Ampheck

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Ampheck, from Greek &#945;&#956;&#966;&#942;&#954;&#951;&#962; 'double-edged' or 'cutting both ways', 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 the subject matter variously described as boolean functions, monadic predicate calculus, propositional calculus, sentential calculus, or zeroth order logic.

The computer used in the spacecraft that first carried humans to the moon, the Apollo Guidance Computer, was constructed entirely using NOR gates with three inputs.

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 &#945;&#956;&#966;&#951;&#954;&#942;&#962;, 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(&#9800;), but set in the text above by means of the 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.

References and further reading

 * Clark, Glenn (1997), &ldquo;New Light on Peirce's Iconic Notation for the Sixteen Binary Connectives&rdquo;, 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), &ldquo;What Is a Number, that a Man May Know It, and a Man, that He May Know a Number?&rdquo; (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.  Online.


 * 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), &ldquo;The Simplest Mathematics&rdquo;. First published as CP 4.227–323 in Collected Papers.


 * Zellweger, Shea (1997), &ldquo;Untapped Potential in Peirce's Iconic Notation for the Sixteen Binary Connectives&rdquo;, pp. 334–386 in Houser, Roberts, Van Evra (eds.), Studies in the Logic of Charles Sanders Peirce, Indiana University Press, Bloomington, IN, 1997.

Focal nodes

 * Inquiry Live
 * Logic Live

Peer nodes

 * Ampheck @ InterSciWiki
 * Ampheck @ Subject Wikis
 * Ampheck @ Wikiversity
 * Ampheck @ Wikiversity Beta

Logical operators

 * Exclusive disjunction
 * Logical conjunction
 * Logical disjunction
 * Logical equality


 * Logical implication
 * Logical NAND
 * Logical NNOR
 * Negation

Related topics

 * Ampheck
 * Boolean domain
 * Boolean function
 * Boolean-valued function
 * Differential logic


 * Logical graph
 * Minimal negation operator
 * Multigrade operator
 * Parametric operator
 * Peirce's law


 * Propositional calculus
 * Sole sufficient operator
 * Truth table
 * Universe of discourse
 * Zeroth order logic

Relational concepts

 * Continuous predicate
 * Hypostatic abstraction
 * Logic of relatives
 * Logical matrix


 * Relation
 * Relation composition
 * Relation construction
 * Relation reduction


 * Relation theory
 * Relative term
 * Sign relation
 * Triadic relation

Information, Inquiry

 * Inquiry
 * Dynamics of inquiry


 * Semeiotic
 * Logic of information


 * Descriptive science
 * Normative science


 * Pragmatic maxim
 * Truth theory

Related articles

 * Cactus Language
 * Futures Of Logical Graphs
 * Propositional Equation Reasoning Systems


 * Differential Logic : Introduction
 * Differential Propositional Calculus
 * Differential Logic and Dynamic Systems


 * Prospects for Inquiry Driven Systems
 * Introduction to Inquiry Driven Systems
 * Inquiry Driven Systems : Inquiry Into Inquiry

Document history
Portions of the above article were adapted from the following sources under the GNU Free Documentation License, under other applicable licenses, or by permission of the copyright holders.


 * Ampheck, InterSciWiki
 * Ampheck, PlanetMath
 * Ampheck, Wikiversity
 * Ampheck, Wikiversity Beta
 * Ampheck, Wikipedia