View source for Sheffer stroke

{{short description|Logical operation}}
{{use list-defined references|date=May 2023}}
{{Infobox logical connective
| title        = Sheffer stroke
| other titles = NAND
| Venn diagram = Venn1110.svg
| definition   = <math>\overline{x \cdot y}</math>
| truth table  = <math>(0111)</math>
| logic gate   = NAND_ANSI.svg
| DNF          = <math>\overline{x} + \overline{y}</math>
| CNF          = <math>\overline{x} + \overline{y}</math>
| Zhegalkin    = <math>1 \oplus xy</math>
| 0-preserving = no
| 1-preserving = no
| monotone     = no
| affine       = no
| self-dual    = no
}}

In [[Boolean function]]s and [[Propositional calculus|propositional calculus]], the '''Sheffer stroke''' denotes a logical operation that is equivalent to the negation of the [[Logical conjunction|conjunction]] operation, expressed in ordinary language as "not both". It is also called '''non-conjunction''', or '''alternative denial''' since it says in effect that at least one of its operands is false, or '''NAND''' ("not and"). In [[Engineering:Digital electronics|digital electronics]], it corresponds to the [[Engineering:NAND gate|NAND gate]]. It is named after Henry Maurice Sheffer and written as <math>\mid</math> or as <math>\uparrow</math> or as <math>\overline{\wedge}</math> or as <math>Dpq</math> in [[Polish notation]] by [[Biography:Jan Łukasiewicz|Łukasiewicz]] (but not as ||, often used to represent [[Logical disjunction|disjunction]]).

Its [[Duality (mathematics)|dual]] is the [[Logical NOR|NOR operator]] (also known as the Peirce arrow, Quine dagger or Webb operator). Like its dual, NAND can be used by itself, without any other logical operator, to constitute a logical [[Formal system|formal system]] (making NAND functionally complete). This property makes the [[Engineering:NAND gate|NAND gate]] crucial to modern [[Engineering:Digital electronics|digital electronics]], including its use in computer processor design.

==Definition==
The '''non-conjunction''' is a logical operation on two logical values.  It produces a value of true, if — and only if — at least one of the [[Proposition|proposition]]s is false.

===Truth table===
The [[Truth table|truth table]] of <math>P \uparrow Q</math> is as follows.

{| class="wikitable" style="text-align:center;"
|- bgcolor="#ddeeff"
| <math> P </math> || <math> Q </math> || <math> P \uparrow Q </math>
|-
| {{yes2|True}} || {{yes2|True}} || {{no2|False}}
|-
| {{yes2|True}} || {{no2|False}} || {{yes2|True}}
|-
| {{no2|False}} || {{yes2|True}} || {{yes2|True}}
|-
| {{no2|False}} || {{no2|False}} || {{yes2|True}}
|}

===Logical equivalences===
The Sheffer stroke of <math>P</math> and <math>Q</math> is the negation of their conjunction

{| style="text-align: center; border: 1px solid darkgray;"
|-
| <math>P \uparrow Q</math>
| &emsp;&emsp;<math>\Leftrightarrow</math>&emsp;&emsp;
| <math>\neg (P \land Q)</math>
|-
| [[File:Venn1110.svg|50px]]
| &emsp;&emsp;<math>\Leftrightarrow</math>&emsp;&emsp;
| <math>\neg</math> [[File:Venn0001.svg|50px]]
|}

By [[De Morgan's laws]], this is also equivalent to the disjunction of the negations of <math>P</math> and <math>Q</math>

{| style="text-align: center; border: 1px solid darkgray;"
|-
| <math>P \uparrow Q</math>
| &emsp;&emsp;<math>\Leftrightarrow</math>&emsp;&emsp;
| <math>\neg P</math>
| <math>\lor</math>
| <math>\neg Q</math>
|-
| [[File:Venn1110.svg|50px]]
| &emsp;&emsp;<math>\Leftrightarrow</math>&emsp;&emsp;
| [[File:Venn1010.svg|50px]]
| <math>\lor</math> 
| [[File:Venn1100.svg|50px]]
|}

