Propositional Calculus stated in simplistic terms is symbolic logic without the quantifiers "For All" and "There Exists".

In order to prove most equivalences in symbolic logic one must usually employ the fundamental equivalences as well as derived rules such as DeMorgan’s Laws. These are stated in the text.

Creating proofs in symbolic logic often seems almost "magical" to students beginning to study the field. A common question is "How did you know to apply that rule at this point?". Generally, the answer is that you do not know for certain what rules to apply beforehand. Oftentimes there are many rules that can be applied at a particular point in a proof. A clever practitioner of the art with an eye toward the ultimate goal (what we are trying to prove) will be better able to apply the correct rule than a novice will. However, often determining the correct rule to apply is just a matter of trial and error. A reasonable approach is to apply a rule that looks reasonable and follow that line of reasoning until you reach a dead end or you begin to generate expressions which look like they are leading nowhere. At that point backtrack to where you could have applied a different rule and try that rule instead of the one you tried originally. Note that it DOES help to keep in mind at all times what you are trying to prove. In the end you will have a proof (usually a very simple one for the problems in this course) and others will look at it and ask "How did you know to apply that rule at this point?"

Excellent examples of formal proofs using logical equivalences are given in the text.

A counterexample is a case that disproves a statement.

You may prove a statement false by finding one case for which it is false.

Prove that p \/ q Þ p is a tautology or find a counterexample.

The statement "p \/ q Þ p" is not a tautology. Let p = F and q = T, then p \/ q = T. This yields T Þ F which is a false implication.

Quite often we do not know if what we are asked to prove is actually true. In such a case you should always try to disprove the statement before you try to prove it. There are, at least, two reasons for this

1. Counterexamples are often easy to find.
2. Attempting to find a counterexample for a statement that is actually true often gives one insight into why the statement is actually true.

In attempting to find a counterexample, always look first for a very simple one.

Counterexamples are nearly always very straightforward cases.

We may often apply logical reasoning to construct a counter example.

For example, the statement p \/ q Þ p is an implication.

The only case for which an implication is false is T Þ F.

Therefore one may reason that if a counterexample exists for p \/ q Þ p then p \/ q must be True and p must be False

If p \/ q is True and p is False then q must be True. And this case is, in fact, a counterexample.