Answer
First, we will use a letter to represent each simple statement of the argument:
$p:$ An argument is valid.
$q:$ An argument produce truth.
\[r:\] An argument is sound.
Now, the above statement can be written in form of premises and conclusion symbolically as:
$\sim p\to \sim q$ If an argument is invalid, it does not produce truth.
$\left( p\wedge \sim r \right)\to \sim q$ A valid unsound argument also does not produce truth
$\sim p\vee \left( p\wedge \sim r \right)$ Arguments are invalid, or they are valid but unsound.
$\therefore \ \sim q$ No arguments produce truth.
Rewriting the conditional statements in symbolic form:
$\left[ \left( \sim p\to \sim q \right)\wedge \left\{ \left( p\wedge \sim r \right)\to \sim q \right\}\wedge \left\{ \sim p\vee \left( p\wedge \sim r \right) \right\} \right]\to \sim q$.
Now, with all the information above, we will construct a truth table for the conditional statement,
Work Step by Step
The entries in the final column of the truth table are all true, so the conditional statement is a tautology. Thus, the given argument is valid but sound.
The argument is valid but sound.
