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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 2 (view source)
Revision as of 18:00, 4 October 2013
, 18:00, 4 October 2013update
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%"
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%"
|+ style="height:30px" |
|+ style="height:30px" | <math>\text{Table 5.} ~~ \text{Alignments of Capacities}\!</math>
<math>\text{Table 5.} ~~ \text{Alignments of Capacities}\!</math>
|- style="height:40px; background:ghostwhite"
|- style="height:40px; background:#f0f0ff"
| <math>\text{Formal}\!</math>
| <math>\text{Formal}\!</math>
| colspan="2" | <math>\text{Formative}\!</math>
| colspan="2" | <math>\text{Formative}\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%"
|+ style="height:30px" |
|+ style="height:30px" | <math>\text{Table 6.} ~~ \text{Alignments of Capacities in Aristotle}\!</math>
<math>\text{Table 6.} ~~ \text{Alignments of Capacities in Aristotle}\!</math>
|- style="height:40px; background:ghostwhite"
|- style="height:40px; background:#f0f0ff"
| <math>\text{Matter}\!</math>
| <math>\text{Matter}\!</math>
| colspan="2" | <math>\text{Form}\!</math>
| colspan="2" | <math>\text{Form}\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%"
|+ style="height:30px" |
|+ style="height:30px" | <math>\text{Table 7.} ~~ \text{Synthesis of Alignments}\!</math>
<math>\text{Table 7.} ~~ \text{Synthesis of Alignments}\!</math>
|- style="height:40px; background:ghostwhite"
|- style="height:40px; background:#f0f0ff"
| <math>\text{Formal}\!</math>
| <math>\text{Formal}\!</math>
| colspan="2" | <math>\text{Formative}\!</math>
| colspan="2" | <math>\text{Formative}\!</math>
====2.2.3. Propositions and Sentences====
====2.2.3. Propositions and Sentences====
The concept of a sign relation is typically extended as a set <math>\mathcal{L} \subseteq \mathcal{O} \times \mathcal{S} \times \mathcal{I}.</math> Because this extensional representation of a sign relation is one of the most natural forms that it can take up, along with being one of the most important forms that it is likely to be encountered in, a good amount of set-theoretic machinery is necessary to carry out a reasonably detailed analysis of sign relations in general.
The concept of a sign relation is typically extended as a set <math>L \subseteq O \times S \times I.\!</math> Because this extensional representation of a sign relation is one of the most natural forms that it can take up, along with being one of the most important forms in which it is likely to be encountered, a good amount of set-theoretic machinery is necessary to carry out a reasonably detailed analysis of sign relations in general.
For the purposes of this discussion, let it be supposed that each set <math>Q,\!</math> that comprises a subject of interest in a particular discussion or that constitutes a topic of interest in a particular moment of discussion, is a subset of a set <math>X,\!</math> one that is sufficiently universal relative to that discussion or big enough to cover everything that is being talked about in that moment. In a setting like this it is possible to make a number of useful definitions, to which we now turn.
For the purposes of this discussion, let it be supposed that each set <math>Q,\!</math> that comprises a subject of interest in a particular discussion or that constitutes a topic of interest in a particular moment of discussion, is a subset of a set <math>X,\!</math> one that is sufficiently universal relative to that discussion or big enough to cover everything that is being talked about in that moment. In a setting like this it is possible to make a number of useful definitions, to which we now turn.
The ''negation'' of a sentence <math>s\!</math>, written as <math>{}^{\backprime\backprime} \, \underline{(} s \underline{)} \, {}^{\prime\prime}</math> and read as <math>{}^{\backprime\backprime} \, \operatorname{not}\ s \, {}^{\prime\prime},\!</math> is a sentence that is true when <math>s\!</math> is false and false when <math>s\!</math> is true.
The ''negation'' of a sentence <math>s\!</math>, written as <math>{}^{\backprime\backprime} \texttt{(} s \texttt{)} \, {}^{\prime\prime}</math> and read as <math>{}^{\backprime\backprime} \, \operatorname{not}\ s \, {}^{\prime\prime},\!</math> is a sentence that is true when <math>s\!</math> is false and false when <math>s\!</math> is true.
