ampheck

ampheck

Ampheck

, from the Greek

α

⁢

μ

⁢

ϕ

⁢

η

⁢

κ

⁢

η

⁢

ς

,

double-edged

, is a term coined by

Charles Sanders Peirce

for either one of the pair of logically dual

operators

, variously referred to as

Peirce arrows

,

Sheffer strokes

, or logical NAND and logical NNOR. Either of these logical operators is a

sole sufficient operator

(

http://

planetmath

.org/SoleSufficientOperator

)

for defining all of the other operators in the subject matter variously described as

boolean functions

, monadic predicate calculus,

propositional calculus

, sentential calculus, or

zeroth order logic

. See

logical connective

(

http://planetmath.org/LogicalConnective

)

for further discussion.

For example,

x

⋏

y

signifies that

x

is

𝐟

and

y

is

𝐟

. Then

(

x

⋏

y

)

⋏

z

, or

x

⋏

y

¯

⋏

z

, will signify that

z

is

𝐟

, but that the statement that

x

and

y

are both

𝐟

is itself

𝐟

, that is, is

false

. Hence, the value of

x

⋏

x

is the same as that of

x

¯

; and the value of

x

⋏

x

¯

⋏

x

is

𝐟

, because it is necessarily false; while the value of

x

⋏

y

¯

⋏

x

⋏

y

¯

is only

𝐟

in case

x

⋏

y

is

𝐯

; and

(

x

⋏

x

¯

⋏

x

)

⋏

(

x

⋏

x

⋏

x

¯

)

is necessarily true, so that its value is

𝐯

.

With these two signs, the vinculum (with its

equivalents

, parentheses, brackets, braces, etc.) and the sign

⋏

, which I will call the

ampheck

(from

α

⁢

μ

⁢

ϕ

⁢

η

⁢

κ

⁢

η

⁢

ς

,

cutting

both ways), all assertions as to the values of quantities can be expressed. (C.S. Peirce, CP 4.264).

In the above passage, Peirce introduces the term

ampheck

for the 2-place logical connective or the binary logical operator that is currently called the

joint denial

in logic, the NNOR operator in computer science, or indicated by means of phrases like “neither-nor” or “both not” in ordinary language. In his handwritten manuscripts Peirce used a cursive symbol for the amphecks that he derived from his

dot-cross

notation for

truth tables

, one that the typographer most likely set by inverting the zodiac symbol for Aries, and that is set in the text above by using the so-called

curly wedge

symbol.

In the same paper, Peirce introduces a symbol for the logically dual operator. This was rendered by the editors of his

Collected Papers

as an inverted Aries symbol with a bar or a serif at the top, in this way denoting the

connective

or logical operator that is currently called the

alternative denial

in logic, the NAND operator in computer science, or invoked by means of phrases like “not-and” or “not both” in ordinary language. It is not clear whether it was Peirce himself or later writers who initiated the practice, but on account of their dual relationship it became common to refer to these two operators in the plural, as the

amphecks

.

1

Bibliography

•

Clark, Glenn (1997), “New Light on Peirce’s Iconic Notation for the Sixteen Binary Connectives”, pp. 304–333 in Houser, Roberts, Van Evra (eds.),

Studies in the Logic of Charles Sanders Peirce

, Indiana University Press, Bloomington, IN.

•

Houser, Nathan; Roberts, Don D.; and Van Evra, James (eds., 1997),

Studies in the Logic of Charles Sanders Peirce

, Indiana University Press, Bloomington, IN.

•

McCulloch, Warren Sturgis (1961), “What Is a Number, that a Man May Know It, and a Man, that He May Know a Number?” (Ninth Alfred Korzybski Memorial Lecture),

General

Semantics

Bulletin

, Nos. 26 & 27, 7–18, Institute of General Semantics, Lakeville, CT. Reprinted, pp. 1–18 in

Embodiments of Mind

.

•

McCulloch, Warren Sturgis (1965),

Embodiments of Mind

, MIT Press, Cambridge, MA.

•

Peirce, Charles Sanders,

Collected Papers of Charles Sanders Peirce

, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958. (Cited as CP volume.paragraph).

•

Peirce, Charles Sanders (1902), “The Simplest Mathematics”. First published as CP 4.227–323 in

Collected Papers

.

•

Zellweger, Shea (1997), “Untapped Potential in Peirce’s Iconic Notation for the Sixteen Binary Connectives”, pp. 334–386 in Houser, Roberts, Van Evra (eds.),

Studies in the Logic of Charles Sanders Peirce

, Indiana University Press, Bloomington, IN,
1997.

Title

ampheck

Canonical

name

Ampheck

Date of creation

2013-03-22 17:47:21

Last modified on

2013-03-22 17:47:21

Owner

Jon Awbrey (15246)

Last modified by

Jon Awbrey (15246)

Numerical id

18

Author

Jon Awbrey (15246)

Entry type

Definition

Classification

msc 03B70

Classification

msc 03B35

Classification

msc 03B22

Classification

msc 03B05

Classification

msc 03-03

Classification

msc 01A55

Related topic

LogicalConnective

Related topic

LogicalGraph

Related topic

LogicalGraphFormalDevelopment

Related topic

SoleSufficientOperator

Defines

NAND

Defines

NNOR

Defines

Peirce arrow

Defines

Sheffer stroke

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