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Directory:Jon Awbrey/Papers/Syntactic Transformations (view source)
Revision as of 22:14, 9 December 2015
, 22:14, 9 December 2015'''Author: Jon Awbrey'''
{{DISPLAYTITLE:Syntactic Transformations}}
{{DISPLAYTITLE:Syntactic Transformations}}
'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''
<div class="nonumtoc">__TOC__</div>
<div class="nonumtoc">__TOC__</div>
==Syntactic Transformations==
====1.3.12. Syntactic Transformations====
To discuss the import of the above definitions in greater depth, it serves to establish a number of logical relations and set-theoretic identities that can be found to hold among this array of conceptions and constructions. Facilitating this task requires in turn a number of auxiliary concepts and notations.
We have been examining several distinct but closely related notions of ''indication''. To discuss the import of these ideas in greater depth, it serves to establish a number of logical relations and set-theoretic identities that can be found to hold among their roughly parallel arrays of conceptions and constructions. Facilitating this task requires in turn a number of auxiliary concepts and notations. The notions of indication in question are expressed in a variety of different notations, enumerated as follows:
The diverse notions of ''indication'' under discussion are expressed in a variety of different notations, in particular, the logical language of sentences, the functional language of propositions, and the geometric language of sets. Thus, one way to explain the relationships that exist among these concepts is to describe the ''translations'' that they induce among the allied families of notation.
# The functional language of propositions
# The logical language of sentences
# The geometric language of sets
===Syntactic Transformation Rules===
Thus, one way to explain the relationships that hold among these concepts is to describe the ''translations'' that are induced among their allied families of notation.
=====1.3.12.1. Syntactic Transformation Rules=====
A good way to summarize these translations and to organize their use in practice is by means of the ''syntactic transformation rules'' (STRs) that partially formalize them. A rudimentary example of a STR is readily mined from the raw materials that are already available in this area of discussion. To begin, let the definition of an indicator function be recorded in the following form:
A good way to summarize these translations and to organize their use in practice is by means of the ''syntactic transformation rules'' (STRs) that partially formalize them. A rudimentary example of a STR is readily mined from the raw materials that are already available in this area of discussion. To begin, let the definition of an indicator function be recorded in the following form:
'''Editing Note.''' Need a transition here. Give a brief description of the Tables of Translation Rules that have now been moved to the Appendices, and then move on to the rest of the Definitions and Proof Schemata.
'''Editing Note.''' Need a transition here. Give a brief description of the Tables of Translation Rules that have now been moved to the Appendices, and then move on to the rest of the Definitions and Proof Schemata.
A rule that allows one to turn equivalent sentences into identical propositions:
A rule that allows one to turn equivalent sentences into identical propositions:
(S <=> T) <=> ([S] = [T])
{| align="center" cellpadding="8" width="90%"
| <math>(S \Leftrightarrow T) \quad \Leftrightarrow \quad (\downharpoonleft S \downharpoonright = \downharpoonleft T \downharpoonright)</math>
|}
Compare:
{| align="center" cellpadding="8" width="90%"
| <math>\downharpoonleft v = w \downharpoonright (v, w)</math>
|-
| <math>\downharpoonleft v(u) = w(u) \downharpoonright (u)</math>
|}
'''Editing Note.''' The last draft I can find has 5 variants for the next box, "Value Rule 1", and I can't tell right off which I meant to use. Until I can get back to this, here's a link to the collection of variants:
* [http://mywikibiz.com/User:Jon_Awbrey/SCRATCHPAD#Value_Rule_1 Value Rule 1]
<br>
<br>
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black" width="90%"
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:50px; text-align:right"
| width="98%" | <math>\operatorname{Evaluation~Rule~1}</math>
| width="2%" |
|}
|-
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:50px"
| width="2%" style="border-top:1px solid black" |
| width="14%" style="border-top:1px solid black" | <math>\text{If}\!</math>
| width="84%" style="border-top:1px solid black" | <math>f, g ~:~ X \to \underline\mathbb{B}</math>
|- style="height:50px"
|
| <math>\text{and}\!</math>
| <math>x ~\in~ X</math>
|- style="height:50px"
|
| <math>\text{then}\!</math>
| <math>\text{the following are equivalent:}\!</math>
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black" width="90%"
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:40px; text-align:center"
| width="80%" |
| width="20%" | <math>\operatorname{Definition~2}</math>
|}
|}
|-
|-
|
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:40px"
|- style="height:10px"
| width="2%" style="border-top:1px solid black" |
| width="2%" style="border-top:1px solid black" |
| width="18%" style="border-top:1px solid black" | <math>\text{If}\!</math>
| width="14%" style="border-top:1px solid black" |
| width="80%" style="border-top:1px solid black" | <math>P, Q ~\subseteq~ X</math>
| width="64%" style="border-top:1px solid black" |
| width="20%" style="border-top:1px solid black; border-left:1px solid black" |
|- style="height:40px"
|
| <math>\operatorname{E1a.}</math>
| <math>f(x) ~=~ g(x)</math>
| style="border-left:1px solid black; text-align:center" | <math>\operatorname{E1a~:~V1a}</math>
|- style="height:20px"
| colspan="3" |
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:40px"
|
| <math>\operatorname{E1b.}</math>
| <math>f(x) ~\Leftrightarrow~ g(x)</math>
| style="border-left:1px solid black; text-align:center" | <math>\operatorname{E1b~:~V1b}</math>
|- style="height:20px"
| colspan="3" |
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:60px"
|
| <math>\operatorname{E1c.}</math>
| <math>\underline{((}~ f(x) ~,~ g(x) ~\underline{))}</math>
| style="border-left:1px solid black; text-align:center" |
<p><math>\operatorname{E1c~:~V1c}</math></p>
<p><math>\operatorname{E1c~:~$1a}</math></p>
|- style="height:20px"
| colspan="3" |
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:40px"
|
| <math>\operatorname{E1d.}</math>
| <math>\underline{((}~ f ~,~ g ~\underline{))}^\$ (x)</math>
| style="border-left:1px solid black; text-align:center" | <math>\operatorname{E1d~:~$1b}</math>
|- style="height:10px"
| colspan="3" |
| style="border-left:1px solid black" |
|}
|}
<br>
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black" width="90%"
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:40px; text-align:center"
| width="80%" |
| width="20%" | <math>\operatorname{Definition~2}</math>
|}
|-
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:40px"
| width="2%" style="border-top:1px solid black" |
| width="18%" style="border-top:1px solid black" | <math>\text{If}\!</math>
| width="80%" style="border-top:1px solid black" | <math>P, Q ~\subseteq~ X</math>
|- style="height:40px"
|- style="height:40px"
|
|
<br>
<br>
===Derived Equivalence Relations===
=====1.3.12.2. Derived Equivalence Relations=====
One seeks a method of general application for approaching the individual sign relation, a way to select an aspect of its form, to analyze it with regard to its intrinsic structure, and to classify it in comparison with other sign relations. With respect to a particular sign relation, one approach that presents itself is to examine the relation between signs and interpretants that is given directly by its connotative component and to compare it with the various forms of derived, indirect, mediate, or peripheral relationships that can be found to exist among signs and interpretants by way of secondary considerations or subsequent studies. Of especial interest are the relationships among signs and interpretants that can be obtained by working through the collections of objects that they commonly or severally denote.
