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| {{about|connectives in logical systems|connectors in natural languages|discourse connective|other logical symbols|List of logic symbols}}
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| In [[logic]], a '''logical connective''' (also called a '''logical operator''') is a [[symbol (formal)|symbol]] or [[word]] used to connect two or more [[sentence (linguistics)|sentences]] (of either a [[formal language|formal]] or a [[natural language|natural]] language) in a [[syntax (logic)|grammatically valid]] way, such that the sense of the compound sentence produced depends only on the original sentences.
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| The most common logical connectives are '''binary connectives''' (also called '''dyadic connectives''') which join two sentences which can be thought of as the function's [[operand]]s. Also commonly, [[negation]] is considered to be a '''unary connective'''.
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| Logical connectives along with [[quantifier]]s are the two main types of [[logical constant]]s used in [[formal system]]s such as [[propositional logic]] and [[predicate logic]]. Semantics of a logical connective is often, but not always, presented as a [[truth function]].
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| ==In language==
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| ===Natural language===
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| In the grammar of natural languages two sentences may be joined by a [[grammatical conjunction]] to form a ''grammatically'' [[compound sentence (linguistics)|compound sentence]]. Some but not all such grammatical conjunctions are [[truth function]]s. For example, consider the following sentences:
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| :A: Jack went up the hill.
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| :B: Jill went up the hill.
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| :C: Jack went up the hill ''and'' Jill went up the hill.
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| :D: Jack went up the hill ''so'' Jill went up the hill.
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| The words ''and'' and ''so'' are ''grammatical'' conjunctions joining the sentences (A) and (B) to form the compound sentences (C) and (D). The ''and'' in (C) is a ''logical'' connective, since the truth of (C) is completely determined by (A) and (B): it would make no sense to affirm (A) and (B) but deny (C). However ''so'' in (D) is not a logical connective, since it would be quite reasonable to affirm (A) and (B) but deny (D): perhaps, after all, Jill went up the hill to fetch a pail of water, not because Jack had gone up the hill at all.
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| Various English words and word pairs express logical connectives, and some of them are synonymous. Examples (with the name of the relationship in parentheses) are:
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| * "and" ([[Logical conjunction|conjunction]])
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| * "and then" ([[Logical conjunction with sequencing |conjunction]])
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| * "and then within" ([[Logical conjunction with sequencing and time window requirement|conjunction]])
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| * "or" ([[Logical disjunction|disjunction]])
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| * "either...or" ([[Exclusive or|exclusive disjunction]])
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| * "implies" ([[Material conditional|implication]])
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| * "if...then" ([[Material conditional|implication]])
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| * "if and only if" ([[logical biconditional|equivalence]])
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| * "only if" ([[Material conditional|implication]])
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| * "just in case" ([[logical biconditional|biconditional]])
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| * "but" ([[Logical conjunction|conjunction]])
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| * "however" ([[Logical conjunction|conjunction]])
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| * "not both" ([[Sheffer stroke|alternative denial]])
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| * "neither...nor" ([[Logical NOR|joint denial]])
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| The word "not" (negation) and the phrases "it is false that" (negation) and "it is not the case that" (negation) also express a logical connective – even though they are applied to a single statement, and do not connect two statements.
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| ===Formal languages===
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| In formal languages, truth functions are represented by unambiguous symbols. These symbols are called "logical connectives", "logical operators", "propositional operators", or, in [[classical logic]], "[[truth function|truth-functional]] connectives". See [[well-formed formula]] for the rules which allow new well-formed formulas to be constructed by joining other well-formed formulas using truth-functional connectives.
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| Logical connectives can be used to link more than two statements, so one can speak about "[[arity|{{var|n}}-ary]] logical connective".
