Back to Basics (2) Tautologies and Contradictions

A formula is a tautology iff it is true for all possible truth values of its components. To find out if a propositional form is a tautology a truth table is a sufficient means of proof. Here is a list of five important tautological formulae:

1.) A ∨ ¬A                                                         Law of Excluded Middle
2.) ¬(¬A ) ⇔ A                                                 Complete Law of Double Negation
3.) (A ⇒ B) ⇔ (¬B ⇒ ¬A)                               Complete Law of Contraposition
4.) (A ⇒ (B ⇒ C) ⇒ [(A ⇒ B) ⇒ (A ⇒ C)]     Transitivity of Logical Implication
5.) (A ∧¬A) ⇒ B                                               A Contradiction Implies Every Consequent

A and B are logical formulae and logically equivalent iff A ⇔ B is a tautology.

A formula is a contradiction iff it is false for all possible truth values of its components. To find out if a propositional form is a contradiction again a truth table can be used. Here are a couple examples of contradictions:

1.) A ∧¬A
2.) (A ∧ B) ∧ (¬A ∧ ¬B)