Editing Logical connective

{{Short description|Symbol connecting sentential formulas in logic}}
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{{For|other logical symbols|List of logic symbols}}
{{Logical connectives sidebar}}
[[File:Logical connectives Hasse diagram.svg|300px|right|thumb|[[Hasse diagram]] of logical connectives.]]

In [[Mathematical logic|logic]], a '''logical connective''' (also called a '''logical operator''', '''sentential connective''', or '''sentential operator''') is a [[logical constant]]. Connectives can be used to connect logical formulas. For instance in the [[syntax (logic)|syntax]] of [[propositional logic]], the [[Binary relation|binary]] connective <math> \lor </math> can be used to join the two [[atomic formula]]s <math> P</math> and <math> Q</math>, rendering the complex formula <math> P \lor Q </math>.

Common connectives include [[negation]], [[disjunction]], [[Logical conjunction|conjunction]], [[material conditional|implication]], and [[Logical biconditional|equivalence]]. In standard systems of [[classical logic]], these connectives are [[semantics of logic|interpreted]] as [[truth function]]s, though they receive a variety of alternative interpretations in [[nonclassical logic]]s. Their classical interpretations are similar to the meanings of natural language expressions such as [[English language|English]] "not", "or", "and", and "if", but not identical. Discrepancies between natural language connectives and those of classical logic have motivated nonclassical approaches to natural language meaning as well as approaches which pair a classical [[formal semantics (natural language)|compositional semantics]] with a robust [[pragmatics]].

== Overview ==
In [[formal language]]s, truth functions are represented by unambiguous symbols. This allows logical statements to not be understood in an ambiguous way. These symbols are called ''logical connectives'', ''logical operators'', ''propositional operators'', or, in [[classical logic]], ''[[truth function|truth-functional]] connectives''. For the rules which allow new well-formed formulas to be constructed by joining other well-formed formulas using truth-functional connectives, see [[well-formed formula]].

Logical connectives can be used to link zero or more statements, so one can speak about ''[[arity|{{mvar|n}}-ary]] logical connectives''. The [[Boolean algebra|boolean]] constants ''True'' and ''False'' can be thought of as zero-ary operators. Negation is a unary connective, and so on.

{| class="floatright" style="margin-left:2em; margin-bottom:1ex; text-align:center; border: 1px solid #a2a9b1; color: black; padding:0.2em; background-color: #f8f9fa; font-size:90%;"
 ! colspan=2 | Symbol, name
 ! colspan=4 | Truth<br/>table
 !           | Venn<br/><small>diagram</small>
 |-
 ! colspan=7 | Zeroary connectives (constants)
 |-
 | <math>\top</math> || style="text-align:left; | [[Truth]]/[[tautology (logic)|tautology]]
 | colspan=4 | 1
 | [[Image:Red Square.svg|32px]]
 |-
 | <math>\bot</math> || style="text-align:left; | [[False (logic)|Falsity]]/[[contradiction]]
 | colspan=4 | 0
 | [[Image:Blank Square.svg|32px]]
 |-
 ! colspan=7 | Unary connectives
 |- style="background-color:#ffff66;"
 | colspan=2 style="text-align:right;" | <math>p</math>&nbsp;=
 | colspan=2 | 0
 | colspan=2 | 1
 |-
 | || style="text-align:left; | Proposition <math>p</math>
 | colspan=2 | 0
 | colspan=2 | 1
 | [[Image:Venn01.svg|32px]]
 |-
 | <math>\neg</math> || style="text-align:left; | [[Negation]]
 | colspan=2 | 1
 | colspan=2 | 0
 | [[Image:Venn10.svg|32px]]

