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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:
<pre>
{| 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>
<pre>
{| 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%"
If v, w C B,
|- style="height:50px; text-align:right"
| width="98%" | <math>\operatorname{Evaluation~Rule~1}</math>
then the following are equivalent:
| width="2%" |
|}
V1a. v = w.
|-
|
V1b. v <=> w.
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:50px"
V1c. (( v , w )).
| width="2%" style="border-top:1px solid black" |
</pre>
| 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" width="100%"
|- style="height:10px"
| width="2%" style="border-top:1px solid black" |
| width="14%" style="border-top:1px solid black" |
| 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" |
|}
|}
<pre>
<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"
</pre>
|
| <math>\text{then}\!</math>
<br>
| <math>\text{the following are equivalent:}\!</math>
|}
<pre>
|-
|
{| 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>\operatorname{D2a.}</math>
| width="80%" style="border-top:1px solid black" | <math>P ~=~ Q</math>
|- style="height:60px"
|
| <math>\operatorname{D2b.}</math>
| <math>\overset{X}{\underset{x}{\forall}}~ (x \in P ~\Leftrightarrow~ x \in Q)</math>
|}
|}
</pre>
<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:40px; text-align:center"
| width="80%" |
| width="20%" | <math>\operatorname{Definition~3}</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>
|}
|}
|-
|-
| 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="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>
| width="80%" style="border-top:1px solid black" | <math>f, g ~:~ X \to Y</math>
|- style="height:40px"
|- style="height:40px"
|
|
|- style="height:40px"
|- style="height:40px"
| 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>\operatorname{D2a.}</math>
| width="18%" style="border-top:1px solid black" | <math>\operatorname{D3a.}</math>
| width="80%" style="border-top:1px solid black" | <math>P ~=~ Q</math>
| width="80%" style="border-top:1px solid black" | <math>f ~=~ g</math>
|- style="height:60px"
|- style="height:60px"
|
|
| <math>\operatorname{D2b.}</math>
| <math>\operatorname{D3b.}</math>
| <math>\overset{X}{\underset{x}{\forall}}~ (x \in P ~\Leftrightarrow~ x \in Q)</math>
| <math>\overset{X}{\underset{x}{\forall}}~ (f(x) ~=~ g(x))</math>
|}
|}
|}
|}
|- style="height:40px; text-align:center"
|- style="height:40px; text-align:center"
| width="80%" |
| width="80%" |
| width="20%" | <math>\operatorname{Definition~3}</math>
| width="20%" | <math>\operatorname{Definition~4}</math>
|}
|}
|-
|-
| 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="18%" style="border-top:1px solid black" | <math>\text{If}\!</math>
| width="80%" style="border-top:1px solid black" | <math>f, g ~:~ X \to Y</math>
| width="80%" style="border-top:1px solid black" | <math>Q ~\subseteq~ X</math>
|- style="height:40px"
|- style="height:40px"
|
|
| <math>\text{then}\!</math>
| <math>\text{then}\!</math>
| <math>\text{the following are equivalent:}\!</math>
| <math>\text{the following are identical subsets of}~ X \times \underline\mathbb{B}:</math>
|}
|}
|-
|-
|- style="height:40px"
|- style="height:40px"
| 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>\operatorname{D3a.}</math>
| width="18%" style="border-top:1px solid black" | <math>\operatorname{D4a.}</math>
| width="80%" style="border-top:1px solid black" | <math>f ~=~ g</math>
| width="80%" style="border-top:1px solid black" | <math>\upharpoonleft Q \upharpoonright</math>
|- style="height:60px"
|- style="height:40px"
|
|
| <math>\operatorname{D3b.}</math>
| <math>\operatorname{D4b.}</math>
| <math>\overset{X}{\underset{x}{\forall}}~ (f(x) ~=~ g(x))</math>
| <math>\{ (x, y) \in X \times \underline\mathbb{B} ~:~ y ~=~ \downharpoonleft x \in Q \downharpoonright</math>
|}
|}
|}
|}
|- style="height:40px; text-align:center"
|- style="height:40px; text-align:center"
| width="80%" |
| width="80%" |
| width="20%" | <math>\operatorname{Definition~4}</math>
| width="20%" | <math>\operatorname{Definition~5}</math>
|}
|}
|-
|-
|
|
| <math>\text{then}\!</math>
| <math>\text{then}\!</math>
| <math>\text{the following are identical subsets of}~ X \times \underline\mathbb{B}:</math>
| <math>\text{the following are identical propositions} ~:~ X \to \underline\mathbb{B}</math>
|}
|}
|-
|-
|- style="height:40px"
|- style="height:40px"
| 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>\operatorname{D4a.}</math>
| width="18%" style="border-top:1px solid black" | <math>\operatorname{D5a.}</math>
| width="80%" style="border-top:1px solid black" | <math>\upharpoonleft Q \upharpoonright</math>
| width="80%" style="border-top:1px solid black" | <math>\upharpoonleft Q \upharpoonright</math>
|- style="height:40px"
|- style="height:40px"
|
|
| <math>\operatorname{D4b.}</math>
| <math>\operatorname{D5b.}</math>
| <math>\{ (x, y) \in X \times \underline\mathbb{B} ~:~ y ~=~ \downharpoonleft x \in Q \downharpoonright</math>
| <math>\downharpoonleft x \in Q \downharpoonright</math>
|}
|}
|}
|}
<br>
Given an indexed set of sentences, <math>s_j\!</math> for <math>j \in J,</math> it is possible to consider the logical conjunction of the corresponding propositions. Various notations for this concept are be useful in various contexts, a sufficient sample of which are recorded in Definition 6.
