Tautologies and Fallacies
The compound statements (or propositions) which are true for any truth value of their components are called “tautologies”. For example “p ∨ ~ p” is a tautology, p being any logical statement. This is illustrated by the truth table given below which shows only T’s in the last column.
Truth table (p ∨ ~ q)


p

~p

p∨ ~ p

T

F

T

F

T

T

The negation of a tautology is called a fallacy or a contradiction i.e. a proposition which is false for any truth value of their components is called a fallacy. For example, “p ∧ ~ p” is a fallacy, p being any logical statement. This is illustrated by the truth table given above which shows only F’s in the last column.
Truth table (p ∧ ~p)


p

~p

p∧ ~ p

T

F

F

F

T

F

Notes: A tautology is usually denoted by “t” and a fallacy by “f ”.
 p ∨ q is true iff at least one of p and q is true.
 p ∨ q is true iff exactly one of p and q is true and the other is false.
 p ∧ q is true iff both p and q are true.
 A tautology is always true.
 A fallacy is always false.