Logical equivalence

Logical equivalences

Logical equivalences involving conditional statements：

1. p ⟹ q ≡ ¬ p ∨ q
2. p ⟹ q ≡ ¬ q ⟹ ¬ p
3. p ∨ q ≡ ¬ p ⟹ q
4. p ∧ q ≡ ¬ ( p ⟹ ¬ q )
5. ¬ ( p ⟹ q ) ≡ p ∧ ¬ q
6. ( p ⟹ q ) ∧ ( p ⟹ r ) ≡ p ⟹ ( q ∧ r )
7. ( p ⟹ q ) ∨ ( p ⟹ r ) ≡ p ⟹ ( q ∨ r )
8. ( p ⟹ r ) ∧ ( q ⟹ r ) ≡ ( p ∧ q ) ⟹ r
9. ( p ⟹ r ) ∨ ( q ⟹ r ) ≡ ( p ∧ q ) ⟹ r

Logical equivalences involving biconditionals：

1. p ⟺ q ≡ ( p ⟹ q ) ∧ ( q ⟹ p )
2. p ⟺ q ≡ ¬ p ⟺ ¬ q
3. p ⟺ q ≡ ( p ∧ q ) ∨ ( ¬ p ∧ ¬ q )
4. ¬ ( p ⟺ q ) ≡ p ⟺ ¬ q

Example

The following statements are logically equivalent:

1. If Lisa is in France, then she is in Europe. (In symbols, f ⟹ e .)
2. 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 p and q (often written p ⟺ q ) is itself another statement, call it r , in the same object language as p and q . r expresses the idea "' p if and only if q '". In particular, the truth value of p ⟺ q can change from one model to another.

The claim that two formulas are logically equivalent is a statement in the metalanguage, expressing a relationship between two statements p and q . The claim that p and q are semantically equivalent does not depend on any particular model; it says that in every possible model, p will have the same truth value as q . The claim that p and q are syntactically equivalent does not depend on models at all; it states that there is a deduction of q from p and a deduction of p from q .

There is a close relationship between material equivalence and logical equivalence. Formulas p and q are syntactically equivalent if and only if p ⟺ q is a theorem, while p and q are semantically equivalent if and only if p ⟺ q is true in every model (that is, p ⟺ q is logically valid).