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| == Common logical connectives ==
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| {| align=right style="margin-left:2em; margin-bottom:1ex"
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| ! rowspan=2 colspan=2 |Name / Symbol
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| ! colspan=5 |Truth table
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| ! rowspan=2 |Venn<small><br/>diagram</small>
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| |-
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| | bgcolor=#FFFF66 |{{mvar|P}} =
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| | bgcolor=#FFFF66 colspan=2 align=center |0
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| | bgcolor=#FFFF66 colspan=2 align=center |1
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| |-
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| |[[Truth]]/[[Tautology (logic)|Tautology]]
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| | align=right |⊤||
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| | colspan=2 align=center |1
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| | colspan=2 align=center |1
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| |[[Image:Venn11.svg|32px]]
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| |-
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| |Proposition {{mvar|P}}|| ||
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| | colspan=2 align=center |0
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| | colspan=2 align=center |1
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| |[[Image:Venn01.svg|32px]]
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| |-
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| |[[False (logic)|False]]/[[Contradiction]]
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| | align=right |⊥||
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| | colspan=2 align=center |0
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| | colspan=2 align=center |0
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| |[[Image:Venn00.svg|32px]]
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| |-
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| |[[Negation]]
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| | align=right |¬||
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| | colspan=2 align=center |1
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| | colspan=2 align=center |0
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| |[[Image:Venn10.svg|32px]]
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| |-
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| ! rowaspan=2 colspan=2 | Binary connectives
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| | bgcolor=#FFFF66 |{{mvar|P}} =
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| | bgcolor=#FFFF66 |0
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| | bgcolor=#FFFF66 |0
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| | bgcolor=#FFFF66 |1
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| | bgcolor=#FFFF66 |1
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| |-
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| ! colspan=2 |
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| | bgcolor=#FFFF66 |{{mvar|Q}} =
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| | bgcolor=#FFFF66 |0
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| | bgcolor=#FFFF66 |1
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| | bgcolor=#FFFF66 |0
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| | bgcolor=#FFFF66 |1
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| |-
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| |[[Logical conjunction|Conjunction]]
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| | align=right |∧|| ||0||0||0||1||[[Image:Venn0001.svg|40px]]
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| |-
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| |[[Sheffer stroke|Alternative denial]]
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| | align=right |↑|| ||1||1||1||0||[[Image:Venn1110.svg|40px]]
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| |-
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| |[[Logical disjunction|Disjunction]]
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| | align=right |∨|| ||0||1||1||1||[[Image:Venn0111.svg|40px]]
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| |-
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| |[[Logical NOR|Joint denial]]
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| | align=right |↓|| ||1||0||0||0||[[Image:Venn1000.svg|40px]]
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| |-
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| |[[Material conditional]]
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| | align=right |→|| ||1||1||0||1||[[Image:Venn1011.svg|40px]]
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| |-
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| |[[Exclusive or]]
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| | align=right |<math>\nleftrightarrow</math>|| ||0||1||1||0||[[Image:Venn0110.svg|40px]]
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| |-
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| |[[logical biconditional|Biconditional]]
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| | align=right |↔|| ||1||0||0||1||[[Image:Venn1001.svg|40px]]
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| |-
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| |[[Converse implication]]
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| | align=right |←|| ||1||0||1||1||[[Image:Venn1101.svg|40px]]
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| |-
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| |Proposition {{mvar|P}}|| || ||0||0||1||1||[[Image:Venn0101.svg|40px]]
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| |-
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| |Proposition {{mvar|Q}}|| || ||0||1||0||1||[[Image:Venn0011.svg|40px]]
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| |-
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| | colspan=8 align=center style="letter-spacing:0.4em" |[[Truth function#Table of binary truth functions|More information]]
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| |}
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| === List of common logical connectives ===
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| Commonly used logical connectives include:
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| *[[negation|Negation (not)]]: ¬ , N''p'', ~
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| *[[logical conjunction|Conjunction (and)]]: <math>\wedge</math> , K''pq'', & , ∙
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| *[[logical disjunction|Disjunction (or)]]: <math>\or</math>, A''pq''
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| *[[material conditional|Material implication (if...then)]]: <math>\rightarrow</math> , C''pq'', <math>\Rightarrow</math> , <math>\supset</math>
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| *[[logical biconditional|Biconditional (if and only if)]]: <math>\leftrightarrow</math> , E''pq'', <math>\equiv</math> , <math>=</math>
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| Alternative names for biconditional are "iff", "xnor" and "bi-implication".
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| For example, the meaning of the statements ''it is raining'' and ''I am indoors'' is transformed when the two are combined with logical connectives:
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| * It is raining '''and ''' I am indoors (''P'' <math>\wedge</math> ''Q'')
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| * '''If''' it is raining, '''then''' I am indoors (''P'' <math>\rightarrow</math> ''Q'')
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| * '''If''' I am indoors, '''then''' it is raining (''Q'' <math>\rightarrow</math> ''P'')
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| * I am indoors '''if and only if''' it is raining (''P'' <math>\leftrightarrow</math> ''Q'')
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| * It is '''not''' raining ({{not}}''P'')
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| For statement ''P'' = ''It is raining'' and ''Q'' = ''I am indoors''.