The ''complement'' of a set <math>Q\!</math> with respect to the universe <math>X\!</math> is denoted by <math>{}^{\backprime\backprime} \, X\!-\!Q \, {}^{\prime\prime}\!</math> and is defined as the set of elements in <math>X\!</math> that do not belong to <math>Q.\!</math> When the universe <math>X\!</math> is fixed throughout a given discussion, the complement <math>X\!-\!Q</math> may be denoted either by <math>{}^{\backprime\backprime} \thicksim \! Q \, {}^{\prime\prime}\!</math> or by <math>{}^{\backprime\backprime} \, \tilde{Q} \, {}^{\prime\prime}.\!</math> Thus we have the following series of equivalences:
The ''complement'' of a set <math>Q\!</math> with respect to the universe <math>X\!</math> is denoted by <math>{}^{\backprime\backprime} \, X\!-\!Q \, {}^{\prime\prime}\!</math> and is defined as the set of elements in <math>X\!</math> that do not belong to <math>Q.\!</math> When the universe <math>X\!</math> is fixed throughout a given discussion, the complement <math>X\!-\!Q</math> may be denoted either by <math>{}^{\backprime\backprime} \thicksim \! Q \, {}^{\prime\prime}\!</math> or by <math>{}^{\backprime\backprime} \, \tilde{Q} \, {}^{\prime\prime}.\!</math> Thus we have the following series of equivalences:
<math>\begin{array}{lllllll}
<math>\begin{array}{lllllll}
\tilde{Q}
\tilde{Q}
& = &
& = & \thicksim\!Q
\thicksim \! Q
& = & X\!-\!Q
& = &
& = & \{ x \in X : \texttt{(} x \in Q \texttt{)} \}.
X\!-\!Q
& = &
\{ \, x \in X : \underline{(} x \in Q \underline{)} \, \}.
\end{array}</math>
\end{array}</math>
|}
|}
<math>\begin{array}{lll}
<math>\begin{array}{lll}
Q\!-\!P
Q\!-\!P
& = &
& = & \{ x \in X : x \in Q ~\operatorname{and}~ \texttt{(} x \in P \texttt{)} \}.
\{ \, x \in X : x \in Q ~\operatorname{and}~ \underline{(} x \in P \underline{)} \, \}.
\end{array}</math>
\end{array}</math>
|}
|}
<math>\begin{array}{lll}
<math>\begin{array}{lll}
P \cap Q
P \cap Q
& = &
& = & \{ x \in X : x \in P ~\operatorname{and}~ x \in Q \}.
\{ \, x \in X : x \in P ~\operatorname{and}~ x \in Q \, \}.
\end{array}</math>
\end{array}</math>
|}
|}
<math>\begin{array}{lll}
<math>\begin{array}{lll}
P \cup Q
P \cup Q
& = &
& = & \{ x \in X : x \in P ~\operatorname{or}~ x \in Q \}.
\{ \, x \in X : x \in P ~\operatorname{or}~ x \in Q \, \}.
\end{array}</math>
\end{array}</math>
|}
|}
<math>\begin{array}{lll}
<math>\begin{array}{lll}
P ~\hat{+}~ Q
P ~\hat{+}~ Q
& = &
& = & \{ x \in X : x \in P\!-\!Q ~\operatorname{or}~ x \in Q\!-\!P \}.
\{ \, x \in X : x \in P\!-\!Q ~\operatorname{or}~ x \in Q\!-\!P \, \}.
\end{array}</math>
\end{array}</math>
|}
|}
Of course, as sets of the same cardinality, the domains <math>\mathbb{B}\!</math> and <math>\underline\mathbb{B}\!</math> and all of the structures that can be built on them become isomorphic at a high enough level of abstraction. Consequently, the main reason for making this distinction in the present context appears to be a matter more of grammar than an issue of logical and mathematical substance, namely, so that the signs <math>{}^{\backprime\backprime} \underline{0} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \underline{1} {}^{\prime\prime}\!</math> can appear with some semblance of syntactic legitimacy in linguistic contexts that call for a grammatical sentence or a sentence surrogate to represent the classes of sentences that are ''always false'' and ''always true'', respectively. The signs <math>{}^{\backprime\backprime} 0 {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} 1 {}^{\prime\prime},\!</math> customarily read as nouns but not as sentences, fail to be suitable for this purpose. Whether these scruples, that are needed to conform to a particular choice of natural language context, are ultimately important, is another thing that remains to be determined.
