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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 ("not and") 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. Image File history File links NAND Logic Gate File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...
Image File history File links NAND Logic Gate File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...
A logic gate is an arrangement of controlled switches used to calculate operations using Boolean logic in digital circuits. ...
In mathematics, a finitary boolean function is a function of the form f : Bk → B, where B = {0, 1} is a boolean domain and where k is a nonnegative integer. ...
Negation (i. ...
Wikibooks has more about Boolean logic, under the somewhat misleading title Boolean Algebra For a basic intro to sets, Boolean operations, Venn diagrams, truth tables, and Boolean applications, see Boolean logic. ...
Digital Electronics is based on a number of discrete voltage levels, usually two, as distinct from analog electronics which uses voltages to represent variables directly. ...
Sole sufficient operator, or sole sufficient connective, is a term used to describe an operator that is sufficient by itself to generate all of the operators in a specified class of operators. ...
In mathematics, a boolean function is usually a function F(b1, b2, ..., bn) of a number n of boolean variables bi from the two-element boolean algebra B = {0, 1}, such that F also takes values in B. A function on an arbitrary set X taking values in B is...
In mathematical logic, propositional logic is the logic of mathematical objects called propositions. ...
The stroke is named for Henry M. Sheffer, who proved (Sheffer 1913) that all the usual operators of propositional logic (not, and, or, implies, and so on), 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. Henry Maurice Sheffer (1882-1964) was an American logician. ...
In mathematical logic, propositional logic is the logic of mathematical objects called propositions. ...
Negation (i. ...
AND Logic Gate In mathematics, logical conjunction (usual symbol and) is a logical operator that results in false if either of the operands is false. ...
OR logic gate. ...
In propositional calculus, or logical calculus in mathematics, the logical conditional is a binary logical operator connecting two statements, if p then q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). ...
Charles Sanders Peirce Charles Sanders Santiago Peirce (pronounced purse), (September 10, 1839, Cambridge, Massachusetts – April 19, 1914, Milford, Pennsylvania) was an American polymath. ...
NOR Logic Gate The logical NOR or joint denial is a boolean logic operator which produces a result that is the inverse of logical or. ...
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 operands are true. In other words, it produces a value of true if and only if at least one operand is false. In mathematics, a finitary boolean function is a function of the form f : Bk → B, where B = {0, 1} is a boolean domain and where k is a nonnegative integer. ...
In logic and mathematics, a logical value, also called a truth value, is a value indicating to what extent a proposition is true. ...
Proposition is a term used in logic to describe the content of assertions. ...
The truth table of p NAND q (also written as p | q or p ↑ q) is as follows: Truth tables are a type of mathematical table used in logic to determine whether an expression is true or whether an argument is valid. ...
Truth Table: 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.
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 "(p NAND q) NAND (p NAND q)" |  | | "p or q" is equivalent to "(p NAND p) NAND (q NAND q)" |  | | "p implies q" is equivalent to "(p NAND q) NAND p" |  | This leads to an alternative axiom system for Boolean algebras, requiring but one operation. Wikibooks has more about Boolean logic, under the somewhat misleading title Boolean Algebra For a basic intro to sets, Boolean operations, Venn diagrams, truth tables, and Boolean applications, see Boolean logic. ...
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. Image File history File links NandFullAdder. ...
Image File history File links NandFullAdder. ...
AND Logic Gate In mathematics, logical conjunction (usual symbol and) is a logical operator that results in false if either of the operands is false. ...
OR logic gate. ...
Negation (i. ...
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: In mathematical logic, propositional logic is the logic of mathematical objects called propositions. ...
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)))) In mathematical logic Freges propositional calculus was the first axiomatization of propositional calculus. ...
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)))) In mathematical logic Freges propositional calculus was the first axiomatization of propositional calculus. ...
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. In Logic, Modus ponens (Latin: mode that affirms) is a valid, simple argument form (often abbreviated to MP): If P, then Q. P. Therefore, Q. or in logical operator notation: P → Q P ⊢ Q where ⊢ represents the logical assertion. ...
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. A lisp is a speech impediment, historically also known as sigmatism. ...
The notation can be simplified further, by letting
- (U) := (UU)
- ((U))
 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. An existential graph is a type of diagrammatic or visual notation for logical expressions, invented by Charles Sanders Peirce, who wrote his first paper on graphical logic in 1882 and continued to develop the method until his death in 1914. ...
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.
Charles Sanders Peirce Charles Sanders Santiago Peirce (pronounced purse), (September 10, 1839, Cambridge, Massachusetts – April 19, 1914, Milford, Pennsylvania) was an American polymath. ...
See also
Other operators Exclusive disjunction (usual symbol XOR occasionally EOR) is a logical operator that results in true if one of the operands, but not both of them, is true. ...
AND Logic Gate In mathematics, logical conjunction (usual symbol and) is a logical operator that results in false if either of the operands is false. ...
OR logic gate. ...
XNOR Logic Gate Symbol Logical equality is a logical operator that corresponds to equality in boolean algebra and to the logical biconditional in propositional calculus. ...
In propositional calculus, or logical calculus in mathematics, the material conditional or the implies operator is a binary truth-functional logical operator yielding the form If a then c, where a and c are statement variables (to be replaced by any meaningful indicative sentence of the language). ...
NOR Logic Gate The logical NOR or joint denial is a boolean logic operator which produces a result that is the inverse of logical or. ...
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