Equivalence, in logic, refers to propositions or formulas that share the same logical meaning. Equivalent propositions or formulas have the same truth value regardless of the valuation of their terms. Equivalent formulas are either both true, or both false, and they imply each other.
If the formulas α and β are equivalent, we formally indicate α ≡ β.
P → Q ≡ ¬Q → ¬P
We can see how these contrapositive statements are equivalent by an English example:
If there’s money in my wallet, then I can buy a movie ticket.
If I can’t buy a movie ticket, then there’s no money in my wallet.
If any two formulas imply each other, they are equivalent to each other. The symbol ⇔ is sometimes used instead of ≡ to indicate equivalence, especially in systems where ≡ indicates a biconditional operator.