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{{DISPLAYTITLE:Cactus Language}}
 
{{DISPLAYTITLE:Cactus Language}}
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<div class="nonumtoc">__TOC__</div>
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'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''
    
==The Cactus Patch==
 
==The Cactus Patch==
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<p>Thus, what looks to us like a sphere of scientific knowledge more accurately should be represented as the inside of a highly irregular and spiky object, like a pincushion or porcupine, with very sharp extensions in certain directions, and virtually no knowledge in immediately adjacent areas.  If our intellectual gaze could shift slightly, it would alter each quill's direction, and suddenly our entire reality would change.</p>
 
<p>Thus, what looks to us like a sphere of scientific knowledge more accurately should be represented as the inside of a highly irregular and spiky object, like a pincushion or porcupine, with very sharp extensions in certain directions, and virtually no knowledge in immediately adjacent areas.  If our intellectual gaze could shift slightly, it would alter each quill's direction, and suddenly our entire reality would change.</p>
 
|-
 
|-
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| align="right" | &mdash; Herbert J. Bernstein, "Idols of Modern Science", [HJB, 38]
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| align="right" | &mdash; Herbert J. Bernstein, &ldquo;Idols of Modern Science&rdquo;, [HJB, 38]
 
|}
 
|}
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{| align="center" cellpadding="4" style="text-align:center" width="90%"
 
{| align="center" cellpadding="4" style="text-align:center" width="90%"
 
|-
 
|-
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| <math>\varepsilon</math>
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| <math>\varepsilon\!</math>
 
| =
 
| =
−
| <math>^{\backprime\backprime\prime\prime}</math>
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| <math>{}^{\backprime\backprime\prime\prime}\!</math>
 
| =
 
| =
 
| align="left" | the empty string.
 
| align="left" | the empty string.
 
|-
 
|-
−
| <math>\underline\varepsilon</math>
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| <math>\underline\varepsilon\!</math>
 
| =
 
| =
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| <math>\{ \varepsilon \}</math>
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| <math>\{ \varepsilon \}\!</math>
 
| =
 
| =
 
| align="left" | the language consisting of a single empty string.
 
| align="left" | the language consisting of a single empty string.
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</ol>
 
</ol>
   −
The easiest way to define the language <math>\mathfrak{C}(\mathfrak{P})</math> is to indicate the general sorts of operations that suffice to construct the greater share of its sentences from the specified few of its sentences that require a special election.  In accord with this manner of proceeding, I introduce a family of operations on strings of <math>\mathfrak{A}^*</math> that are called ''syntactic connectives''.  If the strings on which they operate are exclusively sentences of <math>\mathfrak{C}(\mathfrak{P}),</math> then these operations are tantamount to ''sentential connectives'', and if the syntactic sentences, considered as abstract strings of meaningless signs, are given a semantics in which they denote propositions, considered as indicator functions over some universe, then these operations amount to ''propositional connectives''.
+
The easiest way to define the language <math>\mathfrak{C}(\mathfrak{P})\!</math> is to indicate the general sorts of operations that suffice to construct the greater share of its sentences from the specified few of its sentences that require a special election.  In accord with this manner of proceeding, I introduce a family of operations on strings of <math>\mathfrak{A}^*\!</math> that are called ''syntactic connectives''.  If the strings on which they operate are exclusively sentences of <math>\mathfrak{C}(\mathfrak{P}),\!</math> then these operations are tantamount to ''sentential connectives'', and if the syntactic sentences, considered as abstract strings of meaningless signs, are given a semantics in which they denote propositions, considered as indicator functions over some universe, then these operations amount to ''propositional connectives''.
    
Rather than presenting the most concise description of these languages right from the beginning, it serves comprehension to develop a picture of their forms in gradual stages, starting from the most natural ways of viewing their elements, if somewhat at a distance, and working through the most easily grasped impressions of their structures, if not always the sharpest acquaintances with their details.
 
Rather than presenting the most concise description of these languages right from the beginning, it serves comprehension to develop a picture of their forms in gradual stages, starting from the most natural ways of viewing their elements, if somewhat at a distance, and working through the most easily grasped impressions of their structures, if not always the sharpest acquaintances with their details.
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<p>The ''concatenation'' of one string <math>s_1\!</math> is just the string <math>s_1.\!</math></p>
 
<p>The ''concatenation'' of one string <math>s_1\!</math> is just the string <math>s_1.\!</math></p>
   −
<p>The ''concatenation'' of two strings <math>s_1, s_2\!</math> is the string <math>s_1 \cdot s_2.\!</math></p>
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<p>The ''concatenation'' of two strings <math>s_1, s_2\!</math> is the string <math>{s_1 \cdot s_2}.\!</math></p>
   −
<p>The ''concatenation'' of the <math>k\!</math> strings <math>(s_j)_{j = 1}^k</math> is the string of the form <math>s_1 \cdot \ldots \cdot s_k.\!</math></p></li>
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<p>The ''concatenation'' of the <math>k\!</math> strings <math>(s_j)_{j = 1}^k\!</math> is the string of the form <math>{s_1 \cdot \ldots \cdot s_k}.\!</math></p></li>
    
<li>
 
<li>
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| 6. || <math>s\!</math> is the ''surcatenation'' of the <math>k\!</math> strings <math>s_1, \ldots, s_k\!</math> in <math>\mathfrak{L},</math>
 
| 6. || <math>s\!</math> is the ''surcatenation'' of the <math>k\!</math> strings <math>s_1, \ldots, s_k\!</math> in <math>\mathfrak{L},</math>
 
|-
 
|-
−
| &nbsp; || if and only if <math>s_j\!</math> is a sentence of <math>\mathfrak{L},</math> for all <math>j = 1 \ldots k,\!</math> and
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| &nbsp; || if and only if <math>s_j\!</math> is a sentence of <math>\mathfrak{L},</math> for all <math>{j = 1 \ldots k},\!</math> and
 
|-
 
|-
 
| &nbsp; || <math>s \ = \ \operatorname{Surc}_{j=1}^k s_j \ = \ ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, \ldots \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_k \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math>
 
| &nbsp; || <math>s \ = \ \operatorname{Surc}_{j=1}^k s_j \ = \ ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, \ldots \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_k \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math>
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|}
 
|}
   −
As usual, saying that <math>s\!</math> is a sentence is just a conventional way of stating that the string <math>s\!</math> belongs to the relevant formal language <math>\mathfrak{L}.</math>  An individual sentence of <math>\mathfrak{C} (\mathfrak{P}),</math> for any palette <math>\mathfrak{P},</math> is referred to as a ''painted and rooted cactus expression'' (PARCE) on the palette <math>\mathfrak{P},</math> or a ''cactus expression'', for short.  Anticipating the forms that the parse graphs of these PARCE's will take, to be described in the next Subsection, the language <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P})</math> is also described as the set <math>\operatorname{PARCE} (\mathfrak{P})</math> of PARCE's on the palette <math>\mathfrak{P},</math> more generically, as the PARCE's that constitute the language <math>\operatorname{PARCE}.</math>
+
As usual, saying that <math>s\!</math> is a sentence is just a conventional way of stating that the string <math>s\!</math> belongs to the relevant formal language <math>\mathfrak{L}.</math>  An individual sentence of <math>\mathfrak{C} (\mathfrak{P}),\!</math> for any palette <math>\mathfrak{P},</math> is referred to as a ''painted and rooted cactus expression'' (PARCE) on the palette <math>\mathfrak{P},</math> or a ''cactus expression'', for short.  Anticipating the forms that the parse graphs of these PARCE's will take, to be described in the next Subsection, the language <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P})</math> is also described as the set <math>\operatorname{PARCE} (\mathfrak{P})</math> of PARCE's on the palette <math>\mathfrak{P},</math> more generically, as the PARCE's that constitute the language <math>\operatorname{PARCE}.</math>
    
A ''bare'' PARCE, a bit loosely referred to as a ''bare cactus expression'', is a PARCE on the empty palette <math>\mathfrak{P} = \varnothing.</math>  A bare PARCE is a sentence in the ''bare cactus language'', <math>\mathfrak{C}^0 = \mathfrak{C} (\varnothing) = \operatorname{PARCE}^0 = \operatorname{PARCE} (\varnothing).</math>  This set of strings, regarded as a formal language in its own right, is a sublanguage of every cactus language <math>\mathfrak{C} (\mathfrak{P}).</math>  A bare cactus expression is commonly encountered in practice when one has occasion to start with an arbitrary PARCE and then finds a reason to delete or to erase all of its paints.
 
A ''bare'' PARCE, a bit loosely referred to as a ''bare cactus expression'', is a PARCE on the empty palette <math>\mathfrak{P} = \varnothing.</math>  A bare PARCE is a sentence in the ''bare cactus language'', <math>\mathfrak{C}^0 = \mathfrak{C} (\varnothing) = \operatorname{PARCE}^0 = \operatorname{PARCE} (\varnothing).</math>  This set of strings, regarded as a formal language in its own right, is a sublanguage of every cactus language <math>\mathfrak{C} (\mathfrak{P}).</math>  A bare cactus expression is commonly encountered in practice when one has occasion to start with an arbitrary PARCE and then finds a reason to delete or to erase all of its paints.
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====Grammar 1====
 
====Grammar 1====
   −
Grammar&nbsp;1 is something of a misnomer.  It is nowhere near exemplifying any kind of a standard form and it is only intended as a starting point for the initiation of more respectable grammars.  Such as it is, it uses the terminal alphabet <math>\mathfrak{A} = \mathfrak{M} \cup \mathfrak{P}</math> that comes with the territory of the cactus language <math>\mathfrak{C} (\mathfrak{P}),</math> it specifies <math>\mathfrak{Q} = \varnothing,</math> in other words, it employs no intermediate symbols, and it embodies the ''covering set'' <math>\mathfrak{K}</math> as listed in the following display.
+
Grammar&nbsp;1 is something of a misnomer.  It is nowhere near exemplifying any kind of a standard form and it is only intended as a starting point for the initiation of more respectable grammars.  Such as it is, it uses the terminal alphabet <math>\mathfrak{A} = \mathfrak{M} \cup \mathfrak{P}</math> that comes with the territory of the cactus language <math>\mathfrak{C} (\mathfrak{P}),\!</math> it specifies <math>\mathfrak{Q} = \varnothing,</math> in other words, it employs no intermediate symbols, and it embodies the ''covering set'' <math>\mathfrak{K}</math> as listed in the following display.
    