One seeks a method of general application for approaching the individual sign relation, a way to select an aspect of its form, to analyze it with regard to its intrinsic structure, and to classify it in comparison with other sign relations. With respect to a particular sign relation, one approach that presents itself is to examine the relation between signs and interpretants that is given directly by its connotative component and to compare it with the various forms of derived, indirect, mediate, or peripheral relationships that can be found to exist among signs and interpretants by way of secondary considerations or subsequent studies. Of especial interest are the relationships among signs and interpretants that can be obtained by working through the collections of objects that they commonly or severally denote.
<br>
<br>
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black" width="90%"
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:50px; text-align:center"
| style="width:80%" |
| style="width:20%; border-left:1px solid black" | <math>\operatorname{Fact~2.2}</math>
| <math>\upharpoonleft \operatorname{Der}(L) \upharpoonright (x, y) \quad = \quad \underset{o \in O}{\operatorname{Conj}} ~\underline{((}~ \upharpoonleft L_{SO} \upharpoonright (x, o) ~,~ \upharpoonleft L_{OS} \upharpoonright (o, y) ~\underline{))}~.</math>
|}
|}
|-
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
{| align="center" cellpadding="8" width="90%"
|- style="height:50px"
| <math>\upharpoonleft P \circeq Q \upharpoonright (x, y) \quad = \quad \underset{m \in M}{\operatorname{Conj}} ~\underline{((}~ \upharpoonleft P \upharpoonright (x, m) ~,~ \upharpoonleft Q \upharpoonright (m, y) ~\underline{))}~.</math>
| width="2%" style="border-top:1px solid black" |
|}
| width="12%" style="border-top:1px solid black" | <math>\text{If}\!</math>
| width="66%" style="border-top:1px solid black" | <math>L ~\subseteq~ O \times S \times I</math>
| width="20%" style="border-top:1px solid black; border-left:1px solid black" |
|- style="height:50px"
|
| <math>\upharpoonleft P \circ Q \upharpoonright (x, y) \quad = \quad \underset{m \in M}{\operatorname{Disj}} ~\upharpoonleft P \upharpoonright (x, m) ~\cdot~ \upharpoonleft Q \upharpoonright (m, y)~.</math>
| <math>\text{then}\!</math>
|}
| <math>\text{the following are equivalent:}\!</math>
| style="border-left:1px solid black" |
|
|- style="height:48px; text-align:right"
| width="2%" |
|}
|}
|-
|-
|
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:48px"
|- style="height:10px"
| width="2%" style="border-top:1px solid black" |
| width="2%" style="border-top:1px solid black" |
| width="18%" style="border-top:1px solid black" | <math>\text{If}\!</math>
| width="12%" style="border-top:1px solid black" |
| width="80%" style="border-top:1px solid black" |
| width="66%" style="border-top:1px solid black" |
| width="20%" style="border-top:1px solid black; border-left:1px solid black" |
|- style="height:48px"
|- style="height:100px"
|
|
| <math>\text{and}\!</math>
| valign="top" | <math>\operatorname{F2.2a.}</math>
| <math>p ~\text{is a proposition} ~:~ X \to \underline\mathbb{B}</math>
| valign="top" |
|- style="height:48px"
<math>\begin{array}{cccl}
\operatorname{Der}^L
& = & \{ & (x, y) \in S \times I ~: \\
& & & \begin{array}{ccl}
\underset{o \in O}{\operatorname{Conj}} \\
& ( & \upharpoonleft \operatorname{Den}^L x \upharpoonright (o) \\
& = & \upharpoonleft \operatorname{Den}^L y \upharpoonright (o) \\
& ) & \\
\end{array} \\
& & \} & \\
\end{array}</math>
| style="border-left:1px solid black; text-align:center" | <math>\operatorname{F2.2a~:~R11a}</math>
|- style="height:20px"
| colspan="3" |
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:100px"
|
|
| <math>\text{such that:}\!</math>
| valign="top" | <math>\operatorname{F2.2b.}</math>
| valign="top" |
<math>\begin{array}{ccccl}
\upharpoonleft \operatorname{Der}^L \upharpoonright
& = & \upharpoonleft & \{ & (x, y) \in S \times I ~: \\
& & & & \begin{array}{ccl}
\underset{o \in O}{\operatorname{Conj}} \\
& ( & \upharpoonleft \operatorname{Den}^L x \upharpoonright (o) \\
& = & \upharpoonleft \operatorname{Den}^L y \upharpoonright (o) \\
& ) & \\
\end{array} \\
& & & \} & \\
& & \upharpoonright & & \\
\end{array}</math>
| style="border-left:1px solid black; text-align:center" |
<math>\operatorname{F2.2b~:~R11b}</math>
|- style="height:20px"
| colspan="3" |
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:100px"
|
|
|- style="height:48px"
| valign="top" | <math>\operatorname{F2.2c.}</math>
| valign="top" |
<math>\begin{array}{cccl}
\upharpoonleft \operatorname{Der}^L \upharpoonright
& = & \{ & (x, y, z) \in S \times I \times \underline\mathbb{B} ~:~ z = \\
& & & \begin{array}{cccl}
\downharpoonleft & \underset{o \in O}{\operatorname{Conj}} \\
& & ( & \upharpoonleft \operatorname{Den}^L x \upharpoonright (o) \\
& & = & \upharpoonleft \operatorname{Den}^L y \upharpoonright (o) \\
& & ) & \\
\downharpoonright & & \\
\end{array} \\
& & \} & \\
\end{array}</math>
| style="border-left:1px solid black; text-align:center" |
<math>\operatorname{F2.2c~:~R11c}</math></p>
|- style="height:20px"
| colspan="3" |
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:100px"
|
|
| <math>\text{L1a.}\!</math>
| valign="top" | <math>\operatorname{F2.2d.}</math>
| <math>\downharpoonleft s \downharpoonright ~=~ p</math>
| valign="top" |
|- style="height:48px"
<math>\begin{array}{cccl}
\upharpoonleft \operatorname{Der}^L \upharpoonright
& = & \{ & (x, y, z) \in S \times I \times \underline\mathbb{B} ~:~ z = \\
& & & \begin{array}{cccl}
\underset{o \in O}{\operatorname{Conj}} \\
& \downharpoonleft & ( & \upharpoonleft \operatorname{Den}^L x \upharpoonright (o) \\
& & = & \upharpoonleft \operatorname{Den}^L y \upharpoonright (o) \\
& & ) & \\
& \downharpoonright & & \\
\end{array} \\
& & \} & \\
\end{array}</math>
| style="border-left:1px solid black; text-align:center" |
<math>\operatorname{F2.2d~:~Log}</math>
