Minimal negation operator - Revision history

Jon Awbrey: update from blog

2017-09-03T13:54:20Z

update from blog

Jon Awbrey: update, clarity, less awkward phrasing

2017-08-30T14:48:22Z

update, clarity, less awkward phrasing

Jon Awbrey: update

2017-08-28T03:42:39Z

update

Jon Awbrey: update

2017-08-27T17:56:05Z

update

Jon Awbrey: update

2017-08-27T15:45:17Z

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Jon Awbrey: * '''Note.''' The MathJax parser is not rendering this page properly.
Until it can be fixed please see the [http://intersci.ss.uci.edu/wiki/index.php/Minimal_negation_operator InterSciWiki version].

2016-07-28T18:42:38Z

* '''Note.''' The MathJax parser is not rendering this page properly.<br>Until it can be fixed please see the [http://intersci.ss.uci.edu/wiki/index.php/Minimal_negation_operator InterSciWiki version].

Jon Awbrey: * '''Note.''' The MathJax parser is not rendering this page properly.
Until it can be fixed please see the [http://intersci.ss.uci.edu/wiki/index.php/Minimal_negation_operator InterSciWiki version].

2016-07-28T18:40:29Z

Jon Awbrey: update

2015-11-06T19:16:05Z

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Jon Awbrey: update

2013-12-04T16:36:57Z

update

Jon Awbrey: /* Peer nodes */ update link

2013-06-08T16:35:10Z

Peer nodes: update link

@@ Line 85: / Line 85: @@
 To express the general case of <math>\nu_k~\!</math> in terms of familiar operations, it helps to introduce an intermediary concept:
-'''Definition.'''  Let the function <math>\lnot_j : \mathbb{B}^k \to \mathbb{B}~\!</math> be defined for each integer <math>j~\!</math> in the interval <math>[1, k]~\!</math> by the following equation:
+'''Definition.'''&nbsp; Let the function <math>\lnot_j : \mathbb{B}^k \to \mathbb{B}~\!</math> be defined for each integer <math>j~\!</math> in the interval <math>[1, k]~\!</math> by the following equation:
 {| align="center" cellpadding="8"
@@ Line 97: / Line 97: @@
 |}
-If we think of the point <math>x = (x_1, \ldots, x_k) \in \mathbb{B}^k~\!</math> as indicated by the boolean product <math>x_1 \cdot \ldots \cdot x_k~\!</math> or the logical conjunction <math>x_1 \land \ldots \land x_k,~\!</math> then the minimal negation <math>\texttt{(} x_1, \ldots, x_k \texttt{)}~\!</math> indicates the set of points in <math>\mathbb{B}^k~\!</math> that differ from <math>x~\!</math> in exactly one coordinate.  This makes <math>\texttt{(} x_1, \ldots, x_k \texttt{)}~\!</math> a discrete functional analogue of a ''point omitted neighborhood'' in analysis, more exactly, a ''point omitted distance one neighborhood''.  In this light, the minimal negation operator can be recognized as a differential construction, an observation that opens a very wide field.  It also serves to explain a variety of other names for the same concept, for example, ''logical boundary operator'', ''limen operator'', ''least action operator'', or ''hedge operator'', to name but a few.  The rationale for these names is visible in the venn diagrams of the corresponding operations on sets.
+If we take the boolean product <math>x_1 \cdot \ldots \cdot x_k~\!</math> or the logical conjunction <math>x_1 \land \ldots \land x_k~\!</math> to indicate the point <math>x = (x_1, \ldots, x_k)~\!</math> in the space <math>\mathbb{B}^k~\!</math> then the minimal negation <math>\texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_k \texttt{)}~\!</math> indicates the set of points in <math>\mathbb{B}^k~\!</math> that differ from <math>x~\!</math> in exactly one coordinate.&nbsp; This makes <math>\texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_k \texttt{)}~\!</math> a discrete functional analogue of a point-omitted neighborhood in ordinary real analysis, more exactly, a point-omitted distance-one neighborhood.&nbsp; In this light, the minimal negation operator can be recognized as a differential construction, an observation that opens a very wide field.
-The remainder of this discussion proceeds on the ''algebraic boolean convention'' that the plus sign <math>(+)~\!</math> and the summation symbol <math>(\textstyle\sum)~\!</math> both refer to addition modulo 2.  Unless otherwise noted, the boolean domain <math>\mathbb{B} = \{ 0, 1 \}~\!</math> is interpreted so that <math>0 = \mathrm{false}~\!</math> and <math>1 = \mathrm{true}.~\!</math>  This has the following consequences:
+The remainder of this discussion proceeds on the algebraic convention that the plus sign <math>(+)~\!</math> and the summation symbol <math>(\textstyle\sum)~\!</math> both refer to addition mod 2.&nbsp; Unless otherwise noted, the boolean domain <math>\mathbb{B} = \{ 0, 1 \}~\!</math> is interpreted for logic in such a way that <math>0 = \mathrm{false}~\!</math> and <math>1 = \mathrm{true}.~\!</math>&nbsp; This has the following consequences:
 {| align="center" cellpadding="4" width="90%"
@@ Line 122: / Line 122: @@
 |-
 | valign="top" | <big>&bull;</big>
-| The inclusive disjunctions indicated for the <math>\nu_k~\!</math> of more than one argument may be replaced with exclusive disjunctions without affecting the meaning, since the terms disjoined are already disjoint.
+| The inclusive disjunctions indicated for the <math>\nu_k~\!</math> of more than one argument may be replaced with exclusive disjunctions without affecting the meaning since the terms in disjunction are already disjoint.
 |}

