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I am currently doing the Intro to Mathematical Thinking course on Coursera.

A quiz therein, asks:

For what truth values of Φ and Ψ do we define Φ ⇏ Ψ to be T (true)?

It says the right answer is Φ is true and Ψ is false. I understand that; but I do not follow why the reverse is not right. That is, why does Φ false and Ψ true does not make the statement Φ ⇏ Ψ to be true. I am confused because if Ψ is true while Φ is false, the truth of Ψ does not follow from the truth of Φ and so it should be that Φ ⇏ Ψ.

Could someone please explain?

Screenshot from the quiz

Someone
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1 Answers1

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Presumably your text defines $\not\Rightarrow$ in such a way that $\Phi\not\Rightarrow\Psi$ has the same meaning as $\lnot(\Phi\Rightarrow\Psi)$.

Do you understand that $\Phi\Rightarrow\Psi$ is true when $\Phi$ is false and $\Psi$ is true? See In classical logic, why is $(p\Rightarrow q)$ True if $p$ is False and $q$ is True?

When $\Phi\Rightarrow\Psi$ is true, we must have that $\lnot(\Phi\Rightarrow\Psi)$ is false, not true.

David K
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