|- style="height:20px"
| colspan="3" |
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:100px"
|
|
| <math>\text{then}\!</math>
| valign="top" | <math>\operatorname{F2.2e.}</math>
| <math>\text{the following equations hold:}\!</math>
| valign="top" |
|}
<math>\begin{array}{cccl}
|-
\upharpoonleft \operatorname{Der}^L \upharpoonright
& = & \{ & (x, y, z) \in S \times I \times \underline\mathbb{B} ~:~ z = \\
& & & \begin{array}{ccl}
\underset{o \in O}{\operatorname{Conj}} \\
& \underline{((} & \upharpoonleft \operatorname{Den}^L x \upharpoonright (o) \\
& , & \upharpoonleft \operatorname{Den}^L y \upharpoonright (o) \\
& \underline{))} & \\
\end{array} \\
& & \} & \\
\end{array}</math>
| style="border-left:1px solid black; text-align:center" |
<math>\operatorname{F2.2e~:~Log}</math>
|- style="height:20px"
| colspan="3" |
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:100px"
|
| valign="top" | <math>\operatorname{F2.2f.}</math>
| valign="top" |
<math>\begin{array}{cccl}
\upharpoonleft \operatorname{Der}^L \upharpoonright
& = & \{ & (x, y, z) \in S \times I \times \underline\mathbb{B} ~:~ z = \\
& & & \begin{array}{cll}
\underset{o \in O}{\operatorname{Conj}} \\
& \underline{((} & \upharpoonleft \operatorname{Den}^L x \upharpoonright \\
& , & \upharpoonleft \operatorname{Den}^L y \upharpoonright \\
& \underline{))}^\$ & (o) \\
\end{array} \\
& & \} & \\
\end{array}</math>
| style="border-left:1px solid black; text-align:center" |
<math>\operatorname{F2.2f~:~$~}</math>
|}
|}
<br>
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black" width="90%"
|
|
{| align="center" cellpadding="0" cellspacing="0" style="text-align:center" width="100%"
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:52px"
|- style="height:50px; text-align:center"
| style="width:80%" |
| style="width:20%; border-left:1px solid black" | <math>\operatorname{Fact~2.3}</math>
|}
|-
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:50px"
| width="2%" style="border-top:1px solid black" |
| width="2%" style="border-top:1px solid black" |
| width="18%" style="border-top:1px solid black" align="left" | <math>\text{L1b}_{00}.\!</math>
| width="12%" style="border-top:1px solid black" | <math>\text{If}\!</math>
| width="20%" style="border-top:1px solid black" |
| width="66%" style="border-top:1px solid black" | <math>L ~\subseteq~ O \times S \times I</math>
<math>\downharpoonleft \operatorname{false} \downharpoonright</math>
| width="20%" style="border-top:1px solid black; border-left:1px solid black" |
|- style="height:50px"
| width="20%" style="border-top:1px solid black" | <math>(~)</math>
|- style="height:52px"
|
|
| <math>\text{then}\!</math>
| <math>\downharpoonleft \operatorname{not}~ s \downharpoonright</math>
| <math>\text{the following are equivalent:}\!</math>
| style="border-left:1px solid black" |
| <math>(\downharpoonleft s \downharpoonright)</math>
|
|}
|}
|}
|-
|
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:48px; text-align:right"
|- style="height:10px"
| width="98%" | <math>\text{Geometric Translation Rule 1}\!</math>
| width="2%" style="border-top:1px solid black" |
| width="12%" style="border-top:1px solid black" |
|}
| width="66%" style="border-top:1px solid black" |
| width="20%" style="border-top:1px solid black; border-left:1px solid black" |
|- style="height:100px"
| width="18%" style="border-top:1px solid black" | <math>\text{If}\!</math>
| width="80%" style="border-top:1px solid black" | <math>Q \subseteq X</math>
|- style="height:48px"
|
|
| <math>\text{and}\!</math>
| valign="top" | <math>\operatorname{F2.3a.}</math>
| <math>p ~:~ X \to \underline\mathbb{B}</math>
| valign="top" |
|- style="height:48px"
<math>\begin{array}{cccl}
\operatorname{Der}^L
& = & \{ & (x, y) \in S \times I ~: \\
& & & \begin{array}{ccl}
\underset{o \in O}{\operatorname{Conj}} \\
& ( & \upharpoonleft \operatorname{Den}^L x \upharpoonright (o) \\
& = & \upharpoonleft \operatorname{Den}^L y \upharpoonright (o) \\
& ) & \\
\end{array} \\
& & \} & \\
\end{array}</math>
| style="border-left:1px solid black; text-align:center" | <math>\operatorname{F2.3a~:~R11a}</math>
|- style="height:20px"
| colspan="3" |
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:100px"
|
|
| <math>\text{such that:}\!</math>
| valign="top" | <math>\operatorname{F2.3b.}</math>
| valign="top" |
<math>\begin{array}{ccccl}
\upharpoonleft \operatorname{Der}^L \upharpoonright (x, y)
& = & \downharpoonleft & \underset{o \in O}{\operatorname{Conj}} \\
& & & & \begin{array}{cl}
( & \upharpoonleft \operatorname{Den}^L x \upharpoonright (o) \\
= & \upharpoonleft \operatorname{Den}^L y \upharpoonright (o) \\
) & \\
\end{array} \\
& & \downharpoonright & & \\
\end{array}</math>
| style="border-left:1px solid black; text-align:center" | <math>\operatorname{F2.3b~:~R11d}</math>
|- style="height:20px"
| colspan="3" |
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:100px"
|
|
|- style="height:48px"
| valign="top" | <math>\operatorname{F2.3c.}</math>
| valign="top" |
<math>\begin{array}{ccccl}
\upharpoonleft \operatorname{Der}^L \upharpoonright (x, y)
& = & \underset{o \in O}{\operatorname{Conj}} \\
& & & \begin{array}{ccl}
\downharpoonleft & ( & \upharpoonleft \operatorname{Den}^L x \upharpoonright (o) \\
& = & \upharpoonleft \operatorname{Den}^L y \upharpoonright (o) \\
& ) & \\
\downharpoonright & & \\
\end{array} \\
& & & \\
\end{array}</math>
| style="border-left:1px solid black; text-align:center" | <math>\operatorname{F2.3c~:~Log}</math></p>
|- style="height:20px"
| colspan="3" |
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:100px"
|
|
| <math>\text{G1a.}\!</math>
| valign="top" | <math>\operatorname{F2.3d.}</math>
| <math>\upharpoonleft Q \upharpoonright ~=~ p</math>
| valign="top" |
|- style="height:48px"
<math>\begin{array}{ccccl}
\upharpoonleft \operatorname{Der}^L \upharpoonright (x, y)
& = & \underset{o \in O}{\operatorname{Conj}} \\
& & & \begin{array}{ccl}