 |-
 ! colspan=9 | Binary connectives
 |- style="background-color:#ffff66;"
 | colspan=2 style="text-align:right;" | <math>p</math>&nbsp;=
 | 0 || 0 || 1 || 1
 |- style="background-color:#ffff66;"
 | colspan=2 style="text-align:right;" | <math>q</math>&nbsp;=
 | 0 || 1 || 0 || 1
 |-
 | <math>\and</math> || style="text-align:left;" | [[Logical conjunction|Conjunction]]
 |0||0||0||1|| [[Image:Venn0001.svg|40px]]
 |-
 | <math>\uparrow</math> || style="text-align:left;" | [[Sheffer stroke|Alternative denial]]
 |1||1||1||0|| [[Image:Venn1110.svg|40px]]
 |-
 | <math>\vee</math> || style="text-align:left; | [[Logical disjunction|Disjunction]]
 |0||1||1||1|| [[Image:Venn0111.svg|40px]]
 |-
 | <math>\downarrow</math> || style="text-align:left; | [[Logical NOR|Joint denial]]
 |1||0||0||0|| [[Image:Venn1000.svg|40px]]
 |-
 | <math>\nleftrightarrow</math> || style="text-align:left; | [[Exclusive or]]
 |0||1||1||0|| [[Image:Venn0110.svg|40px]]
 |-
 | <math>\leftrightarrow</math> || style="text-align:left; | [[logical biconditional|Biconditional]]
 |1||0||0||1|| [[Image:Venn1001.svg|40px]]
 |-
 | <math>\rightarrow</math> || style="text-align:left; | [[Material conditional]]
 |1||1||0||1|| [[Image:Venn1011.svg|40px]]
 |-
 | <math>\nrightarrow</math> || style="text-align:left; | [[Material nonimplication]]
 |0||0||1||0|| [[Image:Venn0100.svg|40px]]
 |-
 | <math>\leftarrow</math> || style="text-align:left; | [[Converse implication]]
 |1||0||1||1|| [[Image: Venn1101.svg|40px]]
 |-
 | <math>\nleftarrow</math> || style="text-align:left; | [[Converse nonimplication]]
 |0||1||0||0|| [[Image:Venn0010.svg|40px]]
 |-
 | colspan=7" | [[Truth function#Table of binary truth functions|More information]]
 |}

===List of common logical connectives===
Commonly used logical connectives include the following ones.<ref name="chao2023">{{cite book |last1=Chao |first1=C. |title=数理逻辑：形式化方法的应用 |trans-title=Mathematical Logic: Applications of the Formalization Method |date=2023 |publisher=Preprint. |location=Beijing |pages=15–28 |language=Chinese}}</ref> 
* [[negation|Negation (not)]]: <math>\neg</math>, <math>\sim</math>, <math>N</math> (prefix) in which <math>\neg</math> is the most modern and widely used, and <math>\sim</math> is also common;
* [[logical conjunction|Conjunction (and)]]: <math>\wedge</math>, <math>\&</math>, <math>K</math> (prefix)  in which <math>\wedge</math> is the most modern and widely used;
* [[logical disjunction|Disjunction (or)]]: <math>\vee</math>, <math>A</math> (prefix)  in which <math>\vee</math> is the most modern and widely used;
* [[Material conditional|Implication (if...then)]]: <math>\to</math>, <math>\supset</math>, <math>\Rightarrow</math>, <math>C</math> (prefix)  in which <math>\to</math> is the most modern and widely used, and <math>\supset</math> is also common;
* [[Logical biconditional|Equivalence (if and only if)]]: <math>\leftrightarrow</math>, <math>\subset\!\!\!\supset</math>,  <math>\Leftrightarrow</math>, <math>\equiv</math>, <math>E</math> (prefix) in which <math>\leftrightarrow</math> is the most modern and widely used, and <math>\subset\!\!\!\supset</math> is commonly used where <math>\supset</math> is also used.

For example, the meaning of the statements ''it is raining'' (denoted by <math>p</math>) and ''I am indoors'' (denoted by <math>q</math>) is transformed, when the two are combined with logical connectives:
* It is '''''not''''' raining (<math>\neg p</math>);
* It is raining '''''and''''' I am indoors (<math>p \wedge q</math>);
* It is raining '''''or''''' I am indoors (<math>p \lor q</math>);
* '''''If''''' it is raining, '''''then''''' I am indoors (<math>p \rightarrow q</math>);
* '''''If''''' I am indoors, '''''then''''' it is raining (<math>q \rightarrow p</math>);
* I am indoors '''''if and only if''''' it is raining (<math>p \leftrightarrow q</math>).

