An important class of these rules of reasoning are known as tautologies. A tautology is a sentence which is true no matter what the truth value of its constituent parts.
Example: The sentence 
is a tautology, where P
and Q represent arbitrary mathematical sentences. We can show that this is a
tautology from a truth table.
| P | Q |  | |
| T | T | T | T |
| T | F | T | T |
| F | T | F | T |
| F | F | F | T |
The way in which we do this is to compute the truth values for in
the third column first, and then use columns one and three to compute the
truth values in column four.
Logic Axiom 1: Every tautology is a rule of reasoning.
The following are tautologies that we commonly use. You will find these listed in the Rules of Logic that you have been given.
contrapositive
Modus ponens
Law
of Syllogism
Law of the Excluded Middle
Law
of Syllogism
Proof by Cases
