Applying a rule forwards or backwards is in fact a technique of using Natural Deduction rules in Pandora. Generally, in order to apply a rule forwards in Pandora, you will click on an empty line in your proof at first, and then choose the rule to apply; where in order to apply a rule backwards, you will click on a goal in your proof at first, and then choose the rule to apply (or vice versa).

After applying a rule forwards, you will have a new derived formula in your proof. At the end of the new line can be found numbers of the lines use to derive it. Usually a rule should be applied forwards only when you know the new formula derived will be helpful in proving the existing goal(s) of your proof. Typically, elimination rules are most suitable for forwards application, although occasionally it is helpful to use an introduction rule forwards as well.

There are three types or rules suitable for applying backwards:

• The first type is illustrated by Arrow Elimination: you will be awarded an additional formula (known as the Assumption) at the top of a box and have a new formula (known as the Conculsion/goal) at the bottom of the box. Meanwhile, the old goal will be assumed to be derived from the sub proof in the box (by the assumption and the conclusion). In this case, usually the assumption will play an important part while proving the new goal in the box. The purpose of a box is to give scope to the assumption, which is only to be used within the box. Introduction rules are often automatic, in that the structure is given on applying the rule.
• The second type is illustrated by Not Elimination; after selecting the rule and goal, which should be 'bottom' and a 'not' formula of the form ¬x, you will be given the new goal 'x' from which together with the old goal allows bottom to be derived by Not elimination.
• The Or elimination rule (backwards) is a combination of both types. After selecting the rule and goal you are required to select a disjunction of the form 'x or y', which then yields two boxes, one with assumption x and the other with assumption y.

In other words, a rule should be applied backwards if you want to replace your current goal in the proof with a new goal, which might be easier to prove, and/or if you want some additional "givens"(assumptions) to help in proving the goal. Using a rule backwards sometimes is a very powerful technique, especially when you are stuck in proving a goal, when you can employ the PC rule (for Proof by contradiction).

You can find out the directions in which a rule can be applied from the manual, which can be accessed from the Help tab. In fact, most rules can be applied both forwards and backwards, although you might not use some rules in a particular diretion. Eg using And elimination backwards is possible, but not usually very useful, as the new goal would be more complex than the original goal.

"Proving backwards is a common stratege -- Instead of trying to show the goal from the data directly, you may also try to show the situation in which the goal is true, and then have the goal."