<br>
 
<br>
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|}
 
|}
   −
There is nothing wrong with the more expansive pan of the covered equation, since it follows straightforwardly from the definition of the kleene star operation, but the covering statement to the effect that <math>S :> S^*\!</math> is not a very productive piece of information, in the sense of telling very much about the language that falls under the type of a sentence <math>S.\!</math>  In particular, since it implies that <math>S :> \underline\varepsilon,</math> and since <math>\underline\varepsilon \cdot \mathfrak{L} \, = \, \mathfrak{L} \cdot \underline\varepsilon \, = \, \mathfrak{L},</math> for any formal language <math>\mathfrak{L},</math> the empty string <math>\varepsilon</math> is counted over and over in every term of the union, and every non-empty sentence under <math>S\!</math> appears again and again in every term of the union that follows the initial appearance of <math>S.\!</math>  As a result, this style of characterization has to be classified as ''true but not very informative''.  If at all possible, one prefers to partition the language of interest into a disjoint union of subsets, thereby accounting for each sentence under its proper term, and one whose place under the sum serves as a useful parameter of its character or its complexity.  In general, this form of description is not always possible to achieve, but it is usually worth the trouble to actualize it whenever it is.
+
There is nothing wrong with the more expansive pan of the covered equation, since it follows straightforwardly from the definition of the kleene star operation, but the covering statement to the effect that <math>S :> S^*\!</math> is not a very productive piece of information, in the sense of telling very much about the language that falls under the type of a sentence <math>S.\!</math>  In particular, since it implies that <math>S :> \underline\varepsilon,</math> and since <math>\underline\varepsilon \cdot \mathfrak{L} \, = \, \mathfrak{L} \cdot \underline\varepsilon \, = \, \mathfrak{L},</math> for any formal language <math>\mathfrak{L},</math> the empty string <math>\varepsilon\!</math> is counted over and over in every term of the union, and every non-empty sentence under <math>S\!</math> appears again and again in every term of the union that follows the initial appearance of <math>S.\!</math>  As a result, this style of characterization has to be classified as ''true but not very informative''.  If at all possible, one prefers to partition the language of interest into a disjoint union of subsets, thereby accounting for each sentence under its proper term, and one whose place under the sum serves as a useful parameter of its character or its complexity.  In general, this form of description is not always possible to achieve, but it is usually worth the trouble to actualize it whenever it is.
    
Suppose that one tries to deal with this problem by eliminating each use of the kleene star operation, by reducing it to a purely finitary set of steps, or by finding an alternative way to cover the sublanguage that it is used to generate.  This amounts, in effect, to ''recognizing a type'', a complex process that involves the following steps:
 
Suppose that one tries to deal with this problem by eliminating each use of the kleene star operation, by reducing it to a purely finitary set of steps, or by finding an alternative way to cover the sublanguage that it is used to generate.  This amounts, in effect, to ''recognizing a type'', a complex process that involves the following steps:
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In sum, one introduces a non-terminal symbol for each type of sentence and each ''part of speech'' or sentential component that is generated by means of iteration or recursion under the ruling constraints of the grammar.  In order to do this one needs to analyze the iteration of each grammatical operation in a way that is analogous to a mathematically inductive definition, but further in a way that is not forced explicitly to recognize a distinct and separate type of expression merely to account for and to recount every increment in the parameter of iteration.
 
In sum, one introduces a non-terminal symbol for each type of sentence and each ''part of speech'' or sentential component that is generated by means of iteration or recursion under the ruling constraints of the grammar.  In order to do this one needs to analyze the iteration of each grammatical operation in a way that is analogous to a mathematically inductive definition, but further in a way that is not forced explicitly to recognize a distinct and separate type of expression merely to account for and to recount every increment in the parameter of iteration.
   −
Returning to the case of the cactus language, the process of recognizing an iterative type or a recursive type can be illustrated in the following way.  The operative phrases in the simplest sort of recursive definition are its ''initial part'' and its ''generic part''.  For the cactus language <math>\mathfrak{C} (\mathfrak{P}),</math> one has the following definitions of concatenation as iterated precatenation and of surcatenation as iterated subcatenation, respectively:
+
Returning to the case of the cactus language, the process of recognizing an iterative type or a recursive type can be illustrated in the following way.  The operative phrases in the simplest sort of recursive definition are its ''initial part'' and its ''generic part''.  For the cactus language <math>\mathfrak{C} (\mathfrak{P}),\!</math> one has the following definitions of concatenation as iterated precatenation and of surcatenation as iterated subcatenation, respectively:
    
{| align="center" cellpadding="8" width="90%"
 
{| align="center" cellpadding="8" width="90%"
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When there is no possibility of confusion, the letter <math>^{\backprime\backprime} R \, ^{\prime\prime}</math> can be used either as a string variable that ranges over the set of runes or else as a type name for the class of runes.  The latter reading amounts to the enlistment of a fresh intermediate symbol, <math>^{\backprime\backprime} R \, ^{\prime\prime} \in \mathfrak{Q},</math> as a part of a new grammar for <math>\mathfrak{C} (\mathfrak{P}).</math>  In effect, <math>^{\backprime\backprime} R \, ^{\prime\prime}</math> affords a grammatical recognition for any rune that forms a part of a sentence in <math>\mathfrak{C} (\mathfrak{P}).</math>  In situations where these variant usages are likely to be confused, the types of strings can be indicated by means of expressions like <math>r <: R\!</math> and <math>W <: R.\!</math>
 
When there is no possibility of confusion, the letter <math>^{\backprime\backprime} R \, ^{\prime\prime}</math> can be used either as a string variable that ranges over the set of runes or else as a type name for the class of runes.  The latter reading amounts to the enlistment of a fresh intermediate symbol, <math>^{\backprime\backprime} R \, ^{\prime\prime} \in \mathfrak{Q},</math> as a part of a new grammar for <math>\mathfrak{C} (\mathfrak{P}).</math>  In effect, <math>^{\backprime\backprime} R \, ^{\prime\prime}</math> affords a grammatical recognition for any rune that forms a part of a sentence in <math>\mathfrak{C} (\mathfrak{P}).</math>  In situations where these variant usages are likely to be confused, the types of strings can be indicated by means of expressions like <math>r <: R\!</math> and <math>W <: R.\!</math>
   −
A ''foil'' is a string of the form <math>^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime},</math> where <math>T\!</math> is a tract.  Thus, a typical foil <math>F\!</math> has the form:
+
A ''foil'' is a string of the form <math>{}^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime},\!</math> where <math>T\!</math> is a tract.  Thus, a typical foil <math>F\!</math> has the form:
    
{| align="center" cellpadding="8" width="90%"
 
{| align="center" cellpadding="8" width="90%"
 
|
 
|
−
<math>\begin{array}{lllllllllllllll}
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<math>\begin{array}{*{15}{l}}
 
F
 
F
 
& =
 
& =
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|}
 
|}
   −
This is just the surcatenation of the sentences <math>S_1, \ldots, S_k.\!</math>  Given the possibility that this sequence of sentences is empty, and thus that the tract <math>T\!</math> is the empty string, the minimum foil <math>F\!</math> is the expression <math>^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}.</math>  Explicitly marking each foil <math>F\!</math> that is embodied in a cactus expression is tantamount to recognizing another intermediate symbol, <math>^{\backprime\backprime} F \, ^{\prime\prime} \in \mathfrak{Q},</math> further articulating the structures of sentences and expanding the grammar for the language
+
This is just the surcatenation of the sentences <math>S_1, \ldots, S_k.\!</math>  Given the possibility that this sequence of sentences is empty, and thus that the tract <math>T\!</math> is the empty string, the minimum foil <math>F\!</math> is the expression <math>^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}.</math>  Explicitly marking each foil <math>F\!</math> that is embodied in a cactus expression is tantamount to recognizing another intermediate symbol, <math>^{\backprime\backprime} F \, ^{\prime\prime} \in \mathfrak{Q},</math> further articulating the structures of sentences and expanding the grammar for the language <math>\mathfrak{C} (\mathfrak{P}).\!</math>  All of the same remarks about the versatile uses of the intermediate symbols, as string variables and as type names, apply again to the letter <math>^{\backprime\backprime} F \, ^{\prime\prime}.</math>
−
<math>\mathfrak{C} (\mathfrak{P}).</math>  All of the same remarks about the versatile uses of the intermediate symbols, as string variables and as type names, apply again to the letter <math>^{\backprime\backprime} F \, ^{\prime\prime}.</math>
      
<br>
 
<br>
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& T \, \cdot \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S
 
& T \, \cdot \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S
 
\\
 
\\
−
\end{array}</math>
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\end{array}\!</math>
 
|}
 
|}
    
<br>
 
<br>
   −
In Grammar&nbsp;3, the first three Rules say that a sentence (a string of type <math>S\!</math>), is a rune (a string of type <math>R\!</math>), a foil (a string of type <math>F\!</math>), or an arbitrary concatenation of strings of these two types.  Rules&nbsp;4 through 7 specify that a rune <math>R\!</math> is an empty string <math>\varepsilon,</math> a blank symbol <math>m_1,\!</math> a paint <math>p_j,\!</math> or any concatenation of strings of these three types.  Rule&nbsp;8 characterizes a foil <math>F\!</math> as a string of the form <math>^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime},</math> where <math>T\!</math> is a tract.  The last two Rules say that a tract <math>T\!</math> is either a sentence <math>S\!</math> or else the concatenation of a tract, a comma, and a sentence, in that order.
+
In Grammar&nbsp;3, the first three Rules say that a sentence (a string of type <math>S\!</math>), is a rune (a string of type <math>R\!</math>), a foil (a string of type <math>F\!</math>), or an arbitrary concatenation of strings of these two types.  Rules&nbsp;4 through 7 specify that a rune <math>R\!</math> is an empty string <math>\varepsilon,</math> a blank symbol <math>m_1,\!</math> a paint <math>p_j,\!</math> or any concatenation of strings of these three types.  Rule&nbsp;8 characterizes a foil <math>F\!</math> as a string of the form <math>{}^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime},\!</math> where <math>T\!</math> is a tract.  The last two Rules say that a tract <math>T\!</math> is either a sentence <math>S\!</math> or else the concatenation of a tract, a comma, and a sentence, in that order.
   −
At this point in the succession of grammars for <math>\mathfrak{C} (\mathfrak{P}),</math> the explicit uses of indefinite iterations, like the kleene star operator, are now completely reduced to finite forms of concatenation, but the problems that some styles of analysis have with allowing non-terminal symbols to cover both themselves and the empty string are still present.
+
At this point in the succession of grammars for <math>\mathfrak{C} (\mathfrak{P}),\!</math> the explicit uses of indefinite iterations, like the kleene star operator, are now completely reduced to finite forms of concatenation, but the problems that some styles of analysis have with allowing non-terminal symbols to cover both themselves and the empty string are still present.
    
