I understand there are two ways to determine validity of an argument. The first way is to construct a truth table and if the statement consisting of the premises combined together implying the conclusion is a tautology, then it is valid.

The other method is to use the fact that it no valid arguments have all premises to be true and the conclusion to be false. So we could assume that the conclusion is false, and work are way through the premises, determining their truth. This is the "short cut" method.

With regards to this question, how could you use the short cut method?

$P \lor Q$

$(P \to \sim R) \land [Q\to (S \,\lor \sim S)]$

$(S \to \sim R) \land [\sim S \to (T \lor U)]$

$(T \to \sim R) \land (U \to \sim W)$

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$\sim R \,\lor \sim W$

So we would assume $\sim R \,\lor \sim W$ is false. Where would we take it from here.