# Prove If Statement Is Valid or Invalid

Given:

Every student has an email account Maggie does not have an email account Homer is a student Using E(x): x has an email, S(x): x is a student and M to represent Maggie while H represents Homer, I came up with the following premises:

• 1: ∀x[S(x)→E(x)]
• 2: ¬E(M)
• 3: S(H)

I then have to determine if the two following conclusions are valid. The first conclusion:

Maggie is not a student I determined this not a valid conclusion because you can only reach it by using the 'denying the antecedent' fallacy.

Is my thinking correct?

Thanks

• What's deny the antecedent fallacy. If Maggie is a student she'd have an email account. But she doesn't. So it's impossible for her to be a student. Seems a valid conclusion to me. Jun 4, 2019 at 23:03

$$S(M) \to E(M)$$
This, combined with $$\neg E(M)$$ gives you $$\neg S(M)$$ by Modus Tollens
I think it is not "denying the antecedent" but we can conclude by using contraposition. We know that $$\forall x[S(x) \implies E(x)]$$ and we are given $$\lnot E(M)$$ and implication of $$\lnot E(M)$$ is given as $$\lnot S(M)$$ (not the other way around), therefore conclusion is $$\lnot E(M) \implies \lnot S(M)$$. From the first premise, we know that $$S(M) \implies E(M)$$ is valid. Therefore, contrapositive $$\lnot E(M) \implies \lnot S(M)$$ should also be valid.