Understanding the use of existential quantifier with conditional statement

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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?

• first-order-logic
• predicate-logic
• logic-translation

$\begingroup$ "Some Men are Happy" is the same as "There is someone that is a Man and is Happy". $\endgroup$

$\begingroup$ Right, that would be a translation for $\exists x, M(x) \and H(x)$. But if you were to translate $\exists x, M(x) \implies H(x)$ how would you do so? $\endgroup$

$\begingroup$ @M.Shin It would translate as "There is something such that in case it is a man then it is happy", which is different from saying that "There is something such that it is a man and it is happy". $\endgroup$

2 Answers 2

$\exists x. M(x)\implies H(x)$ 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.