@@ Line 50: / Line 50: @@
 ==Initial definition==
-The '''minimal negation operator''' <math>\nu~\!</math> is a [[multigrade operator]] <math>(\nu_k)_{k \in \mathbb{N}}~\!</math> where each <math>\nu_k~\!</math> is a <math>k~\!</math>-ary [[boolean function]] defined in such a way that <math>\nu_k (x_1, \ldots , x_k) = 1~\!</math> in just those cases where exactly one of the arguments <math>x_j~\!</math> is <math>0.~\!</math>
+The '''minimal negation operator''' <math>\nu~\!</math> is a [[multigrade operator]] <math>(\nu_k)_{k \in \mathbb{N}}~\!</math> where each <math>\nu_k~\!</math> is a <math>k~\!</math>-ary [[boolean function]] defined by the rule that <math>\nu_k (x_1, \ldots , x_k) = 1~\!</math> if and only if exactly one of the arguments <math>x_j~\!</math> is <math>0.~\!</math>
-In contexts where the initial letter <math>\nu~\!</math> is understood, the minimal negation operators can be indicated by argument lists in parentheses.  In the following text a distinctive typeface will be used for logical expressions based on minimal negation operators, for example, <math>\texttt{(x, y, z)}~\!</math> = <math>\nu (x, y, z).~\!</math>
+In contexts where the initial letter <math>\nu~\!</math> is understood, the minimal negation operators can be indicated by argument lists in parentheses.&nbsp; In the following text a distinctive typeface will be used for logical expressions based on minimal negation operators, for example, <math>\texttt{(x, y, z)} = \nu (x, y, z).~\!</math>
-The first four members of this family of operators are shown below, with paraphrases in a couple of other notations, where tildes and primes, respectively, indicate logical negation.
+The first four members of this family of operators are shown below.&nbsp; The third and fourth columns give paraphrases in two other notations, where tildes and primes, respectively, indicate logical negation.
 {| align="center" cellpadding="8" style="text-align:center"