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| It is also common to consider the ''always true'' formula and the ''always false'' formula to be connective:
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| * [[truth|True]] formula (⊤, 1, V''pq'', or T)
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| * [[False (logic)|False]] formula (⊥, 0, O''pq'', or F)
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| === History of notations ===
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| * Negation: the symbol ¬ appeared in [[Arend Heyting|Heyting]] in 1929.<ref name="autogenerated1929">[[Heyting]] (1929) ''Die formalen Regeln der intuitionistischen Logik''.</ref><ref>Denis Roegel (2002), ''[http://www.loria.fr/~roegel/cours/symboles-logiques.pdf Petit panorama des notations logiques du 20e siècle]'' (see chart on page 2).</ref> (compare to [[Gottlob Frege|Frege]]'s symbol [[Image:Begriffsschrift connective1.svg|50px]] in his [[Begriffsschrift]]); the symbol ~ appeared in Russell in 1908;<ref name="autogenerated222">[[Bertrand Russell|Russell]] (1908) ''Mathematical logic as based on the theory of types'' (American Journal of Mathematics 30, p222–262, also in From Frege to Gödel edited by van Heijenoort).</ref> an alternative notation is to add an horizontal line on top of the formula, as in <math>\overline{P}</math>; another alternative notation is to use a [[prime (symbol)|prime symbol]] as in P'.
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| * Conjunction: the symbol ∧ appeared in Heyting in 1929<ref name="autogenerated1929"/> (compare to [[Giuseppe Peano|Peano]]'s use of the set-theoretic notation of [[intersection (set theory)|intersection]] ∩ <ref>[[Giuseppe Peano|Peano]] (1889) ''Arithmetices principia, nova methodo exposita''.</ref>); & appeared at least in [[Moses Schönfinkel|Schönfinkel]] in 1924;<ref name="autogenerated1924">[[Moses Schönfinkel|Schönfinkel]] (1924) '' Über die Bausteine der mathematischen Logik'', translated as ''On the building blocks of mathematical logic'' in From Frege to Gödel edited by van Heijenoort.</ref> '''.''' comes from [[George Boole|Boole]]'s interpretation of logic as an [[elementary algebra]].
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| * Disjunction: the symbol ∨ appeared in [[Bertrand Russell|Russell]] in 1908 <ref name="autogenerated222"/> (compare to [[Giuseppe Peano|Peano]]'s use of the set-theoretic notation of [[union (set theory)|union]] ∪); the symbol + is also used, in spite of the ambiguity coming from the fact that the + of ordinary [[elementary algebra]] is an [[exclusive or]] when interpreted logically in a two-element [[Boolean ring|ring]]; punctually in the history a + together with a dot in the lower right corner has been used by [[Charles Sanders Peirce|Peirce]],<ref>[[Charles Sanders Peirce|Peirce]] (1867) ''On an improvement in Boole's calculus of logic.</ref>
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| * Implication: the symbol → can be seen in [[David Hilbert|Hilbert]] in 1917;<ref>[[David Hilbert|Hilbert]] (1917/1918) ''Prinzipien der Mathematik'' (Bernays' course notes).</ref> ⊃ was used by Russell in 1908<ref name="autogenerated222"/> (compare to Peano's inverted C notation); <math>\Rightarrow</math> was used in Vax.<ref>Vax (1982) ''Lexique logique'', Presses Universitaires de France.</ref>
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| * Biconditional: the symbol ≡ was used at least by Russell in 1908;<ref name="autogenerated222"/> ↔ was used at least by [[Alfred Tarski|Tarski]] in 1940;<ref>[[Alfred Tarski|Tarski]] (1940) ''Introduction to logic and to the methodology of deductive sciences''.</ref> ⇔ was used in Vax; other symbols appeared punctually in the history such as ⊃⊂ in [[Gerhard Gentzen|Gentzen]],<ref>[[Gerhard Gentzen|Gentzen]] (1934) ''Untersuchungen über das logische Schließen''.</ref> ~ in Schönfinkel<ref name="autogenerated1924"/> or ⊂⊃ in Chazal.<ref>Chazal (1996) : Éléments de logique formelle.</ref>
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| * True: the symbol 1 comes from [[George Boole|Boole]]'s interpretation of logic as an [[elementary algebra]] over the [[two-element Boolean algebra]]; other notations include <math>\bigwedge</math> to be found in Peano.
