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How could I understand If p then q = q is necessary for p = p is sufficient for q ???

I am so confused why are they the same.

Andes Lam
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  • If $p$ then $q$ is $p \rightarrow q$. This means that it is sufficient for $p$ to be true for $q$ to also be true. If $q$ is not true, it is impossible for $p$ to be true, that's why it is said that $q$ is necessary, it is not sufficient, since if $q$ is true, it doesn't mean that $p$ is true, it just means that it is possible for $p$ to also be true. – lightxbulb Feb 09 '19 at 01:53

2 Answers2

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They are the same because their meaning in English is the same. Each says, in a slightly different way, that $q$ must be true when $p$ is. In a math paper or text an author can use any of them, choosing which sounds better in a particular context.

Ethan Bolker
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  • But if you are comparing whether the 2 statements are the same, shouldn't comparison base on the truth table? Say in the if p then q case, even if p is False then q is True, the statement is still true. However, from the context of q is necessary for p, p can't be False unless q is False. Correct me if I am wrong in this part since I see contradiction in the truth table in this place. – Andes Lam Feb 09 '19 at 01:55
  • I think each has the same true table. $q$ is necessary for $p$ still allows for $p$ to be false when $q$ is true. – Ethan Bolker Feb 09 '19 at 02:02
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It is best demonstrated with an Euler diagram. 'If p then q' means p is a subset of q:

 -----------
|  q        |
|           |
|    ----   |
|    | p |  |
|    ----   |
------------

q is necessary for p: if q is not true then p cannot be true. (But it is not sufficient, i.e. it is possible that q is true but p is false)

On the other hand, p is sufficient for q. If p is true then q is always true.

Igor Muntyan
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