I stumbled across this solution on the forum and had trouble understanding one part of the explanation.
Angus St. (https://math.stackexchange.com/users/652222/angus-st), English to predicate logic, URL (version: 2019-03-12): https://math.stackexchange.com/q/3144648
shouldn't the translation be:
There is at least one person who is happy only if they are a man
since this is a conditional statement? or is there something I am not understanding correctly?
Best Answer
$\exists x. M(x)\implies H(x)$ reads: There exists a person such that if they are a man, they are happy.
An easier way to say this is: there is a person who is happy when they are a man.
They may still be happy when they are not a man $(F\implies T)$,
but if they are a man then they are happy $(T\implies T)$.
Likewise, when they are not happy, they are not a man $(F\implies F$),
but they are never unhappy when they are a man $\neg(T\implies F)$.
You can also do this:
$$\exists x.M(x)\implies H(x)\quad\to \quad\neg\nexists x.M(x)\implies H(x)\quad\to\quad\neg\forall x.\neg(M(x)\implies H(x))\\\equiv\neg \forall x.M(x)\land\neg H(x)$$
Not every person is unhappy when they are man.