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'''Ampheck''', from [[Ancient Greek|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 arrow]]s, [[Sheffer stroke]]s, or [[logical NAND|NAND]] and [[logical NNOR|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 function]]s, [[propositional logic]], sentential calculus, or [[zeroth-order logic]].

 

'''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 [[logical NAND|NAND]] and [[logical NNOR|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 function]]s, monadic predicate calculus, [[propositional calculus]], sentential calculus, or [[zeroth order logic]].

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<p>For example, <math>x \perp y</math> signifies that <math>x\!</math> is <math>\mathbf{f}</math> and <math>y\!</math> is <math>\mathbf{f}</math>.  Then <math>(x \perp y) \perp z</math>, or <math>\underline {x \perp y} \perp z</math>, will signify that <math>z\!</math> is <math>\mathbf{f}</math>, but that the statement that <math>x\!</math> and <math>y\!</math> are both <math>\mathbf{f}</math> is itself <math>\mathbf{f}</math>, that is, is ''false''.  Hence, the value of <math>x \perp x</math> is the same as that of <math>\overline {x}</math>;  and the value of <math>\underline {x \perp x} \perp x</math> is <math>\mathbf{f}</math>, because it is necessarily false;  while the value of <math>\underline {x \perp y} \perp \underline {x \perp y}</math> is only <math>\mathbf{f}</math> in case <math>x \perp y</math> is <math>\mathbf{v}</math>;  and <math>( \underline {x \perp x} \perp x) \perp (x \perp \underline {x \perp x})</math> is necessarily true, so that its value is <math>\mathbf{v}</math>.</p>

<p>For example, <math>x \curlywedge y</math> signifies that <math>x\!</math> is <math>\mathbf{f}</math> and <math>y\!</math> is <math>\mathbf{f}</math>.  Then <math>(x \curlywedge y) \curlywedge z</math>, or <math>\underline {x \curlywedge y} \curlywedge z</math>, will signify that <math>z\!</math> is <math>\mathbf{f}</math>, but that the statement that <math>x\!</math> and <math>y\!</math> are both <math>\mathbf{f}</math> is itself <math>\mathbf{f}</math>, that is, is ''false''.  Hence, the value of <math>x \curlywedge x</math> is the same as that of <math>\overline {x}</math>;  and the value of <math>\underline {x \curlywedge x} \curlywedge x</math> is <math>\mathbf{f}</math>, because it is necessarily false;  while the value of <math>\underline {x \curlywedge y} \curlywedge \underline {x \curlywedge y}</math> is only <math>\mathbf{f}</math> in case <math>x \curlywedge y</math> is <math>\mathbf{v}</math>;  and <math>( \underline {x \curlywedge x} \curlywedge x) \curlywedge (x \curlywedge \underline {x \curlywedge x})</math> is necessarily true, so that its value is <math>\mathbf{v}</math>.</p>

<p>With these two signs, the [[vinculum]] (with its equivalents, parentheses, brackets, braces, etc.) and the sign <math>\perp</math>, which I will call the ''ampheck'' (from &#945;&#956;&#966;&#951;&#954;&#942;&#962;&nbsp;, cutting both ways), all assertions as to the values of quantities can be expressed. (C.S. Peirce, CP 4.264).</p>

<p>With these two signs, the vinculum (with its equivalents, parentheses, brackets, braces, etc.) and the sign <math>\curlywedge</math>, which I will call the ''ampheck'' (from &#945;&#956;&#966;&#951;&#954;&#942;&#962;&nbsp;, cutting both ways), all assertions as to the values of quantities can be expressed. (C.S. Peirce, CP 4.264).</p>

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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 (order)|lattice]] or [[partially ordered set]].

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''.

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==

==References and further reading==

* [[Glenn Clark|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.

* Clark, Glenn (1997), &ldquo;New Light on Peirce's Iconic Notation for the Sixteen Binary Connectives&rdquo;, pp. 304&ndash;333 in Houser, Roberts, Van Evra (eds.), ''Studies in the Logic of Charles Sanders Peirce'', Indiana University Press, Bloomington, IN, 1997.

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

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

* [[Warren Sturgis McCulloch|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. (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&ndash;18, Institute of General Semantics, Lakeville, CT.  Reprinted, pp. 1&ndash;18 in ''Embodiments of Mind''.  [http://www.vordenker.de/ggphilosophy/mcculloch_what-is-a-number.pdf Online].

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

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

* [[Charles Sanders Peirce (Bibliography)|Peirce, C.S., Bibliography]].

* [[Charles Sanders Peirce (Bibliography)|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., ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, Charles Hartshorne and Paul Weiss (eds.), vols. 7&ndash;8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931&ndash;1935, 1958.

* Peirce, C.S. (1902), "The Simplest Mathematics".  First published as CP 4.227–323 in ''Collected Papers''.

* Peirce, C.S. (1902), &ldquo;The Simplest Mathematics&rdquo;.  First published as CP&nbsp;4.227&ndash;323 in ''Collected Papers''.

* [[Shea Zellweger|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.

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

==See also==

==Syllabus==

* [[Laws of Form]]

 

* [[Inquiry Live]]

 

 

 

 

 

 

* [[Logical graph]]

* [[Logical graph]]

* [[Minimal negation operator]]

* [[Minimal negation operator]]

* [[Sole sufficient operator]]

* [[Sole sufficient operator]]

[[Category:Automata Theory]]

[[Category:Automata Theory]]

[[Category:Combinatorics]]

[[Category:Combinatorics]]

[[Category:Computer Science]]

[[Category:Computer Science]]

[[Category:Mathematics]]

[[Category:Mathematics]]

[[Category:Neural Networks]]

[[Category:Neural Networks]]