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| * False: the symbol 0 comes also from Boole's interpretation of logic as a ring; other notations include <math>\bigvee</math> to be found in Peano.
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| Some authors used letters for connectives at some time of the history: '''u.''' for conjunction (German's "und" for "and") and '''o.''' for disjunction (German's "oder" for "or") in earlier works by Hilbert (1904); '''N''p''''' for negation, '''K''pq''''' for conjunction, '''A''pq''''' for disjunction, '''C''pq''''' for implication, '''E''pq''''' for biconditional in [[Jan Łukasiewicz|Łukasiewicz]] (1929).<ref>See Roegel</ref>
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| === Redundancy ===
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| Such logical connective as [[converse implication]] ← is actually the same as [[material conditional]] with swapped arguments, so the symbol for converse implication is redundant. In some logical calculi (notably, in [[classical logic]]) certain essentially different compound statements are [[logical equivalence|logically equivalent]]. Less trivial example of a redundancy is a classical equivalence between {{math|¬''P'' ∨ ''Q''}} and {{math|''P'' → ''Q''}}. Therefore, a classical-based logical system does not need the conditional operator "→" if "¬" (not) and "∨" (or) are already in use, or may use the "→" only as a [[syntactic sugar]] for a compound having one negation and one disjunction.
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| There are sixteen [[Boolean function]]s associating the input [[truth value]]s {{mvar|P}} and {{mvar|Q}} with four-digit [[binary numeral system|binary]] outputs. These correspond to possible choices of binary logical connectives for [[classical logic]]. Different implementation of classical logic can choose different [[functionally complete]] subsets of connectives.
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| One approach is to choose a ''minimal'' set, and define other connectives by some logical form, like in the example with material conditional above.
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| The following are the minimal functionally complete sets of operators in classical logic whose arities do not exceed 2:
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| ;One element: {↑}, {↓}.
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| ;Two elements: {<math>\vee</math>, ¬}, {<math>\wedge</math>, ¬}, {→, ¬}, {←, ¬}, {→, <math>\bot</math>}, {←, <math>\bot</math>}, {→, <math>\nleftrightarrow</math>}, {←, <math>\nleftrightarrow</math>}, {→, <math>\nrightarrow</math>}, {→, <math>\nleftarrow</math>}, {←, <math>\nrightarrow</math>}, {←, <math>\nleftarrow</math>}, {<math>\nrightarrow</math>, ¬}, {<math>\nleftarrow</math>, ¬}, {<math>\nrightarrow</math>, <math>\top</math>}, {<math>\nleftarrow</math>, <math>\top</math>}, {<math>\nrightarrow</math>, <math>\leftrightarrow</math>}, {<math>\nleftarrow</math>, <math>\leftrightarrow</math>}.
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| ;Three elements: {<math>\lor</math>, <math>\leftrightarrow</math>, <math>\bot</math>}, {<math>\lor</math>, <math>\leftrightarrow</math>, <math>\nleftrightarrow</math>}, {<math>\lor</math>, <math>\nleftrightarrow</math>, <math>\top</math>}, {<math>\land</math>, <math>\leftrightarrow</math>, <math>\bot</math>}, {<math>\land</math>, <math>\leftrightarrow</math>, <math>\nleftrightarrow</math>}, {<math>\land</math>, <math>\nleftrightarrow</math>, <math>\top</math>}.
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| See more details about functional completeness in classical logic at [[Truth function#Functional completeness]].
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| Another approach is to use on equal rights connectives of a certain convenient and functionally complete, but ''not minimal'' set. This approach requires more propositional [[axiom]]s and each equivalence between logical forms must be either an axiom or provable as a theorem.
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| But [[intuitionistic logic]] has the situation more complicated. Of its five connectives {∧, ∨, →, ¬, ⊥} only negation ¬ has to be reduced to other connectives (see [[false (logic)#False, negation and contradiction|details]]). Neither of conjunction, disjunction and material conditional has an equivalent form constructed of other four logical connectives.
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| ==Properties==
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| Some logical connectives possess properties which may be expressed in the theorems containing the connective. Some of those properties that a logical connective may have are:
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| *'''[[Associativity]]''': Within an expression containing two or more of the same associative connectives in a row, the order of the operations does not matter as long as the sequence of the operands is not changed.