Of course, as sets of the same cardinality, the domains <math>\mathbb{B}\!</math> and <math>\underline\mathbb{B}\!</math> and all of the structures that can be built on them become isomorphic at a high enough level of abstraction. Consequently, the main reason for making this distinction in the present context appears to be a matter more of grammar than an issue of logical and mathematical substance, namely, so that the signs <math>{}^{\backprime\backprime} \underline{0} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \underline{1} {}^{\prime\prime}\!</math> can appear with some semblance of syntactic legitimacy in linguistic contexts that call for a grammatical sentence or a sentence surrogate to represent the classes of sentences that are ''always false'' and ''always true'', respectively. The signs <math>{}^{\backprime\backprime} 0 {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} 1 {}^{\prime\prime},\!</math> customarily read as nouns but not as sentences, fail to be suitable for this purpose. Whether these scruples, that are needed to conform to a particular choice of natural language context, are ultimately important, is another thing that remains to be determined.
The ''negation'' of a value <math>x\!</math> in <math>\underline\mathbb{B},</math> written <math>{}^{\backprime\backprime} \underline{(} x \underline{)} {}^{\prime\prime}</math> or <math>{}^{\backprime\backprime} \lnot x {}^{\prime\prime}</math> and read as <math>{}^{\backprime\backprime} \operatorname{not}\ x {}^{\prime\prime},</math> is the boolean value <math>\underline{(} x \underline{)} \in \underline\mathbb{B}</math> that is <math>\underline{1}</math> when <math>x\!</math> is <math>\underline{0}</math> and <math>\underline{0}</math> when <math>x\!</math> is <math>\underline{1}.</math> Negation is a monadic operation on boolean values, that is, a function of the form <math>f : \underline\mathbb{B} \to \underline\mathbb{B},</math> as shown in Table 8.
The ''negation'' of a value <math>x\!</math> in <math>\underline\mathbb{B},</math> written <math>{}^{\backprime\backprime} \texttt{(} x \texttt{)} {}^{\prime\prime}</math> or <math>{}^{\backprime\backprime} \lnot x {}^{\prime\prime}</math> and read as <math>{}^{\backprime\backprime} \operatorname{not}\ x {}^{\prime\prime},</math> is the boolean value <math>\texttt{(} x \texttt{)} \in \underline\mathbb{B}</math> that is <math>\underline{1}</math> when <math>x\!</math> is <math>\underline{0}</math> and <math>\underline{0}</math> when <math>x\!</math> is <math>\underline{1}.</math> Negation is a monadic operation on boolean values, that is, a function of the form <math>f : \underline\mathbb{B} \to \underline\mathbb{B},</math> as shown in Table 8.
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"
|+ '''Table 8. Negation Operation for the Boolean Domain'''
|+ style="height:30px" | <math>\text{Table 8.} ~~ \text{Negation Operation for the Boolean Domain}\!</math>
|- style="background:whitesmoke"
|- style="height:40px; background:ghostwhite"
| <math>x\!</math>
| <math>x\!</math>
| <math>\underline{(} x \underline{)}</math>
| <math>\texttt{(} x \texttt{)}</math>
|-
|-
| <math>\underline{0}</math>
| <math>\underline{0}</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"
|+ '''Table 9. Product Operation for the Boolean Domain'''
|+ style="height:30px" | <math>\text{Table 9.} ~~ \text{Product Operation for the Boolean Domain}\!</math>
|- style="background:whitesmoke"
|- style="height:40px; background:ghostwhite"
| <math>\cdot\!</math>
| <math>\cdot\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}</math>
| <math>\underline{1}</math>
| <math>\underline{1}</math>
|-
|-
| style="background:whitesmoke" | <math>\underline{0}</math>
| style="background:ghostwhite" | <math>\underline{0}</math>
| <math>\underline{0}</math>
| <math>\underline{0}</math>
| <math>\underline{0}</math>
| <math>\underline{0}</math>
|-
|-
| style="background:whitesmoke" | <math>\underline{1}</math>
| style="background:ghostwhite" | <math>\underline{1}</math>
| <math>\underline{0}</math>
| <math>\underline{0}</math>
| <math>\underline{1}</math>
| <math>\underline{1}</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"
|+ '''Table 10. Sum Operation for the Boolean Domain'''
|+ style="height:30px" | <math>\text{Table 10.} ~~ \text{Sum Operation for the Boolean Domain}\!</math>
|- style="background:whitesmoke"
|- style="height:40px; background:ghostwhite"
| <math>+\!</math>
| <math>+\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}</math>
| <math>\underline{1}</math>
| <math>\underline{1}</math>
|-
|-
| style="background:whitesmoke" | <math>\underline{0}</math>
| style="background:ghostwhite" | <math>\underline{0}</math>
| <math>\underline{0}</math>
| <math>\underline{0}</math>
| <math>\underline{1}</math>
| <math>\underline{1}</math>
|-
|-
| style="background:whitesmoke" | <math>\underline{1}</math>
| style="background:ghostwhite" | <math>\underline{1}</math>
| <math>\underline{1}</math>
| <math>\underline{1}</math>
| <math>\underline{0}</math>
| <math>\underline{0}</math>
& = & \{ x \in X ~:~ f(x) = \underline{0} \}
& = & \{ x \in X ~:~ f(x) = \underline{0} \}
\\
\\
& = & \{ x \in X ~:~ \underline{(} f(x) \underline{)} \, \}.
