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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 <math>p \equiv q</math>, Epq, or <math>p \Leftrightarrow q</math>. 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.
Contents
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：
- 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, <math>f \rightarrow e</math>.)
- If Lisa is not in Europe, then she is not in France. (In symbols, <math>\neg e \rightarrow \neg f</math>.)
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
References
- Elliot Mendelson, Introduction to Mathematical Logic, second edition, 1979.