<br>
<br>
|- style="height:40px; text-align:center"
|- style="height:40px; text-align:center"
| width="80%" |
| width="80%" |
| width="20%" | <math>\operatorname{Definition~5}</math>
| width="20%" | <math>\operatorname{Definition~6}</math>
|}
|}
|-
|-
| 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="18%" style="border-top:1px solid black" | <math>\text{If}\!</math>
| width="80%" style="border-top:1px solid black" | <math>Q ~\subseteq~ X</math>
| width="80%" style="border-top:1px solid black" | <math>\text{each string}~ s_j, ~\text{as}~ j ~\text{ranges over the set}~ J,</math>
|- style="height:40px"
|- style="height:20px"
|
|
| <math>\text{is a sentence about things in the universe}~ X~</math>
|- style="height:60px"
|
|
| <math>\text{then}\!</math>
| <math>\text{then}\!</math>
| <math>\text{the following are identical propositions} ~:~ X \to \underline\mathbb{B}</math>
| <math>\text{the following are equivalent:}\!</math>
|}
|}
|-
|-
|
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:40px"
|- style="height:60px"
| 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>\operatorname{D5a.}</math>
| width="18%" style="border-top:1px solid black" | <math>\operatorname{D6a.}</math>
| width="80%" style="border-top:1px solid black" | <math>\upharpoonleft Q \upharpoonright</math>
| width="80%" style="border-top:1px solid black" | <math>\overset{J}{\underset{j}{\forall}}~ s_j</math>
|- style="height:40px"
|- style="height:60px"
|
|
| <math>\operatorname{D5b.}</math>
| <math>\operatorname{D6b.}</math>
| <math>\downharpoonleft x \in Q \downharpoonright</math>
| <math>\operatorname{Conj}_j^J s_j</math>
|}
|}
|}
|}
<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" 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%"
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:40px; text-align:center"
|- style="height:40px; text-align:center"
| width="80%" |
| width="80%" |
| width="20%" | <math>\operatorname{Definition~6}</math>
| width="20%" | <math>\operatorname{Definition~7}</math>
|}
|}
|-
|-
| 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="18%" style="border-top:1px solid black" | <math>\text{If}\!</math>
| width="80%" style="border-top:1px solid black" | <math>\text{each string}~ s_j, ~\text{as}~ j ~\text{ranges over the set}~ J,</math>
| width="80%" style="border-top:1px solid black" | <math>s, t ~\text{are sentences about things in the universe}~ X</math>
|- style="height:40px"
|- style="height:60px"
|
|
| <math>\text{then}\!</math>
| <math>\text{then}\!</math>
|
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:60px"
|- style="height:40px"
| 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>\operatorname{D6a.}</math>
| width="18%" style="border-top:1px solid black" | <math>\operatorname{D7a.}</math>
| width="80%" style="border-top:1px solid black" | <math>\overset{J}{\underset{j}{\forall}}~ s_j</math>
| width="80%" style="border-top:1px solid black" | <math>s ~\Leftrightarrow~ t</math>
|- style="height:60px"
|- style="height:40px"
|
|
| <math>\operatorname{D6b.}</math>
| <math>\operatorname{D7b.}</math>
| <math>\operatorname{Conj}_j^J s_j</math>
| <math>\downharpoonleft s \downharpoonright ~=~ \downharpoonleft t \downharpoonright</math>
|}
|}
|}
|}
|- style="height:40px; text-align:center"
|- style="height:40px; text-align:center"
| width="80%" |
| width="80%" |
| width="20%" | <math>\operatorname{Definition~7}</math>
| width="20%" style="border-left:1px solid black" | <math>\operatorname{Rule~5}</math>
|}
|}
|-
|-
| 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="18%" style="border-top:1px solid black" | <math>\text{If}\!</math>
| width="80%" style="border-top:1px solid black" | <math>s, t ~\text{are sentences about things in the universe}~ X</math>
| width="60%" style="border-top:1px solid black" | <math>P, Q ~\subseteq~ X</math>
| width="20%" style="border-top:1px solid black; border-left:1px solid black" |
|- style="height:40px"
|- style="height:40px"
|
|
| <math>\text{then}\!</math>
| <math>\text{then}\!</math>
| <math>\text{the following are equivalent:}\!</math>
| <math>\text{the following are equivalent:}\!</math>
| style="border-left:1px solid black" |
|}
|}
|-
|-
|- style="height:40px"
|- style="height:40px"
| 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>\operatorname{R5a.}</math>
| width="60%" style="border-top:1px solid black" | <math>P ~=~ Q</math>
| width="20%" style="border-top:1px solid black; border-left:1px solid black; text-align:center" | <math>\operatorname{R5a~:~D2a}</math>
|- style="height:20px"
|
| width="18%" style="border-top:1px solid black" | <math>\operatorname{R5a.}</math>
| width="60%" style="border-top:1px solid black" | <math>P ~=~ Q</math>
| width="20%" style="border-top:1px solid black; border-left:1px solid black; text-align:center" | <math>\operatorname{R5a~:~D2a}</math>
|- style="height:20px"
|
|
|
|
|
<br>
<br>
An application of Rule7anbsp;11 involves the recognition of an antecedent condition as a case under the Rule, that is, as a condition that matches one of the sentences in the Rule's chain of equivalents, and it requires the relegation of the other expressions to the production of a result. Thus, there is the choice of an initial expression that has to be checked on input for whether it fits the antecedent condition, and there is the choice of three types of output that are generated as a consequence, only one of which is generally needed at any given time. More often than not, though, a rule is applied in only a few of its possible ways. The usual antecedent and the usual consequents for Rule 11 can be distinguished in form and specialized in practice as follows:
An application of Rule 11 involves the recognition of an antecedent condition as a case under the Rule, that is, as a condition that matches one of the sentences in the Rule's chain of equivalents, and it requires the relegation of the other expressions to the production of a result. Thus, there is the choice of an initial expression that has to be checked on input for whether it fits the antecedent condition, and there is the choice of three types of output that are generated as a consequence, only one of which is generally needed at any given time. More often than not, though, a rule is applied in only a few of its possible ways. The usual antecedent and the usual consequents for Rule 11 can be distinguished in form and specialized in practice as follows:
{| align="center" cellpadding="8" width="90%"
{| align="center" cellpadding="8" width="90%"
|}
|}
Applying Rule 9, Rule 8, and the Logical Rules to the special case where <math>s \Leftrightarrow (X = Y),</math> one obtains the following general fact:
Applying Rule 9, Rule 8, and the Logical Rules to the special case where <math>s \Leftrightarrow (X = Y),</math> one obtains the following general Fact:
<br>
<br>
<pre>
{| 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%"
Fact 1
|
{| 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~1}</math>
|}
|-
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:50px"
| style="border-top:1px solid black; width:2%" |
| style="border-top:1px solid black; width:18%" | <math>\text{If}\!</math>
| style="border-top:1px solid black; width:60%" | <math>P, Q ~\subseteq~ X</math>
| style="border-top:1px solid black; border-left:1px solid black" |
|- style="height:50px"
|
| <math>\text{then}\!</math>
| <math>\text{the following are equivalent:}\!</math>
| style="border-left:1px solid black" |
|}
|-
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:60px"
///
| style="width:2%; border-top:1px solid black" |
| style="width:14%; border-top:1px solid black" | <math>\operatorname{F1a.}</math>
| style="width:64%; border-top:1px solid black" | <math>s \quad \Leftrightarrow \quad (P ~=~ Q)</math>
| style="width:20%; border-top:1px solid black; border-left:1px solid black; text-align:center" |
<math>\operatorname{F1a~:~R9a}</math>
</pre>
|- style="height:20px"
|
|
|
<pre>
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:40px"
|
| <math>\operatorname{F1b.}</math>
| <math>s \quad \Leftrightarrow \quad (\upharpoonleft P \upharpoonright ~=~ \upharpoonleft Q \upharpoonright)</math>
| style="border-left:1px solid black; text-align:center" |
<math>\operatorname{F1b~:~R9b}</math>
|- style="height:20px"
<=> the indicator functions of these sets are equal as functions
|
<=> the values of these functions are equal on all domain elements.
