Ampheck

Ampheck, from Greek αμφήκης 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 NAND and NNOR. Either of these logical operators is a sole sufficient operator for deriving or generating all of the other operators in what is variously called the subject matter of boolean functions, monadic logic, propositional logic, sentential calculus, or zeroth order logic.

For example, \(x \perp y\) signifies that \(x\!\) is \(\mathbf{f}\) and \(y\!\) is \(\mathbf{f}\). Then \((x \perp y) \perp z\), or \(\underline {x \perp y} \perp z\), 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 \(x \perp x\) is the same as that of \(\overline {x}\); and the value of \(\underline {x \perp x} \perp x\) is \(\mathbf{f}\), because it is necessarily false; while the value of \(\underline {x \perp y} \perp \underline {x \perp y}\) is only \(\mathbf{f}\) in case \(x \perp y\) is \(\mathbf{v}\); and \(( \underline {x \perp x} \perp x) \perp (x \perp \underline {x \perp x})\) 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 \(\perp\), which I will call the ampheck (from αμφηκής , 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. For this operation he employs a symbol that the typographer most likely set by inverting the zodiac symbol for Aries, but set in the text above by commandeering the symbol for the bottom element of a lattice or partially ordered set.

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.

References and further reading

• 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, 1997.

• Houser, N., Roberts, Don D., and Van Evra, James (eds., 1997), Studies in the Logic of Charles Sanders Peirce, Indiana University Press, Bloomington, IN.

• McCulloch, W.S. (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, 1961. Reprinted, pp. 1–18 in Embodiments of Mind.

• McCulloch, W.S. (1965), Embodiments of Mind, MIT Press, Cambridge, MA.

• Peirce, C.S., Bibliography.

• Peirce, C.S., 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.

• Peirce, C.S. (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.

See also

• Laws of Form
• Logical graph
• Logical NAND (Sheffer stroke)
• Logical NNOR (Peirce arrow)
• Minimal negation operator
• Sole sufficient operator

Aficionados

• See Talk:Ampheck for discussions/comments regarding this article.
• See Ampheck/Aficionados for those who have listed Ampheck as an interest.
• See Talk:Ampheck/Aficionados for discussions regarding this interest.

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