Answer
This question is divided in two parts. In each part, we will first analyze each statement in argument and see whether it uses any rules of inference. If it uses valid rules of inference to reach its conclusion then it is a valid argument, if it does not uses valid rules of inference to reach its conclusion then it's a fallacy and argument is invalid. We will also describe which fallacy occurs if an argument is invalid.
Work Step by Step
a.) First we will write argument in their respective argument form involving prepositional variables.
$p(x)$ : $x$ is a positive real number
$q(x)$ : $x^2$ is a positive real number
Given premises:
$\forall x(p(x) \rightarrow q(x))$
$p(a) \rightarrow q(a)$ {By universal instantiation}
Given conclusion:
$q(a) \rightarrow p(a)$
This is an invalid argument. According to truth table of conditional statement $p(a) \rightarrow q(a)$, If $q(a)$ is True that does not necessarily implies $p(a)$ is True, $p(a)$ can be True and it can be False. This is the $Fallacy \hspace {2mm}of \hspace {2mm} affirming \hspace {2mm}the \hspace {2mm}Conclusion$ and hence Invalid.
b.)First we will write argument in their respective argument form involving prepositional variables.
$p(x)$ : $x^2 \neq 0$
$q(x)$ : $x\neq0$
Given premises:
$\forall x(p(x) \rightarrow q(x))$
(Where domain is the set of all real numbers)
Given conclusion:
$p(a) \rightarrow q(a)$, (a is real number)
This is a valid argument, because conclusion can be reached through the universal instantiation of the premise.