Any degree of reflection on this difficulty raises the general question:  What is a practical strategy for accounting for the empty string in the organization of any formal language that counts it among its sentences?  One answer that presents itself is this:  If the empty string belongs to a formal language, it suffices to count it once at the beginning of the formal account that enumerates its sentences and then to move on to more interesting materials.
 
Any degree of reflection on this difficulty raises the general question:  What is a practical strategy for accounting for the empty string in the organization of any formal language that counts it among its sentences?  One answer that presents itself is this:  If the empty string belongs to a formal language, it suffices to count it once at the beginning of the formal account that enumerates its sentences and then to move on to more interesting materials.
   −
Returning to the case of the cactus language <math>\mathfrak{C} (\mathfrak{P}),</math> in other words, the formal language <math>\operatorname{PARCE}</math> of ''painted and rooted cactus expressions'', it serves the purpose of efficient accounting to partition the language into the following couple of sublanguages:
+
Returning to the case of the cactus language <math>\mathfrak{C} (\mathfrak{P}),\!</math> in other words, the formal language <math>\operatorname{PARCE}\!</math> of ''painted and rooted cactus expressions'', it serves the purpose of efficient accounting to partition the language into the following couple of sublanguages:
    
<ol style="list-style-type:decimal">
 
<ol style="list-style-type:decimal">
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{| align="center" cellpadding="12" cellspacing="0" style="border-top:1px solid black" width="90%"
 
{| align="center" cellpadding="12" cellspacing="0" style="border-top:1px solid black" width="90%"
−
| align="left" style="border-left:1px solid black;"  width="50%" |
+
| align="left" style="border-left:1px solid black;"  width="50%" |
−
<math>\mathfrak{C} (\mathfrak{P}) : \text{Grammar 6}\!</math>
+
<math>{\mathfrak{C} (\mathfrak{P}) : \text{Grammar 6}}\!</math>
 
| align="right" style="border-right:1px solid black;" width="50%" |
 
| align="right" style="border-right:1px solid black;" width="50%" |
−
<math>\mathfrak{Q} = \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} F \, ^{\prime\prime}, \, ^{\backprime\backprime} R \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \}</math>
+
<math>\mathfrak{Q} = \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} F \, ^{\prime\prime}, \, ^{\backprime\backprime} R \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \}\!</math>
 
|-
 
|-
 
| colspan="2" style="border-top:1px solid black; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black" |
 
| colspan="2" style="border-top:1px solid black; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black" |
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|}
 
|}
   −
In this usage, the characterization <math>S_1 :> S_2\!</math> is tantamount to a grammatical license to transform a string of the form <math>Q_1 \cdot q \cdot Q_2</math> into a string of the form <math>Q1 \cdot W \cdot Q2,</math> in effect, replacing the non-terminal symbol <math>q\!</math> with the non-initial string <math>W\!</math> in any selected, preserved, and closely adjoining context of the form <math>Q1 \cdot \underline{~~~} \cdot Q2.</math>  In this application the notation <math>S_1 :> S_2\!</math> can be read to say that <math>S_1\!</math> ''produces'' <math>S_2\!</math> or that <math>S_1\!</math> ''transforms into'' <math>S_2.\!</math>
+
In this usage, the characterization <math>S_1 :> S_2\!</math> is tantamount to a grammatical license to transform a string of the form <math>Q_1 \cdot q \cdot Q_2</math> into a string of the form <math>Q1 \cdot W \cdot Q2,</math> in effect, replacing the non-terminal symbol <math>q\!</math> with the non-initial string <math>W\!</math> in any selected, preserved, and closely adjoining context of the form <math>Q1 \cdot \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]])} \cdot Q2.</math>  In this application the notation <math>S_1 :> S_2\!</math> can be read to say that <math>S_1\!</math> ''produces'' <math>S_2\!</math> or that <math>S_1\!</math> ''transforms into'' <math>S_2.\!</math>
   −
An ''immediate derivation'' in <math>\mathfrak{G}</math> is an ordered pair <math>(W, W')\!</math> of sentential forms in <math>\mathfrak{G}</math> such that:
+
An ''immediate derivation'' in <math>\mathfrak{G}\!</math> is an ordered pair <math>(W, W^\prime)\!</math> of sentential forms in <math>\mathfrak{G}\!</math> such that:
    
{| align="center" cellpadding="8" width="90%"
 
{| align="center" cellpadding="8" width="90%"
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Any style of declarative programming, also called ''logic programming'', depends on a capacity, as embodied in a programming language or other formal system, to describe the relation between problems and solutions in logical terms.  A recurring problem in building this capacity is in bridging the gap between ostensibly non-logical orders and the logical orders that are used to describe and to represent them.  For instance, to mention just a couple of the most pressing cases, and the ones that are currently proving to be the most resistant to a complete analysis, one has the orders of dynamic evolution and rhetorical transition that manifest themselves in the process of inquiry and in the communication of its results.
 
Any style of declarative programming, also called ''logic programming'', depends on a capacity, as embodied in a programming language or other formal system, to describe the relation between problems and solutions in logical terms.  A recurring problem in building this capacity is in bridging the gap between ostensibly non-logical orders and the logical orders that are used to describe and to represent them.  For instance, to mention just a couple of the most pressing cases, and the ones that are currently proving to be the most resistant to a complete analysis, one has the orders of dynamic evolution and rhetorical transition that manifest themselves in the process of inquiry and in the communication of its results.
   −
This patch of the ongoing discussion is concerned with describing a particular variety of formal languages, whose typical representative is the painted cactus language <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P}).</math>  It is the intention of this work to interpret this language for propositional logic, and thus to use it as a sentential calculus, an order of reasoning that forms an active ingredient and a significant component of all logical reasoning.  To describe this language, the standard devices of formal grammars and formal language theory are more than adequate, but this only raises the next question:  What sorts of devices are exactly adequate, and fit the task to a "T"?  The ultimate desire is to turn the tables on the order of description, and so begins a process of eversion that evolves to the point of asking:  To what extent can the language capture the essential features and laws of its own grammar and describe the active principles of its own generation?  In other words:  How well can the language be described by using the language itself to do so?
+
This patch of the ongoing discussion is concerned with describing a particular variety of formal languages, whose typical representative is the painted cactus language <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P}).\!</math>  It is the intention of this work to interpret this language for propositional logic, and thus to use it as a sentential calculus, an order of reasoning that forms an active ingredient and a significant component of all logical reasoning.  To describe this language, the standard devices of formal grammars and formal language theory are more than adequate, but this only raises the next question:  What sorts of devices are exactly adequate, and fit the task to a "T"?  The ultimate desire is to turn the tables on the order of description, and so begins a process of eversion that evolves to the point of asking:  To what extent can the language capture the essential features and laws of its own grammar and describe the active principles of its own generation?  In other words:  How well can the language be described by using the language itself to do so?
    
In order to speak to these questions, I have to express what a grammar says about a language in terms of what a language can say on its own.  In effect, it is necessary to analyze the kinds of meaningful statements that grammars are capable of making about languages in general and to relate them to the kinds of meaningful statements that the syntactic ''sentences'' of the cactus language might be interpreted as making about the very same topics.  So far in the present discussion, the sentences of the cactus language do not make any meaningful statements at all, much less any meaningful statements about languages and their constitutions.  As of yet, these sentences subsist in the form of purely abstract, formal, and uninterpreted combinatorial constructions.
 
In order to speak to these questions, I have to express what a grammar says about a language in terms of what a language can say on its own.  In effect, it is necessary to analyze the kinds of meaningful statements that grammars are capable of making about languages in general and to relate them to the kinds of meaningful statements that the syntactic ''sentences'' of the cactus language might be interpreted as making about the very same topics.  So far in the present discussion, the sentences of the cactus language do not make any meaningful statements at all, much less any meaningful statements about languages and their constitutions.  As of yet, these sentences subsist in the form of purely abstract, formal, and uninterpreted combinatorial constructions.
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|}
 
|}
   −
The concatenation <math>\mathfrak{L}_1 \cdot \mathfrak{L}_2</math> of the formal languages <math>\mathfrak{L}_1</math> and <math>\mathfrak{L}_2</math> is just the cartesian product of sets <math>\mathfrak{L}_1 \times \mathfrak{L}_2</math> without the extra <math>\times</math>'s, but the relation of cartesian products to set-theoretic intersections and thus to logical conjunctions is far from being clear.  One way of seeing a type of relation is to focus on the information that is needed to specify each construction, and thus to reflect on the signs that are used to carry this information.  As a first approach to the topic of information, according to a strategy that seeks to be as elementary and as informal as possible, I introduce the following set of ideas, intended to be taken in a very provisional way.
+
The concatenation <math>\mathfrak{L}_1 \cdot \mathfrak{L}_2</math> of the formal languages <math>\mathfrak{L}_1\!</math> and <math>\mathfrak{L}_2\!</math> is just the cartesian product of sets <math>\mathfrak{L}_1 \times \mathfrak{L}_2</math> without the extra <math>\times\!</math>'s, but the relation of cartesian products to set-theoretic intersections and thus to logical conjunctions is far from being clear.  One way of seeing a type of relation is to focus on the information that is needed to specify each construction, and thus to reflect on the signs that are used to carry this information.  As a first approach to the topic of information, according to a strategy that seeks to be as elementary and as informal as possible, I introduce the following set of ideas, intended to be taken in a very provisional way.
    