It is also common to consider the ''always true'' formula and the ''always false'' formula to be connective (in which case they are [[nullary]]).
* [[Truth|True]] formula: <math>\top</math>, <math>1</math>, <math>V</math> (prefix), or <math>\mathrm{T}</math>;
* [[False (logic)|False]] formula: <math>\bot</math>, <math>0</math>, <math>O</math> (prefix), or <math>\mathrm{F}</math>.

This table summarizes the terminology:

{| class="wikitable" style="margin:1em auto; text-align:left;"
|-
! Connective 
! In English 
! Noun for parts 
! Verb phrase 
|-
! Conjunction 
| Both A and B 
| conjunct 
| A and B are conjoined
|-
! Disjunction 
| Either A or B, or both 
| disjunct 
| A and B are disjoined
|-
! Negation 
| It is not the case that A 
| negatum/negand 
| A is negated
|-
! Conditional 
| If A, then B 
| antecedent, consequent 
| B is implied by A
|-
! Biconditional 
| A if, and only if, B 
| equivalents
| A and B are equivalent
|}

===History of notations===
* Negation: the symbol <math>\neg</math> appeared in [[Arend Heyting|Heyting]] in 1930<ref name="heyting1930">{{cite journal |last1=Heyting |first1=A. |title=Die formalen Regeln der intuitionistischen Logik |journal=Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse |date=1930 |pages=42–56 |language=German}}</ref><ref>Denis Roegel (2002), 
''[https://members.loria.fr/Roegel/loc/symboles-logiques-eng.pdf A brief survey of 20th century logical notations]'' (see chart on page 2).</ref> (compare to [[Gottlob Frege|Frege]]'s symbol ⫟ in his [[Begriffsschrift]]<ref name="frege1879a">{{cite book |last1=Frege |first1=G. |title=Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens |date=1879 |publisher=Verlag von Louis Nebert |location=Halle a/S. |page=10}}</ref>); the symbol <math>\sim</math> appeared in [[Bertrand Russell|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 a 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 <math>p'</math>.
* Conjunction: the symbol <math>\wedge</math> appeared in Heyting in 1930<ref name="heyting1930"/> (compare to [[Giuseppe Peano|Peano]]'s use of the set-theoretic notation of [[intersection (set theory)|intersection]] <math>\cap</math><ref>[[Giuseppe Peano|Peano]] (1889) ''[[Arithmetices principia, nova methodo exposita]]''.</ref>); the symbol <math>\&</math> 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> the symbol <math>\cdot</math> comes from [[George Boole|Boole]]'s interpretation of logic as an [[elementary algebra]].
* Disjunction: the symbol <math>\vee</math> 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]] <math>\cup</math>); the symbol <math>+</math> is also used, in spite of the ambiguity coming from the fact that the <math>+</math> of ordinary [[elementary algebra]] is an [[exclusive or]] when interpreted logically in a two-element [[Boolean ring|ring]]; punctually in the history a <math>+</math> 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>
* Implication: the symbol <math>\to</math> appeared in [[David Hilbert|Hilbert]] in 1918;<ref name="hilbert1918">{{cite book |last1=Hilbert |first1=D. |editor1-last=Bernays |editor1-first=P. |title=Prinzipien der Mathematik |date=1918 |others=Lecture notes at Universität Göttingen, Winter Semester, 1917-1918 |postscript=none}}; Reprinted as {{cite encyclopedia |title=Prinzipien der Mathematik |last=Hilbert |first=D. |encyclopedia=David Hilbert's Lectures on the Foundations of Arithmetic and Logic 1917–1933 |date=2013 |editor1-last=Ewald |editor1-first=W. |editor2-last=Sieg |editor2-first=W. |publisher=Springer |location=Heidelberg, New York, Dordrecht and London |pages=59–221}}</ref>{{rp|page=76}} <math>\supset</math> was used by Russell in 1908<ref name="autogenerated222"/> (compare to Peano's Ɔ the inverted C); <math>\Rightarrow</math> appeared in [[Nicolas Bourbaki|Bourbaki]] in 1954.<ref name="bourbaki1954a">{{cite book |last1=Bourbaki |first1=N. |title=Théorie des ensembles |date=1954 |publisher=Hermann & Cie, Éditeurs |location=Paris |page=14}}</ref>
* Equivalence: the symbol <math>\equiv</math> in [[Gottlob Frege|Frege]] in 1879;<ref name="frege1879b">{{cite book |last1=Frege |first1=G. |title=Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens |date=1879 |publisher=Verlag von Louis Nebert |location=Halle a/S. |page=15 |language=German}}</ref> <math>\leftrightarrow</math> in Becker in 1933 (not the first time and for this see the following);<ref name="becker1933">{{cite book |last1=Becker |first1=A. |title=Die Aristotelische Theorie der Möglichkeitsschlösse: Eine logisch-philologische Untersuchung der Kapitel 13-22 von Aristoteles' Analytica priora I |date=1933 |publisher=Junker und Dünnhaupt Verlag |location=Berlin |page=4 |language=German}}</ref> <math>\Leftrightarrow</math> appeared in [[Nicolas Bourbaki|Bourbaki]] in 1954;<ref name="bourbaki1954b">{{cite book |last1=Bourbaki |first1=N. |title=Théorie des ensembles |date=1954 |publisher=Hermann & Cie, Éditeurs |location=Paris |page=32 |language=French}}</ref> other symbols appeared punctually in the history, such as <math>\supset\subset</math> in [[Gerhard Gentzen|Gentzen]],<ref>[[Gerhard Gentzen|Gentzen]] (1934) ''Untersuchungen über das logische Schließen''.</ref> <math>\sim</math> in Schönfinkel<ref name="autogenerated1924"/> or <math>\subset\supset</math> in Chazal, <ref>Chazal (1996) : Éléments de logique formelle.</ref>
* True: the symbol <math>1</math> comes from [[George Boole|Boole]]'s interpretation of logic as an [[elementary algebra]] over the [[two-element Boolean algebra]]; other notations include <math>\mathrm{V}</math> (abbreviation for the Latin word "verum") to be found in Peano in 1889.
* False: the symbol <math>0</math> comes also from Boole's interpretation of logic as a ring; other notations include <math>\Lambda</math> (rotated <math>\mathrm{V}</math>) to be found in Peano in 1889.

