In logic, statements
Contents
Logical equivalences
Logical equivalences involving conditional statements:
-
p ⟹ q ≡ ¬ p ∨ q -
p ⟹ q ≡ ¬ q ⟹ ¬ p -
p ∨ q ≡ ¬ p ⟹ q -
p ∧ q ≡ ¬ ( p ⟹ ¬ q ) -
¬ ( p ⟹ q ) ≡ p ∧ ¬ q -
( p ⟹ q ) ∧ ( p ⟹ r ) ≡ p ⟹ ( q ∧ r ) -
( p ⟹ q ) ∨ ( p ⟹ r ) ≡ p ⟹ ( q ∨ r ) -
( p ⟹ r ) ∧ ( q ⟹ r ) ≡ ( p ∧ q ) ⟹ r -
( p ⟹ r ) ∨ ( q ⟹ r ) ≡ ( p ∧ q ) ⟹ r
Logical equivalences involving biconditionals:
-
p ⟺ q ≡ ( p ⟹ q ) ∧ ( q ⟹ p ) -
p ⟺ q ≡ ¬ p ⟺ ¬ q -
p ⟺ q ≡ ( p ∧ q ) ∨ ( ¬ p ∧ ¬ q ) -
¬ ( p ⟺ q ) ≡ p ⟺ ¬ q
Example
The following statements are logically equivalent:
- If Lisa is in France, then she is in Europe. (In symbols,
f ⟹ e .) - If Lisa is not in Europe, then she is not in France. (In symbols,
¬ e ⟹ ¬ f .)
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 France 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) logically equivalent.)
Relation to material equivalence
Logical equivalence is different from material equivalence. The material equivalence of
The claim that two formulas are logically equivalent is a statement in the metalanguage, expressing a relationship between two statements
There is a close relationship between material equivalence and logical equivalence. Formulas