& = & \{ x \in X ~:~ \texttt{(} f(x) \texttt{)} \, \}.
\end{array}</math>
\end{array}</math>
|}
|}
Taken in a context of communication, an assertion invites the interpreter to consider the things for which the sentence is true, in other words, to find the fiber of truth in the associated proposition, or yet again, to invert the indicator function denoted by the sentence with respect to its possible value of truth.
Taken in a context of communication, an assertion invites the interpreter to consider the things for which the sentence is true, in other words, to find the fiber of truth in the associated proposition, or yet again, to invert the indicator function denoted by the sentence with respect to its possible value of truth.
A ''denial'' of a sentence <math>s\!</math> is an assertion of its negation <math>{}^{\backprime\backprime} \, \underline{(} s \underline{)} \, {}^{\prime\prime}.</math> The denial acts as a request to think about the things for which the sentence is false, in other words, to find the fiber of falsity in the indicted proposition, or to invert the indicator function denoted by the sentence with respect to its possible value of falsity.
A ''denial'' of a sentence <math>s\!</math> is an assertion of its negation <math>{}^{\backprime\backprime} \, \texttt{(} s \texttt{)} \, {}^{\prime\prime}.</math> The denial acts as a request to think about the things for which the sentence is false, in other words, to find the fiber of falsity in the indicted proposition, or to invert the indicator function denoted by the sentence with respect to its possible value of falsity.
According to this manner of definition, any sign that happens to denote a proposition, any sign that is taken as denoting an indicator function, by that very fact alone successfully qualifies as a sentence. That is, a sentence is any sign that actually succeeds in denoting a proposition, any sign that one way or another brings to mind, as its actual object, a function of the form <math>f : X \to \underline\mathbb{B}.</math>
According to this manner of definition, any sign that happens to denote a proposition, any sign that is taken as denoting an indicator function, by that very fact alone successfully qualifies as a sentence. That is, a sentence is any sign that actually succeeds in denoting a proposition, any sign that one way or another brings to mind, as its actual object, a function of the form <math>f : X \to \underline\mathbb{B}.</math>
<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:60%"
|+ '''Table 11. Levels of Indication'''
|+ style="height:30px" | <math>\text{Table 11.} ~~ \text{Levels of Indication}\!</math>
|- style="background:whitesmoke"
|- style="height:40px; background:ghostwhite"
| width="33%" | <math>\operatorname{Object}</math>
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\operatorname{Sign}</math>
| width="33%" | <math>\text{Sign}\!</math>
| width="34%" | <math>\operatorname{Higher~Order~Sign}</math>
| width="34%" | <math>\text{Higher Order Sign}\!</math>
|- style="background:whitesmoke"
|- style="background:ghostwhite"
| <math>\text{Set}\!</math>
| <math>\text{Proposition}\!</math>
| <math>\text{Sentence}\!</math>
|-
|-
| <math>f^{-1} (y)\!</math>
| <math>f^{-1} (y)\!</math>
| <math>f\!</math>
| <math>f\!</math>
| <math>{}^{\backprime\backprime} f \, {}^{\prime\prime}</math>
| <math>{}^{\backprime\backprime} f \, {}^{\prime\prime}\!</math>
|-
|-
| <math>Q\!</math>
| <math>Q\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>{}^{\backprime\backprime} \underline{1} {}^{\prime\prime}</math>
| <math>{}^{\backprime\backprime} \underline{1} {}^{\prime\prime}\!</math>
|-
|-
| <math>{}^{_\sim} Q</math>
| <math>{}^{_\sim} Q\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>{}^{\backprime\backprime} \underline{0} {}^{\prime\prime}</math>