==Alternative notations and names==
[[Biography:Charles Sanders Peirce|Peirce]] was the first to show the functional completeness of non-conjunction (representing this as <math>\overline{\curlywedge}</math>) but didn't publish his result.<ref name="peirce1880">{{cite encyclopedia |last1=Peirce |first1=C. S. |title=A Boolian Algebra with One Constant |encyclopedia=Collected Papers of Charles Sanders Peirce, Volume IV The Simplest Mathematics |editor1-last=Hartshorne |editor1-first=C. |editor2-last=Weiss |editor2-first=P. |orig-date=1880 |date=1933 |pages=13–18 |location=Massachusetts |publisher=Harvard University Press}}</ref><ref name="peirce1902">{{cite encyclopedia |last1=Peirce |first1=C. S. |title=The Simplest Mathematics |encyclopedia=Collected Papers of Charles Sanders Peirce, Volume IV The Simplest Mathematics |editor1-last=Hartshorne |editor1-first=C. |editor2-last=Weiss |editor2-first=P. |orig-date=1902 |date=1933 |pages=189–262 |location=Massachusetts |publisher=Harvard University Press}}</ref> Peirce's editor added <math>\overline{\curlywedge}</math>) for non-disjunction{{citation needed|date=July 2023}}.<ref name="peirce1902"/>

In 1911, {{ill|Stamm|pl|Edward Bronisław Stamm}} was the first to publish a proof of the completeness of non-conjunction, representing this with <math>\sim</math> (the '''Stamm hook''')<ref name="zach2023">{{cite web |last1=Zach |first1=R. |title=Sheffer stroke before Sheffer: Edward Stamm |url=https://richardzach.org/2023/02/sheffer-stroke-before-sheffer-edward-stamm/ |date=18 February 2023|access-date=2 July 2023}}</ref> and non-disjunction in print at the first time and showed their functional completeness.<ref name="Stamm_1911"/>

In 1913, Sheffer described non-disjunction using <math>\mid</math> and showed its functional completeness. Sheffer also used <math>\wedge</math> for non-disjunction.{{citation needed|date=July 2023}} Many people, beginning with Nicod in 1917, and followed by Whitehead, Russell and many others, mistakenly thought Sheffer has described non-conjunction using <math>\mid</math>, naming this the Sheffer Stroke.

In 1928, [[Biography:David Hilbert|Hilbert]] and Ackermann described non-conjunction with the operator <math>/</math>.<ref name="hilbert-ackermann1928">{{cite book |last1=Hilbert |first1=D. |last2=Ackermann |first2=W. |title=Grundzügen der theoretischen Logik |edition=1 |date=1928 |publisher=Verlag von Julius Springer |location=Berlin |page=9 |language=German}}</ref><ref name="hilbert-ackermann1950">{{cite book |last1=Hilbert |first1=D. |last2=Ackermann |first2=W. |editor1-last=Luce |editor1-first=R. E. |translator1-last=Hammond |translator1-first=L. M. |translator2-last=Leckie |translator2-first=G. G. |translator3-last=Steinhardt |translator3-first=F. |title=Principles of Mathematical Logic |date=1950 |publisher=Chelsea Publishing Company |location=New York |page=11}}</ref>

In 1929, [[Biography:Jan Łukasiewicz|Łukasiewicz]] used <math>D</math> in <math>Dpq</math> for non-conjunction in his [[Polish notation]].<ref name="lukasiewicz1929">{{cite book |last1=Łukasiewicz |first1=J. |title=Elementy logiki matematycznej |orig-date=1929|date=1958 |location=Warszawa |publisher=Państwowe Wydawnictwo Naukowe |edition=2 |language=Polish}}</ref>

An alternative notation for non-conjunction is <math>\uparrow</math>. It is not clear who first introduced this notation, although the corresponding <math>\downarrow</math> for non-disjunction was used by Quine in 1940,.<ref name="quine1940">{{cite book |last1=Quine |first1=W. V |title=Mathematical Logic |date=1981 |orig-date=1940 |publisher=Harvard University Press |location=Cambridge, London, New York, New Rochelle, Melbourne and Sydney |edition=Revised |page=45}}</ref>