|
|
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:60px"
|
| <math>\operatorname{F1c.}</math>
| <math>s \quad \Leftrightarrow \quad \overset{X}{\underset{x}{\forall}}~ (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x))</math>
| style="border-left:1px solid black; text-align:center" |
<math>\operatorname{F1c~:~R9c}</math>
|- style="height:20px"
|
|
|
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:60px"
|
| <math>\operatorname{F1d.}</math>
| <math>s \quad \Leftrightarrow \quad \operatorname{Conj_x^X}~ (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x))</math>
| style="border-left:1px solid black; text-align:center" |
<p><math>\operatorname{F1d~:~R9d}</math></p>
<p><math>\operatorname{F1d~:~R8a}</math></p>
|- style="height:20px"
|
|
|
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:60px"
|
| <math>\operatorname{F1e.}</math>
| <math>\downharpoonleft s \downharpoonright \quad = \quad \downharpoonleft \operatorname{Conj_x^X}~ (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x)) \downharpoonright</math>
| style="border-left:1px solid black; text-align:center" |
<p><math>\operatorname{F1e~:~R8b}</math></p>
<p><math>\operatorname{F1e~:~\_\_?\_\_}</math></p>
|- style="height:20px"
|
|
|
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:60px"
|
| <math>\operatorname{F1f.}</math>
| <math>\downharpoonleft s \downharpoonright \quad = \quad \operatorname{Conj_x^X}~ \downharpoonleft (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x)) \downharpoonright</math>
| style="border-left:1px solid black; text-align:center" |
<p><math>\operatorname{F1f~:~\_\_?\_\_}</math></p>
<p><math>\operatorname{F1f~:~\_\_?\_\_}</math></p>
|- style="height:20px"
|
|
|
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:60px"
|
| <math>\operatorname{F1g.}</math>
| <math>\downharpoonleft s \downharpoonright \quad = \quad \operatorname{Conj_x^X}~ \underline{((}~ \upharpoonleft P \upharpoonright (x) ~,~ \upharpoonleft Q \upharpoonright (x) ~\underline{))}</math>
| style="border-left:1px solid black; text-align:center" |
<p><math>\operatorname{F1g~:~\_\_?\_\_}</math></p>
<p><math>\operatorname{F1g~:~\_\_?\_\_}</math></p>
|- style="height:20px"
|
|
|
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:40px"
|
| <math>\operatorname{F1h.}</math>
| <math>\downharpoonleft s \downharpoonright \quad = \quad \operatorname{Conj_x^X}~ \underline{((}~ \upharpoonleft P \upharpoonright ~,~ \upharpoonleft Q \upharpoonright ~\underline{))}^\$ (x)</math>
| style="border-left:1px solid black; text-align:center" |
<math>\operatorname{F1h~:~~\_\_?\_\_}</math>
|}
|}
<br>
=====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.
A classic way of showing that two sets are equal is to show that every element of the first belongs to the second and that every element of the second belongs to the first. The problem with this strategy is that one can exhaust a considerable amount of time trying to prove that two sets are equal before it occurs to one to look for a counterexample, that is, an element of the first that does not belong to the second or an element of the second that does not belong to the first, in cases where that is precisely what one ought to be seeking. It would be nice if there were a more balanced, impartial, or neutral way to go about this task, one that did not require such an undue commitment to either side, a technique that helps to pinpoint the counterexamples when they exist, and a method that keeps in mind the original relation of ''proving that'' and ''showing that'' to probing, testing, and seeing ''whether''.
A different way of seeing that two sets are equal, or of seeing whether two sets are equal, is based on the following observation:
{| align="center" cellpadding="8" style="text-align:center" width="90%"
| Two sets are equal as sets
|-
| <math>\iff</math>
|-
| The indicator functions of the two sets are equal as functions
|-
| <math>\iff</math>
|-
| The values of the two indicator functions are equal to each other on all domain elements.
|}
It is important to notice the hidden quantifier, of a universal kind, that lurks in all three equivalent statements but is only revealed in the last.
In making the next set of definitions and in using the corresponding terminology it is taken for granted that all of the references of signs are relative to a particular sign relation <math>L \subseteq O \times S \times I</math> that either remains to be specified or is already understood. Further, I continue to assume that <math>S = I,\!</math> in which case this set is called the ''syntactic domain'' of <math>L.\!</math>
In the following definitions, let <math>L \subseteq O \times S \times I,</math> let <math>S = I,\!</math> and let <math>x, y \in S.\!</math>
Recall the definition of <math>\operatorname{Con} (L),</math> the connotative component of a sign relation <math>L,\!</math> in the following form:
{| align="center" cellpadding="8" width="90%"
| <math>\operatorname{Con} (L) ~=~ L_{SI} ~=~ \{ (s, i) \in S \times I ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \}.</math>
|}
Equivalent expressions for this concept are recorded in Definition 8.
Equivalent expressions for this concept are recorded in Definition 12.
<br>
If R c OxSxI,
{| 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~8}</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>L ~\subseteq~ O \times S \times I</math>
|- style="height:40px"
|
| <math>\text{then}\!</math>
| <math>\text{the following are identical subsets of}~ S \times I \, :</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>\operatorname{D8a.}</math>
| width="80%" style="border-top:1px solid black" | <math>L_{SI}\!</math>
|- style="height:40px"
|
| <math>\operatorname{D8b.}</math>
| <math>\operatorname{Con}^L</math>
|- style="height:40px"
|
| <math>\operatorname{D8c.}</math>
| <math>\operatorname{Con}(L)</math>
|- style="height:40px"
|
| <math>\operatorname{D8d.}</math>
| <math>\operatorname{proj}_{SI}(L)</math>
|- style="height:40px"
|
| <math>\operatorname{D8e.}</math>
| <math>\{ (s, i) \in S \times I ~:~ (o, s, i) \in L ~\operatorname{for~some}~ o \in O \}</math>
|}
|}
<br>
'''Editing Note.''' Need a discussion of converse relations here. Perhaps it would work to introduce the operators that Peirce used for the converse of a dyadic relative <math>\ell,</math> namely, <math>K\ell ~=~ k\!\cdot\!\ell ~=~ \breve\ell.</math>
The dyadic relation <math>L_{IS}\!</math> that is the converse of the connotative relation <math>L_{SI}\!</math> can be defined directly in the following fashion:
{| align="center" cellpadding="8" width="90%"
| <math>\overset{\smile}{\operatorname{Con}(L)} ~=~ L_{IS} ~=~ \{ (i, s) \in I \times S ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \}.</math>
|}
A few of the many different expressions for this concept are recorded in Definition 9.
