logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...

and

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, statements

p

and

q

are said to be logically equivalent if they have the same

truth value In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values (''true'' or '' false''). Computing In some progr ...

in every

model A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models c ...

. The logical equivalence of

p

and

q

is sometimes expressed as

p \equiv q

p :: q

\textsfpq

, or

p \iff q

, depending on the notation being used. However, these symbols are also used for

material equivalence Material is a substance or mixture of substances that constitutes an object. Materials can be pure or impure, living or non-living matter. Materials can be classified on the basis of their physical and chemical properties, or on their geolog ...

, 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 \implies q \equiv \neg p \vee q

p \implies q \equiv \neg q \implies \neg p

p \vee q \equiv \neg p \implies q

p \wedge q \equiv \neg (p \implies \neg q)

\neg (p \implies q) \equiv p \wedge \neg q

(p \implies q) \wedge (p \implies r) \equiv p \implies (q \wedge r)

(p \implies q) \vee (p \implies r) \equiv p \implies (q \vee r)

(p \implies r) \wedge (q \implies r) \equiv (p \vee q) \implies r

(p \implies r) \vee (q \implies r) \equiv (p \wedge q) \implies r

Logical equivalences involving biconditionals

p \iff q \equiv (p \implies q) \wedge (q \implies p)

p \iff q \equiv \neg p \iff \neg q

p \iff q \equiv (p \wedge q) \vee (\neg p \wedge \neg q)

\neg (p \iff q) \equiv p \iff \neg q

Examples

In logic

The following statements are logically equivalent: #If Lisa is in

Denmark ) , song = ( en, "King Christian stood by the lofty mast") , song_type = National and royal anthem , image_map = EU-Denmark.svg , map_caption = , subdivision_type = Sovereign state , subdivision_name = Danish Realm, Kingdom of Denmark ...

, then she is in

Europe Europe is a large peninsula conventionally considered a continent in its own right because of its great physical size and the weight of its history and traditions. Europe is also considered a Continent#Subcontinents, subcontinent of Eurasia ...

(a statement of the form

d \implies e

). #If Lisa is not in Europe, then she is not in Denmark (a statement of the form

\neg e \implies \neg d

). Syntactically, (1) and (2) are derivable from each other via the rules of

contraposition In logic and mathematics, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as proof by contraposition. The contrapositive of a stateme ...

and

double negation In propositional logic, double negation is the theorem that states that "If a statement is true, then it is not the case that the statement is not true." This is expressed by saying that a proposition ''A'' is logically equivalent to ''not (not ...

. 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 Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this class ...

is assumed. Some

non-classical logic Non-classical logics (and sometimes alternative logics) are formal systems that differ in a significant way from standard logical systems such as propositional and predicate logic. There are several ways in which this is done, including by way of ...

s do not deem (1) and (2) to be logically equivalent.)

Relation to material equivalence

Logical equivalence is different from material equivalence. Formulas

p

and

q

are logically equivalent if and only if the statement of their material equivalence (

p \iff q

) is a tautology. The material equivalence of

p

and

q

(often written as

p \leftrightarrow q

) is itself another statement in the same object language as

p

and

q

. This statement expresses the idea "'

p

if and only if

q

'". In particular, the truth value of

p \leftrightarrow q

can change from one model to another. On the other hand, the claim that two formulas are logically equivalent is a statement in

metalanguage In logic and linguistics, a metalanguage is a language used to describe another language, often called the ''object language''. Expressions in a metalanguage are often distinguished from those in the object language by the use of italics, quot ...

, which expresses a relationship between two statements

p

and

q

. The statements are logically equivalent if, in every model, they have the same truth value.

References

{{DEFAULTSORT:Logical Equivalence Mathematical logic Metalogic Logical consequence Equivalence (mathematics)