ampheck

Ampheck, from the Greek $\alpha\mu\phi\eta\kappa\eta\varsigma$ , 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, $x\curlywedge y$ signifies that $x$ is $\mathbf{f}$ and $y$ is $\mathbf{f}$ . Then $(x\curlywedge y)\curlywedge z$ , or $\underline{x\curlywedge y}\curlywedge z$ , will signify that $z$ is $\mathbf{f}$ , but that the statement that $x$ and $y$ are both $\mathbf{f}$ is itself $\mathbf{f}$ , that is, is false. Hence, the value of $x\curlywedge x$ is the same as that of $\overline{x}$ ; and the value of $\underline{x\curlywedge x}\curlywedge x$ is $\mathbf{f}$ , because it is necessarily false; while the value of $\underline{x\curlywedge y}\curlywedge\underline{x\curlywedge y}$ is only $\mathbf{f}$ in case $x\curlywedge y$ is $\mathbf{v}$ ; and $(\underline{x\curlywedge x}\curlywedge x)\curlywedge(x\curlywedge\underline{x% \curlywedge x})$ is necessarily true, so that its value is $\mathbf{v}$ .
With these two signs, the vinculum (with its equivalents, parentheses, brackets, braces, etc.) and the sign $\curlywedge$ , which I will call the ampheck (from $\alpha\mu\phi\eta\kappa\eta\varsigma$ , 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

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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.
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Houser, Nathan; Roberts, Don D.; and Van Evra, James (eds., 1997), Studies in the Logic of Charles Sanders Peirce, Indiana University Press, Bloomington, IN.
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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.
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McCulloch, Warren Sturgis (1965), Embodiments of Mind, MIT Press, Cambridge, MA.
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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.
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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