Is the following argument valid?

Question

I am trying to determine if the following argument is valid. Assuming the first four statements are true, can we be assured that $\lnot s \land t$ must also be true.

Here is the argument.

$(1)\quad p \land r$

$(2)\quad \lnot r \lor t$

$(3)\quad s \to \lnot(q)$

$(4) \quad p \to q$

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$\therefore \quad \lnot(s) \land t$

Answer 1

1) $\quad p \land r$

2) $\quad\lnot r \lor t$

3) $\quad s \to \lnot(q)$

4) $\quad p \to q$

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5) $\quad p\quad$ (from premise one), simplification

6) $\quad r\quad$ (from premise one), simplification

7) $\quad\lnot (\lnot r)\quad$ (from 6, Double negation).

8) $\quad \color{blue}{t}\quad$ (from premise 2, and 7: disjunctive syllogism.)

9) $\quad q\quad $ (from 4 and 5, modus ponens)

10) $\quad\lnot (\lnot q)\quad$ (from 9, double negation.)

11) $\quad \color{blue}{\lnot s}\quad$ (from 3, 10, modus tollens.)

$\therefore\;12) \quad\color{blue}{\lnot s \land t}\quad $ (from 11, 8, conjunction introduction).

Therefore, $\lnot s \land t$ follows from the given premises. That means, if the premises are all true, so is the conclusion.

Answer 2

Using the contrapositive, observe you have

$$\begin{cases}S\to\neg Q\iff Q\to\neg S\\P\to Q\end{cases}$$

and you already get $\;\neg S\;$ .

Since you have $\;P\wedge R\;$ , from $\;\neg R\lor T\;$ it must be that $\;T\;$ . Complete the little details left and prove what you want.