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?


  • $\begingroup$ 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. $\endgroup$
    – fleablood
    Jun 4, 2019 at 23:03

2 Answers 2


No. It would be 'denying the consequent' (in order to get the denial of the antecedent) .. better known as Modus Tollens... which is perfectly valid.

To be precise, from the universal you can instantitate Maggie:

$S(M) \to E(M)$

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.

If we were given "Maggie is not a student" and conclusion was "Maggie does not have an e-mail account", then it would be denying the antecedent.


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