Some authors used letters for connectives: <math>\operatorname{u.}</math> for conjunction (German's "und" for "and") and <math>\operatorname{o.}</math> for disjunction (German's "oder" for "or") in early works by Hilbert (1904);<ref name="hilbert1904">{{cite encyclopedia |last1=Hilbert |first1=D. |title=Über die Grundlagen der Logik und der Arithmetik |encyclopedia=Verhandlungen des Dritten Internationalen Mathematiker Kongresses in Heidelberg vom 8. bis 13. August 1904 |editor1-last=Krazer |editor1-first=K. |orig-date=1904 |date=1905 |pages=174–185}}</ref> <math>Np</math> for negation, <math>Kpq</math> for conjunction, <math>Dpq</math> for alternative denial, <math>Apq</math> for disjunction, <math>Cpq</math> for implication, <math>Epq</math> for biconditional in [[Jan Łukasiewicz|Łukasiewicz]] in 1929.

===Redundancy===

Such a logical connective as [[converse implication]] "<math>\leftarrow</math>" is actually the same as [[material conditional]] with swapped arguments; thus, 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]]. A less [[Triviality (mathematics)|trivial]] example of a redundancy is the classical equivalence between <math>\neg p\vee q</math> and <math>p\to q</math>. Therefore, a classical-based logical system does not need the conditional operator "<math>\to</math>" if "<math>\neg</math>" (not) and "<math>\vee</math>" (or) are already in use, or may use the "<math>\to</math>" only as a [[syntactic sugar]] for a compound having one negation and one disjunction.