\downharpoonleft & ( & \upharpoonleft \operatorname{Den}^L \upharpoonright (o, x) \\
& = & \upharpoonleft \operatorname{Den}^L \upharpoonright (o, y) \\
& ) & \\
\downharpoonright & & \\
\end{array} \\
& & & \\
\end{array}</math>
| style="border-left:1px solid black; text-align:center" | <math>\operatorname{F2.3d~:~Def}</math>
|- style="height:20px"
| colspan="3" |
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:100px"
|
| valign="top" | <math>\operatorname{F2.3e.}</math>
| valign="top" |
<math>\begin{array}{ccccl}
\upharpoonleft \operatorname{Der}^L \upharpoonright (x, y)
& = & \underset{o \in O}{\operatorname{Conj}} \\
& & & \begin{array}{cl}
\underline{((} & \upharpoonleft \operatorname{Den}^L \upharpoonright (o, x) \\
, & \upharpoonleft \operatorname{Den}^L \upharpoonright (o, y) \\
\underline{))} & \\
\end{array} \\
& & & \\
\end{array}</math>
| style="border-left:1px solid black; text-align:center" |
<p><math>\operatorname{F2.3e~:~Log}</math></p>
<p><math>\operatorname{F2.3e~:~D10b}</math></p>
|- style="height:20px"
| colspan="3" |
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:100px"
|
|
| <math>\text{then}\!</math>
| valign="top" | <math>\operatorname{F2.3f.}</math>
| <math>\text{the following equations hold:}\!</math>
| valign="top" |
<math>\begin{array}{ccccl}
\upharpoonleft \operatorname{Der}^L \upharpoonright (x, y)
& = & \underset{o \in O}{\operatorname{Conj}} \\
& & & \begin{array}{cl}
\underline{((} & \upharpoonleft L_{OS} \upharpoonright (o, x) \\
, & \upharpoonleft L_{OS} \upharpoonright (o, y) \\
\underline{))} & \\
\end{array} \\
& & & \\
\end{array}</math>
| style="border-left:1px solid black; text-align:center" | <math>\operatorname{F2.3f~:~D10a}</math>
|}
|}
|}
<br>
=====1.3.12.3. Digression on Derived Relations=====
A better understanding of derived equivalence relations (DERs) can be achieved by placing their constructions within a more general context and thus comparing the associated type of derivation operation, namely, the one that takes a triadic relation <math>L\!</math> into a dyadic relation <math>\operatorname{Der}(L),</math> with other types of operations on triadic relations. The proper setting would permit a comparative study of all their constructions from a basic set of projections and a full array of compositions on dyadic relations.
To that end, let the derivation <math>\operatorname{Der}(L)</math> be expressed in the following way:
{| align="center" cellpadding="8" width="90%"
| <math>\upharpoonleft \operatorname{Der}(L) \upharpoonright (x, y) \quad = \quad \underset{o \in O}{\operatorname{Conj}} ~\underline{((}~ \upharpoonleft L_{SO} \upharpoonright (x, o) ~,~ \upharpoonleft L_{OS} \upharpoonright (o, y) ~\underline{))}~.</math>
<math>\underline{0} ~:~ X \to \underline\mathbb{B}</math>
|- style="height:52px"
| <math>\upharpoonleft {}^{_\sim} Q \upharpoonright</math>
|}
|}
From this may be abstracted a way of composing two dyadic relations that have a domain in common. For example, let <math>P \subseteq X \times M</math> and <math>Q \subseteq M \times Y</math> be dyadic relations that have the middle domain <math>M\!</math> in common. Then we may define a form of composition, notated <math>P \circeq Q,</math> where <math>P \circeq Q ~\subseteq~ X \times Y</math> is defined as follows:
{| align="center" cellpadding="8" width="90%"
| <math>\upharpoonleft P \circeq Q \upharpoonright (x, y) \quad = \quad \underset{m \in M}{\operatorname{Conj}} ~\underline{((}~ \upharpoonleft P \upharpoonright (x, m) ~,~ \upharpoonleft Q \upharpoonright (m, y) ~\underline{))}~.</math>
|}
|}
<br>
Compare this with the usual form of composition, typically notated <math>P \circ Q</math> and defined as follows:
===Logical Translation Rule 2===
{| align="center" cellpadding="8" width="90%"
| <math>\upharpoonleft P \circ Q \upharpoonright (x, y) \quad = \quad \underset{m \in M}{\operatorname{Disj}} ~\upharpoonleft P \upharpoonright (x, m) ~\cdot~ \upharpoonleft Q \upharpoonright (m, y)~.</math>
|}
==Appendices==
===Logical Translation Rule 1===
<br>
<br>
{| align="center" cellpadding="0" cellspacing="0" width="100%"
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:48px; text-align:right"
|- style="height:48px; text-align:right"
| width="98%" | <math>\text{Logical Translation Rule 2}\!</math>
| width="98%" | <math>\text{Logical Translation Rule 1}\!</math>
| width="2%" |
| width="2%" |
|}
|}
|- style="height:48px"
|- style="height:48px"
| width="2%" style="border-top:1px solid black" |
| width="2%" style="border-top:1px solid black" |
| width="14%" style="border-top:1px solid black" | <math>\text{If}\!</math>
| width="18%" style="border-top:1px solid black" | <math>\text{If}\!</math>
| width="84%" style="border-top:1px solid black" |
| width="80%" style="border-top:1px solid black" |
<math>s, t ~\text{are sentences about things in the universe}~ X</math>
<math>s ~\text{is a sentence about things in the universe X}</math>
|- style="height:48px"
|- style="height:48px"
|
|
| <math>\text{and}\!</math>
| <math>\text{and}\!</math>
| <math>p, q ~\text{are propositions} ~:~ X \to \underline\mathbb{B}</math>
| <math>p ~\text{is a proposition} ~:~ X \to \underline\mathbb{B}</math>
|- style="height:48px"
|- style="height:48px"
|
|
|- style="height:48px"
|- style="height:48px"
|
|
| <math>\text{L2a.}\!</math>
| <math>\text{L1a.}\!</math>
| <math>\downharpoonleft s \downharpoonright ~=~ p \quad \operatorname{and} \quad \downharpoonleft t \downharpoonright ~=~ q</math>
| <math>\downharpoonleft s \downharpoonright ~=~ p</math>
|- style="height:48px"
|- style="height:48px"
|
|
|- style="height:52px"
|- style="height:52px"
| width="2%" style="border-top:1px solid black" |
| width="2%" style="border-top:1px solid black" |
| width="14%" style="border-top:1px solid black" align="left" | <math>\text{L2b}_{0}.\!</math>
| width="18%" style="border-top:1px solid black" align="left" | <math>\text{L1b}_{00}.\!</math>
| width="32%" style="border-top:1px solid black" |