@@ Line 1: / Line 1: @@
 <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].
-A '''minimal negation operator''' <math>(\nu)~\!</math> is a logical connective that says &ldquo;just one false&rdquo; of its logical arguments.&nbsp; The first four cases are as follows:
+A '''minimal negation operator''' <math>(\nu)~\!</math> is a logical connective that says &ldquo;just one false&rdquo; of its logical arguments.&nbsp; The first four cases are described below.
 <ol start="0">
 <li style="padding:8px">
-If the list of arguments is empty, as expressed in the form <math>\nu(),~\!</math> then it cannot be true that exactly one of the arguments is false, so <math>\nu() = \mathrm{false}.~\!</math></li>
+If the list of arguments is empty, as expressed in the form <math>\nu(),~\!</math> then it cannot be true that exactly one of the arguments is false, so <math>\nu() = \mathrm{false}.~\!</math>
 <li style="padding:8px">
-If <math>p~\!</math> is the only argument, then <math>\nu(p)~\!</math> says that <math>p~\!</math> is false, so <math>\nu(p)~\!</math> expresses the logical negation of the proposition <math>p.~\!</math>  Written in several different notations, <math>\nu(p) = \mathrm{not}(p) = \lnot p = \tilde{p} = p^\prime.~\!</math></li>
+If <math>p~\!</math> is the only argument then <math>\nu(p)~\!</math> says that <math>p~\!</math> is false, so <math>\nu(p)~\!</math> expresses the logical negation of the proposition <math>p.~\!</math>&nbsp; Written in several different notations, we have the following equivalent expressions.
 <li style="padding:8px">
-If <math>p~\!</math> and <math>q~\!</math> are the only two arguments, then <math>\nu(p, q)~\!</math> says that exactly one of <math>p, q~\!</math> is false, so <math>\nu(p, q)~\!</math> says the same thing as <math>p \neq q.~\!</math>  Expressing <math>\nu(p, q)~\!</math> in terms of ands <math>(\cdot),~\!</math> ors <math>(\lor),~\!</math> and nots <math>(\tilde{~})~\!</math> gives the following form.
+If <math>p~\!</math> and <math>q~\!</math> are the only two arguments then <math>\nu(p, q)~\!</math> says that exactly one of <math>p, q~\!</math> is false, so <math>\nu(p, q)~\!</math> says the same thing as <math>p \neq q.~\!</math>  Expressing <math>\nu(p, q)~\!</math> in terms of ands <math>(\cdot),~\!</math> ors <math>(\lor),~\!</math> and nots <math>(\tilde{~})~\!</math> gives the following form.
-<p style="padding:8px; text-align:center">
+<p style="padding:8px; text-align:center"><math>\nu(p, q) ~=~ \tilde{p} \cdot q ~\lor~ p \cdot \tilde{q}~\!</math></p>
-As usual, one drops the dot <math>(\cdot)~\!</math> in contexts where it's understood, giving the following form.
+It is permissible to omit the dot <math>(\cdot)~\!</math> in contexts where it is understood, giving the following form.
-<p style="padding:8px; text-align:center">
+<p style="padding:8px; text-align:center"><math>\nu(p, q) ~=~ \tilde{p}q \lor p\tilde{q}~\!</math></p>
 The venn diagram for <math>\nu(p, q)~\!</math> is shown in Figure&nbsp;1.
@@ Line 40: / Line 42: @@
 |}
-The center cell is the region where all three arguments <math>p, q, r~\!</math> hold true, so <math>\nu(p, q, r)~\!</math> holds true in just the three neighboring cells.  In other words:
+The center cell is the region where all three arguments <math>p, q, r~\!</math> hold true, so <math>\nu(p, q, r)~\!</math> holds true in just the three neighboring cells.&nbsp; In other words:
-<p style="padding:8px; text-align:center">
+<p style="padding:8px; text-align:center"><math>\nu(p, q, r) ~=~ \tilde{p}qr \lor p\tilde{q}r \lor pq\tilde{r}~\!</math></p>
 </li></ol>