| <math>{}^{\backprime\backprime} \underline{0} {}^{\prime\prime}\!</math>
|}
|}
<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:60%"
|+ '''Table 12. Ilustrations of Notation'''
|+ style="height:30px" | <math>\text{Table 12.} ~~ \text{Illustrations of Notation}\!</math>
|- style="background:whitesmoke"
|- style="height:40px; background:ghostwhite"
| width="33%" | <math>\operatorname{Object}</math>
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\operatorname{Sign}</math>
| width="33%" | <math>\text{Sign}\!</math>
| width="34%" | <math>\operatorname{Higher~Order~Sign}</math>
| width="34%" | <math>\text{Higher Order Sign}\!</math>
|- style="background:whitesmoke"
|- style="background:ghostwhite"
| <math>\text{Set}\!</math>
| <math>\text{Proposition}\!</math>
| <math>\text{Sentence}\!</math>
|-
|-
| <math>Q\!</math>
| <math>Q\!</math>
| <math>s\!</math>
| <math>s\!</math>
|-
|-
| <math>[| \downharpoonleft s \downharpoonright |]</math>
| <math>[| \downharpoonleft s \downharpoonright |]\!</math>
| <math>\downharpoonleft s \downharpoonright</math>
| <math>\downharpoonleft s \downharpoonright\!</math>
| <math>s\!</math>
| <math>s\!</math>
|-
|-
| <math>[| q |]\!</math>
| <math>[| q |]\!</math>
| <math>q\!</math>
| <math>q\!</math>
| <math>{}^{\backprime\backprime} q \, {}^{\prime\prime}</math>
| <math>{}^{\backprime\backprime} q \, {}^{\prime\prime}~\!</math>
|-
|-
| <math>[| f_Q |]\!</math>
| <math>[| f_Q |]\!</math>
| <math>f_Q\!</math>
| <math>f_Q\!</math>
| <math>{}^{\backprime\backprime} f_Q \, {}^{\prime\prime}</math>
| <math>{}^{\backprime\backprime} f_Q \, {}^{\prime\prime}\!</math>
|-
|-
| <math>Q\!</math>
| <math>Q\!</math>
| <math>\upharpoonleft Q \upharpoonright</math>
| <math>\upharpoonleft Q \upharpoonright\!</math>
| <math>{}^{\backprime\backprime} \upharpoonleft Q \upharpoonright \, {}^{\prime\prime}</math>
| <math>{}^{\backprime\backprime} \upharpoonleft Q \upharpoonright \, {}^{\prime\prime}\!</math>
|}
|}
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
{| align="center" cellpadding="10" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:70%"
|+ '''Table 13. Algorithmic Translation Rules'''
|+ style="height:30px" | <math>\text{Table 13.} ~~ \text{Algorithmic Translation Rules}\!</math>
|- style="background:whitesmoke"
|- style="height:50px; background:ghostwhite"
| width="33%" | <math>\mathrm{Sentence~in~PARCE}\!</math>
| width="33%" | <math>\xrightarrow{\mathrm{Parse}}\!</math>
| width="33%" | <math>\text{Sentence in PARCE}\!</math>
| width="33%" | <math>\mathrm{Graph~in~PARC}\!</math>
| align="center" | <math>\xrightarrow{\operatorname{Parse}}</math>
| width="33%" | <math>\text{Graph in PARC}\!</math>
|-
|-
|
| style="border-top:1px solid black" |
<math>\begin{matrix}
\mathrm{Conc}^0
| align="center" | <math>\xrightarrow{\operatorname{Parse}}</math>
\\[8pt]
| width="33%" | <math>\operatorname{Node}^0</math>
\mathrm{Conc}_{j=1}^k s_j
\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
\xrightarrow{\mathrm{Parse}}
\\[8pt]
\xrightarrow{\mathrm{Parse}}
\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
\mathrm{Node}^0
\\[8pt]
\mathrm{Node}_{j=1}^k \mathrm{Parse}(s_j)
\end{matrix}</math>
|-
|-
| width="33%" | <math>\operatorname{Conc}_{j=1}^k s_j</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
\mathrm{Surc}^0
|}
\\[8pt]
\mathrm{Surc}_{j=1}^k s_j
\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
\xrightarrow{\mathrm{Parse}}
\\[8pt]
|-
\xrightarrow{\mathrm{Parse}}
\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
\mathrm{Lobe}^0
\\[8pt]
\mathrm{Lobe}_{j=1}^k \mathrm{Parse}(s_j)
\end{matrix}</math>
|}
|}
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
{| align="center" cellpadding="10" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:70%"
|+ '''Table 14. Semantic Translation : Functional Form'''