A ''stricture'' is a specification of a certain set in a certain place, relative to a number of other sets, yet to be specified.  It is assumed that one knows enough to tell if two strictures are equivalent as pieces of information, but any more determinate indications, like names for the places that are mentioned in the stricture, or bounds on the number of places that are involved, are regarded as being extraneous impositions, outside the proper concern of the definition, no matter how convenient they are found to be for a particular discussion.  As a schematic form of illustration, a stricture can be pictured in the following shape:
 
A ''stricture'' is a specification of a certain set in a certain place, relative to a number of other sets, yet to be specified.  It is assumed that one knows enough to tell if two strictures are equivalent as pieces of information, but any more determinate indications, like names for the places that are mentioned in the stricture, or bounds on the number of places that are involved, are regarded as being extraneous impositions, outside the proper concern of the definition, no matter how convenient they are found to be for a particular discussion.  As a schematic form of illustration, a stricture can be pictured in the following shape:
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# <math>q_{[2]}\!</math> says that <math>q\!</math> is in the second place of the product element under construction.
 
# <math>q_{[2]}\!</math> says that <math>q\!</math> is in the second place of the product element under construction.
   −
Notice that, in construing the cartesian product of the sets <math>P\!</math> and <math>Q\!</math> or the concatenation of the languages <math>\mathfrak{L}_1</math> and <math>\mathfrak{L}_2</math> in this way, one shifts the level of the active construction from the tupling of the elements in <math>P\!</math> and <math>Q\!</math> or the concatenation of the strings that are internal to the languages <math>\mathfrak{L}_1</math> and <math>\mathfrak{L}_2</math> to the concatenation of the external signs that it takes to indicate these sets or these languages, in other words, passing to a conjunction of indexed propositions, <math>P_{[1]}\!</math> and <math>Q_{[2]},\!</math> or to a conjunction of assertions, <math>(\mathfrak{L}_1)_{[1]}</math> and <math>(\mathfrak{L}_2)_{[2]},</math> that marks the sets or the languages in question for insertion in the indicated places of a product set or a product language, respectively.  In effect, the subscripting by the indices <math>^{\backprime\backprime} [1] ^{\prime\prime}</math> and <math>^{\backprime\backprime} [2] ^{\prime\prime}</math> can be recognized as a special case of concatenation, albeit through the posting of editorial remarks from an external ''mark-up'' language.
+
Notice that, in construing the cartesian product of the sets <math>P\!</math> and <math>Q\!</math> or the concatenation of the languages <math>\mathfrak{L}_1\!</math> and <math>\mathfrak{L}_2\!</math> in this way, one shifts the level of the active construction from the tupling of the elements in <math>P\!</math> and <math>Q\!</math> or the concatenation of the strings that are internal to the languages <math>\mathfrak{L}_1\!</math> and <math>\mathfrak{L}_2\!</math> to the concatenation of the external signs that it takes to indicate these sets or these languages, in other words, passing to a conjunction of indexed propositions, <math>P_{[1]}\!</math> and <math>Q_{[2]},\!</math> or to a conjunction of assertions, <math>(\mathfrak{L}_1)_{[1]}</math> and <math>(\mathfrak{L}_2)_{[2]},</math> that marks the sets or the languages in question for insertion in the indicated places of a product set or a product language, respectively.  In effect, the subscripting by the indices <math>^{\backprime\backprime} [1] ^{\prime\prime}</math> and <math>^{\backprime\backprime} [2] ^{\prime\prime}</math> can be recognized as a special case of concatenation, albeit through the posting of editorial remarks from an external ''mark-up'' language.
    
In order to systematize the relations that strictures and straits placed at higher levels of complexity, constraint, information, and organization have with those that are placed at the associated lower levels, I introduce the following pair of definitions:
 
In order to systematize the relations that strictures and straits placed at higher levels of complexity, constraint, information, and organization have with those that are placed at the associated lower levels, I introduce the following pair of definitions:
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|}
 
|}
   −
Working from a structural description of the cactus language, or any suitable formal grammar for <math>\mathfrak{C} (\mathfrak{P}),</math> it is possible to give a recursive definition of the function called <math>\operatorname{Parse}</math> that maps each sentence in <math>\operatorname{PARCE} (\mathfrak{P})</math> to the corresponding graph in <math>\operatorname{PARC} (\mathfrak{P}).</math>  One way to do this proceeds as follows:
+
Working from a structural description of the cactus language, or any suitable formal grammar for <math>\mathfrak{C} (\mathfrak{P}),\!</math> it is possible to give a recursive definition of the function called <math>\operatorname{Parse}</math> that maps each sentence in <math>\operatorname{PARCE} (\mathfrak{P})\!</math> to the corresponding graph in <math>\operatorname{PARC} (\mathfrak{P}).\!</math>  One way to do this proceeds as follows:
    
<ol style="list-style-type:decimal">
 
<ol style="list-style-type:decimal">
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<br>
 
<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:80%"
−
|+ '''Table 13. Algorithmic Translation Rules'''
+
|+ style="height:30px" | <math>\text{Table 13.} ~~ \text{Algorithmic Translation Rules}\!</math>
−
|- style="background:whitesmoke"
+
|- style="height:40px; background:ghostwhite"
 
|
 
|
−
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:whitesmoke; width:100%"
+
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:ghostwhite; text-align:center; width:100%"
−
| width="33%"   | <math>\text{Sentence in PARCE}\!</math>
+
| width="33%" | <math>\text{Sentence in PARCE}\!</math>
−
| align="center" | <math>\xrightarrow{\operatorname{Parse}}</math>
+
| width="33%" | <math>\xrightarrow{\mathrm{Parse}}\!</math>
−
| width="33%"   | <math>\text{Graph in PARC}\!</math>
+
| width="33%" | <math>\text{Graph in PARC}\!</math>
 
|}
 
|}
 
|-
 
|-
 
|
 
|
−
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
+
{| align="center" border="0" cellpadding="8" cellspacing="0" style="text-align:center; width:100%"
−
| width="33%"   | <math>\operatorname{Conc}^0</math>
+
| width="33%" | <math>\mathrm{Conc}^0\!</math>
−
| align="center" | <math>\xrightarrow{\operatorname{Parse}}</math>
+
| width="33%" | <math>\xrightarrow{\mathrm{Parse}}\!</math>
−
| width="33%"   | <math>\operatorname{Node}^0</math>
+
| width="33%" | <math>\mathrm{Node}^0\!</math>
 
|-
 
|-
−
| width="33%"   | <math>\operatorname{Conc}_{j=1}^k s_j</math>
+
| width="33%" | <math>\mathrm{Conc}_{j=1}^k s_j\!</math>
−
| align="center" | <math>\xrightarrow{\operatorname{Parse}}</math>
+
| width="33%" | <math>\xrightarrow{\mathrm{Parse}}\!</math>
−
| width="33%"   | <math>\operatorname{Node}_{j=1}^k \operatorname{Parse} (s_j)</math>
+
| width="33%" | <math>\mathrm{Node}_{j=1}^k \mathrm{Parse} (s_j)\!</math>
 
|}
 
|}
 
|-
 
|-
 
|
 
|
−
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
+
{| align="center" border="0" cellpadding="8" cellspacing="0" style="text-align:center; width:100%"
−
| width="33%"   | <math>\operatorname{Surc}^0</math>
+
| width="33%" | <math>\mathrm{Surc}^0\!</math>
−
| align="center" | <math>\xrightarrow{\operatorname{Parse}}</math>
+
| width="33%" | <math>\xrightarrow{\mathrm{Parse}}\!</math>
−
| width="33%"   | <math>\operatorname{Lobe}^0</math>
+
| width="33%" | <math>\mathrm{Lobe}^0\!</math>
 
|-
 
|-
−
| width="33%"   | <math>\operatorname{Surc}_{j=1}^k s_j</math>
+
| width="33%" | <math>\mathrm{Surc}_{j=1}^k s_j\!</math>
−
| align="center" | <math>\xrightarrow{\operatorname{Parse}}</math>
+
| width="33%" | <math>\xrightarrow{\mathrm{Parse}}\!</math>
−
| width="33%"   | <math>\operatorname{Lobe}_{j=1}^k \operatorname{Parse} (s_j)</math>
+
| width="33%" | <math>\mathrm{Lobe}_{j=1}^k \mathrm{Parse} (s_j)\!</math>
 
|}
 
|}
 
|}
 
|}
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<br>
 
<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:80%"
−
|+ '''Table 14. Semantic Translation : Functional Form'''
+
|+ style="height:30px" | <math>\text{Table 14.} ~~ \text{Semantic Translation : Functional Form}\!</math>
−
|- style="background:whitesmoke"
+
|- style="height:40px; background:ghostwhite"
 
|
 
|
−
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:whitesmoke; width:100%"
+
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:ghostwhite; width:100%"
−
| width="20%" | <math>\operatorname{Sentence}</math>
+
| width="20%" | <math>\mathrm{Sentence}\!</math>
−
| width="20%" | <math>\xrightarrow[\operatorname{~~~~~~~~~~}]{\operatorname{Parse}}</math>
+
| width="20%" | <math>\xrightarrow[\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}]{\mathrm{Parse}}\!</math>
−
| width="20%" | <math>\operatorname{Graph}</math>
+
| width="20%" | <math>\mathrm{Graph}\!</math>
−
| width="20%" | <math>\xrightarrow[\operatorname{~~~~~~~~~~}]{\operatorname{Denotation}}</math>
+
| width="20%" | <math>\xrightarrow[\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}]{\mathrm{Denotation}}\!</math>
−
| width="20%" | <math>\operatorname{Proposition}</math>
+
| width="20%" | <math>\mathrm{Proposition}\!</math>
 
|}
 
|}
 
|-
 
|-
Line 1,980: Line 1,979:  
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
 