@@ Line 1: / Line 1: @@
 <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].
-A '''minimal negation operator''' <math>(\nu)~\!</math> is a logical connective that says &ldquo;just one false&rdquo; of its logical arguments.
+A '''minimal negation operator''' <math>(\nu)~\!</math> is a logical connective that says &ldquo;just one false&rdquo; of its logical arguments.&nbsp; The first four cases are as follows:
-* If the list of arguments is empty, as expressed in the form <math>\nu(),~\!</math> then it cannot be true that exactly one of the arguments is false, so <math>\nu() = \mathrm{false}.~\!</math>
+<ol start="0">
-* If <math>p~\!</math> is the only argument, then <math>\nu(p)~\!</math> says that <math>p~\!</math> is false, so <math>\nu(p)~\!</math> expresses the logical negation of the proposition <math>p.~\!</math>  Written in several different notations, <math>\nu(p) = \mathrm{not}(p) = \lnot p = \tilde{p} = p^\prime.~\!</math>
+<li style="padding:8px">
-* If <math>p~\!</math> and <math>q~\!</math> are the only two arguments, then <math>\nu(p, q)~\!</math> says that exactly one of <math>p, q~\!</math> is false, so <math>\nu(p, q)~\!</math> says the same thing as <math>p \neq q.~\!</math>  Expressing <math>\nu(p, q)~\!</math> in terms of ands <math>(\cdot),~\!</math> ors <math>(\lor),~\!</math> and nots <math>(\tilde{~})~\!</math> gives the following form.
+<li style="padding:8px">
-{| align="center" cellpadding="8"
+<li style="padding:8px">
-| <math>\nu(p, q) = \tilde{p} \cdot q ~\lor~ p \cdot \tilde{q}.~\!</math>
+If <math>p~\!</math> and <math>q~\!</math> are the only two arguments, then <math>\nu(p, q)~\!</math> says that exactly one of <math>p, q~\!</math> is false, so <math>\nu(p, q)~\!</math> says the same thing as <math>p \neq q.~\!</math>  Expressing <math>\nu(p, q)~\!</math> in terms of ands <math>(\cdot),~\!</math> ors <math>(\lor),~\!</math> and nots <math>(\tilde{~})~\!</math> gives the following form.
-|}
-As usual, one drops the dots <math>(\cdot)~\!</math> in contexts where they are understood, giving the following form.
+As usual, one drops the dot <math>(\cdot)~\!</math> in contexts where it's understood, giving the following form.
-{| align="center" cellpadding="8"
+<p style="padding:8px; text-align:center">
-| <math>\nu(p, q) = \tilde{p}q \lor p\tilde{q}.~\!</math>
+<math>\nu(p, q) = \tilde{p}q \lor p\tilde{q}.~\!</math></p>
 The venn diagram for <math>\nu(p, q)~\!</math> is shown in Figure&nbsp;1.
@@ Line 26: / Line 29: @@
 <p><math>\text{Figure 1.}~~\nu(p, q)~\!</math></p>
 |}
 The venn diagram for <math>\nu(p, q, r)~\!</math> is shown in Figure&nbsp;2.
@@ Line 35: / Line 40: @@
 |}
-The center cell is the region where all three arguments <math>p, q, r~\!</math> hold true, so <math>\nu(p, q, r)~\!</math> holds true in just the three neighboring cells.  In other words, <math>\nu(p, q, r) = \tilde{p}qr \lor p\tilde{q}r \lor pq\tilde{r}.~\!</math>
+The center cell is the region where all three arguments <math>p, q, r~\!</math> hold true, so <math>\nu(p, q, r)~\!</math> holds true in just the three neighboring cells.  In other words:
 ==Initial definition==

← Older revision		Revision as of 18:42, 28 July 2016
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		+	* '''Note.''' The MathJax parser is not rendering this page properly.<br>Until it can be fixed please see the [http://intersci.ss.uci.edu/wiki/index.php/Minimal_negation_operator InterSciWiki version].
		+
	<font size="3">☞</font> This page belongs to resource collections on [[Logic Live\|Logic]] and [[Inquiry Live\|Inquiry]].		<font size="3">☞</font> This page belongs to resource collections on [[Logic Live\|Logic]] and [[Inquiry Live\|Inquiry]].
−
−	* '''Note.''' The MathJax parser is not rendering this page properly.<br>Until it can be fixed please see the [http://intersci.ss.uci.edu/wiki/index.php/Minimal_negation_operator InterSciWiki version].