<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~9}</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>L ~\subseteq~ O \times S \times I</math>
|- style="height:40px"
|
| <math>\text{then}\!</math>
| <math>\text{the following are identical subsets of}~ I \times S \, :</math>
|}
|-
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:50px"
| width="2%" style="border-top:1px solid black" |
| width="18%" style="border-top:1px solid black" | <math>\operatorname{D9a.}</math>
| width="80%" style="border-top:1px solid black" | <math>L_{IS}\!</math>
|- style="height:50px"
|
| <math>\operatorname{D9b.}</math>
| <math>\overset{\smile}{L_{SI}}</math>
|- style="height:50px"
|
| <math>\operatorname{D9c.}</math>
| <math>\overset{\smile}{\operatorname{Con}^L}</math>
|- style="height:50px"
|
| <math>\operatorname{D9d.}</math>
| <math>\overset{\smile}{\operatorname{Con}(L)}</math>
|- style="height:50px"
|
| <math>\operatorname{D9e.}</math>
| <math>\operatorname{proj}_{IS}(L)</math>
|- style="height:50px"
|
| <math>\operatorname{D9f.}</math>
| <math>\operatorname{Conv}(\operatorname{Con}(L))</math>
|- style="height:50px"
|
| <math>\operatorname{D9g.}</math>
| <math>\{ (i, s) \in I \times S ~:~ (o, s, i) \in L ~\operatorname{for~some}~ o \in O \}</math>
|}
|}
<br>
Recall the definition of <math>\operatorname{Den} (L),</math> the denotative component of <math>L,\!</math> in the following form:
{| align="center" cellpadding="8" width="90%"
| <math>\operatorname{Den} (L) ~=~ L_{OS} ~=~ \{ (o, s) \in O \times S ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \}.</math>
|}
Equivalent expressions for this concept are recorded in Definition 10.
<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~10}</math>
|}
|-
If R c OxSxI,
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
then the following are identical subsets of SxI:
|- 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>L ~\subseteq~ O \times S \times I</math>
|- style="height:40px"
|
| <math>\text{then}\!</math>
| <math>\text{the following are identical subsets of}~ O \times S \, :</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>\operatorname{D10a.}</math>
| width="80%" style="border-top:1px solid black" | <math>L_{OS}\!</math>
|- style="height:40px"
|
| <math>\operatorname{D10b.}</math>
| <math>\operatorname{Den}^L</math>
|- style="height:40px"
|
| <math>\operatorname{D10c.}</math>
| <math>\operatorname{Den}(L)</math>
|- style="height:40px"
|
| <math>\operatorname{D10d.}</math>
| <math>\operatorname{proj}_{OS}(L)</math>
|- style="height:40px"
|
| <math>\operatorname{D10e.}</math>
| <math>\{ (o, s) \in O \times S ~:~ (o, s, i) \in L ~\operatorname{for~some}~ i \in I \}</math>
|}
|}
<br>
The dyadic relation <math>L_{SO}\!</math> that is the converse of the denotative relation <math>L_{OS}\!</math> can be defined directly in the following fashion:
{| align="center" cellpadding="8" width="90%"
| <math>\overset{\smile}{\operatorname{Den}(L)} ~=~ L_{SO} ~=~ \{ (s, o) \in S \times O ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \}.</math>
|}
A few of the many different expressions for this concept are recorded in Definition 11.
<br>
2. If R is a sign relation such that RSI is a SER on S = I,
{| 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%"
then the semiotic equation "x =R y" means that <x, y> C RSI.
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:40px; text-align:center"
| width="80%" |
| width="20%" | <math>\operatorname{Definition~11}</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>L ~\subseteq~ O \times S \times I</math>
|- style="height:40px"
|
| <math>\text{then}\!</math>
| <math>\text{the following are identical subsets of}~ S \times O \, :</math>
|}
|-
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:50px"
| width="2%" style="border-top:1px solid black" |
| width="18%" style="border-top:1px solid black" | <math>\operatorname{D11a.}</math>
| width="80%" style="border-top:1px solid black" | <math>L_{SO}\!</math>
|- style="height:50px"
|
| <math>\operatorname{D11b.}</math>
| <math>\overset{\smile}{L_{OS}}</math>
|- style="height:50px"
|
| <math>\operatorname{D11c.}</math>
| <math>\overset{\smile}{\operatorname{Den}^L}</math>
|- style="height:50px"
|
| <math>\operatorname{D11d.}</math>
| <math>\overset{\smile}{\operatorname{Den}(L)}</math>
|- style="height:50px"
|
| <math>\operatorname{D11e.}</math>
| <math>\operatorname{proj}_{SO}(L)</math>
|- style="height:50px"
|
| <math>\operatorname{D11f.}</math>
| <math>\operatorname{Conv}(\operatorname{Den}(L))</math>
|- style="height:50px"
|
| <math>\operatorname{D11g.}</math>
| <math>\{ (s, o) \in S \times O ~:~ (o, s, i) \in L ~\operatorname{for~some}~ i \in I \}</math>
|}
|}
<br>
The ''denotation of <math>x\!</math> in <math>L,\!</math>'' written <math>\operatorname{Den}(L, x),</math> is defined as follows:
{| align="center" cellpadding="8" width="90%"
| <math>\operatorname{Den}(L, x) ~=~ \{ o \in O ~:~ (o, x) \in \operatorname{Den}(L) \}.</math>
|}
In other words:
{| align="center" cellpadding="8" width="90%"
| <math>\operatorname{Den}(L, x) ~=~ \{ o \in O ~:~ (o, x, i) \in L ~\text{for some}~ i \in I \}.</math>
|}
Equivalent expressions for this concept are recorded in Definition 12.
