Editing Logical connective (section)

===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.