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In
logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...

logic
and
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, statements p and q are said to be logically equivalent if they are provable from each other under a set of axioms, or have the same
truth value In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ...
in every
model In general, 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. ...
. 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 A material is a substance Substance may refer to: * Substance (Jainism), a term in Jain ontology to denote the base or owner of attributes * Chemical substance, a material with a definite chemical composition * Matter, anything that has mass and ...
, 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 Denmark ( da, Danmark, ) is a Nordic country The Nordic countries, or the Nordics, are a geographical and cultural region In geography, regions are areas that are broadly divided by physical characteristics ( physical geography), hu ...

Denmark
, then she is in
Europe Europe is a continent A continent is any of several large landmass A landmass, or land mass, is a large region In geography Geography (from Greek: , ''geographia'', literally "earth description") is a field of scienc ...

Europe
(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 Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ac ...
and
double negation Double may refer to: * Look-alike, a person who closely resembles another person * Body double, someone who substitutes for the credited actor of a character * Doppelgänger, ghostly double of a living person * Polish Enigma doubles, replicatin ...

double negation
. 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) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy, the type of philosophy most often found in the English-speaking world. Char ...
is assumed. Some
non-classical logicNon-classical logics (and sometimes alternative logics) are formal system A formal system is used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are ...
s do not deem (1) and (2) to be logically equivalent.)


In mathematics

In mathematics, two statements p and q are often said to be logically equivalent, if they are provable from each other given a set of axioms and presuppositions. For example, the statement "n is divisible by 6" can be regarded as equivalent to the statement "n is divisible by 2 and 3", since one can prove the former from the latter (and vice versa) using some knowledge from basic
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

number theory
.


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 An object language is a language A language is a structured system of communication used by humans, including speech (spoken language), gestures (Signed language, sign language) and writing. Most languages have a writing system composed of gly ...
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 Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument ...
, 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.


See also

*
Entailment Logical consequence (also entailment) is a fundamental concept Concepts are defined as abstract ideas A mental representation (or cognitive representation), in philosophy of mind Philosophy of mind is a branch of philosophy that studies th ...
* Equisatisfiability *
If and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
*
Logical biconditional In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents stat ...
*
Logical equality Logical equality is a logical operator Logic (from Greek: grc, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argument In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογ ...
*

≡
the iff symbol (U+2261 ''IDENTICAL TO'') *

∷
the ''a'' is to ''b'' as ''c'' is to ''d'' symbol (U+2237 ''PROPORTION'') *

⇔
the
double struck
double struck
biconditional (U+21D4 ''LEFT RIGHT DOUBLE ARROW'') * the bidirectional arrow (U+2194 ''LEFT RIGHT ARROW'')


References

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