Logical Equivalence in Propositional Logic

We need to understand the concept of logical equivalence to better understand the laws or propositional logic.

What is a Tautology and a Contradiction in Logic?

In logic, a tautology occurs when a compound proposition is always true, and a contradiction occurs when a compound proposition is always false.

In English, tautology is saying the same thing using different words.

Example of tautology and contradiction in logic
S ∨ ¬S is a tautology because the value is always true.
S ∧ ¬S is a contradiction because the value is always false.

What Are Logically Equivalent Compound Propositions?

When two compound propositions have the same value, they are considered logically equivalent. This symbol ≡ indicates that
Q ≡ Q is logically equivalent regardless of their inner statements’ value.

Example of logically equivalent compound propositions
R →¬R ≡¬R both sides are logically equivalent
T →¬T ≡¬T assuming R = True
T →¬F ≡¬T first negation evaluates to False
T →¬F ≡¬F second negation evaluates to False
F ≡¬F the logical proposition evaluates to false because the hypothesis is True while the conclusion is False
F ≡¬F both statements are logically equivalent