|+ style="height:30px" | <math>\text{Table 14.} ~~ \text{Semantic Translation : Functional Form}\!</math>
|- style="background:whitesmoke"
|- style="height:50px; background:ghostwhite"
| width="20%" | <math>\mathrm{Sentence}\!</math>
| width="20%" | <math>\xrightarrow[~~~~~~~~~~]{\mathrm{Parse}}\!</math>
| width="20%" | <math>\operatorname{Sentence}</math>
| width="20%" | <math>\mathrm{Graph}\!</math>
| width="20%" | <math>\xrightarrow[\operatorname{~~~~~~~~~~}]{\operatorname{Parse}}</math>
| width="20%" | <math>\xrightarrow[~~~~~~~~~~]{\mathrm{Denotation}}\!</math>
| width="20%" | <math>\operatorname{Graph}</math>
| width="20%" | <math>\mathrm{Proposition}\!</math>
| width="20%" | <math>\xrightarrow[\operatorname{~~~~~~~~~~}]{\operatorname{Denotation}}</math>
| width="20%" | <math>\operatorname{Proposition}</math>
|-
|-
|
| style="border-top:1px solid black" | <math>s_j\!</math>
| style="border-top:1px solid black" | <math>\xrightarrow{~~~~~~~~~~}</math>
| style="border-top:1px solid black" | <math>C_j\!</math>
| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
| style="border-top:1px solid black" | <math>\xrightarrow{~~~~~~~~~~}</math>
| width="20%" | <math>C_j\!</math>
| style="border-top:1px solid black" | <math>q_j\!</math>
| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
| width="20%" | <math>q_j\!</math>
|-
|-
|
| style="border-top:1px solid black" |
<math>\begin{matrix}
\mathrm{Conc}^0
| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
\\[8pt]
| width="20%" | <math>\operatorname{Node}^0</math>
\mathrm{Conc}_{j=1}^k s_j
| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
\end{matrix}</math>
| width="20%" | <math>\underline{1}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
\xrightarrow{~~~~~~~~~~}
\\[8pt]
\xrightarrow{~~~~~~~~~~}
\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
\mathrm{Node}^0
\\[8pt]
\mathrm{Node}_{j=1}^k C_j
\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
\xrightarrow{~~~~~~~~~~}
\\[8pt]
\xrightarrow{~~~~~~~~~~}
\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
\underline{1}
\\[8pt]
\mathrm{Conj}_{j=1}^k q_j
\end{matrix}</math>
|-
|-
| width="20%" | <math>\operatorname{Conc}^k_j s_j</math>
| style="border-top:1px solid black" |
| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
<math>\begin{matrix}
\mathrm{Surc}^0
\\[8pt]
| width="20%" | <math>\operatorname{Conj}^k_j q_j</math>
\mathrm{Surc}_{j=1}^k s_j
\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
\xrightarrow{~~~~~~~~~~}
\\[8pt]
\xrightarrow{~~~~~~~~~~}
\end{matrix}</math>
| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
\mathrm{Lobe}^0
\\[8pt]
\mathrm{Lobe}_{j=1}^k C_j
\end{matrix}</math>
| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
\xrightarrow{~~~~~~~~~~}
\\[8pt]
\xrightarrow{~~~~~~~~~~}
\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
\underline{0}
\\[8pt]
\mathrm{Surj}_{j=1}^k q_j
\end{matrix}</math>
|}
|}
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
{| align="center" cellpadding="10" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:70%"
|+ '''Table 15. Semantic Translation : Equational Form'''
|+ style="height:30px" | <math>\text{Table 15.} ~~ \text{Semantic Translation : Equational Form}\!</math>
|- style="background:whitesmoke"
|- style="height:50px; background:ghostwhite"
| width="20%" | <math>\downharpoonleft \mathrm{Sentence} \downharpoonright\!</math>
| width="20%" | <math>\stackrel{\mathrm{Parse}}{=}\!</math>
| width="20%" | <math>\downharpoonleft \operatorname{Sentence} \downharpoonright</math>
| width="20%" | <math>\downharpoonleft \mathrm{Graph} \downharpoonright\!</math>
| width="20%" | <math>\stackrel{\operatorname{Parse}}{=}</math>
| width="20%" | <math>\stackrel{\mathrm{Denotation}}{=}\!</math>
| width="20%" | <math>\downharpoonleft \operatorname{Graph} \downharpoonright</math>