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
 
| width="20%" | <math>s_j\!</math>
 
| width="20%" | <math>s_j\!</math>
−
| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
+
| width="20%" | <math>\xrightarrow{\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}}\!</math>
 
| width="20%" | <math>C_j\!</math>
 
| width="20%" | <math>C_j\!</math>
−
| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
+
| width="20%" | <math>\xrightarrow{\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}}\!</math>
 
| width="20%" | <math>q_j\!</math>
 
| width="20%" | <math>q_j\!</math>
 
|}
 
|}
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|
 
|
 
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
 
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
−
| width="20%" | <math>\operatorname{Conc}^0</math>
+
| width="20%" | <math>\mathrm{Conc}^0\!</math>
−
| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
+
| width="20%" | <math>\xrightarrow{\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}}\!</math>
−
| width="20%" | <math>\operatorname{Node}^0</math>
+
| width="20%" | <math>\mathrm{Node}^0\!</math>
−
| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
+
| width="20%" | <math>\xrightarrow{\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}}\!</math>
−
| width="20%" | <math>\underline{1}</math>
+
| width="20%" | <math>\underline{1}\!</math>
 
|-
 
|-
−
| width="20%" | <math>\operatorname{Conc}^k_j s_j</math>
+
| width="20%" | <math>\mathrm{Conc}^k_j s_j\!</math>
−
| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
+
| width="20%" | <math>\xrightarrow{\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}}\!</math>
−
| width="20%" | <math>\operatorname{Node}^k_j C_j</math>
+
| width="20%" | <math>\mathrm{Node}^k_j C_j\!</math>
−
| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
+
| width="20%" | <math>\xrightarrow{\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}}\!</math>
−
| width="20%" | <math>\operatorname{Conj}^k_j q_j</math>
+
| width="20%" | <math>\mathrm{Conj}^k_j q_j\!</math>
 
|}
 
|}
 
|-
 
|-
 
|
 
|
 
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
 
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
−
| width="20%" | <math>\operatorname{Surc}^0</math>
+
| width="20%" | <math>\mathrm{Surc}^0\!</math>
−
| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
+
| width="20%" | <math>\xrightarrow{\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}}\!</math>
−
| width="20%" | <math>\operatorname{Lobe}^0</math>
+
| width="20%" | <math>\mathrm{Lobe}^0\!</math>
−
| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
+
| width="20%" | <math>\xrightarrow{\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}}\!</math>
−
| width="20%" | <math>\underline{0}</math>
+
| width="20%" | <math>\underline{0}\!</math>
 
|-
 
|-
−
| width="20%" | <math>\operatorname{Surc}^k_j s_j</math>
+
| width="20%" | <math>\mathrm{Surc}^k_j s_j~\!</math>
−
| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
+
| width="20%" | <math>\xrightarrow{\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}}\!</math>
−
| width="20%" | <math>\operatorname{Lobe}^k_j C_j</math>
+
| width="20%" | <math>\mathrm{Lobe}^k_j C_j\!</math>
−
| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
+
| width="20%" | <math>\xrightarrow{\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}}\!</math>
−
| width="20%" | <math>\operatorname{Surj}^k_j q_j</math>
+
| width="20%" | <math>\mathrm{Surj}^k_j q_j\!</math>
 
|}
 
|}
 
|}
 
|}
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<br>
 
<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:80%"
−
|+ '''Table 15. Semantic Translation : Equational Form'''
+
|+ style="height:30px" | <math>\text{Table 15.} ~~ \text{Semantic Translation : Equational Form}\!</math>
−
|- style="background:whitesmoke"
+
|- style="height:40px; background:ghostwhite"
 
|
 
|
−
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:whitesmoke; width:100%"
+
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:ghostwhite; width:100%"
−
| width="20%" | <math>\downharpoonleft \operatorname{Sentence} \downharpoonright</math>
+
| width="20%" | <math>\downharpoonleft \mathrm{Sentence} \downharpoonright\!</math>
−
| width="20%" | <math>\stackrel{\operatorname{Parse}}{=}</math>
+
| width="20%" | <math>\stackrel{\mathrm{Parse}}{=}\!</math>
−
| width="20%" | <math>\downharpoonleft \operatorname{Graph} \downharpoonright</math>
+
| width="20%" | <math>\downharpoonleft \mathrm{Graph} \downharpoonright\!</math>
−
| width="20%" | <math>\stackrel{\operatorname{Denotation}}{=}</math>
+
| width="20%" | <math>\stackrel{\mathrm{Denotation}}{=}\!</math>
−
| width="20%" | <math>\operatorname{Proposition}</math>
+
| width="20%" | <math>\mathrm{Proposition}\!</math>
 
|}
 
|}
 
|-
 
|-
 
|
 
|
 
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
 
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
−
| width="20%" | <math>\downharpoonleft s_j \downharpoonright</math>
+
| width="20%" | <math>\downharpoonleft s_j \downharpoonright\!</math>
 
| width="20%" | <math>=\!</math>
 
| width="20%" | <math>=\!</math>
−
| width="20%" | <math>\downharpoonleft C_j \downharpoonright</math>
+
| width="20%" | <math>\downharpoonleft C_j \downharpoonright\!</math>
 
| width="20%" | <math>=\!</math>
 
| width="20%" | <math>=\!</math>
 
| width="20%" | <math>q_j\!</math>
 
| width="20%" | <math>q_j\!</math>
Line 2,042: Line 2,041:  
|
 
|
 
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
 
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
−
| width="20%" | <math>\downharpoonleft \operatorname{Conc}^0 \downharpoonright</math>
+
| width="20%" | <math>\downharpoonleft \mathrm{Conc}^0 \downharpoonright\!</math>
 
| width="20%" | <math>=\!</math>
 
| width="20%" | <math>=\!</math>
−
| width="20%" | <math>\downharpoonleft \operatorname{Node}^0 \downharpoonright</math>
+
| width="20%" | <math>\downharpoonleft \mathrm{Node}^0 \downharpoonright\!</math>
 
| width="20%" | <math>=\!</math>
 
| width="20%" | <math>=\!</math>
−
| width="20%" | <math>\underline{1}</math>
+
| width="20%" | <math>\underline{1}\!</math>
 
|-
 
|-
−
| width="20%" | <math>\downharpoonleft \operatorname{Conc}^k_j s_j \downharpoonright</math>
+
| width="20%" | <math>\downharpoonleft \mathrm{Conc}^k_j s_j \downharpoonright\!</math>
 
| width="20%" | <math>=\!</math>
 
| width="20%" | <math>=\!</math>
−
| width="20%" | <math>\downharpoonleft \operatorname{Node}^k_j C_j \downharpoonright</math>
+
| width="20%" | <math>\downharpoonleft \mathrm{Node}^k_j C_j \downharpoonright\!</math>
 
| width="20%" | <math>=\!</math>
 
| width="20%" | <math>=\!</math>
−
| width="20%" | <math>\operatorname{Conj}^k_j q_j</math>
+
| width="20%" | <math>\mathrm{Conj}^k_j q_j\!</math>
 
|}
 
|}
 
|-
 
|-
 
|
 
|
 
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
 
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
−
| width="20%" | <math>\downharpoonleft \operatorname{Surc}^0 \downharpoonright</math>
+
| width="20%" | <math>\downharpoonleft \mathrm{Surc}^0 \downharpoonright\!</math>
 
| width="20%" | <math>=\!</math>
 
| width="20%" | <math>=\!</math>
−
| width="20%" | <math>\downharpoonleft \operatorname{Lobe}^0 \downharpoonright</math>
+
| width="20%" | <math>\downharpoonleft \mathrm{Lobe}^0 \downharpoonright\!</math>
 
| width="20%" | <math>=\!</math>
 
| width="20%" | <math>=\!</math>
−
| width="20%" | <math>\underline{0}</math>
+
| width="20%" | <math>\underline{0}\!</math>
 
|-
 
|-
−
| width="20%" | <math>\downharpoonleft \operatorname{Surc}^k_j s_j \downharpoonright</math>
+
| width="20%" | <math>\downharpoonleft \mathrm{Surc}^k_j s_j \downharpoonright\!</math>
 
| width="20%" | <math>=\!</math>
 
| width="20%" | <math>=\!</math>
−
| width="20%" | <math>\downharpoonleft \operatorname{Lobe}^k_j C_j \downharpoonright</math>
+
| width="20%" | <math>\downharpoonleft \mathrm{Lobe}^k_j C_j \downharpoonright\!</math>
 
| width="20%" | <math>=\!</math>
 
| width="20%" | <math>=\!</math>
−
| width="20%" | <math>\operatorname{Surj}^k_j q_j</math>
+
| width="20%" | <math>\mathrm{Surj}^k_j q_j\!</math>
 
|}
 
|}
 
|}
 
|}
Line 2,081: Line 2,080:  
<br>
 
<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:80%"
−
|+ '''Table 16. Boolean Functions on Zero Variables'''
+
|+ style="height:30px" | <math>\text{Table 16.} ~~ \text{Boolean Functions on Zero Variables}\!</math>
−
|- style="background:whitesmoke"
+
|- style="height:40px; background:ghostwhite"
 
| width="14%" | <math>F\!</math>
 
| width="14%" | <math>F\!</math>
 
| width="14%" | <math>F\!</math>
 
| width="14%" | <math>F\!</math>
Line 2,089: Line 2,088:  
| width="24%" | <math>F\!</math>
 
| width="24%" | <math>F\!</math>
 
|-
 
|-
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
 
| <math>F_0^{(0)}\!</math>
 
| <math>F_0^{(0)}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
−
| <math>(~)</math>
+
| <math>\texttt{(~)}\!</math>
 
|-
 
|-
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
 
| <math>F_1^{(0)}\!</math>
 
| <math>F_1^{(0)}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>((~))</math>
+
| <math>\texttt{((~))}\!</math>
 
|}
 
|}
   Line 2,114: Line 2,113:  
<br>
 
<br>
   −
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"
+
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"
−
|+ '''Table 17. Boolean Functions on One Variable'''
+
|+ style="height:30px" | <math>\text{Table 17.} ~~ \text{Boolean Functions on One Variable}\!</math>
−
|- style="background:whitesmoke"
+
|- style="height:40px; background:ghostwhite"
 
| width="14%" | <math>F\!</math>
 
| width="14%" | <math>F\!</math>
 
| width="14%" | <math>F\!</math>
 
| width="14%" | <math>F\!</math>
 
| colspan="2" | <math>F(x)\!</math>
 
| colspan="2" | <math>F(x)\!</math>
 
| width="24%" | <math>F\!</math>
 
| width="24%" | <math>F\!</math>
−
|- style="background:whitesmoke"
+
|- style="height:40px; background:ghostwhite"
 
| width="14%" | &nbsp;
 
| width="14%" | &nbsp;
 
| width="14%" | &nbsp;
 
| width="14%" | &nbsp;
Line 2,130: Line 2,129:  
| <math>F_0^{(1)}\!</math>
 
| <math>F_0^{(1)}\!</math>
 
| <math>F_{00}^{(1)}\!</math>
 
| <math>F_{00}^{(1)}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
−
| <math>(~)</math>
+
| <math>\texttt{(~)}\!</math>
 
