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Consider this conditional statement: If the safety net is broken, then I will fix it.

We have a rule in logic that says ( true implies false is false) . That is if the 1st part were true that the safety net was really broken , and the 2nd part were false that I'm not going to fix it, then the whole statement becomes false. This is clear till now.

We also have that: false implies true is true. That means that if the safety net was not broken, then I'm going to fix it , but that shouldn't turn the whole statement true , because if it weren't broken then certainly I am not going to fix it..

So.. any help?

  • any help? With what? Once you proved "P(x) implies Q(x)" in general, its truth value will not change depending on "x". Since you posted to MSE, compare your example to "if $n \in \mathbb{Z}$ is even, then $n+2$ is even". That's as true for $n = 42$ as it is for $n=1001$. – dxiv Dec 24 '17 at 07:06
  • Ty. I face the same problem with your example. If n is even( was considered false) then n+1 is even (considered true ) then the whole statement becomes false not true. I mean these rules are supposed to be compatible with the reality not that reality should follow them –  Dec 24 '17 at 08:40
  • If n is even ... then n+1 is even That is a provably false statement, always. ... then the whole statement becomes false not true So you changed a statement that was always true to something entirely different, and the new statement happens to be false after your change. Sorry, don't know what you find to be surprising here. – dxiv Dec 24 '17 at 08:47
  • See my answer at https://math.stackexchange.com/questions/2576180/how-does-one-know-if-a-implies-b-an-implication-is-true-without-knowing-if/2576428#2576428 where I prove that $\neg A \implies [A\implies B]$ – Dan Christensen Dec 24 '17 at 14:22

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Statements of the form false implies true are known as vacuously true. Because the antecedent is false we cannot infer anything about the consequent. Such statement are typically interpreted as discussing members of the empty set and you can say anything you like about such members because they don't exist. In your example the net is not broken, so no broken net exists and therefore we can say anything we like about it without contradiction.

CyclotomicField
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  • Ty. But if the net weren't broken that doesn't mean that it no longer exist. It is still present but not broken –  Dec 24 '17 at 08:25
  • You say "That means that if the safety net was not broken, then I'm going to fix it". What it really means is: if the safety net is not broken, then the statement "if the safety net is broken then I'm going to fix it" is true. – Michael Behrend Dec 24 '17 at 09:22
  • @ZahraaKhalife if the net is not broken, then a "broken net" does not exist. You cannot remove the condition from the statement without altering the truth value. For another example consider the statement "Sunflower seeds are safe to eat" does not mean "Seeds are safe to eat". Removing the adjective changes the meaning entirely. – CyclotomicField Dec 25 '17 at 06:56