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| *'''[[Commutativity]]''': The operands of the connective may be swapped preserving logical equivalence to the original expression.
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| *'''[[Distributivity]]''': A connective denoted by · distributes over another connective denoted by +, if {{math|1=''a'' · (''b'' + ''c'') = (''a'' · ''b'') + (''a'' · ''c'')}} for all operands {{mvar|a}}, {{mvar|b}}, {{mvar|c}}.
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| *'''[[Idempotence]]''': Whenever the operands of the operation are the same, the compound is logically equivalent to the operand.
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| *'''[[Absorption Law|Absorption]]''': A pair of connectives <math>\land</math>, <math>\lor</math> satisfies the absorption law if <math>a\land(a\lor b)=a</math> for all operands {{mvar|a}}, {{mvar|b}}.
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| *'''[[Monotonicity]]''': If ''f''(''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>) ≤ ''f''(''b''<sub>1</sub>, ..., ''b''<sub>''n''</sub>) for all ''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>, ''b''<sub>1</sub>, ..., ''b''<sub>''n''</sub> ∈ {0,1} such that ''a''<sub>1</sub> ≤ ''b''<sub>1</sub>, ''a''<sub>2</sub> ≤ ''b''<sub>2</sub>, ..., ''a''<sub>''n''</sub> ≤ ''b''<sub>''n''</sub>. E.g., <math>\vee</math>, <math>\wedge</math>, <math>\top</math>, <math>\bot</math>.
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| *'''[[affine transformation|Affinity]]''': Each variable always makes a difference in the truth-value of the operation or it never makes a difference.<!-- has this an appropriate generalization to non-classical logics? --> E.g., <math>\neg</math>, <math>\leftrightarrow</math>, <math>\nleftrightarrow</math>, <math>\top</math>, <math>\bot</math>.
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| *'''[[Duality (mathematics)|Duality]]''': To read the truth-value assignments for the operation from top to bottom on its [[truth table]] is the same as taking the complement of reading the table of the same or another connective from bottom to top. Without resorting to truth tables it may be formulated as {{math|1=''g̃''(¬''a''<sub>1</sub>, ..., ¬''a''<sub>''n''</sub>) = ¬''g''(''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>)}}. E.g., <math>\neg</math>.
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| *'''Truth-preserving''': The compound all those argument are tautologies is a tautology itself. E.g., <math>\vee</math>, <math>\wedge</math>, <math>\top</math>, <math>\rightarrow</math>, <math>\leftrightarrow</math>, ⊂. (see [[validity]])
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| *'''Falsehood-preserving''': The compound all those argument are [[contradiction]]s is a contradiction itself. E.g., <math>\vee</math>, <math>\wedge</math>, <math>\nleftrightarrow</math>, <math>\bot</math>, ⊄, ⊅. (see [[validity]])
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| *'''[[Involution (mathematics)|Involutivity]]''' (for unary connectives): {{math|1=''f''(''f''(''a'')) = ''a''}}. E.g. negation in classical logic.
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| For classical and intuitionistic logic, the "="<!-- BTW why not "⇔"? --> symbol means that corresponding implications "…→…" and "…←…" for logical compounds can be both proved as theorems, and the "≤"<!-- BTW why not "⇒"/"→"? --> symbol means that "…→…" for logical compounds is a consequence of corresponding "…→…" connectives for propositional variables. Some of [[many-valued logic]]s may have incompatible definitions of equivalence and order (entailment).
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| Both conjunction and disjunction are associative, commutative and idempotent in classical logic, most varieties of many-valued logic and intuitionistic logic. The same is true about distributivity of conjunction over disjunction and disjunction over conjunction, as well as for the absorption law.
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| In classical logic and some varieties of many-valued logic, conjunction and disjunction are dual, and negation is self-dual, the latter is also self-dual in intuitionistic logic. <!-- I am not sure about ∧ and ∨. Aforementioned definition of duality does not imply that one connective is equivalent to a form with two-layer negation, so such intuitionistic duality is plausible. But one should carefully verify such additions, at least because intuitionistic negation is not an involution and hence the duality relation is not symmetric. -->
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| {{expand section|date=March 2012}}
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| ==Order of precedence==
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| As a way of reducing the number of necessary parentheses, one may introduce [[Order of operations|precedence rules]]: ¬ has higher precedence than <math>\wedge</math>, <math>\wedge</math> higher than <math>\vee</math>, and <math>\vee</math> higher than →. So for example, ''P'' <math>\vee</math> ''Q'' <math>\wedge</math> ¬''R'' → ''S'' is short for (''P'' <math>\vee</math> (''Q'' <math>\wedge</math> (¬''R''))) → ''S''.