<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~12}</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%"
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:48px; text-align:right"
|- style="height:40px"
| 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="18%" style="border-top:1px solid black" | <math>\text{If}\!</math>
| width="80%" style="border-top:1px solid black" |
| width="80%" style="border-top:1px solid black" | <math>L ~\subseteq~ O \times S \times I</math>
<math>s ~\text{is a sentence about things in the universe X}</math>
|- style="height:40px"
|- style="height:48px"
|
|
| <math>\text{and}\!</math>
| <math>\text{and}\!</math>
| <math>p ~\text{is a proposition} ~:~ X \to \underline\mathbb{B}</math>
| <math>x ~\in~ S</math>
|- style="height:40px"
|- style="height:48px"
|
|
| <math>\text{then}\!</math>
| <math>\text{then}\!</math>
| <math>\text{the following equations hold:}\!</math>
| <math>\text{the following are identical subsets of}~ O \, :</math>
|}
|}
|-
|-
|
|
{| 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:40px"
| 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="18%" style="border-top:1px solid black" | <math>\operatorname{D12a.}</math>
| width="20%" style="border-top:1px solid black" |
| width="80%" style="border-top:1px solid black" | <math>L_{OS} \cdot x</math>
<math>\downharpoonleft \operatorname{false} \downharpoonright</math>
|- style="height:40px"
| width="5%" style="border-top:1px solid black" | <math>=\!</math>
|
| width="20%" style="border-top:1px solid black" | <math>(~)</math>
| <math>\operatorname{D12b.}</math>
| <math>\operatorname{Den}^L \cdot x</math>
| width="30%" style="border-top:1px solid black" |
|- style="height:40px"
<math>\underline{0} ~:~ X \to \underline\mathbb{B}</math>
|
|- style="height:52px"
| <math>\operatorname{D12c.}</math>
| <math>\operatorname{Den}^L |_x</math>
|- style="height:40px"
|
| <math>\operatorname{D12d.}</math>
| <math>\operatorname{Den}^L (-, x)</math>
|- style="height:40px"
|
|
| <math>\operatorname{D12e.}</math>
| <math>\downharpoonleft \operatorname{not}~ s \downharpoonright</math>
| <math>\operatorname{Den}(L, x)</math>
|- style="height:40px"
|- style="height:52px"
|
|
| <math>\operatorname{D12f.}</math>
| <math>\downharpoonleft s \downharpoonright</math>
| <math>\operatorname{Den}(L) \cdot x</math>
|- style="height:40px"
|- style="height:52px"
|
|
| <math>\operatorname{D12g.}</math>
| <math>\downharpoonleft \operatorname{true} \downharpoonright</math>
| <math>\{ o \in O ~:~ (o, x) \in \operatorname{Den}(L) \}</math>
| <math>=\!</math>
|- style="height:40px"
| <math>((~))</math>
|
| <math>=\!</math>
| <math>\operatorname{D12h.}</math>
| <math>\underline{1} ~:~ X \to \underline\mathbb{B}</math>
| <math>\{ o \in O ~:~ (o, x, i) \in L ~\operatorname{for~some}~ i \in I \}</math>
|}
|}
|}
|}
<br>
<br>
===Geometric Translation Rule 1===
Signs are ''equiferent'' if they refer to all and only the same objects, that is, if they have exactly the same denotations. In other language for the same relation, signs are said to be ''denotatively equivalent'' or ''referentially equivalent'', but it is probably best to check whether the extension of this concept over the syntactic domain is really a genuine equivalence relation before jumping to the conclusions that are implied by these latter terms.
To define the ''equiference'' of signs in terms of their denotations, one says that ''<math>x\!</math> is equiferent to <math>y\!</math> under <math>L,\!</math>'' and writes <math>x ~\overset{L}{=}~ y,\!</math> to mean that <math>\operatorname{Den}(L, x) = \operatorname{Den}(L, y).</math> Taken in extension, this notion of a relation between signs induces an ''equiference relation'' on the syntactic domain.
For each sign relation <math>L,\!</math> this yields a binary relation <math>\operatorname{Der}(L) \subseteq S \times I</math> that is defined as follows:
{| align="center" cellpadding="8" width="90%"
| <math>\operatorname{Der}(L) ~=~ Der^L ~=~ \{ (x, y) \in S \times I ~:~ \operatorname{Den}(L, x) = \operatorname{Den}(L, y) \}.</math>
|}
These definitions and notations are recorded in the following display.
<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:40px; text-align:center"
| width="98%" | <math>\text{Geometric Translation Rule 1}\!</math>
| width="80%" |
| width="20%" | <math>\operatorname{Definition~13}</math>
|}
|}
|-
|-
|
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:48px"
|- style="height:40px"
| 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="18%" style="border-top:1px solid black" | <math>\text{If}\!</math>
| width="80%" style="border-top:1px solid black" | <math>Q \subseteq X</math>
| width="80%" style="border-top:1px solid black" | <math>L ~\subseteq~ O \times S \times I</math>
|- style="height:48px"
|- style="height:40px"
|
|
| <math>\text{and}\!</math>
| <math>\text{then}\!</math>
| <math>p ~:~ X \to \underline\mathbb{B}</math>
| <math>\text{the following are identical subsets of}~ S \times I \, :</math>
|- style="height:48px"
|}
|
|-
| <math>\text{such that:}\!</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>\operatorname{D13a.}</math>
| width="80%" style="border-top:1px solid black" | <math>\operatorname{Der}^L</math>
|- style="height:40px"
|
|
|- style="height:48px"
| <math>\operatorname{D13b.}</math>
| <math>\operatorname{Der}(L)</math>
|- style="height:40px"
|
|
| <math>\text{G1a.}\!</math>
| <math>\operatorname{D13c.}</math>
| <math>\upharpoonleft Q \upharpoonright ~=~ p</math>
| <math>\{ (x, y) \in S \times I ~:~ \operatorname{Den}^L|_x = \operatorname{Den}^L|_y \}</math>
|- style="height:48px"
|- style="height:40px"
|
|
| <math>\text{then}\!</math>
| <math>\operatorname{D13d.}</math>
| <math>\text{the following equations hold:}\!</math>
| <math>\{ (x, y) \in S \times I ~:~ \operatorname{Den}(L, x) = \operatorname{Den}(L, y) \}</math>
|}
|}
|}
|}
<br>
<br>
The relation <math>\operatorname{Der}(L)</math> is defined and the notation <math>x ~\overset{L}{=}~ y</math> is meaningful in every situation where the corresponding denotation operator <math>\operatorname{Den}(-,-)</math> makes sense, but it remains to check whether this relation enjoys the properties of an equivalence relation.
<ol style="list-style-type:decimal">
<li>
<p>'''Reflexive property.'''</p>
<p>Is it true that <math>x ~\overset{L}{=}~ x</math> for every <math>x \in S = I</math>?</p>
<p> By definition, <math>x ~\overset{L}{=}~ x</math> if and only if <math>\operatorname{Den}(L, x) = \operatorname{Den}(L, x).</math></p>
<p>Thus, the reflexive property holds in any setting where the denotations <math>\operatorname{Den}(L, x)</math> are defined for all signs <math>x\!</math> in the syntactic domain of <math>R.\!</math></p></li>
<li>
<p>'''Symmetric property.'''</p>
<p>Does <math>x ~\overset{L}{=}~ y</math> imply <math>y ~\overset{L}{=}~ x</math> for all <math>x, y \in S</math>?</p>
===Logical Translation Rule 2===
<p>In effect, does <math>\operatorname{Den}(L, x) = \operatorname{Den}(L, y)</math> imply <math>\operatorname{Den}(L, y) = \operatorname{Den}(L, x)</math> for all signs <math>x\!</math> and <math>y\!</math> in the syntactic domain <math>S\!</math>?</p>
<br>
<p>Yes, so long as the sets <math>\operatorname{Den}(L, x)</math> and <math>\operatorname{Den}(L, y)</math> are well-defined, a fact which is already being assumed.</p></li>
<li>
<p>'''Transitive property.'''</p>
<p>Does <math>x ~\overset{L}{=}~ y</math> and <math>y ~\overset{L}{=}~ z</math> imply <math>x ~\overset{L}{=}~ z</math> for all <math>x, y, z \in S</math>?</p>
<p>To belabor the point, does <math>\operatorname{Den}(L, x) = \operatorname{Den}(L, y)</math> and <math>\operatorname{Den}(L, y) = \operatorname{Den}(L, z)</math> imply <math>\operatorname{Den}(L, x) = \operatorname{Den}(L, z)</math> for all <math>x, y, z \in S</math>?</p>
<p>Yes, once again, under the stated conditions.</p></li>
</ol>
It should be clear at this point that any question about the equiference of signs reduces to a question about the equality of sets, specifically, the sets that are indexed by these signs. As a result, so long as these sets are well-defined, the issue of whether equiference relations induce equivalence relations on their syntactic domains is almost as trivial as it initially appears.