| width="20%" | <math>\mathrm{Proposition}\!</math>
| width="20%" | <math>\stackrel{\operatorname{Denotation}}{=}</math>
| width="20%" | <math>\operatorname{Proposition}</math>
|-
|-
|
| style="border-top:1px solid black" | <math>\downharpoonleft s_j \downharpoonright\!</math>
| style="border-top:1px solid black" | <math>=\!</math>
| style="border-top:1px solid black" | <math>\downharpoonleft C_j \downharpoonright\!</math>
| width="20%" | <math>=\!</math>
| style="border-top:1px solid black" | <math>=\!</math>
| width="20%" | <math>\downharpoonleft C_j \downharpoonright</math>
| style="border-top:1px solid black" | <math>q_j\!</math>
| width="20%" | <math>=\!</math>
| width="20%" | <math>q_j\!</math>
|-
|-
|
| style="border-top:1px solid black" |
<math>\begin{matrix}
\downharpoonleft \mathrm{Conc}^0 \downharpoonright
| width="20%" | <math>=\!</math>
\\[8pt]
| width="20%" | <math>\downharpoonleft \operatorname{Node}^0 \downharpoonright</math>
\downharpoonleft \mathrm{Conc}_{j=1}^k s_j \downharpoonright
| width="20%" | <math>=\!</math>
\end{matrix}</math>
| width="20%" | <math>\underline{1}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}=\\[8pt]=\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
\downharpoonleft \mathrm{Node}^0 \downharpoonright
\\[8pt]
\downharpoonleft \mathrm{Node}_{j=1}^k C_j \downharpoonright
\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}=\\[8pt]=\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
\underline{1}
\\[8pt]
\mathrm{Conj}_{j=1}^k q_j
\end{matrix}</math>
|-
|-
| width="20%" | <math>\downharpoonleft \operatorname{Conc}^k_j s_j \downharpoonright</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
\downharpoonleft \mathrm{Surc}^0 \downharpoonright
| width="20%" | <math>=\!</math>
\\[8pt]
\downharpoonleft \mathrm{Surc}_{j=1}^k s_j \downharpoonright
|}
\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}=\\[8pt]=\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
\downharpoonleft \mathrm{Lobe}^0 \downharpoonright
\\[8pt]
\downharpoonleft \mathrm{Lobe}_{j=1}^k C_j \downharpoonright
\end{matrix}</math>
| width="20%" | <math>\underline{0}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}=\\[8pt]=\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
| width="20%" | <math>=\!</math>
\underline{0}
\\[8pt]
\mathrm{Surj}_{j=1}^k q_j
\end{matrix}</math>
|}
|}
<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:70%"
|+ '''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>
| width="48%" | <math>F()~\!</math>
| width="48%" | <math>F()\!</math>
| 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>\texttt{(~)}\!</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>\texttt{((~))}\!</math>
| <math>\texttt{((~))}\!</math>
|}
|}
<br>
<br>
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:70%"
|+ '''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="background:ghostwhite"
| width="14%" |
| width="14%" |
| width="14%" |
| width="14%" |
| width="24%" | <math>F(\underline{1})</math>
| width="24%" | <math>F(\underline{1})~\!</math>
| width="24%" | <math>F(\underline{0})</math>
| width="24%" | <math>F(\underline{0})~\!</math>
| width="24%" |
| width="24%" |
|-
|-
| <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>\texttt{(~)}\!</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>\texttt{(} x \texttt{)}\!</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>\texttt{((~))}\!</math>
| <math>\texttt{((~))}\!</math>
|}
|}
<br>
<br>
{| align="center" border="1" cellpadding="4" cellspacing="0" style="text-align:center; width:90%"
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:70%"
|+ '''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="background:ghostwhite"
| width="14%" |
| width="14%" |
| width="14%" |
| width="14%" |
| <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>
|}
|}