There are sixteen [[Boolean function]]s associating the input [[truth value]]s <math>p</math> and <math>q</math> with four-digit [[binary numeral system|binary]] outputs.<ref>[[Józef Maria Bocheński|Bocheński]] (1959), ''A Précis of Mathematical Logic'', passim.</ref> These correspond to possible choices of binary logical connectives for [[classical logic]]. Different implementations of classical logic can choose different [[Functional completeness|functionally complete]] subsets of connectives.

One approach is to choose a ''minimal'' set, and define other connectives by some logical form, as in the example with the material conditional above.
The following are the [[Functional completeness#Minimal functionally complete operator sets|minimal functionally complete sets of operators]] in classical logic whose arities do not exceed 2:
;One element: <math>\{\uparrow\}</math>, <math>\{\downarrow\}</math>.
;Two elements: <math>\{\vee, \neg\}</math>, <math>\{\wedge, \neg\}</math>, <math>\{\to, \neg\}</math>, <math>\{\gets, \neg\}</math>, <math>\{\to, \bot\}</math>, <math>\{\gets, \bot\}</math>, <math>\{\to, \nleftrightarrow\}</math>, <math>\{\gets, \nleftrightarrow\}</math>, <math>\{\to, \nrightarrow\}</math>, <math>\{\to, \nleftarrow\}</math>, <math>\{\gets, \nrightarrow\}</math>, <math>\{\gets, \nleftarrow\}</math>, <math>\{\nrightarrow, \neg\}</math>, <math>\{\nleftarrow, \neg\}</math>, <math>\{\nrightarrow, \top\}</math>, <math>\{\nleftarrow, \top\}</math>, <math>\{\nrightarrow, \leftrightarrow\}</math>, <math>\{\nleftarrow, \leftrightarrow\}</math>.
;Three elements: <math>\{\lor, \leftrightarrow, \bot\}</math>, <math>\{\lor, \leftrightarrow, \nleftrightarrow\}</math>, <math>\{\lor, \nleftrightarrow, \top\}</math>, <math>\{\land, \leftrightarrow, \bot\}</math>, <math>\{\land, \leftrightarrow, \nleftrightarrow\}</math>, <math>\{\land, \nleftrightarrow, \top\}</math>.

Another approach is to use with 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.

The situation, however, is more complicated in [[intuitionistic logic]]. Of its five connectives, {∧, ∨, →, ¬, ⊥}, only negation "¬" can be reduced to other connectives (see {{Section link|False (logic)|False, negation and contradiction}} for more). Neither conjunction, disjunction, nor material conditional has an equivalent form constructed from the other four logical connectives.

==Natural language==

The standard logical connectives of classical logic have rough equivalents in the grammars of natural languages. In [[English language|English]], as in many languages, such expressions are typically [[grammatical conjunction]]s. However, they can also take the form of [[complementizer]]s, [[verb]] [[suffix]]es, and [[grammatical particle|particle]]s. The [[denotation]]s of natural language connectives is a major topic of research in [[formal semantics (natural language)|formal semantics]], a field that studies the logical structure of natural languages.

The meanings of natural language connectives are not precisely identical to their nearest equivalents in classical logic. In particular, disjunction can receive an [[exclusive disjunction|exclusive interpretation]] in many languages. Some researchers have taken this fact as evidence that natural language [[semantics (natural language)|semantics]] is [[nonclassical logic|nonclassical]]. However, others maintain classical semantics by positing [[pragmatics|pragmatic]] accounts of exclusivity which create the illusion of nonclassicality. In such accounts, exclusivity is typically treated as a [[scalar implicature]]. Related puzzles involving disjunction include [[free choice inference]]s, [[Hurford disjunction|Hurford's Constraint]], and the contribution of disjunction in [[alternative question]]s.

Other apparent discrepancies between natural language and classical logic include the [[paradoxes of material implication]], [[donkey anaphora]] and the problem of [[counterfactual conditionals]]. These phenomena have been taken as motivation for identifying the denotations of natural language conditionals with logical operators including the [[strict conditional]], the [[variably strict conditional]], as well as various [[dynamic semantics|dynamic]] operators.