Taken in its set-theoretic extension, a relation of equiference induces a ''denotative equivalence relation'' (DER) on its syntactic domain <math>S = I.\!</math> This leads to the formation of ''denotative equivalence classes'' (DECs), ''denotative partitions'' (DEPs), and ''denotative equations'' (DEQs) on the syntactic domain. But what does it mean for signs to be equiferent?
Notice that this is not the same thing as being ''semiotically equivalent'', in the sense of belonging to a single ''semiotic equivalence class'' (SEC), falling into the same part of a ''semiotic partition'' (SEP), or having a ''semiotic equation'' (SEQ) between them. It is only when very felicitous conditions obtain, establishing a concord between the denotative and the connotative components of a sign relation, that these two ideas coalesce.
In general, there is no necessity that the equiference of signs, that is, their denotational equivalence or their referential equivalence, induces the same equivalence relation on the syntactic domain as that defined by their semiotic equivalence, even though this state of accord seems like an especially desirable situation. This makes it necessary to find a distinctive nomenclature for these structures, for which I adopt the term ''denotative equivalence relations'' (DERs). In their train they bring the allied structures of ''denotative equivalence classes'' (DECs) and ''denotative partitions'' (DEPs), while the corresponding statements of ''denotative equations'' (DEQs) are expressible in the form <math>x ~\overset{L}{=}~ y.</math>
The uses of the equal sign for denoting equations or equivalences are recalled and extended in the following ways:
# If <math>E\!</math> is an arbitrary equivalence relation, then the equation <math>x =_E y\!</math> means that <math>(x, y) \in E.</math>
# If <math>L\!</math> is a sign relation such that <math>L_{SI}\!</math> is a SER on <math>S = I,\!</math> then the semiotic equation <math>x =_L y\!</math> means that <math>(x, y) \in L_{SI}.</math>
# If <math>L\!</math> is a sign relation such that <math>F\!</math> is its DER on <math>S = I,\!</math> then the denotative equation <math>x ~\overset{L}{=}~ y</math> means that <math>(x, y) \in F,</math> in other words, that <math>\operatorname{Den}(L, x) = \operatorname{Den}(L, y).</math>
The use of square brackets for denoting equivalence classes is recalled and extended in the following ways:
# If <math>E\!</math> is an arbitrary equivalence relation, then <math>[x]_E\!</math> is the equivalence class of <math>x\!</math> under <math>E.\!</math>
# If <math>L\!</math> is a sign relation such that <math>\operatorname{Con}(L)</math> is a SER on <math>S = I,\!</math> then <math>[x]_L\!</math> is the SEC of <math>x\!</math> under <math>\operatorname{Con}(L).</math>
# If <math>L\!</math> is a sign relation such that <math>\operatorname{Der}(L)</math> is a DER on <math>S = I,\!</math> then <math>[x]^L\!</math> is the DEC of <math>x\!</math> under <math>\operatorname{Der}(L).</math>
By applying the form of Fact 1 to the special case where <math>X = \operatorname{Den}(L, x)</math> and <math>Y = \operatorname{Den}(L, y),</math> one obtains the following facts.
<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="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%"
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:48px; text-align:right"
|- style="height:50px; text-align:center"
| width="98%" | <math>\text{Logical Translation Rule 2}\!</math>
| style="width:80%" |
| style="width:20%; border-left:1px solid black" | <math>\operatorname{Fact~2.1}</math>
|}
|}
|-
|-
|
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:48px"
|- style="height:50px"
| 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="10%" style="border-top:1px solid black" | <math>\text{If}\!</math>
| width="84%" style="border-top:1px solid black" |
| width="68%" style="border-top:1px solid black" | <math>L ~\subseteq~ O \times S \times I</math>
<math>s, t ~\text{are sentences about things in the universe}~ X</math>
| width="20%" style="border-top:1px solid black; border-left:1px solid black" |
|- style="height:48px"
|- style="height:50px"
|
|
| <math>\text{and}\!</math>
| <math>\text{then}\!</math>
| <math>p, q ~\text{are propositions} ~:~ X \to \underline\mathbb{B}</math>
| <math>\text{the following are identical subsets of}~ S \times I :</math>
|- style="height:48px"
| style="border-left:1px solid black" |
|}
|-
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:60px"
| width="2%" style="border-top:1px solid black" |
| width="10%" style="border-top:1px solid black" | <math>\operatorname{F2.1a.}</math>
| width="68%" style="border-top:1px solid black" | <math>\operatorname{Der}^L</math>
| width="20%" style="border-top:1px solid black; border-left:1px solid black; text-align:center" | <math>\operatorname{F2.1a~:~D13a}</math>
|- style="height:20px"
|
|
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|
|
|
| <math>\text{L2a.}\!</math>
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
| <math>\downharpoonleft s \downharpoonright ~=~ p \quad \operatorname{and} \quad \downharpoonleft t \downharpoonright ~=~ q</math>
|- style="height:40px"
|- style="height:48px"
|
|
| <math>\text{then}\!</math>
| valign="top" | <math>\operatorname{F2.1b.}</math>
| valign="top" | <math>\operatorname{Der}(L)</math>
| style="border-left:1px solid black; text-align:center" |
<math>\operatorname{F2.1b~:~D13b}</math>
|- style="height:20px"
| width="32%" style="border-top:1px solid black" |
<math>\downharpoonleft \operatorname{false} \downharpoonright</math>
| width="4%" style="border-top:1px solid black" | <math>=\!</math>
|- style="height:52px"
|
|
|
|
|
|
| align="left" | <math>\text{L2b}_{3}.\!</math>
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:60px"
|- style="height:52px"
|
|
| align="left" | <math>\text{L2b}_{4}.\!</math>
| valign="top" | <math>\operatorname{F2.1c.}</math>
| <math>\downharpoonleft s ~\operatorname{and~not}~ t \downharpoonright</math>
| valign="top" |
| <math>=\!</math>
