Sheffer stroke

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 propositions is false.

The truth table of $A\; \backslash uparrow\; B$ is as follows.

{{2-ary truth table|1|1|1|0|$A\; \backslash uparrow\; B$}}

The Sheffer stroke of $P$ and $Q$ is the negation of their conjunction

By De Morgan's laws, this is also equivalent to the disjunction of the negations of $P$ and $Q$

Peirce was the first to show the functional completeness of non-conjunction (representing this as $\backslash overline\{\backslash curlywedge\}$) but didn't publish his result.{{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}}{{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}} Peirce's editor added $\backslash overline\{\backslash curlywedge\}$) for non-disjunction.

In 1911, {{ill|Stamm|pl|Edward Bronisław Stamm}} was the first to publish a proof of the completeness of non-conjunction, representing this with $\backslash sim$ (the Stamm hook){{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}} and non-disjunction in print at the first time and showed their functional completeness.

In 1913, Sheffer described non-disjunction using $\backslash mid$ and showed its functional completeness. Sheffer also used $\backslash wedge$ for non-disjunction. Many people, beginning with Nicod in 1917, and followed by Whitehead, Russell and many others{{Who|date=May 2025}}, mistakenly thought Sheffer had described non-conjunction using $\backslash mid$, naming this symbol the Sheffer stroke.{{Citation needed|date=May 2025}}

In 1928, Hilbert and Ackermann described non-conjunction with the operator $/$.{{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}}{{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}}

In 1929, Łukasiewicz used $D$ in $Dpq$ for non-conjunction in his Polish notation.{{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}}

An alternative notation for non-conjunction is $\backslash uparrow$. It is not clear who first introduced this notation, although the corresponding $\backslash downarrow$ for non-disjunction was used by Quine in 1940.{{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}}

The stroke is named after Henry Maurice Sheffer, who in 1913 published a paper in the Transactions of the American Mathematical Society providing an axiomatization of Boolean algebras using the stroke, and proved its equivalence to a standard formulation thereof by Huntington employing the familiar operators of propositional logic (AND, OR, NOT). Because of self-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 (NOR) in his paper, mentioning non-conjunction only in a footnote and without a special sign for it. It was 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. 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.

Charles Sanders Peirce (1880) had discovered the 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.

NAND is commutative but not associative, which means that $P\; \backslash uparrow\; Q\; \backslash leftrightarrow\; Q\; \backslash uparrow\; P$ but $(P\; \backslash uparrow\; Q)\; \backslash uparrow\; R\; \backslash not\backslash leftrightarrow\; P\; \backslash uparrow\; (Q\; \backslash uparrow\; R)$.{{Cite book |last=Rao |first=G. Shanker |url=https://books.google.com/books?id=M-5m_EdvxuIC |title=Mathematical Foundations of Computer Science |date=2006 |publisher=I. K. International Pvt Ltd |isbn=978-81-88237-49-4 |pages=21 |language=en}}

The Sheffer stroke, taken by itself, is a functionally complete set of connectives.{{Cite web |last=Weisstein |first=Eric W. |title=Propositional Calculus |url=https://mathworld.wolfram.com/ |access-date=2024-03-22 |website=mathworld.wolfram.com |language=en}}{{Citation |last=Franks |first=Curtis |title=Propositional Logic |date=2023 |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/archives/fall2023/entries/logic-propositional/ |access-date=2024-03-22 |edition=Fall 2023 |publisher=Metaphysics Research Lab, Stanford University |editor2-last=Nodelman |editor2-first=Uri |encyclopedia=The Stanford Encyclopedia of Philosophy}} This can be seen from the fact that 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, linearity, monotonicity, self-duality. (An operator is truth-preserving if its value is truth whenever all of its arguments are truth, or falsity-preserving if its value is falsity whenever all of its arguments are falsity.){{cite book | url=https://dokumen.pub/qdownload/the-two-valued-iterative-systems-of-mathematical-logic-am-5-volume-5-9781400882366.html | isbn=9781400882366 | doi=10.1515/9781400882366 | author=Emil Leon Post | title=The Two-Valued Iterative Systems of Mathematical Logic | location=Princeton | publisher=Princeton University Press | series=Annals of Mathematics studies | volume=5 | date=1941 }}

It can also be proved by first showing, with a truth table, that $\backslash neg\; A$ is truth-functionally equivalent to $A\; \backslash uparrow\; A$.{{Cite book |last=Howson |first=Colin |title=Logic with trees: an introduction to symbolic logic |date=1997 |publisher=Routledge |isbn=978-0-415-13342-5 |location=London; New York |pages=41–43}} Then, since $A\; \backslash uparrow\; B$ is truth-functionally equivalent to $\backslash neg\; (A\; \backslash land\; B)$, and $A\; \backslash lor\; B$ is equivalent to $\backslash neg(\backslash neg\; A\; \backslash land\; \backslash neg\; B)$, the Sheffer stroke suffices to define the set of connectives $\backslash \{\backslash land,\; \backslash lor,\; \backslash neg\backslash \}$, which is shown to be truth-functionally complete by the Disjunctive Normal Form Theorem.

Expressed in terms of NAND $\backslash uparrow$, the usual operators of propositional logic are:

|   

|

|   

|

|-

| 

|-

|

|   

|

|}

{{Div col|colwidth=30em}}

• Boolean domain
• CMOS
• Gate equivalent (GE)
• Logical graph
• Minimal axioms for Boolean algebra
• NAND flash memory
• NAND logic
• Peirce's law
• Peirce arrow = NOR
• Sole sufficient operator

{{div col end}}

{{reflist|refs=

{{cite journal |author-first=Edward Bronisław |author-last=Stamm |author-link=:pl:Edward Bronisław 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 }}

{{cite journal |author-first=Henry Maurice |author-last=Sheffer |author-link=Henry Maurice Sheffer |date=1913 |title=A set of five independent postulates for Boolean algebras, with application to logical constants |journal=Transactions of the American Mathematical Society |volume=14 |issue=4 |jstor=1988701 |doi=10.2307/1988701 |doi-access=free |pages=481–488}}

{{cite book |author-last=Church |author-first=Alonzo |author-link=Alonzo Church |title=Introduction to mathematical logic |volume=1 |date=1956 |publisher=Princeton University Press |page=134}}

{{cite journal |author-last=Nicod |author-first=Jean George Pierre |author-link=Jean George Pierre Nicod |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}}

}}

• {{cite book |author-first1=Józef Maria |author-last1=Bocheński |author-link1=Józef Maria 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 Algebra with One Constant |author-first=Charles Sanders |author-last=Peirce |author-link=Charles Sanders 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-link1=Charles Hartshorne |editor-first2=Paul |editor-last2=Weiss |editor-link2=Paul Weiss (philosopher) |date=1931–1935}}

• [https://iep.utm.edu/sheffers/ 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}}

NAND gate

Category:Logical connectives

Category:Logic symbols