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Tautology

The word "tautology" was first used by the ancient people of Greeks. In the year of 1930s, there was formation of the propositional logic's semantics in the term of true assignments, which was developed. So, the tautology is used to applied in the propositional formula, which may be true or false regardless of false or truth of their given propositional variables. In boolean algebra, a tautology is a formula that is always true in each and every possible condition.

If a formula is true in one interpretation, then it is called as satisfiable. So, we can say that tautology is a formula, the negation of that will be unsatisfiable. The definition of Tautology has big meaning in the concepts in propositional logic, where we can define tautology as a propositional formula which is true in each and every possible Boolean valuation. The logic of propositions begins with propositional or propositional variables, atomic units which indicate the propositions.

 

Verifying Tautologies

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For example, consider the formula

((A?B)`->` C)`harr` (A`->` (B`|->` C))

For the given expression, we are going to draw a table which is known as truth table.

There are three variable a, b and c. So, there will be 8 possible valuations for them. So, we will put them in first three columns and then, we have to put the subformula of the given main formula in the remaining columns.

ABCA?B(A?B)`|->` C(B`|->` C)A`->` (B`->` C)((A?B)`->` C)`harr` (A`|->` (B`|->` C))
TTTTTTTT
TTFTFFFT
TFTFTTTT
TFFFTTTT
FTTFTTTT
FTFFTFTT
FFTFTTTT
FFFFTTTT

Examples of Tautology

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Given below are some of the examples of tautology.

Example 1: Is ~b`->` b a tautological statement?

b ~b ~b`->` b
T F T
F T F


Solution: No; the truth values of ~b`->` b are {T, F}.

Example2 : Is [(p`->` q) `^^` p]`->` p a tautological statement?

p q p`->` q (p`->` q)`^^` p [(p`->` q)`^^` p]`->` p
T T T T T
T F F F T
F T T F T
F F T F T


Solution: Yes; the truth values of [(p`->` q)`^^` p]`->` p are {T, T, T, T}.

Example: 3 Is (r`->` s)`harr` (s`->` r) a tautological statement?

r s r`->` s s`->` r (r`->` s)`harr` (s`->` r)
T T T T T
T F F T F
F T T F F
F F T T T

Solution: No; the truth values of (r`->` s)`harr` (s`->` r) are {T, F, F, T}.

More topics in Tautology
Arguments and Validity
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