==History==
The stroke is named after Henry Maurice Sheffer, who in 1913 published a paper in the ''[[Organization:Transactions of the American Mathematical Society|Transactions of the American Mathematical Society]]''<ref name="Sheffer_1913"/> providing an axiomatization of [[Boolean algebra (structure)|Boolean algebra]]s using the stroke, and proved its equivalence to a standard formulation thereof by Huntington employing the familiar operators of propositional logic ([[Logical conjunction|AND]], [[Logical disjunction|OR]], [[Negation|NOT]]). Because of self-[[Duality (order theory)|duality]] of Boolean algebras, Sheffer's axioms are equally valid for either of the NAND or NOR operations in place of the stroke. Sheffer interpreted the stroke as a sign for nondisjunction ([[Logical NOR|NOR]]) in his paper, mentioning non-conjunction only in a footnote and without a special sign for it. It was [[Biography:Jean Nicod|Jean Nicod]] who first used the stroke as a sign for non-conjunction (NAND) in a paper of 1917 and which has since become current practice.<ref name="Nicod_1917"/><ref name="Church_1956"/> Russell and Whitehead used the Sheffer stroke in the 1927 second edition of ''[[Principia Mathematica]]'' and suggested it as a replacement for the "OR" and "NOT" operations of the first edition.

[[Biography:Charles Sanders Peirce|Charles Sanders Peirce]] (1880) had discovered the [[Functional completeness|functional completeness]] of NAND or NOR more than 30 years earlier, using the term ''ampheck'' (for 'cutting both ways'), but he never published his finding. Two years before Sheffer, {{ill|Edward Stamm|pl|Edward Bronisław Stamm}} also described the NAND and NOR operators and showed that the other Boolean operations could be expressed by it.<ref name="Stamm_1911"/>

==Properties==
NAND does not possess any of the following five properties, each of which is required to be absent from, and the absence of all of which is sufficient for, at least one member of a set of functionally complete operators: truth-preservation, falsity-preservation, [[Affine transformation|linearity]], monotonicity, self-duality. (An operator is truth- (or falsity-)preserving if its value is truth (falsity) whenever all of its arguments are truth (falsity).) Therefore {NAND} is a functionally complete set.

This can also be realized as follows: All three elements of the functionally complete set {AND, OR, NOT} can be constructed using only NAND. Thus the set {NAND} must be functionally complete as well.

==Other Boolean operations in terms of the Sheffer stroke==
Expressed in terms of NAND <math>\uparrow</math>, the usual operators of propositional logic are:

{|
|-
|<!--- not --->
{| style="text-align: center; border: 1px solid darkgray;"
|-
|<math>\neg P</math>
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|<math>P</math>
|<math>\uparrow</math>
|<math>P</math>
|-
|[[File:Venn10.svg|36px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn01.svg|36px]]
|<math>\uparrow</math>
|[[File:Venn01.svg|36px]]
|}<!--- end not--->

|&nbsp;&nbsp;&nbsp;
|<!--- arrow --->
{| style="text-align: center; border: 1px solid darkgray;"
|-
|<math>P \rightarrow Q</math>
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|<math>~P</math>
|<math>\uparrow</math>
|<math>(Q \uparrow Q)</math>
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|<math>~P</math>
|<math>\uparrow</math>
|<math>(P \uparrow Q)</math>
|-
|[[File:Venn1011.svg|50px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn0101.svg|50px]]
|<math>\uparrow</math>
|[[File:Venn1100.svg|50px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn0101.svg|50px]]
|<math>\uparrow</math>
|[[File:Venn1110.svg|50px]]
|}<!--- end arrow --->

|&nbsp;&nbsp;&nbsp;
|<!--- bi-arrow/equivalence --->
{| style="text-align: center; border: 1px solid darkgray;"
|-
|<math>P \leftrightarrow Q</math>
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|<math>(P \uparrow Q)</math>
|<math>\uparrow</math>
|<math>((P \uparrow P) \uparrow (Q \uparrow Q))</math>
|-
|[[File:Venn1001.svg|50px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn1110.svg|50px]]
|<math>\uparrow</math>
|[[File:Venn0111.svg|50px]]
|}<!--- end bi-arrow/equivalence --->

