Sheffer stroke
The Sheffer stroke, written "|" 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. It is one of several
sole sufficient operators that can be used to express all of the boolean functions that are the subject matter of propositional logic.
Encyclopedia
The
Sheffer stroke, written "|" 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. It is one of several
sole sufficient operators that can be used to express all of the boolean functions that are the subject matter of propositional logic. Along with
NOR, it is one of the two functionally complete binary operators of propositional logic.
The stroke is named for Henry M. Sheffer, who proved that all the usual operators of propositional logic , could be expressed in terms of it.
Charles Peirce 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.
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.
The truth table of
p NAND q is as follows:
Logical NAND
| p | q | p ↑ q |
|---|
| F | F | T |
| F | T | T |
| T | F | T |
| T | T | F |
One way of expressing
p NAND
q is a , where the symbol signifies
AND and the line over the expression signifies
not, the logical negation of that expression.
NAND is not used in everyday sentences because it exhibits an inherent inversion, which makes it confusing like a double negative. Here's an example of a sentence using the NAND operator:
- We will surely die if we have food nand water.
Expressed in terms of NAND, the usual operators of propositional logic are:
"not p" is equivalent to "p NAND p" |
|
"p and q" is equivalent to " NAND " |
|
"p or q" is equivalent to " NAND " |
|
"p implies q" is equivalent to " NAND p" |
|
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:
The following truth table semantically defines |:
The other logical operators can be defined in terms of '|', like so:
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:
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
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 .
Construction Rule I:
An atom is a well-formed formula .
Construction Rule II:
If X and Y are wffs, then 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-1:4. 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: =
Duality: If strings of the forms X and 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
Mimesis: =
THEN-3: =
MP-1: U, |- V
MP-2: U, |- X
Note. The formula 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
- ,
- .
The resemblance to the syntax of LISP is evident.
The notation can be simplified further, by letting
- :=
- U
for any U. This simplification causes the need to change some rules: more than two letters are allowed within parentheses, letters or wffs within parentheses are allowed to commute, 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
, was an American [i] polymath [i], born in Cambridge, Massachusetts [i] ...
, 1880. 'A Boolean Algebra with One Constant'. In Hartshorne, C, and Weiss, P., eds.,
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
Logical operators
Related topics
...
- Logic gate
- Logical graph
- Peirce's law
- Propositional logic
- Sole sufficient operator
- Zeroth order logic
