Editing Logical connective (section)

== 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.