First, we need to say what reductio ad absurdum is. It is a argumentative strategy to attack a thesis, hypothesis, suggestion, etc. In other words, you are trying to show that some statement is false. You do this by proving the statement in question has a logical consequence that is absurd (i.e., obviously false). Since contradictions are obviously false, they will serve the purpose nicely. The stages for the reductio ad absurdum strategy are:

Step 1: Given an hypothesis (call it H) you wish to show false.

Step 2: Deduce, using valid reasoning, a logical implication (call it I) from H.

Step 3: Point out that I is absurd (i.e., obviously false, preferably a contradiction).

Step 4: Infer that H is false.

Step 2 establishes the conditional 'if H, then I.' So, steps 2, 3, and 4 are just modus tollens, which is why reductio ad absurdum is a valid argument (assuming the argument used in step 2 is also valid).

When we use the short cut method, we first assume the argument is invalid, which is the hypothesis H, and then fill in truth values for statement letters that this assumption forces. If we are forced into a contradiction, we use reductio ad absurdum to infer that H is false (i.e, the argument is valid). Otherwise, we will (eventually) find a row of the argument's truth table that shows it to be invalid.

The steps of the short cut method are:

1. Assume the argument is invalid (i.e, there is at least one row of its truth table that has each premise true and the conclusion false).

2. Put a T under the main connective of each premise (or the premise itself if it is a statement letter) and an F under the main connective for the conclusion (or the conclusion itself if it is a statement letter).

3. Fill in any further truth values that are forced by the truth table definitions of the logical connectives.

4. If you are lucky, step 3 will force you into a contradiction. In which case, invoke reductio ad absurdum, declare the argument valid, and quit.

5. Another way you can get lucky is that step 3 forces you to assign a truth value to every statement letter in the argument, you fill in the appropriate truth values for all the connectives, and there are no contradictions. In this case, you have found a row of the argument's truth table where the premises are true and the conclusion false. So, declare the argument invalid and quit.

6. If you are not lucky, you will "run out of gas," i.e., you are not forced to do any more truth value assignments and there are still statement letters to which you have not given truth values. In this event, you choose one of these statement letters (any one) and give it a truth value (either one). This is an extra assumption that is equivalent to restricting your search through the truth table to one half of the rows you were looking at previous to making the extra assumption.

7. Make a note of the extra assumption, give it a number, and fill in the truth value you chose for each occurrence of the statement letter you chose using a numerical subscript to keep track of what this extra assumption forced you to do.

8. Fill in any further truth value assignments the extra assumption forces, again using numerical subscripts to keep track.

9. If the extra assumption forces you into a contradiction, you do not know that the argument is valid, but only that a row showing invalidity (if there is one) is not in the one half rows you are currently searching. In this case, you backtrack to the most recent extra assumption, switch to the other truth value, and reproduce all the truth values you have (on another line) using the other truth value (keeping the same numerical subscript for it). This is equivalent to looking in the other one half rows.

10. If you keep getting contradictions, you backtrack until you are working with the first extra assumption. If you still get a contradiction, you have shown that none of the rows show invalidity, so declare validity and quit.

11. If the extra assumption forces you to assign truth values to all the remaining statement letters, you fill in the truth values for the logical connectives, and there are no contradictions, you will have found a row that shows invalidity. Declare invalidity and quit.

12. If you "run out of gas" again, make another extra assumption and return to step 7.

If you follow this procedure carefully, keeping notes on which extra assumptions you make, the sequence in which you make them, the truth value assignments that depend on them, and backtracking to the most recent extra assumption, you are guaranteed to either find a row of the argument's truth table that shows invalidity or to show that no such row exists. In practice, the procedure is usually pretty straightforward, only occasionally forcing you to backtrack and even then you are typically dealing with just one or two extra assumptions.

When you find invalidity, you should write out the truth value assignments to the statements letters corresponding to the row you found that shows invalidity.