| width="20%" style="border-top:1px solid black" |
<math>\downharpoonleft \operatorname{false} \downharpoonright</math>
<math>\downharpoonleft \operatorname{false} \downharpoonright</math>
| width="4%" style="border-top:1px solid black" | <math>=\!</math>
| width="5%" style="border-top:1px solid black" | <math>=\!</math>
| width="28%" style="border-top:1px solid black" | <math>(~)</math>
| width="20%" style="border-top:1px solid black" | <math>(~)</math>
| width="4%" style="border-top:1px solid black" | <math>=\!</math>
| width="5%" style="border-top:1px solid black" | <math>=\!</math>
| width="16%" style="border-top:1px solid black" | <math>(~)</math>
| width="30%" style="border-top:1px solid black" |
<math>\underline{0} ~:~ X \to \underline\mathbb{B}</math>
|- style="height:52px"
|- style="height:52px"
|
|
| align="left" | <math>\text{L2b}_{1}.\!</math>
| align="left" | <math>\text{L1b}_{01}.\!</math>
| <math>\downharpoonleft \operatorname{not}~ s \downharpoonright</math>
| <math>\downharpoonleft \operatorname{not}~ s \downharpoonright</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>(\downharpoonleft s \downharpoonright)</math>
| <math>(\downharpoonleft s \downharpoonright)</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>(p)\!</math>
| <math>(p) ~:~ X \to \underline\mathbb{B}</math>
|- style="height:52px"
|- style="height:52px"
|
|
| align="left" | <math>\text{L2b}_{4}.\!</math>
| align="left" | <math>\text{L1b}_{10}.\!</math>
| <math>\downharpoonleft s ~\operatorname{and~not}~ t \downharpoonright</math>
| <math>\downharpoonleft s \downharpoonright</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>\downharpoonleft s \downharpoonright (\downharpoonleft t \downharpoonright)</math>
| <math>\downharpoonleft s \downharpoonright</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>p (q)\!</math>
| <math>p ~:~ X \to \underline\mathbb{B}</math>
|- style="height:52px"
|- style="height:52px"
|
|
| align="left" | <math>\text{L2b}_{5}.\!</math>
| align="left" | <math>\text{L1b}_{11}.\!</math>
| <math>\downharpoonleft \operatorname{not}~ t \downharpoonright</math>
| <math>\downharpoonleft \operatorname{true} \downharpoonright</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>(\downharpoonleft t \downharpoonright)</math>
| <math>((~))</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>(q)\!</math>
| <math>\underline{1} ~:~ X \to \underline\mathbb{B}</math>
|- style="height:52px"
|}
|}
<br>
===Geometric Translation Rule 1===
<br>
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black" width="90%"
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:48px; text-align:right"
| width="98%" | <math>\text{Geometric Translation Rule 1}\!</math>
| width="2%" |
|}
|-
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:48px"
| width="2%" style="border-top:1px solid black" |
| width="18%" style="border-top:1px solid black" | <math>\text{If}\!</math>
| width="80%" style="border-top:1px solid black" | <math>Q \subseteq X</math>
|- style="height:48px"
|
| <math>\text{and}\!</math>
| <math>p ~:~ X \to \underline\mathbb{B}</math>
|- style="height:48px"
|
|
| align="left" | <math>\text{L2b}_{6}.\!</math>
| <math>\text{such that:}\!</math>
| <math>\downharpoonleft s ~\operatorname{or}~ t, ~\operatorname{not~both} \downharpoonright</math>
|
| <math>=\!</math>
|- style="height:48px"
| <math>(\downharpoonleft s \downharpoonright ~,~ \downharpoonleft t \downharpoonright)</math>
|
| <math>=\!</math>
| <math>\text{G1a.}\!</math>
| <math>(p, q)\!</math>
| <math>\upharpoonleft Q \upharpoonright ~=~ p</math>
|- style="height:48px"
|
| <math>\text{then}\!</math>
| <math>\text{the following equations hold:}\!</math>
|}
|-
|
{| align="center" cellpadding="0" cellspacing="0" style="text-align:center" width="100%"
|- style="height:52px"
| width="2%" style="border-top:1px solid black" |
| width="18%" style="border-top:1px solid black" align="left" | <math>\text{G1b}_{00}.\!</math>
| width="20%" style="border-top:1px solid black" |
<math>\upharpoonleft \varnothing \upharpoonright</math>
| width="5%" style="border-top:1px solid black" | <math>=\!</math>
| width="20%" style="border-top:1px solid black" | <math>(~)</math>
| width="5%" style="border-top:1px solid black" | <math>=\!</math>
| width="30%" style="border-top:1px solid black" |
<math>\underline{0} ~:~ X \to \underline\mathbb{B}</math>
|- style="height:52px"
|- style="height:52px"
|
|
| align="left" | <math>\text{L2b}_{7}.\!</math>
| align="left" | <math>\text{G1b}_{01}.\!</math>
| <math>\downharpoonleft \operatorname{not~both}~ s ~\operatorname{and}~ t \downharpoonright</math>
| <math>\upharpoonleft {}^{_\sim} Q \upharpoonright</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>(\downharpoonleft s \downharpoonright ~ \downharpoonleft t \downharpoonright)</math>
| <math>(\upharpoonleft Q \upharpoonright)</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>(p q)\!</math>
| <math>(p) ~:~ X \to \underline\mathbb{B}</math>
|- style="height:52px"
|- style="height:52px"
|
|
| align="left" | <math>\text{L2b}_{8}.\!</math>
| align="left" | <math>\text{G1b}_{10}.\!</math>
| <math>\downharpoonleft s ~\operatorname{and}~ t \downharpoonright</math>
| <math>\upharpoonleft Q \upharpoonright</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>\downharpoonleft s \downharpoonright ~ \downharpoonleft t \downharpoonright</math>
| <math>\upharpoonleft Q \upharpoonright</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>p q\!</math>
| <math>p ~:~ X \to \underline\mathbb{B}</math>
|- style="height:52px"
|- style="height:52px"
|
|
| align="left" | <math>\text{L2b}_{9}.\!</math>
| align="left" | <math>\text{G1b}_{11}.\!</math>
| <math>\downharpoonleft s ~\operatorname{is~equivalent~to}~ t \downharpoonright</math>
| <math>\upharpoonleft X \upharpoonright</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>((\downharpoonleft s \downharpoonright ~,~ \downharpoonleft t \downharpoonright))</math>
| <math>((~))</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>((p, q))\!</math>
