Logical equivalence

In logic and mathematics, two statements are called logically equivalent when they always have the same truth value, no matter what situation you are looking at. This means that if one statement is true, the other must also be true, and if one is false, the other must be false too.

We can show that two statements are logically equivalent in different ways. Sometimes we use special symbols like p ≡ q, p :: q, E p q, or p ⟺ q to express this relationship. Each of these symbols tells us that the statements p and q mean the same thing in logic, even if they look different when written out.

It is important to know that logical equivalence is not the same as another idea called material equivalence, even though they are related. Understanding logical equivalence helps us see when different statements are really saying the same thing, which is very useful in solving problems and thinking clearly.

Logical equivalences

In logic, many common equivalences exist and are often listed as laws or properties. These show how different statements can mean the same thing even if they look different.

Logical equivalences involving conditional statements

Some key relationships include:

If p leads to q, it is the same as saying if q is not true, then p is not true.
p or q is the same as saying if p is not true, then q must be true.
p and q together is the same as saying it is not true that p leads to not q.
Saying it is not true that p leads to q is the same as p being true and q being false.

Logical equivalences involving biconditionals

For statements that must both be true or both be false, we have:

p if and only if q is the same as p leads to q and q leads to p.
p if and only if q is also the same as not p if and only if not q.
p if and only if q can be written as either p and q are both true, or both are false.
It is not true that p if and only if q when not p if and only if q.
It is not true that p if and only if q when p if and only if not q.
It is not true that p if and only if q can also be shown using a special symbol called XOR, which means one is true but not both.

Main article: XOR

Equivalence	Name
p ∧ ⊤ ≡ p {\displaystyle p\wedge \top \equiv p} p ∨ ⊥ ≡ p {\displaystyle p\vee \bot \equiv p}	Identity laws
p ∨ ⊤ ≡ ⊤ {\displaystyle p\vee \top \equiv \top } p ∧ ⊥ ≡ ⊥ {\displaystyle p\wedge \bot \equiv \bot }	Domination laws
p ∨ p ≡ p {\displaystyle p\vee p\equiv p} p ∧ p ≡ p {\displaystyle p\wedge p\equiv p}	Idempotent or tautology laws
¬ ( ¬ p ) ≡ p {\displaystyle \neg (\neg p)\equiv p}	Double negation law
p ∨ q ≡ q ∨ p {\displaystyle p\vee q\equiv q\vee p} p ∧ q ≡ q ∧ p {\displaystyle p\wedge q\equiv q\wedge p}	Commutative laws
( p ∨ q ) ∨ r ≡ p ∨ ( q ∨ r ) {\displaystyle (p\vee q)\vee r\equiv p\vee (q\vee r)} ( p ∧ q ) ∧ r ≡ p ∧ ( q ∧ r ) {\displaystyle (p\wedge q)\wedge r\equiv p\wedge (q\wedge r)}	Associative laws
p ∨ ( q ∧ r ) ≡ ( p ∨ q ) ∧ ( p ∨ r ) {\displaystyle p\vee (q\wedge r)\equiv (p\vee q)\wedge (p\vee r)} p ∧ ( q ∨ r ) ≡ ( p ∧ q ) ∨ ( p ∧ r ) {\displaystyle p\wedge (q\vee r)\equiv (p\wedge q)\vee (p\wedge r)}	Distributive laws
¬ ( p ∧ q ) ≡ ¬ p ∨ ¬ q {\displaystyle \neg (p\wedge q)\equiv \neg p\vee \neg q} ¬ ( p ∨ q ) ≡ ¬ p ∧ ¬ q {\displaystyle \neg (p\vee q)\equiv \neg p\wedge \neg q}	De Morgan's laws
p ∨ ( p ∧ q ) ≡ p {\displaystyle p\vee (p\wedge q)\equiv p} p ∧ ( p ∨ q ) ≡ p {\displaystyle p\wedge (p\vee q)\equiv p}	Absorption laws
p ∨ ¬ p ≡ ⊤ {\displaystyle p\vee \neg p\equiv \top } p ∧ ¬ p ≡ ⊥ {\displaystyle p\wedge \neg p\equiv \bot }	Negation laws

Examples

In logic

Here are two statements that mean the same thing:

If Lisa is in Denmark, then she is in Europe.
If Lisa is not in Europe, then she is not in Denmark.

These two statements are connected by special logic rules. They are true in the same situations — either Lisa is not in Denmark, or she is in Europe. This shows how different ways of speaking can still carry the same meaning.

Relation to material equivalence

Logical equivalence is different from material equivalence. Two statements are logically equivalent if they always have the same truth value, no matter what situation you look at. Material equivalence is a statement that says, "p if and only if q," but this statement's truth can change depending on the situation.

Logical equivalence, however, is a way to describe a relationship between two statements. We say they are logically equivalent if they always have the same truth value in every possible situation.