Question

Determine whether this argument, taken from Kalish and Montague (KaMo64), is valid.If Superman were able and willing to prevent evil, he would do so. If Superman were unable to prevent evil, he would be impotent; if he were unwilling to prevent evil, he would be malevolent. Superman does not prevent evil. If Superman exists, he is neither impotent nor malevolent. Therefore, Superman does not exist.

Short answer

The argument is valid, Superman does not exist.

Step-by-step solution

Assign propositional variables

Proposition	Notation	Negation
Superman is able to prevent evil.	p	\(\neg p\)
Superman is willing to prevent evil.	q	\(\neg q\)
Superman prevents evil.	r	\(\neg r\)
Supeman is impotent.	s	\(\neg s\)
Superman is malevolent.	t	\(\neg t\)
Superman exists.	u	\(\neg u\)

Form Logical Connectives

(1) If Superman were able and willing to prevent evil, he would do so.	\((p \wedge q) \to r\)
(2)If Superman were unable to prevent evil, he would be impotent.	\(\neg p \to s\)
(3)If Superman were unwilling to prevent evil, he would be malevolent.	\(\neg q \to t\)
(4)Superman does not prevent evil.	\(\neg r\)
(5)If Superman exists, he is neither impotent nor malevolent.	\(u \to (\neg s \wedge \neg t)\)

Proof by contradiction

Let u be true, then from (5)

\(u \to (\neg s \wedge \neg t)\)

This implies \(\neg s\) and \(\neg t\)both are true.

Taking contrapositive of (2) and (3)

\(\neg s \to p\)and \(\neg t \to q\)

This implies propositions pand qaretrue.

Using (1), \((p \wedge q) \to r\)

\( \Rightarrow r\)is true, which contradicts (4).

Hence, the assumption was incorrect.

uis not true.

\( \Rightarrow \neg u\) is true, Superman does not exist.