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I am just being introduced to logic in mathematics. Let $P$ be "He is an honest person." and $Q$ be "He can be the President.". My lecturer was saying that "only if" means $P\rightarrow Q$, while "if" means $Q\rightarrow P$. I simply cannot wrap my head around this.

So let us consider the following two sentences.

  1. One can be the president only if one is honest.
  2. One can be the president if one is honest.

How are the two statements any different in meaning?

However, according to the definitions, the first statement is of the form $P\rightarrow Q$, while the second is of the form $Q\rightarrow P$ and I know that $P\rightarrow Q$ and $Q\rightarrow P$ clearly have different meanings.

Any explanations regarding the difference between both terms and how/when to use which will be greatly appreciated!

Ethan Mark
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    Talk about maths classes having contrived and unrealistic examples... –  Jan 25 '20 at 09:27

3 Answers3

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I'll use a different example, just because I find it easier to explain:

  1. "I wear a jacket if it is cold"

  2. "I wear a jacket only if it is cold"

In the first case, there is nothing stopping you from wearing a jacket when it is not cold; the "if" does not preclude your wearing a jacket under different circumstances. However, "only if" implies that you will only wear a jacket when it is cold, and never any other time.

The "if" case can be though of as "if... then,", i.e. "If it is cold, then I will wear a jacket." Again, we can see how this does not stop you from wearing a jacket during other times.

The second statement also allows the following: "If it is not cold, then I will not wear a jacket." We've said that the first statement does not preclude such cases, but the contrapositive of the second does.

Hopefully that helps clear things up.

scoopfaze
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Let me rephrase the examples you gave:

  1. One can be the president only if one is honest. $P\implies Q$

    Is the same as:

    If one can be the president then one is honest. (And therefore: If one is president one has to be honest.)

  2. One can be the president if one is honest. $Q\implies P$

    Is the same as:

    If one is honest then one can be the president. (And therefore: If one is honest one has not to be the president necessarily, though he could be)

powerline
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  • Suggested rephrasing: statement 1 should be “if one is eligible to be president, one is honest”. Right now $P$ has two different meanings in (1) and (2). – SvanN Jan 25 '20 at 09:26
  • @SvanN Thanks for the correction, P truly had two different meanings. I edited a bit differently than you suggested, check it out – powerline Jan 25 '20 at 09:35
  • I think the explanation to (2) is a bit vague now; the statement $P \to Q$ is “if one is honest, one can be president”, but the second explanation boils down to $P \to \lnot Q$ (“if one is honest, one need not be president”). – SvanN Jan 25 '20 at 09:39
  • @SvanN alright, I added a few words to clarify it, check it out. – powerline Jan 25 '20 at 09:41
  • Thank you all for your comments :) They are really insightful! – Ethan Mark Jan 25 '20 at 13:40
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I think you are wrong to find no difference in meaning between your statements 1 and 2. Part of the difficulty is that "can be" is a rather slippery phrase.

Suppose $X$ was born in England and $Y$ is born in the US.

"$X$ is eligible to be president if $X$ is honest" is false, because $X$ fails the born in the US requirement.

"$Y$ is eligible to be president only if $Y$ is honest" is true, because honesty is a requirement (we hope and assume).

almagest
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