The following table shows the standard classically definable approximations for the English connectives.

{| class="wikitable sortable"
|-
! English word !! Connective !! Symbol !! Logical gate
|-
| not || [[negation]] || <math>\neg</math> || [[Inverter (logic gate)|NOT]]
|-
| and || [[Logical conjunction|conjunction]] || <math>\and</math> || [[AND gate|AND]]
|-
| or || [[Logical disjunction|disjunction]] || <math>\vee</math> || [[OR gate|OR]]
|-
| if...then || [[Material conditional|material implication]]  || <math>\rightarrow</math> || [[IMPLY gate|IMPLY]]
|-
| ...if || [[converse implication]] || <math>\leftarrow</math> ||
|-
| either...or || [[Exclusive or|exclusive disjunction]] || <math>\nleftrightarrow</math> || [[XOR gate|XOR]]
|-
| if and only if || [[logical biconditional|biconditional]] || <math>\leftrightarrow</math> || [[XNOR gate|XNOR]]
|-
| not both || [[Sheffer stroke|alternative denial]] || <math>\uparrow</math> || [[NAND gate|NAND]]
|-
| neither...nor || [[Logical NOR|joint denial]] || <math>\downarrow</math> || [[NOR gate|NOR]]
|-
| but not || [[material nonimplication]] || <math>\nrightarrow</math> || [[NIMPLY gate|NIMPLY]]
|-
| not...but || [[converse nonimplication]] || <math>\nleftarrow</math>
|}

==Properties==
Some logical connectives possess properties that may be expressed in the theorems containing the connective. Some of those properties that a logical connective may have are:

; [[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.
; [[Commutativity]]:The operands of the connective may be swapped, preserving logical equivalence to the original expression.
; [[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}}.
; [[Idempotence]]: Whenever the operands of the operation are the same, the compound is logically equivalent to the operand.
; [[Absorption Law|Absorption]]: A pair of connectives &and;, &or; satisfies the absorption law if <math>a\land(a\lor b)=a</math> for all operands {{mvar|a}}, {{mvar|b}}.
; [[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., &or;, &and;, ⊤, ⊥.
; [[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., &not;, ↔,  <math>\nleftrightarrow</math>, ⊤, ⊥.
; [[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=''g&#771;''(¬''a''<sub>1</sub>, ..., ¬''a''<sub>''n''</sub>) = ¬''g''(''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>)}}. E.g., &not;.
; Truth-preserving: The compound all those arguments are tautologies is a tautology itself. E.g., &or;, &and;, ⊤, →, ↔, ⊂ (see [[Validity (logic)|validity]]).
; Falsehood-preserving: The compound all those argument are [[contradiction]]s is a contradiction itself. E.g., &or;, &and;, <math>\nleftrightarrow</math>, ⊥, ⊄, ⊅ (see [[Validity (logic)|validity]]).
; [[Involution (mathematics)|Involutivity]] (for unary connectives): {{math|1=''f''(''f''(''a'')) = ''a''}}. E.g. negation in classical logic.

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 [[many-valued logic]]s may have incompatible definitions of equivalence and order (entailment).

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.

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. -->

{{expand section|date=March 2012}}

==Order of precedence==
As a way of reducing the number of necessary parentheses, one may introduce [[precedence rule]]s: &not; has higher precedence than &and;, &and; higher than &or;, and &or; higher than →. So for example, <math>P \vee Q \and{\neg R} \rightarrow S</math> is short for <math>(P \vee (Q \and (\neg R))) \rightarrow S</math>.

