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'''Ampheck''' (from the Greek ''{{polytonic|ἀμφήκης}}'', "double-edged") is a term coined by [[Charles Sanders Peirce]] for either one of the pair of logical operators now known as [[logical NAND|NAND]] and [[logical NOR|NOR]].<ref name="McCulloch">[[Warren Sturgis McCulloch|McCulloch, W.S.]] (1961), "[http://time-binding.org/misc/akml/akmls/26-27-mcculloch.pdf 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'', MIT Press, Cambridge, MA, 1965. "These logics really began in the latter part of the last century with Charles Pierce [''sic''] as their great pioneer. As with most pioneers, many of the trails he blazed were not followed for a score of years. For example, he discovered the amphecks—that is, 'not both ... and ...' and 'neither ... nor ...'."</ref> Either of these operators is a [[sole sufficient operator]] for deriving all of the [[boolean functions]] of [[propositional logic]]. |
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«Removing Article Acquired Under False Pretenses By Wikipedia (See http://wikipediareview.com/index.php?s=&showtopic=14534&view=findpost&p=66085)» |
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In his paper "The Simplest Mathematics" (1902), Peirce first introduces the term ''ampheck'' for the logical operator now called the NOR operator. For this operation he employs a symbol that the typographer most likely set by inverting the [[zodiac]] symbol for [[Aries (astrology)|Aries]] (♈). |
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: For example, <math> x \curlywedge y </math> signifies that ''x'' is '''f''' and ''y'' is '''f'''. Then <math> ( x \curlywedge y ) \curlywedge z </math>, or <math> \underline {x \curlywedge y} \curlywedge z </math>, will signify that ''z'' is '''f''', but that the statement that ''x'' and ''y'' are both '''f''' is itself '''f''', 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 '''f''', because it is necessarily false; while the value of <math> \underline {x \curlywedge y} \curlywedge \underline {x \curlywedge y} </math> is only '''f''' in case <math> x \curlywedge y </math> is '''v'''; and <math> ( \underline {x \curlywedge x} \curlywedge x ) \curlywedge ( x \curlywedge \underline {x \curlywedge x} ) </math> is necessarily true, so that its value is '''v'''. |
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: With these two signs, the [[vinculum (symbol)|vinculum]] (with its equivalents, parentheses, brackets, braces, etc.) and the sign <math> \curlywedge </math>, which I will call the ''ampheck'' (from {{polytonic|ἀμφηκής}}, cutting both ways), all assertions as to the values of quantities can be expressed.<ref name="Peirce">Peirce, C.S. (1902), "The Simplest Mathematics". First published as CP 4.227–323 in ''Collected Papers of Charles Sanders Peirce'', vol. 4, [[Charles Hartshorne]] and [[Paul Weiss]] (eds.), Harvard University Press, Cambridge, MA, 1933. Excerpt from CP 4.264.</ref> |
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In the same paper, Peirce introduces a symbol for the NAND operator, rendered as the inverted Aries symbol with a bar over it. It is not clear whether it was Peirce himself or later writers who initiated the practice, but due to their dual relationship, it became common to refer to these two operators in the plural as the ''amphecks''. |
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==References== |
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<references/> |
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==Further reading== |
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* Houser, Roberts, Van Evra (eds.), ''Studies in the Logic of Charles Sanders Peirce'', Indiana University Press, Bloomington, IN, 1997. |
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[[Category:Boolean algebra]] |
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[[Category:History of logic]] |
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[[pt:Anfeque]] |
Revision as of 06:19, 23 December 2007
«Article Withdrawn By Request Of Author (See http://wikipediareview.com/index.php?s=&showtopic=14534&view=findpost&p=66085)»
«Removing Article Acquired Under False Pretenses By Wikipedia (See http://wikipediareview.com/index.php?s=&showtopic=14534&view=findpost&p=66085)»