|-
 
|-
 
| <math>F_1^{(1)}\!</math>
 
| <math>F_1^{(1)}\!</math>
 
| <math>F_{01}^{(1)}\!</math>
 
| <math>F_{01}^{(1)}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>(x)\!</math>
+
| <math>\texttt{(} x \texttt{)}\!</math>
 
|-
 
|-
 
| <math>F_2^{(1)}\!</math>
 
| <math>F_2^{(1)}\!</math>
−
| <math>F_{10}^{(1)}\!</math>
+
| <math>F_{10}^{(1)}~\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
 
| <math>x\!</math>
 
| <math>x\!</math>
 
|-
 
|-
 
| <math>F_3^{(1)}\!</math>
 
| <math>F_3^{(1)}\!</math>
 
| <math>F_{11}^{(1)}\!</math>
 
| <math>F_{11}^{(1)}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>((~))</math>
+
| <math>\texttt{((~))}\!</math>
 
|}
 
|}
   Line 2,181: Line 2,180:  
<br>
 
<br>
   −
{| align="center" border="1" cellpadding="4" cellspacing="0" style="text-align:center; width:90%"
+
{| align="center" border="1" cellpadding="4" cellspacing="0" style="text-align:center; width:80%"
−
|+ '''Table 18. Boolean Functions on Two Variables'''
+
|+ style="height:30px" | <math>\text{Table 18.} ~~ \text{Boolean Functions on Two Variables}\!</math>
−
|- style="background:whitesmoke"
+
|- style="height:40px; background:ghostwhite"
 
| width="14%" | <math>F\!</math>
 
| width="14%" | <math>F\!</math>
 
| width="14%" | <math>F\!</math>
 
| width="14%" | <math>F\!</math>
 
| colspan="4" | <math>F(x, y)\!</math>
 
| colspan="4" | <math>F(x, y)\!</math>
 
| width="24%" | <math>F\!</math>
 
| width="24%" | <math>F\!</math>
−
|- style="background:whitesmoke"
+
|- style="height:40px; background:ghostwhite"
 
| width="14%" | &nbsp;
 
| width="14%" | &nbsp;
 
| width="14%" | &nbsp;
 
| width="14%" | &nbsp;
Line 2,198: Line 2,197:  
|-
 
|-
 
| <math>F_{0}^{(2)}\!</math>
 
| <math>F_{0}^{(2)}\!</math>
−
| <math>F_{0000}^{(2)}\!</math>
+
| <math>F_{0000}^{(2)}~\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
−
| <math>(~)</math>
+
| <math>\texttt{(~)}\!</math>
 
|-
 
|-
 
| <math>F_{1}^{(2)}\!</math>
 
| <math>F_{1}^{(2)}\!</math>
 
| <math>F_{0001}^{(2)}\!</math>
 
| <math>F_{0001}^{(2)}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>(x)(y)\!</math>
+
| <math>\texttt{(} x \texttt{)(} y \texttt{)}\!</math>
 
|-
 
|-
 
| <math>F_{2}^{(2)}\!</math>
 
| <math>F_{2}^{(2)}\!</math>
 
| <math>F_{0010}^{(2)}\!</math>
 
| <math>F_{0010}^{(2)}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
−
| <math>(x) y\!</math>
+
| <math>\texttt{(} x \texttt{)} y\!</math>
 
|-
 
|-
 
| <math>F_{3}^{(2)}\!</math>
 
| <math>F_{3}^{(2)}\!</math>
 
| <math>F_{0011}^{(2)}\!</math>
 
| <math>F_{0011}^{(2)}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>(x)\!</math>
+
| <math>\texttt{(} x \texttt{)}\!</math>
 
|-
 
|-
 
| <math>F_{4}^{(2)}\!</math>
 
| <math>F_{4}^{(2)}\!</math>
 
| <math>F_{0100}^{(2)}\!</math>
 
| <math>F_{0100}^{(2)}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
−
| <math>x (y)\!</math>
+
| <math>x \texttt{(} y \texttt{)}\!</math>
 
|-
 
|-
 
| <math>F_{5}^{(2)}\!</math>
 
| <math>F_{5}^{(2)}\!</math>
 
| <math>F_{0101}^{(2)}\!</math>
 
| <math>F_{0101}^{(2)}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>(y)\!</math>
+
| <math>\texttt{(} y \texttt{)}\!</math>
 
|-
 
|-
 
| <math>F_{6}^{(2)}\!</math>
 
| <math>F_{6}^{(2)}\!</math>
 
| <math>F_{0110}^{(2)}\!</math>
 
| <math>F_{0110}^{(2)}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
−
| <math>(x, y)\!</math>
+
| <math>\texttt{(} x \texttt{,} y \texttt{)}\!</math>
 
|-
 
|-
 
| <math>F_{7}^{(2)}\!</math>
 
| <math>F_{7}^{(2)}\!</math>
 
| <math>F_{0111}^{(2)}\!</math>
 
| <math>F_{0111}^{(2)}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>(x y)\!</math>
+
| <math>\texttt{(} x y \texttt{)}\!</math>
 
|-
 
|-
 
| <math>F_{8}^{(2)}\!</math>
 
| <math>F_{8}^{(2)}\!</math>
 
| <math>F_{1000}^{(2)}\!</math>
 
| <math>F_{1000}^{(2)}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
 
| <math>x y\!</math>
 
| <math>x y\!</math>
 
|-
 
|-
 
| <math>F_{9}^{(2)}\!</math>
 
| <math>F_{9}^{(2)}\!</math>
 
| <math>F_{1001}^{(2)}\!</math>
 
| <math>F_{1001}^{(2)}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>((x, y))\!</math>
+
| <math>\texttt{((} x \texttt{,} y \texttt{))}\!</math>
 
|-
 
|-
 
| <math>F_{10}^{(2)}\!</math>
 
| <math>F_{10}^{(2)}\!</math>
 
| <math>F_{1010}^{(2)}\!</math>
 
| <math>F_{1010}^{(2)}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
 
| <math>y\!</math>
 
| <math>y\!</math>
 
|-
 
|-
 
| <math>F_{11}^{(2)}\!</math>
 
| <math>F_{11}^{(2)}\!</math>
 
| <math>F_{1011}^{(2)}\!</math>
 
| <math>F_{1011}^{(2)}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>(x (y))\!</math>
+
| <math>\texttt{(} x \texttt{(} y \texttt{))}\!</math>
 
|-
 
|-
 
| <math>F_{12}^{(2)}\!</math>
 
| <math>F_{12}^{(2)}\!</math>
−
| <math>F_{1100}^{(2)}\!</math>
+
| <math>F_{1100}^{(2)}~\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
 
| <math>x\!</math>
 
| <math>x\!</math>
 
|-
 
|-
 
| <math>F_{13}^{(2)}\!</math>
 
| <math>F_{13}^{(2)}\!</math>
 
| <math>F_{1101}^{(2)}\!</math>
 
| <math>F_{1101}^{(2)}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>((x)y)\!</math>
+
| <math>\texttt{((} x \texttt{)} y \texttt{)}\!</math>
 
|-
 
|-
 
| <math>F_{14}^{(2)}\!</math>
 
| <math>F_{14}^{(2)}\!</math>
 
| <math>F_{1110}^{(2)}\!</math>
 
| <math>F_{1110}^{(2)}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>\underline{0}</math>
+
| <math>\underline{0}\!</math>
−
| <math>((x)(y))\!</math>
+
| <math>\texttt{((} x \texttt{)(} y \texttt{))}\!</math>
 
|-
 
|-
 
| <math>F_{15}^{(2)}\!</math>
 
| <math>F_{15}^{(2)}\!</math>
 
| <math>F_{1111}^{(2)}\!</math>
 
| <math>F_{1111}^{(2)}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>\underline{1}</math>
+
| <math>\underline{1}\!</math>
−
| <math>((~))</math>
+
| <math>\texttt{((~))}\!</math>
 
|}
 
|}
   Line 2,364: Line 2,363:  
For example, suppose that <math>F\!</math> is a connection of the form <math>F : \underline\mathbb{B}^2 \to \underline\mathbb{B},</math> that is, any one of the sixteen possibilities in Table&nbsp;18, while <math>p\!</math> and <math>q\!</math> are propositions of the form <math>p, q : X \to \underline\mathbb{B},</math> that is, propositions about things in the universe <math>X,\!</math> or else the indicators of sets contained in <math>X.\!</math>
 
For example, suppose that <math>F\!</math> is a connection of the form <math>F : \underline\mathbb{B}^2 \to \underline\mathbb{B},</math> that is, any one of the sixteen possibilities in Table&nbsp;18, while <math>p\!</math> and <math>q\!</math> are propositions of the form <math>p, q : X \to \underline\mathbb{B},</math> that is, propositions about things in the universe <math>X,\!</math> or else the indicators of sets contained in <math>X.\!</math>
   −
Then one has the imagination <math>\underline{f} = (f_1, f_2) = (p, q) : (X \to \underline\mathbb{B})^2,</math> and the stretch of the connection <math>F\!</math> to <math>\underline{f}</math> on <math>X\!</math> amounts to a proposition <math>F^\$ (p, q) : X \to \underline\mathbb{B}</math> that may be read as the ''stretch of <math>F\!</math> to <math>p\!</math> and <math>q.\!</math>''  If one is concerned with many different propositions about things in <math>X,\!</math> or if one is abstractly indifferent to the particular choices for <math>p\!</math> and <math>q,\!</math> then one may detach the operator <math>F^\$ : (X \to \underline\mathbb{B}))^2 \to (X \to \underline\mathbb{B})),</math> called the ''stretch of <math>F\!</math> over <math>X,\!</math>'' and consider it in isolation from any concrete application.
+
Then one has the imagination <math>\underline{f} = (f_1, f_2) = (p, q) : (X \to \underline\mathbb{B})^2,</math> and the stretch of the connection <math>F\!</math> to <math>\underline{f}\!</math> on <math>X\!</math> amounts to a proposition <math>F^\$ (p, q) : X \to \underline\mathbb{B}</math> that may be read as the ''stretch of <math>F\!</math> to <math>p\!</math> and <math>q.\!</math>''  If one is concerned with many different propositions about things in <math>X,\!</math> or if one is abstractly indifferent to the particular choices for <math>p\!</math> and <math>q,\!</math> then one may detach the operator <math>F^\$ : (X \to \underline\mathbb{B}))^2 \to (X \to \underline\mathbb{B})),</math> called the ''stretch of <math>F\!</math> over <math>X,\!</math>'' and consider it in isolation from any concrete application.
    
