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I did a logic exercise about which I have a question : considering the booleans A : "Peter is a student" and B : "Julia is a student", what formula represents the statement "Peter is a student if Julia is a student"? I answered $B\Rightarrow A$ but I was said that the statement does not mean that Julia being a student implies Peter being one, but instead that Peter cannot be a student if Julia is not one and therefore the correct answer should be $\lnot B \Rightarrow \lnot A$. However, I assumed that the verb "is" in the first clause implies that whatever happens, if Julia is a student then it is also Peter's case. I would have phrased the second formula as such : "Peter could be a student if Julia were one".

How should the statement be written?

user8171079
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    Assuming I was said means I was told (i.e. someone who is not you said this), I disagree with the other person and I agree with you. The usage "Q if P" is VERY well known (and standard) to mean "P implies Q", at least unless something very subtle or unusual is involved (such as might arise in relevance logic, although even here I don't see why "Q if P" would be translated differently). See Section 2.6 (pp. 26-30) of Ivan Niven's Numbers: Rational and Irrational. – Dave L. Renfro Dec 06 '19 at 11:37
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    My recommendation is change course. Your answer was fine. I can see no good reason to prefer the "correct answer". – almagest Dec 06 '19 at 11:37

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If you consider that "Peter is a student if Julia is a student" implies that : "If Julia is a student, then Peter is a student" , but Peter may be a student even if Julia isn't , which I think is the most correct, $B⇒A $ is the apropriate representation.

If you defend that the statement means $¬B⇒¬A\ $ then by the reciprocal (you can prove it using a truth table or using De Morgan laws) we can write $A⇒B$ and the english representation may be :

"Peter can't be a student if Julia isn't." , or "Only if Julia is a student Peter is a student", or even "If Peter is a student, then Julia is one".

If you want to convey both meanings, that "Peter is a student if and only if Julia is a student", you can even represent it as $A \Leftrightarrow B$ , obviously because if and only if is the logical "equivalent", and also mathematicly because $B⇒A$ and $A⇒B$.

But how should the statement be written ? Like Dave Renfro said, it is standard to translate "$A$ if $B$" in english into "$B⇒A$" in mathematical logic. The only thing I can think of to improve understanding is that you include a "then" in the statement, "If Julia is a student, then Peter is a student" , and that should reinforce the idea of $B⇒A$.

P.S. Your english representation of the second formula is not on point, and you commit the same "error" as the people who formulated the 2nd formula. "Peter could be a student if Julia were one" doesn't implie that Peter couldn't be a student if Julia wasn't one. It can't be represented logicly as $¬B⇒¬A\ $, or even $B⇒A\ $, because it doesn't refer to what Peter is or can't be, just what he could be. It is sort of a tautology, because we can always "assign" Truth value to it (without any more statements). A better representation would be "Peter could only be a student if Julia were one"

RicardoMM
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