	A '''minimal negation operator''' <math>(\texttt{Mno})\!</math> is a logical connective that says “just one false” of its logical arguments.		A '''minimal negation operator''' <math>(\texttt{Mno})\!</math> is a logical connective that says “just one false” of its logical arguments.

← Older revision		Revision as of 18:40, 28 July 2016
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	<font size="3">☞</font> This page belongs to resource collections on [[Logic Live\|Logic]] and [[Inquiry Live\|Inquiry]].		<font size="3">☞</font> This page belongs to resource collections on [[Logic Live\|Logic]] and [[Inquiry Live\|Inquiry]].
		+
		+	* '''Note.''' The MathJax parser is not rendering this page properly.<br>Until it can be fixed please see the [http://intersci.ss.uci.edu/wiki/index.php/Minimal_negation_operator InterSciWiki version].

	A '''minimal negation operator''' <math>(\texttt{Mno})\!</math> is a logical connective that says “just one false” of its logical arguments.		A '''minimal negation operator''' <math>(\texttt{Mno})\!</math> is a logical connective that says “just one false” of its logical arguments.

@@ Line 1: / Line 1: @@
 <font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].
-A '''minimal negation operator''' <math>(\texttt{Mno})</math> is a logical connective that says “just one false” of its logical arguments.
+A '''minimal negation operator''' <math>(\texttt{Mno})</math> is a logical connective that says &ldquo;just one false&rdquo; of its logical arguments.
 If the list of arguments is empty, as expressed in the form <math>\texttt{Mno}(),</math> then it cannot be true that exactly one of the arguments is false, so <math>\texttt{Mno}() = \texttt{False}.</math>
@@ Line 120: / Line 120: @@
 <br>
-{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:70%"
-|+ <math>\text{Table 3.}~~\text{Logical Boundaries and Their Complements}</math>
+|+ <math>\text{Table 3.} ~~ \text{Logical Boundaries and Their Complements}\!</math>
-|- style="background:#f0f0ff"
+|- style="background:ghostwhite"
 | <math>\mathcal{L}_1</math>
 | <math>\mathcal{L}_2</math>
 | <math>\mathcal{L}_3</math>
 | <math>\mathcal{L}_4</math>
-|- style="background:#f0f0ff"
+|- style="background:ghostwhite"
 | &nbsp;
 | align="right" | <math>p\colon\!</math>
 | <math>1~1~1~1~0~0~0~0</math>
 | &nbsp;
-|- style="background:#f0f0ff"
+|- style="background:ghostwhite"
 | &nbsp;
 | align="right" | <math>q\colon\!</math>
 | <math>1~1~0~0~1~1~0~0</math>
 | &nbsp;
-|- style="background:#f0f0ff"
+|- style="background:ghostwhite"
 | &nbsp;
 | align="right" | <math>r\colon\!</math>
@@ Line 420: / Line 420: @@
 {{col-begin}}
 {{col-break}}
 * [http://mywikibiz.com/Minimal_negation_operator Minimal Negation Operator @ MyWikiBiz]
 {{col-break}}
-* [http://p2pfoundation.net/Minimal_Negation_Operator Minimal Negation Operator @ P2P Foundation]
+* [http://beta.wikiversity.org/wiki/Minimal_negation_operator Minimal Negation Operator @ Wikiversity Beta]
 * [http://ref.subwiki.org/wiki/Minimal_negation_operator Minimal Negation Operator @ Subject Wikis]
 {{col-end}}
@@ Line 506: / Line 504: @@
 ===Related articles===
-* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;]
+{{col-begin}}
+{{col-break}}
-* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;]
+* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
+* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
-* [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;]
+* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
+{{col-break}}
-* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;]
+* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
+* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
-* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;]
+* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
+{{col-break}}
-* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;]
+* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
+* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
-* [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;]
+* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
+{{col-end}}
 ==Document history==
@@ Line 530: / Line 527: @@
 * [http://mywikibiz.com/Minimal_negation_operator Minimal Negation Operator], [http://mywikibiz.com/ MyWikiBiz]
 * [http://planetmath.org/MinimalNegationOperator Minimal Negation Operator], [http://planetmath.org/ PlanetMath]
 * [http://beta.wikiversity.org/wiki/Minimal_negation_operator Minimal Negation Operator], [http://beta.wikiversity.org/ Wikiversity Beta]
 {{col-break}}
-* [http://getwiki.net/-Minimal_Negation_Operator Minimal Negation Operator], [http://getwiki.net/ GetWiki]
+* [http://web.archive.org/web/20070703045600/http://wikinfo.org/index.php/Minimal_negation_operator Minimal Negation Operator], [http://wikinfo.org/ Wikinfo]
 * [http://textop.org/wiki/index.php?title=Minimal_negation_operator Minimal Negation Operator], [http://textop.org/wiki/ Textop Wiki]
-* [http://en.wikipedia.org/w/index.php?title=Minimal_negation_operator&oldid=75156728 Minimal Negation Operator], [http://en.wikipedia.org/ Wikipedia]
+* [http://web.archive.org/web/20060913000000/http://en.wikipedia.org/wiki/Minimal_negation_operator Minimal Negation Operator], [http://en.wikipedia.org/ Wikipedia]
 {{col-end}}
 [[Category:Automata Theory]]
 [[Category:Charles Sanders Peirce]]
 [[Category:Combinatorics]]
@@ Line 551: / Line 544: @@
 [[Category:Formal Sciences]]
 [[Category:Formal Systems]]
 [[Category:Linguistics]]
 [[Category:Logic]]
 [[Category:Mathematics]]
 [[Category:Neural Networks]]
 [[Category:Philosophy]]
 [[Category:Semiotics]]

