I was taking the coursera of "Introduction for Mathematical thinking". In one lecture, we are given the following truth table about:

$$\begin{array}{c|c|c} φ &ψ &φ \implies ψ\\ \hline T &T &T\\ T &F &F\\ F &T &T\\ F &F &T \end{array}$$

The first two lines are easy to understand.

But what about third and fourth lines?

Let me make two examples:

"I am a dog"(False) implies that "I am a student"(True); this is true.

"I am a dog"(False) implies that "I am an American"(False); this is true.

The above two claims are ridiculous!

How can they be true? Did I make any mistakes for these two claims?

Edit: Thanks for answering the questions! It seems all about the definition of vacuous truth.

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    $\begingroup$ The third and fourth lines are vacuously true. Try a search too: math.stackexchange.com/search?q=vacuous $\endgroup$ – Alexis Olson Mar 17 '17 at 5:28
  • $\begingroup$ @AlexisOlson many thanks I am having a look. $\endgroup$ – zlin Mar 17 '17 at 5:30
  • $\begingroup$ Really, you find the first line easy to understand? How about: 'if snow is white then grass is green'? $\endgroup$ – Bram28 Mar 17 '17 at 12:29
  • $\begingroup$ @Bram28 This deduction should not be true but still it can be understood. $\endgroup$ – zlin Mar 17 '17 at 12:58
  • $\begingroup$ @zlin So how do you understand it? Anyway, I recommend you search for 'material conditional' or 'material implication'... Many people have asked your question before, and there are some really good answers there. $\endgroup$ – Bram28 Mar 17 '17 at 13:35

The key point is that an implication makes no claim when the antecedent is false. When you say "if I am a dog, then I am an American," how would I prove you wrong? I would have to first argue that you are a dog, and then that you are not an American. I take you at your word that you are not indeed a dog, so this is a hopeless task.

So the implication $\phi \implies \psi$ says that whenever $\phi$ is true, $\psi$ is also true. If $\phi$ is false, there is nothing to check, and the implication holds vacuously.


You made no mistake, your truth tables are correct. But the claims are not as ridiculous as you might think. Imagine the statement "If $x>1$ then $x^2>1$." You'd like this to be true, because you can prove it. But what if you set $x=-400$? The statement says "If $-400>1$ then $(-400)^2>1$". This is an example of false implies true.

Or if you plug in $x=0$: "If $0>1$ then $0^2>1$". This is an example of false implies false. The point is that with the "If-Then" structure, we only really care about what happens if the "If" part is true, because that's all the statement is concerning itself with. It doesn't worry about what happens if the "If" part is false; no matter what comes after, the statement is still true.


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