Logical equivalence

In logic, statements p and q are logically equivalent if they have the same logical content. This is a semantic concept; two statements are equivalent if they have the same truth value in every model (Mendelson 1979:56). The logical equivalence of p and q is sometimes expressed as $p \equiv q$, Epq, or $p \Leftrightarrow q$. However, these symbols are also used for material equivalence; the proper interpretation depends on the context. Logical equivalence is different from material equivalence, although the two concepts are closely related.

Logical equivalences

Equivalence	Name
p∧T≡p p∨F≡p	Identity laws
p∨T≡T p∧F≡F	Domination laws
p∨p≡p p∧p≡p	Idempotent laws
¬(¬p)≡p	Double negation law
p∨q≡q∨p p∧q≡q∧p	Commutative laws
(p∨q)∨r≡p∨(q∨r) (p∧q)∧r≡p∧(q∧r)	Associative laws
p∨(q∧r)≡(p∨q)∧(p∨r) p∧(q∨r)≡(p∧q)∨(p∧r)	Distributive laws
¬(p∧q)≡¬p∨¬q ¬(p∨q)≡¬p∧¬q	De Morgan's laws
p∨(p∧q)≡p p∧(p∨q)≡p	Absorption laws
p∨¬p≡T p∧¬p≡F	Negation laws

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, $f \rightarrow e$.)
2. If Lisa is not in Europe, then she is not in France. (In symbols, $\neg e \rightarrow \neg f$.)

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).

See also

• Entailment
• Equisatisfiability
• If and only if
• Logical biconditional
• Logical equality

References

• Elliot Mendelson, Introduction to Mathematical Logic, second edition, 1979.