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Handout 6:

Rules of Thumb for Proofs in Propositional Logic

January 28, 2004

Here are some rules of thumb based on my experience:

1. Practice

Practice again and again. Prove directly and then prove by contradiction. Write down some propositions as hypothesis, and then derive as many conclusions as possible.

2. “Populate”

Derive as many as possible simpler propositions. For example, if you have , you can break it down to and using the simplification rule. If you have , you can transform the implication to a disjunction: using the implication rule. Some of the propositions that you derive may not be useful, but in general, breaking a complex proposition down to simpler ones help you see the reasoning process better.

3. Backtrack

If you are stuck, then backtrack. Start another branch of derivation, for example.

4. Predict some end states

If you are stuck, take a look at the conclusion that you want to prove. May be work backwards from there and see may be there are some intermediate propositions that you already have proved. That way, it gives you more flexibility.

5. Proof by Contradiction

If you are stuck, then try proof by contradiction. The beauty of proof by contradiction is that it introduces the conclusion that you want to prove into the mix. But, be very careful, you can only assume the negation of the conclusion and your goal is to reach a contradiction. So, if you introduce the negation of the conclusion and you do not reach a contradiction, then your proof by contradiction fails. Proof by contradiction may take longer in terms of the number of steps, but it can be very very helpful.

6. Purge your steps

Take out redundant steps to make your proof more concise. Doing this will help you think better and prove better.

7. Understand thoroughly what all the logical equivalences and implications mean.

Understand them from the standpoint of intuition and common sense. You must do this to appreciate what each rule or law does.