Here is a table that shows a commonly used precedence of logical operators.<ref>{{cite book|title=Discrete Mathematics Using a Computer|first1=John|last1=O'Donnell|first2=Cordelia|last2=Hall|first3=Rex|last3=Page| publisher=Springer| year=2007| isbn=9781846285981|page=120|url=https://books.google.com/books?id=KKxyQQWQam4C&pg=PA120}}.</ref><ref name=":35">{{Cite book |last1=Allen |first1=Colin |title=Logic primer |last2=Hand |first2=Michael |date=2022 |publisher=The MIT Press |isbn=978-0-262-54364-4 |edition=3rd |location=Cambridge, Massachusetts}}</ref>
{| class="wikitable" style="text-align: center;"
!Operator !!Precedence
|-
| <math>\neg</math> || 1
|-
| <math>\and</math> || 2
|-
| <math>\vee</math> || 3
|-
| <math>\rightarrow</math> || 4
|-
| <math>\leftrightarrow</math> || 5
|}
However, not all compilers use the same order; for instance, an ordering in which disjunction is lower precedence than implication or bi-implication has also been used.<ref>{{cite book|title=Software Abstractions: Logic, Language, and Analysis | first=Daniel|last=Jackson|publisher=MIT Press|year=2012|isbn=9780262017152|page=263| url=https://books.google.com/books?id=DDv8Ie_jBUQC&pg=PA263}}.</ref> Sometimes precedence between conjunction and disjunction is unspecified requiring to provide it explicitly in given formula with parentheses. The order of precedence determines which connective is the "main connective" when interpreting a non-atomic formula.

==Table and Hasse diagram==

The 16 logical connectives can be [[Partial order|partially ordered]] to produce the following [[Hasse diagram]]. 
The partial order is defined by declaring that <math>x \leq y</math> if and only if whenever <math>x</math> holds then so does <math>y.</math> 

{{Logical connectives table and Hasse diagram}}

==Applications==
Logical connectives are used in [[computer science]] and in [[set theory]].

===Computer science===
{{Main article|Logic gate}}
A 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|Truth function in computer science]]. Logical operators over [[bit array|bit vectors]] (corresponding to finite [[Boolean algebra (structure)|Boolean algebras]]) are [[bitwise operation]]s.

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 either or both of the expressions {{mvar|P}}, {{mvar|Q}} have [[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&nbsp;Q is not executed if the [[antecedent (logic)|antecedent]]&nbsp;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 views.

===Set theory===
{{Main article|Set theory|Axiomatic set theory}}
Logical connectives are used to define the fundamental operations of [[set theory]],<ref>{{Cite book |last=Pinter |first=Charles C. |title=A book of set theory |date=2014 |publisher=Dover Publications, Inc |isbn=978-0-486-49708-2 |location=Mineola, New York |pages=26–29}}</ref> as follows:

{| class="wikitable" style="margin:1em auto; text-align:left;"
|+Set theory operations and connectives
|-
! Set operation
! Connective 
! Definition
|-
| [[Intersection (set theory)|Intersection]]
| [[Logical conjunction|Conjunction]]
| <math>A \cap B = \{x : x \in A \land x \in B \}</math><ref name=":0">{{Cite web |title=Set operations |url=https://www.siue.edu/~jloreau/courses/math-223/notes/sec-set-operations.html |access-date=2024-06-11 |website=www.siue.edu}}</ref><ref name=":1">{{Cite web |title=1.5 Logic and Sets |url=https://www.whitman.edu/mathematics/higher_math_online/section01.05.html |access-date=2024-06-11 |website=www.whitman.edu}}</ref><ref>{{Cite web |title=Theory Set |url=https://mirror.clarkson.edu/isabelle/dist/library/HOL/HOL/Set.html |access-date=2024-06-11 |website=mirror.clarkson.edu}}</ref>
|-
| [[Union (set theory)|Union]]
| [[Logical disjunction|Disjunction]]  
| <math>A \cup B = \{x : x \in A \lor x \in B \}</math><ref>{{Cite web |title=Set Inclusion and Relations |url=https://autry.sites.grinnell.edu/csc208/readings/set-inclusion.html |access-date=2024-06-11 |website=autry.sites.grinnell.edu}}</ref><ref name=":0" /><ref name=":1" />
|-
| [[Complement (set theory)|Complement]]
| [[Negation]] 
| <math>\overline{A} = \{x : x \notin A \}</math><ref>{{Cite web |title=Complement and Set Difference |url=https://web.mnstate.edu/peil/MDEV102/U1/S6/Complement3.htm |access-date=2024-06-11 |website=web.mnstate.edu}}</ref><ref name=":1" /><ref>{{Cite web |last=Cooper |first=A. |title=Set Operations and Subsets – Foundations of Mathematics |url=https://ma225.wordpress.ncsu.edu/set-operations-and-subsets/ |access-date=2024-06-11 |language=en-US}}</ref>
|-
| [[Subset]]
| [[Material conditional|Implication]]
| <math>A \subseteq B \leftrightarrow (x \in A \rightarrow x \in B)</math><ref name=":2">{{Cite web |title=Basic concepts |url=https://www.siue.edu/~jloreau/courses/math-223/notes/sec-set-basics.html |access-date=2024-06-11 |website=www.siue.edu}}</ref><ref name=":1" /><ref>{{Cite web |last=Cooper |first=A. |title=Set Operations and Subsets – Foundations of Mathematics |url=https://ma225.wordpress.ncsu.edu/set-operations-and-subsets/ |access-date=2024-06-11 |language=en-US}}</ref> 
|-
| [[Equality (mathematics)|Equality]]
| [[Logical biconditional|Biconditional]] 
| <math>A = B \leftrightarrow (\forall X)[A \in X \leftrightarrow B \in X]</math><ref name=":2" /><ref name=":1" /><ref>{{Cite web |last=Cooper |first=A. |title=Set Operations and Subsets – Foundations of Mathematics |url=https://ma225.wordpress.ncsu.edu/set-operations-and-subsets/ |access-date=2024-06-11 |language=en-US}}</ref>
|}
This definition of set equality is equivalent to the [[axiom of extensionality]].