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| Here is a table that shows a commonly used precedence of logical operators.
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| :{| class="wikitable" style="text-align:center"
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| |-
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| ! ''Operator'' !! ''Precedence''
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| |-
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| | ¬ || 1
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| |-
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| | <math>\wedge</math> || 2
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| |-
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| | <math>\vee</math> || 3
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| |-
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| | → || 4
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| |-
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| | {{eqv}} || 5
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| |}
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| The order of precedence determines which connective is the "main connective" when interpreting a non-atomic formula.
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| ==Computer science==
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| Truth-functional approach to logical operators is implemented as [[logic gate]]s in [[digital circuit]]s. Practically all digital circuits (the major exception is [[DRAM]]) are built up from [[logical nand|NAND]], [[logical nor|NOR]], [[negation|NOT]], and [[logic gate|transmission gate]]s; see more details in [[Truth function#Computer science]]. Logical operators over [[bit array|bit vectors]] (corresponding to finite [[Boolean algebra (structure)|Boolean algebras]]) are [[bitwise operation]]s.
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| But not every usage of a logical connective in [[computer programming]] has a Boolean semantic. For example, [[lazy evaluation]] is sometimes implemented for {{math|''P'' ∧ ''Q''}} and {{math|''P'' ∨ ''Q''}}, so these connectives are not commutative if some of expressions {{mvar|P}}, {{mvar|Q}} has [[side effect (computer science)|side effect]]s. Also, a [[conditional (programming)|conditional]], which in some sense corresponds to the [[material conditional]] connective, is essentially non-Boolean because for <code>if (P) then Q;</code> the consequent Q is not executed if the [[antecedent (logic)|antecedent]] P is false (although a compound as a whole is successful ≈ "true" in such case). This is closer to intuitionist and [[constructive mathematics|constructivist]] views on the material conditional, rather than to classical logic's ones.
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| {{expand section|date=March 2012}}
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| ==See also==
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| {{Col-begin}}
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| {{Col-break}}
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| * [[Boolean domain]]
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| * [[Boolean function]]
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| * [[Boolean logic]]
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| * [[Boolean-valued function]]
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| * [[List of Boolean algebra topics]]
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| {{Col-break}}
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| {{Portal|Logic|Thinking}}
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| * [[Logical constant]]
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| * [[Modal operator]]
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| * [[Propositional calculus]]
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| * [[Truth function]]
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| * [[Truth table]]
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| * [[Truth value]]s
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| {{Col-end}}
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * [[Józef Maria Bocheński|Bocheński, Józef Maria]] (1959), ''A Précis of Mathematical Logic'', translated from the French and German editions by Otto Bird, D. Reidel, Dordrecht, South Holland.
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| * {{Citation | last1=Enderton | first1=Herbert |author1-link=Herbert Enderton| title=A Mathematical Introduction to Logic | publisher=Academic Press | location=Boston, MA | edition=2nd | isbn=978-0-12-238452-3 | year=2001}}
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| * {{citation|last=Gamut|first=L.T.F|authorlink=L. T. F. Gamut|title=Logic, Language and Meaning|publisher=University of Chicago Press|year=1991|volume=1|pages=54–64|contribution=Chapter 2|oclc=21372380}}
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| == Further reading ==
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| * {{cite book|author=Lloyd Humberstone|title=The Connectives|year=2011|publisher=MIT Press|isbn=978-0-262-01654-4}}
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| ==External links==
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| * {{springer|title=Propositional connective|id=p/p075490}}
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| * Lloyd Humberstone (2010), "[http://plato.stanford.edu/entries/connectives-logic/ Sentence Connectives in Formal Logic]", [[Stanford Encyclopedia of Philosophy]] (An [[abstract algebraic logic]] approach to connectives.)
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| * John MacFarlane (2005), "[http://plato.stanford.edu/entries/logical-constants/ Logical constants]", [[Stanford Encyclopedia of Philosophy]].
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| {{Logical connectives}}
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| {{Logic}}
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| {{DEFAULTSORT:Logical Connective}}
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| [[Category:Logical connectives]]
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| [[Category:Logic symbols]]
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| [[da:Logisk konnektiv]]
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| [[he:קשר לוגי]]
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