true statement "if $5$ is smaller than $10$ then it is also smaller than $100$". This is an example of 'true implies true'.

Using the number $500$, we get "if $500$ is smaller than $10$ then it is also smaller than $100$". This is also a true statement, of the form 'false implies false'.

Finally, if we use the number $50$, we get "if $50$ is smaller than $10$ then it is also smaller then $100$". This is an example of 'false implies true', and it still should be a true statement.

So the reason for the convention 'false implies true is true' is that it makes statements like $x < 10 \rightarrow x<100$ true for all values of $x$, as one would expect.

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    $\begingroup$ Very well done, this is actually the most convincing explanation I've seen for this law. And I've seen a lot... $\endgroup$ Feb 5, 2017 at 10:32
  • $\begingroup$ those who learned programming before learning logic theory will still have problem with your examples..we are used to the form if x > 10 && x < 100 {print ("number between 10 and 100")} else if x < 10 {print ("number less than 10 (and also 100)")} $\endgroup$
    – senseiwu
    Feb 22, 2021 at 14:58
  • $\begingroup$ So are these laws just conventions? As in things we humans have collectively agreed on? $\endgroup$ Mar 13, 2022 at 6:59
  • $\begingroup$ @BoredComedy Yes. The ones that are the most useful are the ones that are kept. $\endgroup$ Nov 10, 2022 at 5:32
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You want "real life", eh?

Let (P) be the statement

If the policeman sees you speeding, then you will have to pay a fine.

This is true. But it could happen you have to pay a fine because you failed to shovel the snow from your sidewalk. So you have to pay a fine even though you did not speed. But this does not mean that (P) is false.

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  • $\begingroup$ But how have you deduced that if the policeman didn't see me speeding then I will pay the fine? You don't have any knowledge about me not shoveling the snow; all what you know is that the policeman didn't see me speeding $\endgroup$
    – YoussefDir
    Dec 20, 2015 at 14:38
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    $\begingroup$ Not "the fine" (for speeding), but "a fine." $\endgroup$ Mar 27, 2018 at 19:30
  • $\begingroup$ Not sure if my understanding is correct, but because the policeman doesn't see you speeding (making p false) doesn't mean you'll pay a fine. q here (paying the fine) can also be false. If the policeman see's you speeding (false), then you will have to pay a fine (false). This is still a true statement. It is also true that if the police see's you speeding (false), then you will have to pay a fine (true), is also a true statement. p being false doesn't mean no fine. The "q" part could be either true or false, hence both forms are true. Is this valid thinking? $\endgroup$
    – user4779
    Jul 8, 2020 at 6:02
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A conditional statement $p\to q$ is false only if the hypothesis $p$ is true and the conclusion $q$ is false.

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The formula $p \to q$ is logically equivalent to $q \vee \neg p$ ($q$ or not $p$ in English). As you can see, if $p$ is false, then $\neg p$ is true and $q \vee \neg p$ is also true. Thus $p \to q$ is true.

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    $\begingroup$ This seems like a bit of begging the question. Replacing implication by disjunction is only possible after you've decided false implies true. $\endgroup$
    – ziggurism
    Dec 20, 2015 at 15:26
  • $\begingroup$ @ziggurism Replacing implication by disjunction is only possible after someone decided that false implies true, and George Boole did that about 170 years ago. $\endgroup$
    – Caleb
    Jul 23, 2018 at 19:25
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    $\begingroup$ @Caleb might as well say "it's true because it's true" $\endgroup$
    – ziggurism
    Jul 25, 2018 at 18:15
  • $\begingroup$ @ziggurism Or you could say "it's true because you were taught Boolean logic." Either your version or mine would be acceptable if the question were "why is T∨F true?" so why is it any different to rely on the definition of implication? $\endgroup$
    – Caleb
    Jul 25, 2018 at 18:25
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    $\begingroup$ @Caleb to understand why $p \to q$ is equivalent to $\not p \vee q$ you must understand the truth value of $p \to q$ when $p$ is false. So using that formula to understand why "false implies true" is true is begging the question. Or at least it's as useless as "it's true because George Boole liked it 170 years ago". It gives absolutely no intuition or motivation for this formula. It does not even try to answer the "why?" question. $\endgroup$
    – ziggurism
    Jul 25, 2018 at 18:31
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I agree. That is an impication. If you suppose a lie, you won't make a mistake whatever the result.

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