Sheffer stroke
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- This article is about NAND in the logical sense. For the electronic NAND gates see NAND gate.
The Sheffer stroke, written "|", denotes the negation of the conjunction operation. It is also called alternative denial, and in Boolean algebra and digital circuitry, it is known as the NAND ("Not And") operation. It alone suffices to express all of Boolean algebra or sentential logic. It is named for Henry M. Sheffer, who proved (Sheffer 1913) that all the usual operators of the propositional calculus (not, and, or, implies), etc. could be expressed in terms of it. Charles Peirce (1880) had discovered this fact more than 30 years earlier, but never published his finding. Peirce also observed that all Boolean operators could be defined in terms of the NOR operator, the dual of NAND.
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NAND
A common means of writing p NAND q is <math>\overline{p \cdot q}</math>, where the symbol <math>\cdot</math> signifies AND and the line over the expression signifies not, the logical negation of that expression.
The two-input logical NAND operator is commonly described by a truth table, describing the output state for all possible input combinations:
| A | B | A NAND B |
|---|---|---|
| F | F | T |
| F | T | T |
| T | F | T |
| T | T | F |
Expressed in terms of NAND, the usual operators of propositional calculus are:
| "not p" is equivalent to "p NAND p" | <math>\overline{p} \equiv \overline{p \cdot p}</math> |
| "p and q" is equivalent to "(p NAND q) NAND (p NAND q)" | <math>p \cdot q \equiv \overline{\overline{(p \cdot q)} \cdot \overline{(p \cdot q)}}</math> |
| "p or q" is equivalent to "(p NAND p) NAND (q NAND q)" | <math>p + q \equiv \overline{\overline{(p \cdot p)} \cdot \overline{(q \cdot q)}}</math> |
| "p implies q" is equivalent to "(p NAND q) NAND p" | <math>p \rightarrow q \equiv \overline{\overline{(p \cdot q)} \cdot p}</math> |
This leads to an alternative axiom system for Boolean algebras, requiring but one operation.
Digital systems employing certain logic circuits take advantage of this property. In complicated logical expressions, normally written in terms of other logic functions such as AND, OR, and NOT, writing these in terms of NAND saves on cost, because implementing such circuits using NAND gate yields a more compact result than the alternatives.
The dual of NAND, the operator NOR, also suffices to express all Boolean operations.
Sheffer stroke
The Sheffer stroke | is equivalent to the negation of conjunction:
- <math> A | B = \neg (A \wedge B). </math>
The following truth table semantically defines |:
| </math> | F | T |
|---|---|---|
| F | T | T |
| T | T | F |
The other logical operators can be defined in terms of '|', like so:
- <math> \neg A = A | A, </math>
- <math> A \wedge B = (A | B) | (A | B), </math>
- <math> A \vee B = (A | A) | (B | B), </math>
- <math> A \rightarrow B = A | (B | B) = A | (A | B). </math>
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 calculus:
1. Symbols
A B C D E F G '
( | )
The Sheffer stroke commutes but does not associate. Hence any formal system including the Sheffer stroke must also include a means of indicating grouping. We shall employ '(' and ')' to this effect.
2. Grammar
The letters A, B, C, D, E, F and G are atoms.
Any of these letters primed once or several times is also an atom (e.g. A', B′′, C′′′, D′′′′ are atoms).
Construction Rule I: An atom is a well-formed formula (wff).
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 is not a wff.
The letters U, V, 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.
3. Axiom
The following wffs are axiom schemata, which become axioms upon replacing all metavariables with wffs.
THEN-1: (U|(U|(V|(U|U))))
4. Inference rules
Substitution of equivalents. Let the wff X contain one more instances of the subformula U. If U=V, then replacing one ore more instances of U in X by V does not alter the truth value of X. In particular, if X=Y is a theorem, this remains the case after any substitution of V for U.
Commutativity: (X|Y) = (Y|X)
Duality: If strings of the forms X and (X|X) both show up in a theorem, then if these two strings are swapped wherever they appear in the theorem, then the result will also be a theorem.
Double Negation: ((X|X)|(X|X)) = X
Mimesis: (U|(X|X)) = (U|(U|X))
THEN-3: (U|(U|(V|(V|X)))) = (V|(V|(U|(U|X))))
MP-1: U, (U|(V|X)) |- V
MP-2: U, (U|(V|X)) |- X
Note. The formula (U|(V|X)) has the interpretation U→V∧X. Modus ponens is the special case of MP-1 and MP-2 when V and X are identical.
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
- (A(A(B(B((AB)(AB)))))),
- (A(A((BB)(AA)))).
The resemblance to the syntax of LISP is evident.
The notation can be simplified further, by letting
- (U) := (UU)
- ((U)) <math>\equiv</math> 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.
References
- Charles Peirce, 1880. 'A Boolean Algebra with One Constant'. In Hartshorne, C, and Weiss, P., eds., (1931-35) Collected Papers of Charles Sanders Peirce, Vol. 4: 12-20. Harvard University Press.
- H. M. Sheffer, 1913. "A set of five independent postulates for Boolean algebras, with application to logical constants," Transactions of the American Mathematical Society 14: 481-488.
See also
he:NAND לוגי nl:NAND-poort no:Eksklusjon (logikk) pl:NAND tr:VEDEĞİL kapısı