| <math>\underline{1} ~:~ X \to \underline\mathbb{B}</math>
|- style="height:52px"
|}
|
|}
| align="left" | <math>\text{L2b}_{10}.\!</math>
| <math>\downharpoonleft t \downharpoonright</math>
<br>
| <math>=\!</math>
| <math>\downharpoonleft t \downharpoonright</math>
===Logical Translation Rule 2===
| <math>=\!</math>
| <math>q\!</math>
<br>
|- style="height:52px"
|
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black" width="90%"
| align="left" | <math>\text{L2b}_{11}.\!</math>
|
| <math>\downharpoonleft s ~\operatorname{implies}~ t \downharpoonright</math>
{| align="center" cellpadding="0" cellspacing="0" width="100%"
| <math>=\!</math>
|- style="height:48px; text-align:right"
| <math>(\downharpoonleft s \downharpoonright (\downharpoonleft t \downharpoonright))</math>
| width="98%" | <math>\text{Logical Translation Rule 2}\!</math>
| <math>=\!</math>
| width="2%" |
| <math>(p (q))\!</math>
|}
|- style="height:52px"
|-
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:48px"
| width="2%" style="border-top:1px solid black" |
| width="14%" style="border-top:1px solid black" | <math>\text{If}\!</math>
| width="84%" style="border-top:1px solid black" |
<math>s, t ~\text{are sentences about things in the universe}~ X</math>
|- style="height:48px"
|
| <math>\text{and}\!</math>
| <math>p, q ~\text{are propositions} ~:~ X \to \underline\mathbb{B}</math>
|- style="height:48px"
|
|
| <math>\text{such that:}\!</math>
|
|
| align="left" | <math>\text{L2b}_{13}.\!</math>
|- style="height:48px"
| <math>\downharpoonleft s ~\operatorname{is~implied~by}~ t \downharpoonright</math>
|
| <math>=\!</math>
| <math>\text{L2a.}\!</math>
| <math>((\downharpoonleft s \downharpoonright) \downharpoonleft t \downharpoonright)</math>
| <math>\downharpoonleft s \downharpoonright ~=~ p \quad \operatorname{and} \quad \downharpoonleft t \downharpoonright ~=~ q</math>
| <math>=\!</math>
|- style="height:48px"
| <math>((p) q)\!</math>
|
| <math>\text{then}\!</math>
| <math>\text{the following equations hold:}\!</math>
|}
|-
|
{| align="center" cellpadding="0" cellspacing="0" style="text-align:center" width="100%"
|- style="height:52px"
| width="2%" style="border-top:1px solid black" |
| width="14%" style="border-top:1px solid black" align="left" | <math>\text{L2b}_{0}.\!</math>
| width="32%" style="border-top:1px solid black" |
<math>\downharpoonleft \operatorname{false} \downharpoonright</math>
| width="4%" style="border-top:1px solid black" | <math>=\!</math>
| width="28%" style="border-top:1px solid black" | <math>(~)</math>
| width="4%" style="border-top:1px solid black" | <math>=\!</math>
| width="16%" style="border-top:1px solid black" | <math>(~)</math>
|- style="height:52px"
|- style="height:52px"
|
|
| align="left" | <math>\text{L2b}_{14}.\!</math>
| align="left" | <math>\text{L2b}_{1}.\!</math>
| <math>\downharpoonleft s ~\operatorname{or}~ t \downharpoonright</math>
| <math>\downharpoonleft \operatorname{neither}~ s ~\operatorname{nor}~ t \downharpoonright</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>((\downharpoonleft s \downharpoonright)(\downharpoonleft t \downharpoonright))</math>
| <math>(\downharpoonleft s \downharpoonright)(\downharpoonleft t \downharpoonright)</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>((p)(q))\!</math>
| <math>(p)(q)\!</math>
|- style="height:52px"
|- style="height:52px"
|
|
| align="left" | <math>\text{L2b}_{15}.\!</math>
| align="left" | <math>\text{L2b}_{2}.\!</math>
| <math>\downharpoonleft \operatorname{true} \downharpoonright</math>
| <math>\downharpoonleft \operatorname{not}~ s ~\operatorname{but}~ t \downharpoonright</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>((~))</math>
| <math>(\downharpoonleft s \downharpoonright) \downharpoonleft t \downharpoonright</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>((~))</math>
| <math>(p) q\!</math>
|- style="height:52px"
|- style="height:48px"
|
|
| <math>\text{and}\!</math>
| align="left" | <math>\text{L2b}_{3}.\!</math>
| <math>p, q ~:~ X \to \underline\mathbb{B}</math>
| <math>\downharpoonleft \operatorname{not}~ s \downharpoonright</math>
|- style="height:48px"
| <math>=\!</math>
| <math>(\downharpoonleft s \downharpoonright)</math>
| <math>=\!</math>
| <math>(p)\!</math>
|- style="height:52px"
|
|
| <math>\text{such that:}\!</math>
| align="left" | <math>\text{L2b}_{4}.\!</math>
| <math>\downharpoonleft s ~\operatorname{and~not}~ t \downharpoonright</math>
| <math>=\!</math>
| <math>\downharpoonleft s \downharpoonright (\downharpoonleft t \downharpoonright)</math>
| <math>=\!</math>
| <math>p (q)\!</math>
|- style="height:52px"
|
|
|- style="height:48px"
| align="left" | <math>\text{L2b}_{5}.\!</math>
| <math>\downharpoonleft \operatorname{not}~ t \downharpoonright</math>
| <math>=\!</math>
| <math>(\downharpoonleft t \downharpoonright)</math>
| <math>=\!</math>
| <math>(q)\!</math>
|- style="height:52px"
|
|
| <math>\text{G2a.}\!</math>
| align="left" | <math>\text{L2b}_{6}.\!</math>
| <math>\upharpoonleft P \upharpoonright ~=~ p \quad \operatorname{and} \quad \upharpoonleft Q \upharpoonright ~=~ q</math>
| <math>\downharpoonleft s ~\operatorname{or}~ t, ~\operatorname{not~both} \downharpoonright</math>
|- style="height:48px"
| <math>=\!</math>
| <math>(\downharpoonleft s \downharpoonright ~,~ \downharpoonleft t \downharpoonright)</math>
| <math>=\!</math>
| <math>(p, q)\!</math>
|- style="height:52px"
|
|
| <math>\text{then}\!</math>
| align="left" | <math>\text{L2b}_{7}.\!</math>
| <math>\downharpoonleft \operatorname{not~both}~ s ~\operatorname{and}~ t \downharpoonright</math>
| <math>=\!</math>
| <math>(\downharpoonleft s \downharpoonright ~ \downharpoonleft t \downharpoonright)</math>
| <math>=\!</math>
| <math>(p q)\!</math>
| width="32%" style="border-top:1px solid black" |
<math>\upharpoonleft \varnothing \upharpoonright</math>
|- style="height:52px"
|- style="height:52px"
|
|
| align="left" | <math>\text{G2b}_{1}.\!</math>