<math>\begin{array}{ll}
\{ & (x, y) \in S \times I ~: \\
& \operatorname{Den}(L, x) ~=~ \operatorname{Den}(L, y) \\
\} & \\
|- style="height:52px"
\end{array}</math>
| style="border-left:1px solid black; text-align:center" |
<p><math>\operatorname{F2.1c~:~D13c}</math></p>
<p><math>\operatorname{F2.1c~:~R9a}</math></p>
|- style="height:20px"
|
|
|
|
|
|
| align="left" | <math>\text{L2b}_{7}.\!</math>
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
| <math>\downharpoonleft \operatorname{not~both}~ s ~\operatorname{and}~ t \downharpoonright</math>
|- style="height:60px"
| <math>=\!</math>
|
| valign="top" | <math>\operatorname{F2.1d.}</math>
| <math>=\!</math>
| valign="top" |
<math>\begin{array}{ll}
|- style="height:52px"
\{ & (x, y) \in S \times I ~: \\
& \upharpoonleft \operatorname{Den}(L, x) \upharpoonright ~=~ \upharpoonleft \operatorname{Den}(L, y) \upharpoonright \\
\} & \\
\end{array}</math>
| style="border-left:1px solid black; text-align:center" |
<math>\operatorname{F2.1d~:~R9b}</math>
|- style="height:20px"
|
|
|
|
|
| align="left" | <math>\text{L2b}_{9}.\!</math>
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
| <math>\downharpoonleft s ~\operatorname{is~equivalent~to}~ t \downharpoonright</math>
|- style="height:60px"
| <math>=\!</math>
|
| valign="top" | <math>\operatorname{F2.1e.}</math>
| <math>=\!</math>
| valign="top" |
<math>\begin{array}{ll}
|- style="height:52px"
\{ & (x, y) \in S \times I ~: \\
& \overset{O}{\underset{o}{\forall}}~ \upharpoonleft \operatorname{Den}(L, x) \upharpoonright (o) ~=~ \upharpoonleft \operatorname{Den}(L, y) \upharpoonright (o) \\
\} & \\
\end{array}</math>
| style="border-left:1px solid black; text-align:center" |
<math>\operatorname{F2.1e~:~R9c}</math>
|- style="height:20px"
|
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| align="left" | <math>\text{L2b}_{12}.\!</math>
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
| <math>\downharpoonleft s \downharpoonright</math>
|- style="height:60px"
| <math>=\!</math>
|
| valign="top" | <math>\operatorname{F2.1f.}</math>
| <math>=\!</math>
| valign="top" |
<math>\begin{array}{ll}
|- style="height:52px"
\{ & (x, y) \in S \times I ~: \\
& \underset{o \in O}{\operatorname{Conj}}~ \upharpoonleft \operatorname{Den}(L, x) \upharpoonright (o) ~=~ \upharpoonleft \operatorname{Den}(L, y) \upharpoonright (o) \\
\} & \\
\end{array}</math>
| style="border-left:1px solid black; text-align:center" |
<math>\operatorname{F2.1f~:~R9d}</math>
|- style="height:20px"
|
|
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| align="left" | <math>\text{L2b}_{14}.\!</math>
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:60px"
|- style="height:52px"
|
|
| align="left" | <math>\text{L2b}_{15}.\!</math>
| valign="top" | <math>\operatorname{F2.1g.}</math>
| <math>\downharpoonleft \operatorname{true} \downharpoonright</math>
| valign="top" |
| <math>=\!</math>
<math>\begin{array}{ll}
| <math>((~))</math>
\{ & (x, y) \in S \times I ~: \\
| <math>=\!</math>
& \underset{o \in O}{\operatorname{Conj}}~ \underline{((}~ \upharpoonleft \operatorname{Den}(L, x) \upharpoonright (o) ~,~ \upharpoonleft \operatorname{Den}(L, y) \upharpoonright (o) ~\underline{))} \\
| <math>((~))</math>
\} & \\
\end{array}</math>
| style="border-left:1px solid black; text-align:center" |
<math>\operatorname{F2.1g~:~R9e}</math>
|- style="height:20px"
|
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:60px"
|
| valign="top" | <math>\operatorname{F2.1h.}</math>
| valign="top" |
<math>\begin{array}{ll}
\{ & (x, y) \in S \times I ~: \\
& \underset{o \in O}{\operatorname{Conj}}~ \underline{((}~ \upharpoonleft \operatorname{Den}(L, x) \upharpoonright ~,~ \upharpoonleft \operatorname{Den}(L, y) \upharpoonright ~\underline{))}^\$ (o) \\
\} & \\
\end{array}</math>
| style="border-left:1px solid black; text-align:center" |
<p><math>\operatorname{F2.1h~:~R9f}</math></p>
<p><math>\operatorname{F2.1h~:~D12e}</math></p>
|- style="height:20px"
|
|
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:60px"
|
| valign="top" | <math>\operatorname{F2.1i.}</math>
| valign="top" |
<math>\begin{array}{ll}
\{ & (x, y) \in S \times I ~: \\
& \underset{o \in O}{\operatorname{Conj}}~ \underline{((}~ \upharpoonleft L_{OS} \cdot x \upharpoonright ~,~ \upharpoonleft L_{OS} \cdot y \upharpoonright ~\underline{))}^\$ (o) \\
\} & \\
\end{array}</math>
| style="border-left:1px solid black; text-align:center" |
<math>\operatorname{F2.1i~:~D12a}</math>
|}
|}
|}
|}
<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:50px; text-align:center"
| width="98%" | <math>\text{Geometric Translation Rule 2}\!</math>
| style="width:80%" |
| style="width:20%; border-left:1px solid black" | <math>\operatorname{Fact~2.2}</math>
|}
|}
|-
|-
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|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:48px"
|- style="height:50px"
| 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="12%" style="border-top:1px solid black" | <math>\text{If}\!</math>
| width="84%" style="border-top:1px solid black" | <math>P, Q \subseteq X</math>
| width="66%" style="border-top:1px solid black" | <math>L ~\subseteq~ O \times S \times I</math>
|- style="height:48px"
| width="20%" style="border-top:1px solid black; border-left:1px solid black" |
|- style="height:50px"
|
|- style="height:48px"
|
|
| <math>\text{then}\!</math>
| <math>\text{then}\!</math>
| <math>\text{the following equations hold:}\!</math>
| <math>\text{the following are equivalent:}\!</math>
| style="border-left:1px solid black" |
|}
|}
|-
|-
|
|
{| 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:10px"
| 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{G2b}_{0}.\!</math>
| width="12%" style="border-top:1px solid black" |
| width="32%" 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" |
| width="4%" style="border-top:1px solid black" | <math>=\!</math>
|- style="height:100px"
|- style="height:52px"
|
|
| align="left" | <math>\text{G2b}_{1}.\!</math>
| valign="top" | <math>\operatorname{F2.2a.}</math>
