In logic and mathematics, statements p {\displaystyle p} and q {\displaystyle q} are said to be logically equivalent if they have the same truth value in every model. The logical equivalence of p {\displaystyle p} and q {\displaystyle q} is sometimes expressed as p ≡ q {\displaystyle p\equiv q} , p :: q {\displaystyle p::q} , E p q {\displaystyle {\textsf {E}}pq} , or p ⟺ q {\displaystyle p\iff q} , depending on the notation being used. However, these symbols are also used for material equivalence, so proper interpretation would depend on the context. Logical equivalence is different from material equivalence, although the two concepts are intrinsically related.
Logical equivalences In logic, many common logical equivalences exist and are often listed as laws or properties. The following tables illustrate some of these.
General logical equivalences
Logical equivalences involving conditional statements
p → q ≡ ¬ p ∨ q {\displaystyle p\rightarrow q\equiv \neg p\vee q}
p → q ≡ ¬ q → ¬ p {\displaystyle p\rightarrow q\equiv \neg q\rightarrow \neg p}
p ∨ q ≡ ¬ p → q {\displaystyle p\vee q\equiv \neg p\rightarrow q}
p ∧ q ≡ ¬ ( p → ¬ q ) {\displaystyle p\wedge q\equiv \neg (p\rightarrow \neg q)}
¬ ( p → q ) ≡ p ∧ ¬ q {\displaystyle \neg (p\rightarrow q)\equiv p\wedge \neg q}
( p → q ) ∧ ( p → r ) ≡ p → ( q ∧ r ) {\displaystyle (p\rightarrow q)\wedge (p\rightarrow r)\equiv p\rightarrow (q\wedge r)}
( p → q ) ∨ ( p → r ) ≡ p → ( q ∨ r ) {\displaystyle (p\rightarrow q)\vee (p\rightarrow r)\equiv p\rightarrow (q\vee r)}
( p → r ) ∧ ( q → r ) ≡ ( p ∨ q ) → r {\displaystyle (p\rightarrow r)\wedge (q\rightarrow r)\equiv (p\vee q)\rightarrow r}
( p → r ) ∨ ( q → r ) ≡ ( p ∧ q ) → r {\displaystyle (p\rightarrow r)\vee (q\rightarrow r)\equiv (p\wedge q)\rightarrow r}
Logical equivalences involving biconditionals
p ↔ q ≡ ( p → q ) ∧ ( q → p ) {\displaystyle p\leftrightarrow q\equiv (p\rightarrow q)\wedge (q\rightarrow p)}
p ↔ q ≡ ¬ p ↔ ¬ q {\displaystyle p\leftrightarrow q\equiv \neg p\leftrightarrow \neg q}
p ↔ q ≡ ( p ∧ q ) ∨ ( ¬ p ∧ ¬ q ) {\displaystyle p\leftrightarrow q\equiv (p\wedge q)\vee (\neg p\wedge \neg q)}
¬ ( p ↔ q ) ≡ ¬ p ↔ q {\displaystyle \neg (p\leftrightarrow q)\equiv \neg p\leftrightarrow q}
¬ ( p ↔ q ) ≡ p ↔ ¬ q {\displaystyle \neg (p\leftrightarrow q)\equiv p\leftrightarrow \neg q}
¬ ( p ↔ q ) ≡ p ⊕ q {\displaystyle \neg (p\leftrightarrow q)\equiv p\oplus q}
Where ⊕ {\displaystyle \oplus } represents XOR.
Examples
In logic The following statements are logically equivalent:
If Lisa is in Denmark, then she is in Europe (a statement of the form d → e {\displaystyle d\rightarrow e} ). If Lisa is not in Europe, then she is not in Denmark (a statement of the form ¬ e → ¬ d {\displaystyle \neg e\rightarrow \neg d} ). Syntactically, (1) and (2) are derivable from each other via the rules of contraposition and double negation. Semantically, (1) and (2) are true in exactly the same models (interpretations, valuations); namely, those in which either Lisa is in Denmark is false or Lisa is in Europe is true. (Note that in this example, classical logic is assumed. Some non-classical logics do not deem (1) and (2) to be logically equivalent.)
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