| align="left" | <math>\text{L2b}_{8}.\!</math>
| <math>\upharpoonleft \overline{P} ~\cap~ \overline{Q} \upharpoonright</math>
| <math>\downharpoonleft s ~\operatorname{and}~ t \downharpoonright</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>(\upharpoonleft P \upharpoonright)(\upharpoonleft Q \upharpoonright)</math>
| <math>\downharpoonleft s \downharpoonright ~ \downharpoonleft t \downharpoonright</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>(p)(q)\!</math>
| <math>p q\!</math>
|- style="height:52px"
|- style="height:52px"
|
|
| align="left" | <math>\text{G2b}_{2}.\!</math>
| align="left" | <math>\text{L2b}_{9}.\!</math>
| <math>\upharpoonleft \overline{P} ~\cap~ Q \upharpoonright</math>
| <math>\downharpoonleft s ~\operatorname{is~equivalent~to}~ t \downharpoonright</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>(\upharpoonleft P \upharpoonright) \upharpoonleft Q \upharpoonright</math>
| <math>((\downharpoonleft s \downharpoonright ~,~ \downharpoonleft t \downharpoonright))</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>(p) q\!</math>
| <math>((p, q))\!</math>
|- style="height:52px"
|- style="height:52px"
|
|
| align="left" | <math>\text{G2b}_{3}.\!</math>
| align="left" | <math>\text{L2b}_{10}.\!</math>
| <math>\upharpoonleft \overline{P} \upharpoonright</math>
| <math>\downharpoonleft t \downharpoonright</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>(\upharpoonleft P \upharpoonright)</math>
| <math>\downharpoonleft t \downharpoonright</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>(p)\!</math>
| <math>q\!</math>
|- style="height:52px"
|- style="height:52px"
|
|
| align="left" | <math>\text{G2b}_{4}.\!</math>
| align="left" | <math>\text{L2b}_{11}.\!</math>
| <math>\upharpoonleft P ~\cap~ \overline{Q} \upharpoonright</math>
| <math>\downharpoonleft s ~\operatorname{implies}~ t \downharpoonright</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>\upharpoonleft P \upharpoonright (\upharpoonleft Q \upharpoonright)</math>
| <math>(\downharpoonleft s \downharpoonright (\downharpoonleft t \downharpoonright))</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>p (q)\!</math>
| <math>(p (q))\!</math>
|- style="height:52px"
|- style="height:52px"
|
|
| align="left" | <math>\text{G2b}_{5}.\!</math>
| align="left" | <math>\text{L2b}_{12}.\!</math>
| <math>\upharpoonleft \overline{Q} \upharpoonright</math>
| <math>\downharpoonleft s \downharpoonright</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>(\upharpoonleft Q \upharpoonright)</math>
| <math>\downharpoonleft s \downharpoonright</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>(q)\!</math>
| <math>p\!</math>
|- style="height:52px"
|- style="height:52px"
|
|
| align="left" | <math>\text{G2b}_{6}.\!</math>
| align="left" | <math>\text{L2b}_{13}.\!</math>
| <math>\upharpoonleft P ~+~ Q \upharpoonright</math>
| <math>\downharpoonleft s ~\operatorname{is~implied~by}~ t \downharpoonright</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>(\upharpoonleft P \upharpoonright ~,~ \upharpoonleft Q \upharpoonright)</math>
| <math>((\downharpoonleft s \downharpoonright) \downharpoonleft t \downharpoonright)</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>(p, q)\!</math>
| <math>((p) q)\!</math>
|- style="height:52px"
|- style="height:52px"
|
|
| align="left" | <math>\text{G2b}_{7}.\!</math>
| align="left" | <math>\text{L2b}_{14}.\!</math>
| <math>\upharpoonleft \overline{P ~\cap~ Q} \upharpoonright</math>
| <math>\downharpoonleft s ~\operatorname{or}~ t \downharpoonright</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>(\upharpoonleft P \upharpoonright ~ \upharpoonleft Q \upharpoonright)</math>
| <math>((\downharpoonleft s \downharpoonright)(\downharpoonleft t \downharpoonright))</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>(p q)\!</math>
| <math>((p)(q))\!</math>
|- style="height:52px"
|- style="height:52px"
|
|
| align="left" | <math>\text{G2b}_{8}.\!</math>
| align="left" | <math>\text{L2b}_{15}.\!</math>
| <math>\upharpoonleft P ~\cap~ Q \upharpoonright</math>
| <math>\downharpoonleft \operatorname{true} \downharpoonright</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>\upharpoonleft P \upharpoonright ~ \upharpoonleft Q \upharpoonright</math>
| <math>((~))</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>p q\!</math>
| <math>((~))</math>
|- style="height:52px"
|}
|
|}
| align="left" | <math>\text{G2b}_{9}.\!</math>
| <math>\upharpoonleft \overline{P ~+~ Q} \upharpoonright</math>
<br>
| <math>=\!</math>
| <math>((\upharpoonleft P \upharpoonright ~,~ \upharpoonleft Q \upharpoonright))</math>
===Geometric Translation Rule 2===
| <math>=\!</math>
<br>
|- style="height:52px"
|
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black" width="90%"
| align="left" | <math>\text{G2b}_{10}.\!</math>
|
| <math>\upharpoonleft Q \upharpoonright</math>
{| align="center" cellpadding="0" cellspacing="0" width="100%"
| <math>=\!</math>
|- style="height:48px; text-align:right"
| <math>\upharpoonleft Q \upharpoonright</math>
| width="98%" | <math>\text{Geometric Translation Rule 2}\!</math>
| <math>=\!</math>
| width="2%" |
| <math>q\!</math>
|}
|-
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:48px"
| width="2%" style="border-top:1px solid black" |
| width="14%" style="border-top:1px solid black" | <math>\text{If}\!</math>
| width="84%" style="border-top:1px solid black" | <math>P, Q \subseteq X</math>
|- style="height:48px"
|
| <math>\text{and}\!</math>
| <math>p, q ~:~ X \to \underline\mathbb{B}</math>
|- style="height:48px"
|
| <math>\text{such that:}\!</math>
|
|
|- style="height:48px"
|- style="height:52px"
|
|
| align="left" | <math>\text{G2b}_{12}.\!</math>
| <math>\text{G2a.}\!</math>
| <math>\upharpoonleft P \upharpoonright</math>
| <math>\upharpoonleft P \upharpoonright ~=~ p \quad \operatorname{and} \quad \upharpoonleft Q \upharpoonright ~=~ q</math>
| <math>=\!</math>
|- style="height:48px"