|-
|&nbsp;
|-
|<!--- and --->
{| style="text-align: center; border: 1px solid darkgray;"
|-
|<math>P \land Q</math>
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|<math>(P \uparrow Q)</math>
|<math>\uparrow</math>
|<math>(P \uparrow Q)</math>
|-
|[[File:Venn0001.svg|50px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn1110.svg|50px]]
|<math>\uparrow</math>
|[[File:Venn1110.svg|50px]]
|}<!--- end and --->
|&nbsp;&nbsp;&nbsp;
|<!--- or --->
{| style="text-align: center; border: 1px solid darkgray;"
|-
|<math>P \lor Q</math>
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|<math>(P \uparrow P)</math>
|<math>\uparrow</math>
|<math>(Q \uparrow Q)</math>
|-
|[[File:Venn0111.svg|50px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn1010.svg|50px]]
|<math>\uparrow</math>
|[[File:Venn1100.svg|50px]]
|}<!--- end or --->
|}

==Formal system based on the Sheffer stroke==
The following is an example of a [[Formal system|formal system]] based entirely on the Sheffer stroke, yet having the functional expressiveness of the propositional logic:

===Symbols===
''p<sub>n</sub>'' for natural numbers ''n'':
:( | )

The Sheffer stroke commutes but does not associate (e.g., <span class="texhtml">(T | T) | F = T</span>, but <span class="texhtml">T | (T | F) = F</span>). Hence any formal system including the Sheffer stroke as an infix symbol must also include a means of indicating grouping (grouping is automatic if the stroke is used as a prefix, thus: <span class="texhtml">|| TTF = T</span> and <span class="texhtml">| T | TF = F</span>). We shall employ '(' and ')' to this effect.

We also write ''p'', ''q'', ''r'', … instead of ''p''<sub>0</sub>, ''p''<sub>1</sub>, ''p''<sub>2</sub>.

===Syntax===
''Construction rule I:'' For each natural number ''n'', the symbol ''p<sub>n</sub>'' is a [[Well-formed formula|well-formed formula]] (WFF), called an atom.

''Construction rule II:'' If ''X'' and ''Y'' are WFFs, then (''X''&nbsp;|&nbsp;''Y'') is a WFF.

''Closure rule:'' Any formulae which cannot be constructed by means of the first two construction rules are not WFFs.

The letters ''U'', ''V'', ''W'', ''X'', and ''Y'' are metavariables standing for WFFs.

A decision procedure for determining whether a formula is well-formed goes as follows: "deconstruct" the formula by applying the construction rules backwards, thereby breaking the formula into smaller subformulae. Then repeat this recursive deconstruction process to each of the subformulae. Eventually the formula should be reduced to its atoms, but if some subformula cannot be so reduced, then the formula is not a WFF.

===Calculus===
All WFFs of the form
:<span class="texhtml">((''U'' | (''V'' | ''W'')) | ((''Y'' | (''Y'' | ''Y'')) | ((''X'' | ''V'') | ((''U'' | ''X'') | (''U'' | ''X'')))))</span>
are axioms. Instances of
:<math>(U | (V | W)), U \vdash W</math>
are inference rules.

===Simplification===
Since the only connective of this logic is |, the symbol | could be discarded altogether, leaving only the parentheses to group the letters. A pair of parentheses must always enclose a pair of ''WFF''s. Examples of theorems in this simplified notation are

: (''p''(''p''(''q''(''q''((''pq'')(''pq'')))))),

: (''p''(''p''((''qq'')(''pp'')))).

The notation can be simplified further, by letting
: {{math|1=(''U'') := (''UU'')}}
: <math>((U)) \equiv U</math>
for any ''U''. This simplification causes the need to change some rules:
# More than two letters are allowed within parentheses.
# Letters or WFFs within parentheses are allowed to commute.
# Repeated letters or WFFs within a same set of parentheses can be eliminated.
The result is a parenthetical version of the Peirce [[Existential graph|existential graph]]s.

Another way to simplify the notation is to eliminate parentheses by using [[Polish notation]] (PN). For example, the earlier examples with only parentheses could be rewritten using only strokes as follows

: (''p''(''p''(''q''(''q''((''pq'')(''pq'')))))) becomes
: <span class="texhtml">| ''p'' | ''p'' | ''q'' | ''q'' || ''pq'' | ''pq''</span>, and

: (''p''(''p''((''qq'')(''pp'')))) becomes,
: <span class="texhtml">| ''p'' | ''p'' || ''qq'' | ''pp''</span>.