When the cactus notation is used to represent boolean functions, a single <math>\$</math> sign at the end of the expression is enough to remind the reader that the connections are meant to be stretched to several propositions on a universe <math>X.\!</math>
 
When the cactus notation is used to represent boolean functions, a single <math>\$</math> sign at the end of the expression is enough to remind the reader that the connections are meant to be stretched to several propositions on a universe <math>X.\!</math>
Line 2,370: Line 2,369:  
For example, take the connection <math>F : \underline\mathbb{B}^2 \to \underline\mathbb{B}</math> such that:
 
For example, take the connection <math>F : \underline\mathbb{B}^2 \to \underline\mathbb{B}</math> such that:
   −
: <math>F(x, y) ~=~ F_{6}^{(2)} (x, y) ~=~ \underline{(}~x~,~y~\underline{)}</math>
+
: <math>F(x, y) ~=~ F_{6}^{(2)} (x, y) ~=~ \underline{(}~x~,~y~\underline{)}\!</math>
   −
The connection in question is a boolean function on the variables <math>x, y\!</math> that returns a value of <math>\underline{1}</math> just when just one of the pair <math>x, y\!</math> is not equal to <math>\underline{1},</math> or what amounts to the same thing, just when just one of the pair <math>x, y\!</math> is equal to <math>\underline{1}.</math>  There is clearly an isomorphism between this connection, viewed as an operation on the boolean domain <math>\underline\mathbb{B} = \{ \underline{0}, \underline{1} \},</math> and the dyadic operation on binary values <math>x, y \in \mathbb{B} = \operatorname{GF}(2)</math> that is otherwise known as <math>x + y\!.</math>
+
The connection in question is a boolean function on the variables <math>x, y\!</math> that returns a value of <math>\underline{1}</math> just when just one of the pair <math>x, y\!</math> is not equal to <math>\underline{1},</math> or what amounts to the same thing, just when just one of the pair <math>x, y\!</math> is equal to <math>\underline{1}.</math>  There is clearly an isomorphism between this connection, viewed as an operation on the boolean domain <math>\underline\mathbb{B} = \{ \underline{0}, \underline{1} \},</math> and the dyadic operation on binary values <math>x, y \in \mathbb{B} = \operatorname{GF}(2)\!</math> that is otherwise known as <math>x + y.\!</math>
    
The same connection <math>F : \underline\mathbb{B}^2 \to \underline\mathbb{B}</math> can also be read as a proposition about things in the universe <math>X = \underline\mathbb{B}^2.</math>  If <math>s\!</math> is a sentence that denotes the proposition <math>F,\!</math> then the corresponding assertion says exactly what one states in uttering the sentence <math>^{\backprime\backprime} \, x ~\operatorname{is~not~equal~to}~ y \, ^{\prime\prime}.</math>  In such a case, one has <math>\downharpoonleft s \downharpoonright \, = F,</math> and all of the following expressions are ordinarily taken as equivalent descriptions of the same set:
 
The same connection <math>F : \underline\mathbb{B}^2 \to \underline\mathbb{B}</math> can also be read as a proposition about things in the universe <math>X = \underline\mathbb{B}^2.</math>  If <math>s\!</math> is a sentence that denotes the proposition <math>F,\!</math> then the corresponding assertion says exactly what one states in uttering the sentence <math>^{\backprime\backprime} \, x ~\operatorname{is~not~equal~to}~ y \, ^{\prime\prime}.</math>  In such a case, one has <math>\downharpoonleft s \downharpoonright \, = F,</math> and all of the following expressions are ordinarily taken as equivalent descriptions of the same set:
Line 2,541: Line 2,540:  
</pre>
 
</pre>
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===Zeroth Order Logic===
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===Aug 2000 &bull; Extensions Of Logical Graphs===
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====Aug 2000 &mdash; Extensions Of Logical Graphs====
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====CG List &bull; Lost Links====
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# http://www.virtual-earth.de/CG/cg-list/old/msg03351.html
 
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====Sep 2000 &mdash; Zeroth Order Logic====
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====CG List &bull; New Archive====
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# http://web.archive.org/web/20020129134132/http://www.virtual-earth.de/CG/cg-list/msg03381.html
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===Sep 2000 &bull; Zeroth Order Logic===
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* http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/thrd241.html#01246
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* http://web.archive.org/web/20130306202443/http://suo.ieee.org/email/thrd242.html#01406
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# http://web.archive.org/web/20070401102902/http://suo.ieee.org/email/msg01988.html
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===Oct 2000 &bull; All Liar, No Paradox===
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* http://web.archive.org/web/20130306202504/http://suo.ieee.org/email/thrd236.html#01739
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# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg01739.html
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===Nov 2000 &bull; Sowa's Top Level Categories===
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====What Language To Use====
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* http://web.archive.org/web/20070218222218/http://suo.ieee.org/email/threads.html#01956
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# http://web.archive.org/web/20070320012929/http://suo.ieee.org/email/msg01956.html
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====Zeroth Order Logic====
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* http://web.archive.org/web/20070218222218/http://suo.ieee.org/email/threads.html#01966
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# http://web.archive.org/web/20070320012940/http://suo.ieee.org/email/msg01966.html
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====TLC In KIF====
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* http://web.archive.org/web/20130304163442/http://suo.ieee.org/ontology/thrd110.html#00048
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# http://web.archive.org/web/20081204195421/http://suo.ieee.org/ontology/msg00048.html
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# http://web.archive.org/web/20070320014557/http://suo.ieee.org/ontology/msg00051.html
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===Dec 2000 &bull; Sequential Interactions Generating Hypotheses===
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* http://web.archive.org/web/20130306202621/http://suo.ieee.org/email/thrd217.html#02607
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# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg02607.html
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# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg02608.html
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# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg03183.html
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===Jan 2001 &bull; Differential Analytic Turing Automata===
 +
 
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====DATA &bull; Arisbe List====
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* http://web.archive.org/web/20150107163000/http://stderr.org/pipermail/arisbe/2001-January/thread.html#182
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====DATA &bull; Ontology List====
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* http://web.archive.org/web/20130304165332/http://suo.ieee.org/ontology/thrd95.html#00596
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# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg00618.html
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===Mar 2001 &bull; Propositional Equation Reasoning Systems===
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* http://web.archive.org/web/20150107212028/http://stderr.org/pipermail/arisbe/2001-April/thread.html#407
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# http://web.archive.org/web/20150107210011/http://stderr.org/pipermail/arisbe/2001-March/000380.html
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# http://web.archive.org/web/20050920031758/http://stderr.org/pipermail/arisbe/2001-April/000407.html
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====PERS &bull; Arisbe List &bull; Discussion====
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* http://web.archive.org/web/20150107212028/http://stderr.org/pipermail/arisbe/2001-April/thread.html#397
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# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04476.html
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# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04517.html
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# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04533.html
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# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04536.html
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# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04542.html
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# http://web.archive.org/web/20050824231950/http://suo.ieee.org/email/msg04546.html
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 +
===Jul 2001 &bull; Reflective Extension Of Logical Graphs===
 +
 