==See also==

{{Portal|Philosophy|Psychology}}
{{div col|colwidth=22em}}
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean logic]]
* [[Boolean-valued function]]
* [[Catuṣkoṭi]]
* [[Dialetheism]]
* [[Four-valued logic]]
* [[List of Boolean algebra topics]]
* [[Logical conjunction]]
* [[Logical constant]]
* [[Modal operator]]
* [[Propositional calculus]]
* [[Term logic]]
* [[Tetralemma]]
* [[Truth function]]
* [[Truth table]]
* [[Truth value]]s
{{div col end}}

==References==
{{Reflist}}

==Sources==

* [[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.
* {{cite book |last1=Chao |first1=C. |title=数理逻辑：形式化方法的应用 |trans-title=Mathematical Logic: Applications of the Formalization Method |date=2023 |publisher=Preprint. |location=Beijing |pages=15–28 |language=Chinese}}
* {{cite book | 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}}
* {{cite book |last=Gamut|first=L.T.F|author-link=L. T. F. Gamut|title=Logic, Language and Meaning|publisher=University of Chicago Press|year=1991|volume=1|pages=54&ndash;64|contribution=Chapter 2|oclc=21372380}}
* {{cite book |author=Rautenberg, W.|author-link=Wolfgang Rautenberg|doi=10.1007/978-1-4419-1221-3|title=A Concise Introduction to Mathematical Logic|publisher=[[Springer Science+Business Media]] |location=[[New York City|New York]]|edition=3rd|isbn=978-1-4419-1220-6|year=2010}}.
* {{cite book|first=Lloyd|last=Humberstone|title=The Connectives|year=2011|publisher=MIT Press|isbn=978-0-262-01654-4}}

==External links==

{{Commons category}}
*{{springer|title=Propositional connective|id=p/p075490}}
*Lloyd Humberstone (2010), "[https://plato.stanford.edu/entries/connectives-logic/ Sentence Connectives in Formal Logic]", [[Stanford Encyclopedia of Philosophy]] (An [[abstract algebraic logic]] approach to connectives.)
*John MacFarlane (2005), "[https://plato.stanford.edu/entries/logical-constants/ Logical constants]", [[Stanford Encyclopedia of Philosophy]].

{{Logical connectives}}
{{Mathematical logic}}
{{Formal semantics}}
{{Authority control}}

{{DEFAULTSORT:Logical Connective}}
[[Category:Logical connectives| ]]
[[Category:Logic symbols|Connective]]

[[da:Logisk konnektiv]]