Sheffer stroke: Difference between revisions

The Sheffer stroke, written "|" (see vertical bar) or "↑", in the subject matter of boolean functions or propositional calculus, 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.

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 (also written as p | q or p ↑ q) is as follows:

The Venn Diagram of "A nand B" (the red area is the area covered by NAND).

The stroke is named after 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 Sanders 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.

Nand does not possess any of the following five properties, each of which is required to be absent from 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 its arguments are truth (falsity).

One way of expressing p NAND q is as ${\displaystyle {\overline {p\cdot q}}}$, where the symbol ${\displaystyle \cdot }$ 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:

NAND operator: We will surely die if we have food nand water.

Common terms: We will surely die if we do not have both food and water.

Or, by DeMorgan's Law: We will surely die if we lack either food or water.

The Sheffer stroke "|" is equivalent to the negation of conjunction:

${\displaystyle P|Q\equiv \neg (P\wedge Q)}$

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

"not p" is equivalent to "p NAND p"	${\displaystyle \neg P\equiv P|P,}$
"p and q" is equivalent to "(p NAND q) NAND (p NAND q)"	${\displaystyle P\wedge Q\equiv (P|Q)|(P|Q),}$
"p or q" is equivalent to "(p NAND p) NAND (q NAND q)"	${\displaystyle P\vee Q\equiv (P|P)|(Q|Q),}$
"p implies q" is equivalent to "p NAND (q NAND q)"	${\displaystyle P\rightarrow Q\equiv P|(Q|Q)\equiv P|(P|Q)}$

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

p_n for natural numbers n
( | )

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.

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

Construction Rule I: For each natural number n, the symbol p_n 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.

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 ${\displaystyle \vdash }$ W

are inference rules.

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)) ${\displaystyle \equiv }$ 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.

• 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.

• http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/nand.html
• implementations of 2 and 4-input NAND gates
• Proofs of some axioms by Stroke function by Yasuo Setô @ Project Euclid