In the full truth table method, we construct all the possible rows and then look to see whether there are any invalidating rows. In the short-cut method, we try to make an invalidating row or counterexample. The short-cut method is usually simple and easy. This is called forking , and Copi does not explain how to make it systematic. The chief purpose of this hand-out is to provide directions for forking.
Forking is easier to describe roughly than with perfect precision and particularity.
Here is a rough description. Afterwards I try to give all the precision you might ever need. If assuming invalidity doesn't give you enough information to finish making truth-value assignments to all the simple statements in the argument, then you must fork. Take one of the simple statements to which you have so far assigned no truth-value. Assume it to be true, and write down somewhere that you are doing so.