Answer
$\forall x$, if $x$ is a good car, then $x$ is not cheap.
a. valid by universal modus ponens
b. invalid by converse error
c. valid by universal modus tollens
d. invalid by inverse error
Work Step by Step
For all of the cases, P(x) is: x is a good car. Q(x) is: x is not cheap.
a. Universal modus ponens: $\forall x, $ if $P(x)$ then $Q(x)$.
$P(a)$ for a particular $a$.
$\therefore Q(a)$.
b. Converse Error: $\forall x$ if $P(x)$ then $Q(x)$.
$Q(a)$ for a particular $a$.
$\therefore P(a)$. (invalid conclusion)
c. Universal modus tollens: $\forall x, $ if $P(x)$ then $Q(x)$.
~$Q(a)$ for a particular $a$.
~$\therefore P(a)$.
d. Inverse Error: $\forall x$ if $P(x)$ then $Q(x)$.
~$P(a)$ for a particular $a$.
$\therefore$ ~$Q(a)$. (invalid conclusion)