 +
====RefLog &bull; Arisbe List====
 +
 
 +
* http://web.archive.org/web/20150109141200/http://stderr.org/pipermail/arisbe/2001-July/thread.html#711
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====RefLog &bull; SUO List====
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* http://suo.ieee.org/email/thrd244.html#01246
+
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* http://suo.ieee.org/email/thrd241.html#01406
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# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg05694.html
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# http://suo.ieee.org/email/msg01246.html
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# http://suo.ieee.org/email/msg01406.html
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# http://suo.ieee.org/email/msg01561.html
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# http://suo.ieee.org/email/msg01670.html
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# http://suo.ieee.org/email/msg01985.html
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# http://suo.ieee.org/email/msg01988.html
     −
====Oct 2000 &mdash; All Liar, No Paradox====
+
===Dec 2001 &bull; Functional Conception Of Quantificational Logic===
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* http://suo.ieee.org/email/thrd235.html#01739
+
====FunLog &bull; Arisbe List====
−
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====Nov 2000 &mdash; Sowa's Top Level Categories====
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* http://web.archive.org/web/20141005034441/http://stderr.org/pipermail/arisbe/2001-December/thread.html#1212
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# http://web.archive.org/web/20141005034614/http://stderr.org/pipermail/arisbe/2001-December/001212.html
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# http://web.archive.org/web/20141005034615/http://stderr.org/pipermail/arisbe/2001-December/001213.html
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# http://web.archive.org/web/20051201235028/http://stderr.org/pipermail/arisbe/2001-December/001222.html
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# http://web.archive.org/web/20050827214411/http://stderr.org/pipermail/arisbe/2001-December/001224.html
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# http://web.archive.org/web/20051202092500/http://stderr.org/pipermail/arisbe/2001-December/001225.html
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# http://web.archive.org/web/20051202051942/http://stderr.org/pipermail/arisbe/2001-December/001226.html
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# http://web.archive.org/web/20050425140213/http://stderr.org/pipermail/arisbe/2001-December/001227.html
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* http://suo.ieee.org/email/thrd231.html#01956
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# http://suo.ieee.org/ontology/msg00048.html
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# http://suo.ieee.org/ontology/msg00051.html
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====Dec 2000 &mdash; Sequential Interactions Generating Hypotheses====
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* http://web.archive.org/web/20120222171225/http://suo.ieee.org/ontology/thrd38.html#03562
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# http://web.archive.org/web/20110608022546/http://suo.ieee.org/ontology/msg03562.html
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# http://web.archive.org/web/20110608022712/http://suo.ieee.org/ontology/msg03563.html
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# http://web.archive.org/web/20110608023312/http://suo.ieee.org/ontology/msg03564.html
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# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03565.html
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# http://web.archive.org/web/20070812011325/http://suo.ieee.org/ontology/msg03569.html
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# http://web.archive.org/web/20110608023228/http://suo.ieee.org/ontology/msg03570.html
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# http://web.archive.org/web/20110608022616/http://suo.ieee.org/ontology/msg03568.html
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# http://web.archive.org/web/20110608023557/http://suo.ieee.org/ontology/msg03572.html
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# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03577.html
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# http://web.archive.org/web/20070317021141/http://suo.ieee.org/ontology/msg03578.html
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# http://web.archive.org/web/20110608021549/http://suo.ieee.org/ontology/msg03579.html
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# http://web.archive.org/web/20110608021332/http://suo.ieee.org/ontology/msg03580.html
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# http://web.archive.org/web/20110608020250/http://suo.ieee.org/ontology/msg03581.html
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# http://web.archive.org/web/20110608021344/http://suo.ieee.org/ontology/msg03582.html
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# http://web.archive.org/web/20110608021557/http://suo.ieee.org/ontology/msg03583.html
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# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04247.html
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* http://suo.ieee.org/email/thrd216.html#02607
+
===Dec 2001 &bull; Cactus Language===
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# http://suo.ieee.org/email/msg02607.html
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# http://suo.ieee.org/email/msg02608.html
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# http://suo.ieee.org/email/msg03183.html
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====Jan 2001 &mdash; Differential Analytic Turing Automata====
+
====Cactus Town Cartoons &bull; Arisbe List====
   −
* http://stderr.org/pipermail/arisbe/2001-January/thread.html#182
+
* http://web.archive.org/web/20141005034441/http://stderr.org/pipermail/arisbe/2001-December/thread.html#1214
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* http://suo.ieee.org/ontology/thrd95.html#00596
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# http://web.archive.org/web/20050825005438/http://stderr.org/pipermail/arisbe/2001-December/001214.html
 +
# http://web.archive.org/web/20051202101235/http://stderr.org/pipermail/arisbe/2001-December/001217.html
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====Mar 2001 &mdash; Propositional Equation Reasoning Systems====
+
====Cactus Town Cartoons &bull; Ontology List====
   −
* http://stderr.org/pipermail/arisbe/2001-March/thread.html#380
+
* http://web.archive.org/web/20120222171225/http://suo.ieee.org/ontology/thrd38.html#03567
−
* http://stderr.org/pipermail/arisbe/2001-April/thread.html#407
+
# http://web.archive.org/web/20110608023426/http://suo.ieee.org/ontology/msg03567.html
−
* http://suo.ieee.org/ontology/thrd74.html#01779
+
# http://web.archive.org/web/20110608024449/http://suo.ieee.org/ontology/msg03571.html
−
* http://suo.ieee.org/email/thrd186.html#04187
     −
====Jul 2001 &mdash; Reflective Extension Of Logical Graphs====
+
===Jan 2002 &bull; Zeroth Order Theories===
   −
* http://stderr.org/pipermail/arisbe/2001-July/000711.html
+
====ZOT &bull; Arisbe List====
−
* http://suo.ieee.org/email/thrd159.html#05694
     −
====Dec 2001 &mdash; Functional Conception Of Quantificational Logic====
+
* http://web.archive.org/web/20150109041904/http://stderr.org/pipermail/arisbe/2002-January/thread.html#1293
 +
# http://web.archive.org/web/20150109042401/http://stderr.org/pipermail/arisbe/2002-January/001293.html
 +
# http://web.archive.org/web/20150109042402/http://stderr.org/pipermail/arisbe/2002-January/001294.html
 +
# http://web.archive.org/web/20050503213326/http://stderr.org/pipermail/arisbe/2002-January/001295.html
 +
# http://web.archive.org/web/20050503213330/http://stderr.org/pipermail/arisbe/2002-January/001296.html
 +
# http://web.archive.org/web/20050504070444/http://stderr.org/pipermail/arisbe/2002-January/001299.html
 +
# http://web.archive.org/web/20050504070430/http://stderr.org/pipermail/arisbe/2002-January/001300.html
 +
# http://web.archive.org/web/20050504070700/http://stderr.org/pipermail/arisbe/2002-January/001301.html
 +
# http://web.archive.org/web/20050504070704/http://stderr.org/pipermail/arisbe/2002-January/001302.html
 +
# http://web.archive.org/web/20050504070712/http://stderr.org/pipermail/arisbe/2002-January/001304.html
 +
# http://web.archive.org/web/20050504070717/http://stderr.org/pipermail/arisbe/2002-January/001305.html
 +
# http://web.archive.org/web/20050504070722/http://stderr.org/pipermail/arisbe/2002-January/001306.html
 +
# http://web.archive.org/web/20050504070726/http://stderr.org/pipermail/arisbe/2002-January/001308.html
 +
# http://web.archive.org/web/20050504070730/http://stderr.org/pipermail/arisbe/2002-January/001309.html
 +
# http://web.archive.org/web/20050504070434/http://stderr.org/pipermail/arisbe/2002-January/001310.html
 +
# http://web.archive.org/web/20050504070742/http://stderr.org/pipermail/arisbe/2002-January/001313.html
 +
# http://web.archive.org/web/20050504070746/http://stderr.org/pipermail/arisbe/2002-January/001314.html
 +
# http://web.archive.org/web/20050504070438/http://stderr.org/pipermail/arisbe/2002-January/001315.html
 +
# http://web.archive.org/web/20050504070540/http://stderr.org/pipermail/arisbe/2002-January/001316.html
 +
# http://web.archive.org/web/20050504070750/http://stderr.org/pipermail/arisbe/2002-January/001317.html
   −
* http://stderr.org/pipermail/arisbe/2001-December/thread.html#1212
+
====ZOT &bull; Arisbe List &bull; Discussion====
−
* http://suo.ieee.org/ontology/thrd38.html#03562
     −
====Dec 2001 &mdash; Cactus Town Cartoons====
+
* http://web.archive.org/web/20150109041904/http://stderr.org/pipermail/arisbe/2002-January/thread.html#1293
 +
# http://web.archive.org/web/20050503213334/http://stderr.org/pipermail/arisbe/2002-January/001297.html
 +
# http://web.archive.org/web/20050504070656/http://stderr.org/pipermail/arisbe/2002-January/001298.html
 +
# http://web.archive.org/web/20050504070708/http://stderr.org/pipermail/arisbe/2002-January/001303.html
 +
# http://web.archive.org/web/20050504070544/http://stderr.org/pipermail/arisbe/2002-January/001307.html
 +
# http://web.archive.org/web/20050504070734/http://stderr.org/pipermail/arisbe/2002-January/001311.html
 +
# http://web.archive.org/web/20050504070738/http://stderr.org/pipermail/arisbe/2002-January/001312.html
 +
# http://web.archive.org/web/20050504070755/http://stderr.org/pipermail/arisbe/2002-January/001318.html
   −
* http://stderr.org/pipermail/arisbe/2001-December/thread.html#1214
+
====ZOT &bull; Ontology List====
−
* http://suo.ieee.org/ontology/thrd38.html#03567
     −
====Jan 2002 &mdash; Zeroth Order Theories====
+
* http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/thrd35.html#03680
 +
# http://web.archive.org/web/20070323210742/http://suo.ieee.org/ontology/msg03680.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03681.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03682.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03683.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03691.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03693.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03695.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03696.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03701.html
 +
# http://web.archive.org/web/20070329211521/http://suo.ieee.org/ontology/msg03702.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03703.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03706.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03707.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03708.html
 +
# http://web.archive.org/web/20080620074722/http://suo.ieee.org/ontology/msg03712.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03715.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03716.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03717.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03718.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03721.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03722.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03723.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03724.html
   −
* http://stderr.org/pipermail/arisbe/2002-January/thread.html#1293
+
====ZOT &bull; Ontology List &bull; Discussion====
−
* http://suo.ieee.org/ontology/thrd35.html#03680
     −
===Cactus Language===
+
* http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/thrd35.html#03680
 +
* http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/thrd35.html#03697
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03684.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03685.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03686.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03687.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03689.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03690.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03694.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03697.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03698.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03699.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03700.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03704.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03705.html
 +
# http://web.archive.org/web/20070330093628/http://suo.ieee.org/ontology/msg03709.html
 +
# http://web.archive.org/web/20080705071714/http://suo.ieee.org/ontology/msg03710.html
 +
# http://web.archive.org/web/20080620010020/http://suo.ieee.org/ontology/msg03711.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03713.html
 +
# http://web.archive.org/web/20080620074749/http://suo.ieee.org/ontology/msg03714.html
 +
# http://web.archive.org/web/20061005100254/http://suo.ieee.org/ontology/msg03719.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03720.html
   −
====Mar 2003 &mdash; Theme One Program : Logical Cacti====
+
===Mar 2003 &bull; Theme One Program &bull; Logical Cacti===
   −
* http://stderr.org/pipermail/inquiry/2003-March/thread.html#102
+
* http://web.archive.org/web/20150224210000/http://stderr.org/pipermail/inquiry/2003-March/thread.html#102
−
* http://stderr.org/pipermail/inquiry/2003-March/000114.html
+
* http://web.archive.org/web/20150224210000/http://stderr.org/pipermail/inquiry/2003-March/thread.html#114
 +
# http://web.archive.org/web/20081007043317/http://stderr.org/pipermail/inquiry/2003-March/000114.html
 +
# http://web.archive.org/web/20080908075558/http://stderr.org/pipermail/inquiry/2003-March/000115.html
 +
# http://web.archive.org/web/20080908080336/http://stderr.org/pipermail/inquiry/2003-March/000116.html
   −
====Feb 2005 &mdash; Theme One Program : Logical Cacti====
+
===Feb 2005 &bull; Theme One Program &bull; Logical Cacti===
   −
* http://stderr.org/pipermail/inquiry/2005-February/thread.html#2348
+
* http://web.archive.org/web/20150109155110/http://stderr.org/pipermail/inquiry/2005-February/thread.html#2348
−
* http://stderr.org/pipermail/inquiry/2005-February/002360.html
+
* http://web.archive.org/web/20150109155110/http://stderr.org/pipermail/inquiry/2005-February/thread.html#2360
 +
# http://web.archive.org/web/20150109152359/http://stderr.org/pipermail/inquiry/2005-February/002360.html
 +
# http://web.archive.org/web/20150109152401/http://stderr.org/pipermail/inquiry/2005-February/002361.html
 +
# http://web.archive.org/web/20061013233259/http://stderr.org/pipermail/inquiry/2005-February/002362.html
 +
# http://web.archive.org/web/20081121103109/http://stderr.org/pipermail/inquiry/2005-February/002363.html
    
[[Category:Artificial Intelligence]]
 
[[Category:Artificial Intelligence]]
 +
[[Category:Charles Sanders Peirce]]
 
[[Category:Combinatorics]]
 
[[Category:Combinatorics]]
 
[[Category:Computer Science]]
 
[[Category:Computer Science]]
Jon Awbrey
12,080

edits

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