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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 the subject matter variously described as [[boolean function]]s, [[monadic predicate calculus]], [[propositional logic]], sentential calculus, or [[zeroth order logic]].
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<font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].
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'''Ampheck''', from Greek &#945;&#956;&#966;&#942;&#954;&#951;&#962; 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 \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>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>
  
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<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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<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 means of the ''curly wedge'' symbol.
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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('''&#9800;'''), but set in the text above by means of the ''curly wedge'' symbol.
  
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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''.
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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==
  
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* [[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.
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* 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.
  
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* [[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.
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* Houser, N., 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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* [[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''.
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* 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.
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* [[Charles Sanders Peirce (Bibliography)|Peirce, C.S., Bibliography]].
 
* [[Charles Sanders Peirce (Bibliography)|Peirce, C.S., Bibliography]].
  
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* 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.
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* 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.
  
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* Peirce, C.S. (1902), "The Simplest Mathematics".  First published as CP 4.227–323 in ''Collected Papers''.
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* Peirce, C.S. (1902), &ldquo;The Simplest Mathematics&rdquo;.  First published as CP&nbsp;4.227&ndash;323 in ''Collected Papers''.
  
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* [[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.
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* 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.
  
 
==Syllabus==
 
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* [http://intersci.ss.uci.edu/wiki/index.php/Ampheck Ampheck @ InterSciWiki]
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* [http://beta.wikiversity.org/wiki/Ampheck Ampheck @ Beta Wikiversity]
 
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* [http://mywikibiz.com/Ampheck Ampheck @ MyWikiBiz]
 
* [http://mywikibiz.com/Ampheck Ampheck @ MyWikiBiz]
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* [http://ref.subwiki.org/wiki/Ampheck Ampheck @ Subject Wikis]
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* [http://www.netknowledge.org/wiki/Ampheck Ampheck @ NetKnowledge]
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* [http://en.wikiversity.org/wiki/Ampheck Ampheck @ Wikiversity]
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* [http://beta.wikiversity.org/wiki/Ampheck Ampheck @ Wikiversity Beta]
  
 
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* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;]
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* [[Inquiry]]
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* [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;]
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* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;]
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* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
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* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
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* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
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* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
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* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
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* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
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* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
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* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
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* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
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==Document history==
 
==Document history==
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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.
 
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.
  
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* [http://intersci.ss.uci.edu/wiki/index.php/Ampheck Ampheck], [http://intersci.ss.uci.edu/ InterSciWiki]
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* [http://mywikibiz.com/Ampheck Ampheck], [http://mywikibiz.com/ MyWikiBiz]
 
* [http://mywikibiz.com/Ampheck Ampheck], [http://mywikibiz.com/ MyWikiBiz]
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* [http://beta.wikiversity.org/wiki/Ampheck Ampheck], [http://beta.wikiversity.org/ Beta Wikiversity]
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* [http://planetmath.org/Ampheck Ampheck], [http://planetmath.org/ PlanetMath]
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* [http://www.mathweb.org/wiki/Ampheck Ampheck], [http://www.mathweb.org/wiki/ MathWeb Wiki]
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* [http://wikinfo.org/w/index.php/Ampheck Ampheck], [http://wikinfo.org/w/ Wikinfo]
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* [http://www.netknowledge.org/wiki/Ampheck Ampheck], [http://www.netknowledge.org/ NetKnowledge]
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* [http://en.wikiversity.org/wiki/Ampheck Ampheck], [http://en.wikiversity.org/ Wikiversity]
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* [http://planetmath.org/encyclopedia/Ampheck.html Ampheck], [http://planetmath.org/ PlanetMath]
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* [http://beta.wikiversity.org/wiki/Ampheck Ampheck], [http://beta.wikiversity.org/ Wikiversity Beta]
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* [http://semanticweb.org/wiki/Ampheck Ampheck], [http://semanticweb.org/ SemanticWeb]
 
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* [http://www.getwiki.net/-Ampheck Ampheck], [http://www.getwiki.net/ GetWiki]
 
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* [http://www.wikinfo.org/index.php/Ampheck Ampheck], [http://www.wikinfo.org/ Wikinfo]
 
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* [http://www.textop.org/wiki/index.php?title=Ampheck Ampheck], [http://www.textop.org/wiki/ Textop Wiki]
 
 
* [http://en.wikipedia.org/w/index.php?title=Ampheck&oldid=62218032 Ampheck], [http://en.wikipedia.org/ Wikipedia]
 
* [http://en.wikipedia.org/w/index.php?title=Ampheck&oldid=62218032 Ampheck], [http://en.wikipedia.org/ Wikipedia]
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Latest revision as of 15:04, 5 November 2015

☞ This page belongs to resource collections on Logic and Inquiry.

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 the subject matter variously described as boolean functions, monadic predicate calculus, propositional calculus, sentential calculus, or zeroth order logic.

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 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 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), “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. Reprinted, pp. 1–18 in Embodiments of Mind. Online.
  • McCulloch, W.S. (1965), Embodiments of Mind, MIT Press, Cambridge, MA.
  • 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.

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Logical operators

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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.

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