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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.




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