This follows the same rules as the parenthesis version, with the opening parenthesis replaced with a Sheffer stroke and the (redundant) closing parenthesis removed.

Or (for some formulas) one could omit both parentheses ''and'' strokes and allow the order of the arguments to determine the order of [[Function application|function application]] so that for example, applying the function from right to left (reverse Polish notation – any other unambiguous convention based on ordering would do)

: <math>
\begin{align}
& pqr \equiv (p\mid (q\mid r)), \text{ whereas} \\
& rqp \equiv (r \mid (q\mid p)).
\end{align}
</math>

==See also==
{{Div col|colwidth=30em}}
* [[Boolean domain]]
* [[Engineering:CMOS|CMOS]]
* [[Physics:Gate equivalent|Gate equivalent]] (GE)
* [[Logical graph]]
* [[Minimal axioms for Boolean algebra]]
* NAND [[Flash memory|flash memory]]
* [[Engineering:NAND logic|NAND logic]]
* [[Peirce's law]]
* Peirce arrow = NOR
* Sole sufficient operator
{{div col end}}

==References==
{{reflist|refs=
<ref name="Stamm_1911">{{cite journal |author-first=Edward Bronisław |author-last=Stamm |journal=Monatshefte für Mathematik und Physik |title=Beitrag zur Algebra der Logik |language=de |volume=22 |issue=1 |date=1911 |doi=10.1007/BF01742795 |pages=137–149|s2cid=119816758 }}</ref>
<ref name="Sheffer_1913">{{cite journal |author-first=Henry Maurice |author-last=Sheffer |date=1913 |title=A set of five independent postulates for Boolean algebras, with application to logical constants |journal=[[Organization:Transactions of the American Mathematical Society|Transactions of the American Mathematical Society]] |volume=14 |issue=4 |jstor=1988701 |doi=10.2307/1988701 |doi-access=free |pages=481–488}}</ref>
<ref name="Church_1956">{{cite book |author-last=Church |author-first=Alonzo |title=Introduction to mathematical logic |volume=1 |date=1956 |publisher=Princeton University Press |page=134}}</ref>
<ref name="Nicod_1917">{{cite journal |author-last=Nicod |author-first=Jean George Pierre |date=1917 |title=A Reduction in the Number of Primitive Propositions of Logic |journal=Proceedings of the Cambridge Philosophical Society |volume=19 |pages=32–41}}</ref>
}}

==Further reading==
* {{cite book |author-first1=Józef Maria |author-last1=Bocheński |date=1960 |title=Precis of Mathematical Logic |author-first2=Albert Heinrich |author-last2=Menne |author-link2=:de:Albert Heinrich Menne |edition=revised |translator-first=Otto |translator-last=Bird |publisher=D. Reidel |publication-place=Dordrecht, South Holland, Netherlands}} (NB. Edited and translated from the French and German editions: Précis de logique mathématique)
* {{cite book |chapter=A Boolian<!-- sic! --> Algebra with One Constant |author-first=Charles Sanders |author-last=Peirce |orig-date=1880 |title=Collected Papers of Charles Sanders Peirce |title-link=Charles Sanders Peirce bibliography#CP |volume=4 |pages=12–20 |publication-place=Cambridge |publisher=Harvard University Press |editor-first1=Charles |editor-last1=Hartshorne |editor-first2=Paul |editor-last2=Weiss |editor-link2=Paul Weiss (philosopher) |date=1931–1935}}

==External links==
* [http://www.iep.utm.edu/Sheffer_stroke/ Sheffer Stroke] article in the ''[[Internet Encyclopedia of Philosophy]]''
* http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/nand.html
* [http://www.sccs.swarthmore.edu/users/06/adem/engin/e77vlsi/lab3/ Implementations of 2- and 4-input NAND gates]
* [https://web.archive.org/web/20090526075041/http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.pja/1195520940&page=record Proofs of some axioms by Stroke function by Yasuo Setô] @ [http://projecteuclid.org Project Euclid]

{{Logical connectives}}
{{Common logical symbols}}

[[Category:Logic gates|NAND gate]]
[[Category:Logical connectives]]
[[Category:Logic symbols]]

{{Sourceattribution|Sheffer stroke}}