Argument Valid or Not

Question

Premise $1$ : $\neg p \wedge q$

Premise $2$ : $\neg p \implies \neg r$

Premise $3$ : $\neg r \implies s$

Premise $4$ : $s \implies t$

Conclusion : $t$

How to check this argument is valid or not?

Thanks

Answer 1

Consider the premisses written in argument form as follows.

$$\begin{align} (1) \quad & \neg p \wedge q \\ (2) \quad & \neg p \implies \neg r \\ (3) \quad & \neg r \implies s \\ (4) \quad & s \implies t\\ \end{align}$$

Now we will use some Rules of Inference to deduce the conclusion from the premisses. And we do it as follows.

$$\begin{align} & (5) \quad \neg p & \text{($(1),$ Simplification)}\\ & (6) \quad \neg p \implies s & \text{($(2),(3),$ Hypothetical & Syllogism)}\\ & (7) \quad \neg p \implies t & \text{($(6),(4),$ Hypothetical Syllogism)}\\ \therefore \quad & (8) \quad t & \text{($(7),(5),$ Modus Ponens)}\\ \end{align}$$

We have just prove that

$$\big( (\neg p \wedge q) \wedge (\neg p \implies \neg r) \wedge (\neg r \implies s) \wedge (s \implies t) \big) \implies t \quad \square$$