# Sheffer stroke

Sheffer stroke
NAND
Definition${\displaystyle {\overline {x\cdot y}}}$
Truth table${\displaystyle (1110)}$
Logic gate
Normal forms
Disjunctive${\displaystyle {\overline {x}}+{\overline {y}}}$
Conjunctive${\displaystyle {\overline {x}}+{\overline {y}}}$
Zhegalkin polynomial${\displaystyle 1\oplus xy}$
Post's lattices
0-preservingno
1-preservingno
Monotoneno
Affineno
Self-dualno

In

Henry Maurice Sheffer
and written as ${\displaystyle \mid }$ or as ${\displaystyle \uparrow }$ or as ${\displaystyle {\overline {\wedge }}}$ or as ${\displaystyle Dpq}$ in Polish notation by Łukasiewicz (but not as ||, often used to represent disjunction).

Its

computer processor
design.

## Definition

The non-conjunction is a

logical values. It produces a value of true, if — and only if — at least one of the propositions
is false.

### Truth table

The truth table of ${\displaystyle A\uparrow B}$ is as follows.

${\displaystyle A}$${\displaystyle B}$${\displaystyle A\uparrow B}$
FFT
FTT
TFT
TTF

### Logical equivalences

The Sheffer stroke of ${\displaystyle P}$ and ${\displaystyle Q}$ is the negation of their conjunction

 ${\displaystyle P\uparrow Q}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle \neg (P\land Q)}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle \neg }$

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

 ${\displaystyle P\uparrow Q}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle \neg P}$ ${\displaystyle \lor }$ ${\displaystyle \neg Q}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle \lor }$

## Alternative notations and names

Peirce was the first to show the functional completeness of non-conjunction (representing this as ${\displaystyle {\overline {\curlywedge }}}$) but didn't publish his result.[2][3] Peirce's editor added ${\displaystyle {\overline {\curlywedge }}}$) for non-disjunction[citation needed].[3]

In 1911, Stamm was the first to publish a proof of the completeness of non-conjunction, representing this with ${\displaystyle \sim }$ (the Stamm hook)[4] and non-disjunction in print at the first time and showed their functional completeness.[5]

In 1913,

Sheffer
described non-disjunction using ${\displaystyle \mid }$ and showed its functional completeness. Sheffer also used ${\displaystyle \wedge }$ for non-disjunction.[citation needed] Many people, beginning with Nicod in 1917, and followed by Whitehead, Russell and many others, mistakenly thought Sheffer has described non-conjunction using ${\displaystyle \mid }$, naming this the Sheffer Stroke.

In 1928, Hilbert and Ackermann described non-conjunction with the operator ${\displaystyle /}$.[6][7]

In 1929, Łukasiewicz used ${\displaystyle D}$ in ${\displaystyle Dpq}$ for non-conjunction in his Polish notation.[8]

An alternative notation for non-conjunction is ${\displaystyle \uparrow }$. It is not clear who first introduced this notation, although the corresponding ${\displaystyle \downarrow }$ for non-disjunction was used by Quine in 1940,.[9]

## History

The stroke is named after

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.[11][12] 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.

ampheck (for 'cutting both ways'), but he never published his finding. Two years before Sheffer, Edward Stamm [pl] also described the NAND and NOR operators and showed that the other Boolean operations could be expressed by it.[5]

## 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

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 ${\displaystyle \uparrow }$, the usual operators of propositional logic are:

 ${\displaystyle \neg P}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle P}$ ${\displaystyle \uparrow }$ ${\displaystyle P}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle \uparrow }$

 ${\displaystyle P\rightarrow Q}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle ~P}$ ${\displaystyle \uparrow }$ ${\displaystyle (Q\uparrow Q)}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle ~P}$ ${\displaystyle \uparrow }$ ${\displaystyle (P\uparrow Q)}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle \uparrow }$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle \uparrow }$

 ${\displaystyle P\leftrightarrow Q}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle (P\uparrow Q)}$ ${\displaystyle \uparrow }$ ${\displaystyle ((P\uparrow P)\uparrow (Q\uparrow Q))}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle \uparrow }$

 ${\displaystyle P\land Q}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle (P\uparrow Q)}$ ${\displaystyle \uparrow }$ ${\displaystyle (P\uparrow Q)}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle \uparrow }$

 ${\displaystyle P\lor Q}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle (P\uparrow P)}$ ${\displaystyle \uparrow }$ ${\displaystyle (Q\uparrow Q)}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle \uparrow }$

## Functional completeness

The Sheffer stroke, taken by itself, is a functionally complete set of connectives.[13][14] This can be proved by first showing, with a truth table, that ${\displaystyle \neg A}$ is truth-functionally equivalent to ${\displaystyle A\uparrow A}$.[15] Then, since ${\displaystyle A\uparrow B}$ is truth-functionally equivalent to ${\displaystyle \neg (A\land B)}$,[15] and ${\displaystyle A\lor B}$ is equivalent to ${\displaystyle \neg (\neg A\land \neg B)}$,[15] the Sheffer stroke suffices to define the set of connectives ${\displaystyle \{\land ,\lor ,\neg \}}$,

Disjunctive Normal Form Theorem.[15]

## References

1. ^ .
2. ^ Peirce, C. S. (1933) [1880]. "A Boolian Algebra with One Constant". In Hartshorne, C.; Weiss, P. (eds.). Collected Papers of Charles Sanders Peirce, Volume IV The Simplest Mathematics. Massachusetts: Harvard University Press. pp. 13–18.
3. ^ a b Peirce, C. S. (1933) [1902]. "The Simplest Mathematics". In Hartshorne, C.; Weiss, P. (eds.). Collected Papers of Charles Sanders Peirce, Volume IV The Simplest Mathematics. Massachusetts: Harvard University Press. pp. 189–262.
4. ^ Zach, R. (2023-02-18). "Sheffer stroke before Sheffer: Edward Stamm". Retrieved 2023-07-02.
5. ^
S2CID 119816758
.
6. ^ Hilbert, D.; Ackermann, W. (1928). Grundzügen der theoretischen Logik (in German) (1 ed.). Berlin: Verlag von Julius Springer. p. 9.
7. ^ Hilbert, D.; Ackermann, W. (1950). Luce, R. E. (ed.). Principles of Mathematical Logic. Translated by Hammond, L. M.; Leckie, G. G.; Steinhardt, F. New York: Chelsea Publishing Company. p. 11.
8. ^ Łukasiewicz, J. (1958) [1929]. Elementy logiki matematycznej (in Polish) (2 ed.). Warszawa: Państwowe Wydawnictwo Naukowe.
9. ^ Quine, W. V (1981) [1940]. Mathematical Logic (Revised ed.). Cambridge, London, New York, New Rochelle, Melbourne and Sydney: Harvard University Press. p. 45.
10. JSTOR 1988701
.
11. Proceedings of the Cambridge Philosophical Society
. 19: 32–41.
12. ^ Church, Alonzo (1956). Introduction to mathematical logic. Vol. 1. Princeton University Press. p. 134.
13. ^ Weisstein, Eric W. "Propositional Calculus". mathworld.wolfram.com. Retrieved 2024-03-22.
14. ^ Franks, Curtis (2023), "Propositional Logic", in Zalta, Edward N.; Nodelman, Uri (eds.), The Stanford Encyclopedia of Philosophy (Fall 2023 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-03-22
15. ^ .