Sheffer stroke

Sheffer stroke

In Boolean functions and propositional calculus, the Sheffer stroke, named after Henry M. Sheffer, written "|" (see vertical bar, not to be confused with "||" which is often used to represent disjunction), "Dpq", or "↑", denotes a logical operation that is equivalent to the negation of the conjunction operation, expressed in ordinary language as "not both". It is also called the alternative denial, since it says in effect that at least one of its operands is false. In Boolean algebra and digital electronics it is known as the NAND operation ("not and").

Like its dual, the NOR operator (a.k.a. the Peirce arrow or Quine dagger), NAND can be used by itself, without any other logical operator, to constitute a logical formal system (making NAND functionally complete). This property makes the NAND gate crucial to modern digital electronics, including its use in NAND flash memory and computer processor design.

Contents

Definition

The NAND operation is a logical operation on two logical values, typically the values of two propositions, that produces a value of false if and only if both of its operands are true. In other words, it produces a value of true if and only if at least one of its operands is false.

Truth table

The truth table of A NAND B (also written as A | B, Dpq, or A ↑ B) is as follows:

INPUT OUTPUT
A B A NAND B
0 0 1
0 1 1
1 0 1
1 1 0

History

The stroke is named after Henry M. Sheffer, who provided (Sheffer 1913) 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 non-disjunction (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.[1]

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.

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, linearity, monotonicity, self-duality. (An operator is truth- (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.

Introduction, elimination, and equivalencies

The Sheffer stroke \uparrow is the negation of the conjunction:

P \uparrow Q     \Leftrightarrow     \neg (P \and Q)
Venn1110.svg     \Leftrightarrow     Venn0001.svg

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

\neg P     \Leftrightarrow     P \uparrow P
Venn01.svg     \Leftrightarrow     Venn10.svg
   
P \rightarrow Q     \Leftrightarrow     ~P \uparrow (Q \uparrow Q)     \Leftrightarrow     ~P \uparrow (P \uparrow Q)
Venn1011.svg     \Leftrightarrow     Venn0101.svg \uparrow Venn1100.svg     \Leftrightarrow     Venn0101.svg \uparrow Venn1110.svg
 
P \and Q     \Leftrightarrow     (P \uparrow Q) \uparrow (P \uparrow Q)
Venn0001.svg     \Leftrightarrow     Venn1110.svg \uparrow Venn1110.svg
   
P \or Q     \Leftrightarrow     (P \uparrow P) \uparrow (Q \uparrow Q)
Venn0111.svg     \Leftrightarrow     Venn1010.svg \uparrow Venn1100.svg

Formal system based on the Sheffer stroke

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

Symbols

pn for natural numbers n
( | )

The Sheffer stroke commutes but does not associate (e.g., (T|T)|F = T, but T|(T|F) = F). Hence any formal system including the Sheffer stroke must also include a means of indicating grouping. We shall employ '(' and ')' to this effect.

We also write p, q, r, … instead of p0, p1, p2.

Syntax

Construction Rule I: For each natural number n, the symbol pn is a well-formed formula (wff), called an atom.

Construction Rule II: If X and Y are wffs, then (X|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

((U|(V|W))|((Y|(Y|Y))|((X|V)|((U|X)|(U|X)))))

are axioms. Instances of

(U|(V|W)), U \vdash W

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 wffs. 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

(U) := (UU)
((U)) \equiv U

for any U. This simplification causes the need to change some rules:

  1. More than two letters are allowed within parentheses.
  2. Letters or wffs within parentheses are allowed to commute.
  3. Repeated letters or wffs within a same set of parentheses can be eliminated.

The result is a parenthetical version of the Peirce existential graphs.

Another way to simplify the notation is to eliminate parenthesis by using Polish Notation. For example, the earlier examples with only parenthesis could be rewritten using only strokes as follows

(p(p(q(q((pq)(pq)))))) becomes
|p|p|q|q||pq|pq, and
(p(p((qq)(pp)))) becomes,
|p|p||qq|pp.

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

See also

Notes

References

External links


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • Sheffer — Henry Maurice Sheffer Henry Maurice Sheffer (1882 1964) est un logicien américain. Sommaire 1 Vie 2 Travaux 3 Références 4 Voir aussi …   Wikipédia en Français

  • stroke function — Sheffer io funkcija statusas T sritis automatika atitikmenys: angl. NOT AND function; Sheffer function; stroke function vok. NICHT UND Funktion, f; Sheffer Funktion, f rus. функция НЕ И, f; функция Шеффера, f pranc. fonction de Sheffer, f;… …   Automatikos terminų žodynas

  • Sheffer function — Sheffer io funkcija statusas T sritis automatika atitikmenys: angl. NOT AND function; Sheffer function; stroke function vok. NICHT UND Funktion, f; Sheffer Funktion, f rus. функция НЕ И, f; функция Шеффера, f pranc. fonction de Sheffer, f;… …   Automatikos terminų žodynas

  • Sheffer'io funkcija — statusas T sritis automatika atitikmenys: angl. NOT AND function; Sheffer function; stroke function vok. NICHT UND Funktion, f; Sheffer Funktion, f rus. функция НЕ И, f; функция Шеффера, f pranc. fonction de Sheffer, f; fonction NON OT, f ryšiai …   Automatikos terminų žodynas

  • Sheffer-Funktion — Sheffer io funkcija statusas T sritis automatika atitikmenys: angl. NOT AND function; Sheffer function; stroke function vok. NICHT UND Funktion, f; Sheffer Funktion, f rus. функция НЕ И, f; функция Шеффера, f pranc. fonction de Sheffer, f;… …   Automatikos terminų žodynas

  • Sheffer's stroke — Both Peirce in 1880 and the American logician H. M. Sheffer in 1913 realized that the truth functions of elementary logic could all be defined from a single function. Defining p / q to mean neither p nor q, p / p is equivalent to not p, (p / q )/ …   Philosophy dictionary

  • sheffer's stroke — ˈshefə(r)z noun Usage: usually capitalized 1st S Etymology: after Henry M. Sheffer died 1964 American philosopher 1. : joint denial 2. : alternative denial …   Useful english dictionary

  • Henry M. Sheffer — Henry Maurice Sheffer (1882 1964) was an American logician.Sheffer was a Polish Jew born in the Ukraine, who immigrated to the USA with his parents. He was educated at Harvard University, learning logic from Josiah Royce. Sheffer spent most of… …   Wikipedia

  • Henry M. Sheffer — Henry Maurice Sheffer Henry Maurice Sheffer (1882 1964) est un logicien américain. Sommaire 1 Vie 2 Travaux 3 Références 4 Voir aussi …   Wikipédia en Français

  • Henry Sheffer — Henry Maurice Sheffer Henry Maurice Sheffer (1882 1964) est un logicien américain. Sommaire 1 Vie 2 Travaux 3 Références 4 Voir aussi …   Wikipédia en Français

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”