| <math>\upharpoonleft P \upharpoonright</math>
|
| <math>\text{then}\!</math>
| <math>\text{the following equations hold:}\!</math>
|}
|-
|
{| align="center" cellpadding="0" cellspacing="0" style="text-align:center" width="100%"
|- style="height:52px"
| width="2%" style="border-top:1px solid black" |
| width="14%" style="border-top:1px solid black" align="left" | <math>\text{G2b}_{0}.\!</math>
| width="32%" style="border-top:1px solid black" |
<math>\upharpoonleft \varnothing \upharpoonright</math>
| width="4%" style="border-top:1px solid black" | <math>=\!</math>
| width="28%" style="border-top:1px solid black" | <math>(~)</math>
| width="4%" style="border-top:1px solid black" | <math>=\!</math>
| width="16%" style="border-top:1px solid black" | <math>(~)</math>
|- style="height:52px"
|
| align="left" | <math>\text{G2b}_{1}.\!</math>
| <math>\upharpoonleft \overline{P} ~\cap~ \overline{Q} \upharpoonright</math>
| <math>=\!</math>
| <math>(\upharpoonleft P \upharpoonright)(\upharpoonleft Q \upharpoonright)</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>p\!</math>
| <math>(p)(q)\!</math>
|- style="height:52px"
|- style="height:52px"
|
|
| align="left" | <math>\text{G2b}_{13}.\!</math>
| align="left" | <math>\text{G2b}_{2}.\!</math>
| <math>\upharpoonleft \overline{\overline{P} ~\cap~ Q} \upharpoonright</math>
| <math>\upharpoonleft \overline{P} ~\cap~ Q \upharpoonright</math>
| <math>=\!</math>
| <math>(\upharpoonleft P \upharpoonright) \upharpoonleft Q \upharpoonright</math>
| <math>=\!</math>
| <math>(p) q\!</math>
|- style="height:52px"
|
| align="left" | <math>\text{G2b}_{3}.\!</math>
| <math>\upharpoonleft \overline{P} \upharpoonright</math>
| <math>=\!</math>
| <math>(\upharpoonleft P \upharpoonright)</math>
| <math>=\!</math>
| <math>(p)\!</math>
|- style="height:52px"
|
| align="left" | <math>\text{G2b}_{4}.\!</math>
| <math>\upharpoonleft P ~\cap~ \overline{Q} \upharpoonright</math>
| <math>=\!</math>
| <math>\upharpoonleft P \upharpoonright (\upharpoonleft Q \upharpoonright)</math>
| <math>=\!</math>
| <math>p (q)\!</math>
|- style="height:52px"
|
| align="left" | <math>\text{G2b}_{5}.\!</math>
| <math>\upharpoonleft \overline{Q} \upharpoonright</math>
| <math>=\!</math>
| <math>(\upharpoonleft Q \upharpoonright)</math>
| <math>=\!</math>
| <math>(q)\!</math>
|- style="height:52px"
|
| align="left" | <math>\text{G2b}_{6}.\!</math>
| <math>\upharpoonleft P ~+~ Q \upharpoonright</math>
| <math>=\!</math>
| <math>(\upharpoonleft P \upharpoonright ~,~ \upharpoonleft Q \upharpoonright)</math>
| <math>=\!</math>
| <math>(p, q)\!</math>
|- style="height:52px"
|
| align="left" | <math>\text{G2b}_{7}.\!</math>
| <math>\upharpoonleft \overline{P ~\cap~ Q} \upharpoonright</math>
| <math>=\!</math>
| <math>(\upharpoonleft P \upharpoonright ~ \upharpoonleft Q \upharpoonright)</math>
| <math>=\!</math>
| <math>(p q)\!</math>
|- style="height:52px"
|
| align="left" | <math>\text{G2b}_{8}.\!</math>
| <math>\upharpoonleft P ~\cap~ Q \upharpoonright</math>
| <math>=\!</math>
| <math>\upharpoonleft P \upharpoonright ~ \upharpoonleft Q \upharpoonright</math>
| <math>=\!</math>
| <math>p q\!</math>
|- style="height:52px"
|
| align="left" | <math>\text{G2b}_{9}.\!</math>
| <math>\upharpoonleft \overline{P ~+~ Q} \upharpoonright</math>
| <math>=\!</math>
| <math>((\upharpoonleft P \upharpoonright ~,~ \upharpoonleft Q \upharpoonright))</math>
| <math>=\!</math>
| <math>((p, q))\!</math>
|- style="height:52px"
|
| align="left" | <math>\text{G2b}_{10}.\!</math>
| <math>\upharpoonleft Q \upharpoonright</math>
| <math>=\!</math>
| <math>\upharpoonleft Q \upharpoonright</math>
| <math>=\!</math>
| <math>q\!</math>
|- style="height:52px"
|
| align="left" | <math>\text{G2b}_{11}.\!</math>
| <math>\upharpoonleft \overline{P ~\cap~ \overline{Q}} \upharpoonright</math>
| <math>=\!</math>
| <math>(\upharpoonleft P \upharpoonright (\upharpoonleft Q \upharpoonright))</math>
| <math>=\!</math>
| <math>(p (q))\!</math>
|- style="height:52px"
|
| align="left" | <math>\text{G2b}_{12}.\!</math>
| <math>\upharpoonleft P \upharpoonright</math>
| <math>=\!</math>
| <math>\upharpoonleft P \upharpoonright</math>
| <math>=\!</math>
| <math>p\!</math>
|- style="height:52px"
|
| align="left" | <math>\text{G2b}_{13}.\!</math>
| <math>\upharpoonleft \overline{\overline{P} ~\cap~ Q} \upharpoonright</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>((\upharpoonleft P \upharpoonright) \upharpoonleft Q \upharpoonright)</math>
| <math>((\upharpoonleft P \upharpoonright) \upharpoonleft Q \upharpoonright)</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>((p) q)\!</math>
| <math>((p) q)\!</math>
|- style="height:52px"
|- style="height:52px"
|
|
| align="left" | <math>\text{G2b}_{14}.\!</math>
| align="left" | <math>\text{G2b}_{14}.\!</math>
| <math>\upharpoonleft P ~\cup~ Q \upharpoonright</math>
| <math>\upharpoonleft P ~\cup~ Q \upharpoonright</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>((\upharpoonleft P \upharpoonright)(\upharpoonleft Q \upharpoonright))</math>
| <math>((\upharpoonleft P \upharpoonright)(\upharpoonleft Q \upharpoonright))</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>((p)(q))\!</math>
| <math>((p)(q))\!</math>
|- style="height:52px"
|- style="height:52px"
|
|
| align="left" | <math>\text{G2b}_{15}.\!</math>
| align="left" | <math>\text{G2b}_{15}.\!</math>
| <math>\upharpoonleft X \upharpoonright</math>
| <math>\upharpoonleft X \upharpoonright</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>((~))</math>
| <math>((~))</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>((~))</math>
| <math>((~))</math>
|}
|}
|}
|}
<br>
==Document History==
<pre>
| Subject: Inquiry Driven Systems : An Inquiry Into Inquiry
| Contact: Jon Awbrey
| Version: Draft 8.70
| Created: 23 Jun 1996
| Revised: 06 Jan 2002
| Advisor: M.A. Zohdy
| Setting: Oakland University, Rochester, Michigan, USA
| Excerpt: Section 1.3.10 (Recurring Themes)
| Excerpt: Subsections 1.3.10.8 - 1.3.10.13
</pre>
***