@@ Line 422: / Line 422: @@
 * [http://mywikibiz.com/Minimal_negation_operator Minimal Negation Operator @ MyWikiBiz]
 * [http://intersci.ss.uci.edu/wiki/index.php/Minimal_negation_operator Minimal Negation Operator @ InterSciWiki]
-* [http://wiki.oercommons.org/mediawiki/index.php/Minimal_negation_operator Minimal Negation Operator @ OER Commons]
+* [http://wiki.oercommons.org/index.php/Minimal_negation_operator Minimal Negation Operator @ OER Commons]
 {{col-break}}
 * [http://p2pfoundation.net/Minimal_Negation_Operator Minimal Negation Operator @ P2P Foundation]

Minimal negation operator - Revision history

Jon Awbrey: update from blog

Jon Awbrey: update, clarity, less awkward phrasing

Jon Awbrey: update

Jon Awbrey: update

Jon Awbrey: update

Jon Awbrey: * '''Note.''' The MathJax parser is not rendering this page properly.Until it can be fixed please see the [http://intersci.ss.uci.edu/wiki/index.php/Minimal_negation_operator InterSciWiki version].

Jon Awbrey: * '''Note.''' The MathJax parser is not rendering this page properly.Until it can be fixed please see the [http://intersci.ss.uci.edu/wiki/index.php/Minimal_negation_operator InterSciWiki version].

Jon Awbrey: update

Jon Awbrey: update

Jon Awbrey: /* Peer nodes */ update link

Jon Awbrey: * '''Note.''' The MathJax parser is not rendering this page properly.
Until it can be fixed please see the [http://intersci.ss.uci.edu/wiki/index.php/Minimal_negation_operator InterSciWiki version].

Jon Awbrey: * '''Note.''' The MathJax parser is not rendering this page properly.
Until it can be fixed please see the [http://intersci.ss.uci.edu/wiki/index.php/Minimal_negation_operator InterSciWiki version].