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Directory:Jon Awbrey/Papers/Cactus Language (view source)

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→‎The Cactus Language : Syntax: set display as ordered list
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The alphabet <math>\mathfrak{A} = \mathfrak{M} \cup \mathfrak{P}</math> is the disjoint union of two sets of symbols:
 
The alphabet <math>\mathfrak{A} = \mathfrak{M} \cup \mathfrak{P}</math> is the disjoint union of two sets of symbols:
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{| align="center" cellpadding="8" width="90%"
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<ol style="list-style-type:decimal">
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|-
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| valign="top" | 1.
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<li>
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| <math>\mathfrak{M}</math> is the alphabet of ''measures'', the set of ''punctuation marks'', or the collection of ''syntactic constants'' that is common to all of the languages <math>\mathfrak{C}(\mathfrak{P}).</math>  This set of signs is given as follows:
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<p><math>\mathfrak{M}</math> is the alphabet of ''measures'', the set of ''punctuation marks'', or the collection of ''syntactic constants'' that is common to all of the languages <math>\mathfrak{C}(\mathfrak{P}).</math>  This set of signs is given as follows:</p>
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<p><math>\begin{array}{lccccccccccc}
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<math>\begin{array}{lccccccccccc}
   
\mathfrak{M}
 
\mathfrak{M}
 
& = &
 
& = &
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\operatorname{right} &
 
\operatorname{right} &
 
\} \\
 
\} \\
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\end{array}</math>
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\end{array}</math></p></li>
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|-
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| valign="top" | 2.
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<li>
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| <math>\mathfrak{P}</math> is the ''palette'', the alphabet of ''paints'', or the collection of ''syntactic variables'' that is peculiar to the language <math>\mathfrak{C}(\mathfrak{P}).</math>  This set of signs is given as follows:
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<p><math>\mathfrak{P}</math> is the ''palette'', the alphabet of ''paints'', or the collection of ''syntactic variables'' that is peculiar to the language <math>\mathfrak{C}(\mathfrak{P}).</math>  This set of signs is given as follows:</p>
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<p><math>\mathfrak{P} = \{ \mathfrak{p}_j  :  j \in J \}.</math></p></li>
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| <math>\mathfrak{P} = \{ \mathfrak{p}_j  :  j \in J \}.</math>
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</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''.
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<ol style="list-style-type:decimal">
 
<ol style="list-style-type:decimal">
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<li><p>The ''concatenation'' of one string <math>s_1\!</math> is just the string <math>s_1.\!</math></p>
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<li>
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<p>The ''concatenation'' of one string <math>s_1\!</math> is just the string <math>s_1.\!</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>
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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>
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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>
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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>
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<li><p>The ''surcatenation'' of one string <math>s_1\!</math> is the string <math>^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></p>
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<li>
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<p>The ''surcatenation'' of one string <math>s_1\!</math> is the string <math>^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></p>
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    <p>The ''surcatenation'' of two strings <math>s_1, s_2\!</math> is <math>^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_2 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></p>
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<p>The ''surcatenation'' of two strings <math>s_1, s_2\!</math> is <math>^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_2 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></p>
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    <p>The ''surcatenation'' of the <math>k\!</math> strings <math>(s_j)_{j = 1}^k</math> is the string of the form <math>^{\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></p></li>
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<p>The ''surcatenation'' of the <math>k\!</math> strings <math>(s_j)_{j = 1}^k</math> is the string of the form <math>^{\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></p></li>
    
</ol>
 
</ol>
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<li><math>\operatorname{Conc}_{j = 1}^1 (s_j)_{j = 1}^k \ = \ s_1.</math></li>
 
<li><math>\operatorname{Conc}_{j = 1}^1 (s_j)_{j = 1}^k \ = \ s_1.</math></li>
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<li><p>For <math>\ell > 1,\!</math></p>
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<li>
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<p>For <math>\ell > 1,\!</math></p>
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    <p><math>\operatorname{Conc}_{j=1}^\ell (s_j)_{j = 1}^k \ = \ (\operatorname{Conc}_{j=1}^{\ell - 1} (s_j)_{j = 1}^k) \, \cdot \, s_\ell.</math></p></li>
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<p><math>\operatorname{Conc}_{j=1}^\ell (s_j)_{j = 1}^k \ = \ (\operatorname{Conc}_{j=1}^{\ell - 1} (s_j)_{j = 1}^k) \, \cdot \, s_\ell.</math></p></li>
    
</ol>
 
</ol>
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<li><math>\operatorname{Surc}_{j=1}^1 (s_j)_{j = 1}^k \ = \ ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></li>
 
<li><math>\operatorname{Surc}_{j=1}^1 (s_j)_{j = 1}^k \ = \ ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></li>
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<li><p>For <math>\ell > 1,\!</math></p>
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<li>
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<p>For <math>\ell > 1,\!</math></p>
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    <p><math>\operatorname{Surc}_{j=1}^\ell (s_j)_{j = 1}^k \ = \ (\operatorname{Surc}_{j=1}^{\ell - 1} (s_j)_{j = 1}^k) \, \cdot \, ( \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \, )^{-1} \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_\ell \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></p></li>
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<p><math>\operatorname{Surc}_{j=1}^\ell (s_j)_{j = 1}^k \ = \ (\operatorname{Surc}_{j=1}^{\ell - 1} (s_j)_{j = 1}^k) \, \cdot \, ( \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \, )^{-1} \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_\ell \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></p></li>
    
</ol></ol>
 
</ol></ol>
Jon Awbrey
12,080

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