Sheffer stroke
NAND | |
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Definition | |
Truth table | |
Logic gate | ![]() |
Normal forms | |
Disjunctive | |
Conjunctive | |
Zhegalkin polynomial | |
Post's lattices | |
0-preserving | no |
1-preserving | no |
Monotone | no |
Affine | no |
Self-dual | no |
Logical connectives | ||||||||||||||||||||||||||
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Related concepts | ||||||||||||||||||||||||||
Applications | ||||||||||||||||||||||||||
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In
Its
Definition
The non-conjunction is a
Truth table
The truth table of is as follows.
F | F | T |
F | T | T |
T | F | T |
T | T | F |
Logical equivalences
The Sheffer stroke of and is the negation of their conjunction
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By De Morgan's laws, this is also equivalent to the disjunction of the negations of and
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Alternative notations and names
Peirce was the first to show the functional completeness of non-conjunction (representing this as ) but didn't publish his result.[2][3] Peirce's editor added ) 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 (the Stamm hook)[4] and non-disjunction in print at the first time and showed their functional completeness.[5]
In 1913,
In 1928, Hilbert and Ackermann described non-conjunction with the operator .[6][7]
In 1929, Łukasiewicz used in for non-conjunction in his Polish notation.[8]
An alternative notation for non-conjunction is . It is not clear who first introduced this notation, although the corresponding for non-disjunction was used by Quine in 1940.[9]
History
The stroke is named after
Properties
NAND is commutative but not associative, which means that but .[13]
Functional completeness
The Sheffer stroke, taken by itself, is a
It can also be proved by first showing, with a truth table, that is truth-functionally equivalent to .[17] Then, since is truth-functionally equivalent to ,[17] and is equivalent to ,[17] the Sheffer stroke suffices to define the set of connectives ,
Other Boolean operations in terms of the Sheffer stroke
Expressed in terms of NAND , the usual operators of propositional logic are:
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See also
- 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
References
- ^ ISBN 978-0-415-13342-5.
- ^ 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.
- ^ 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.
- ^ Zach, R. (2023-02-18). "Sheffer stroke before Sheffer: Edward Stamm". Retrieved 2023-07-02.
- ^ S2CID 119816758.
- ^ Hilbert, D.; Ackermann, W. (1928). Grundzügen der theoretischen Logik (in German) (1 ed.). Berlin: Verlag von Julius Springer. p. 9.
- ^ 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.
- ^ Łukasiewicz, J. (1958) [1929]. Elementy logiki matematycznej (in Polish) (2 ed.). Warszawa: Państwowe Wydawnictwo Naukowe.
- ^ Quine, W. V (1981) [1940]. Mathematical Logic (Revised ed.). Cambridge, London, New York, New Rochelle, Melbourne and Sydney: Harvard University Press. p. 45.
- JSTOR 1988701.
- Proceedings of the Cambridge Philosophical Society. 19: 32–41.
- ^ Church, Alonzo (1956). Introduction to mathematical logic. Vol. 1. Princeton University Press. p. 134.
- ISBN 978-81-88237-49-4.
- ^ Weisstein, Eric W. "Propositional Calculus". mathworld.wolfram.com. Retrieved 2024-03-22.
- ^ 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
- ISBN 9781400882366.
- ^ ISBN 978-0-415-13342-5.
Further reading
- Précis de logique mathématique)
- Peirce, Charles Sanders (1931–1935) [1880]. "A Boolian Algebra with One Constant". In Hartshorne, Charles; Weiss, Paul (eds.). Collected Papers of Charles Sanders Peirce. Vol. 4. Cambridge: Harvard University Press. pp. 12–20.