Establishing the validity of an argument. I've been trying to determine the validity of a particular argument for some time now and I've had no luck in figuring it out. The argument in question goes as follows:
\begin{align}
& p \wedge q \\
& p \rightarrow(r\wedge q) \\
& r \rightarrow(s \vee t) \\
& \sim s \\
\hline  & \therefore t
\end{align}
Perhaps I'm misunderstanding the rules of inference but I don't see how this is a valid argument.
**Edit
I've started to use a step-by-step approach to determining the validity and have come up with the following:
\begin{align}
1. & & & p\rightarrow (r\wedge 1) \\
2. & & & p \wedge q \\
3. & & & (r\wedge q)\wedge q \\
4. & & & r \rightarrow (s \vee t)
\end{align}
And thats as far as I've gotten... Where should I go from here?
 A: The argument is valid. You can see it informally as follows. If $t$ were false, then both $s$ and $t$ would be false. Since $r\to(s\lor t)$, this means that $r$ must be false, and since $p\to(r\land q)$, this in turn means that $p$ must be false. But then $p\land q$ is false, contrary to the first assumption.
If all else fails, you can work it out by a truth table, verifying that in every line in which all four of the hypotheses are true, $t$ is true. We don’t even have to look at all $32$ possible lines: from the hypothesis $p\land q$ we know that $p$ and $q$ are true, and from the hypothesis $\neg s$ we know that $s$ is false. Thus, only $r$ and $t$ are free to vary, and we have only four lines to consider:
$$\begin{array}{ccc}
p&q&s&r&t&r\land q&p\to(r\land q)&s\lor t&r\to(s\lor t)\\ \hline
T&T&F&T&\color{brown}T&T&\color{brown}T&T&\color{brown}T\\
T&T&F&T&F&T&T&F&F\\
T&T&F&F&T&F&F&T&T\\
T&T&F&F&F&F&F&F&T
\end{array}$$
The only row in which the second and third hypotheses are both true is the first, and in it $t$ is also true.
A: In words:
By the first statement, $p$ and $q$ are both true.
By the second statement, the fact that $p$ is true forces both $r$ and $q$ to be true.
By the third statement, the fact that $r$ is true forces $s \vee t$ to be true. Since $s$ is false (by the fourth statement), $t$ must be true.