| <math>\upharpoonleft \overline{P} ~\cap~ \overline{Q} \upharpoonright</math>
| valign="top" |
<math>\begin{array}{cccl}
\operatorname{Der}^L
| <math>=\!</math>
& = & \{ & (x, y) \in S \times I ~: \\
| <math>(p)(q)\!</math>
& & & \begin{array}{ccl}
|- style="height:52px"
\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"
|
|
| align="left" | <math>\text{G2b}_{2}.\!</math>
| valign="top" | <math>\operatorname{F2.2b.}</math>
| <math>\upharpoonleft \overline{P} ~\cap~ Q \upharpoonright</math>
| valign="top" |
<math>\begin{array}{ccccl}
\upharpoonleft \operatorname{Der}^L \upharpoonright
| <math>=\!</math>
& = & \upharpoonleft & \{ & (x, y) \in S \times I ~: \\
| <math>(p) q\!</math>
& & & & \begin{array}{ccl}
|- style="height:52px"
\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"
|
|
| align="left" | <math>\text{G2b}_{3}.\!</math>
| valign="top" | <math>\operatorname{F2.2c.}</math>
| <math>\upharpoonleft \overline{P} \upharpoonright</math>
| valign="top" |
<math>\begin{array}{cccl}
\upharpoonleft \operatorname{Der}^L \upharpoonright
| <math>=\!</math>
& = & \{ & (x, y, z) \in S \times I \times \underline\mathbb{B} ~:~ z = \\
| <math>(p)\!</math>
& & & \begin{array}{cccl}
|- style="height:52px"
\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"
|
|
| align="left" | <math>\text{G2b}_{4}.\!</math>
| valign="top" | <math>\operatorname{F2.2d.}</math>
| <math>\upharpoonleft P ~\cap~ \overline{Q} \upharpoonright</math>
| valign="top" |
<math>\begin{array}{cccl}
\upharpoonleft \operatorname{Der}^L \upharpoonright
| <math>=\!</math>
& = & \{ & (x, y, z) \in S \times I \times \underline\mathbb{B} ~:~ z = \\
| <math>p (q)\!</math>
& & & \begin{array}{cccl}
|- style="height:52px"
\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"
|
|
| align="left" | <math>\text{G2b}_{5}.\!</math>
| valign="top" | <math>\operatorname{F2.2e.}</math>
| <math>\upharpoonleft \overline{Q} \upharpoonright</math>
| valign="top" |
<math>\begin{array}{cccl}
\upharpoonleft \operatorname{Der}^L \upharpoonright
| <math>=\!</math>
& = & \{ & (x, y, z) \in S \times I \times \underline\mathbb{B} ~:~ z = \\
| <math>(q)\!</math>
& & & \begin{array}{ccl}
|- style="height:52px"
\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"
|
|
| align="left" | <math>\text{G2b}_{6}.\!</math>
| valign="top" | <math>\operatorname{F2.2f.}</math>
| <math>\upharpoonleft P ~+~ Q \upharpoonright</math>
| valign="top" |
| <math>=\!</math>
<math>\begin{array}{cccl}
| <math>(\upharpoonleft P \upharpoonright ~,~ \upharpoonleft Q \upharpoonright)</math>
\upharpoonleft \operatorname{Der}^L \upharpoonright
| <math>=\!</math>
& = & \{ & (x, y, z) \in S \times I \times \underline\mathbb{B} ~:~ z = \\
| <math>(p, q)\!</math>
& & & \begin{array}{cll}
|- style="height:52px"
\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" width="100%"
|- 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="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>\text{then}\!</math>
| <math>\upharpoonleft \overline{P ~\cap~ Q} \upharpoonright</math>
| <math>\text{the following are equivalent:}\!</math>
| <math>=\!</math>
| style="border-left:1px solid black" |
| <math>(\upharpoonleft P \upharpoonright ~ \upharpoonleft Q \upharpoonright)</math>
|}
| <math>=\!</math>
|-
| <math>(p q)\!</math>
|
|- style="height:52px"
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:10px"
| 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"
|
|
| align="left" | <math>\text{G2b}_{8}.\!</math>
| valign="top" | <math>\operatorname{F2.3a.}</math>
| <math>\upharpoonleft P ~\cap~ Q \upharpoonright</math>
| valign="top" |
<math>\begin{array}{cccl}
\operatorname{Der}^L
| <math>=\!</math>
& = & \{ & (x, y) \in S \times I ~: \\
| <math>p q\!</math>
& & & \begin{array}{ccl}
|- style="height:52px"
\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"
|
|
| align="left" | <math>\text{G2b}_{9}.\!</math>
| valign="top" | <math>\operatorname{F2.3b.}</math>
| <math>\upharpoonleft \overline{P ~+~ Q} \upharpoonright</math>
| valign="top" |
<math>\begin{array}{ccccl}
\upharpoonleft \operatorname{Der}^L \upharpoonright (x, y)
| <math>=\!</math>
& = & \downharpoonleft & \underset{o \in O}{\operatorname{Conj}} \\
| <math>((p, q))\!</math>
& & & & \begin{array}{cl}
|- style="height:52px"
( & \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"
|
|
| align="left" | <math>\text{G2b}_{10}.\!</math>
| valign="top" | <math>\operatorname{F2.3c.}</math>
| <math>\upharpoonleft Q \upharpoonright</math>
| valign="top" |
<math>\begin{array}{ccccl}
\upharpoonleft \operatorname{Der}^L \upharpoonright (x, y)
| <math>=\!</math>
& = & \underset{o \in O}{\operatorname{Conj}} \\
| <math>q\!</math>
& & & \begin{array}{ccl}
|- style="height:52px"
\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"
|
|
| align="left" | <math>\text{G2b}_{11}.\!</math>
| valign="top" | <math>\operatorname{F2.3d.}</math>
| <math>\upharpoonleft \overline{P ~\cap~ \overline{Q}} \upharpoonright</math>
| valign="top" |
<math>\begin{array}{ccccl}
\upharpoonleft \operatorname{Der}^L \upharpoonright (x, y)
| <math>=\!</math>
& = & \underset{o \in O}{\operatorname{Conj}} \\
| <math>(p (q))\!</math>
& & & \begin{array}{ccl}
|- style="height:52px"
\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"
|
|
| align="left" | <math>\text{G2b}_{12}.\!</math>
| valign="top" | <math>\operatorname{F2.3e.}</math>
| <math>\upharpoonleft P \upharpoonright</math>
| valign="top" |
| <math>=\!</math>
<math>\begin{array}{ccccl}
\upharpoonleft \operatorname{Der}^L \upharpoonright (x, y)
| <math>=\!</math>
& = & \underset{o \in O}{\operatorname{Conj}} \\
| <math>p\!</math>
& & & \begin{array}{cl}
|- style="height:52px"
\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"
|
|
| align="left" | <math>\text{G2b}_{13}.\!</math>
| valign="top" | <math>\operatorname{F2.3f.}</math>
| <math>\upharpoonleft \overline{\overline{P} ~\cap~ Q} \upharpoonright</math>
| valign="top" |
| <math>=\!</math>
<math>\begin{array}{ccccl}
| <math>((\upharpoonleft P \upharpoonright) \upharpoonleft Q \upharpoonright)</math>
\upharpoonleft \operatorname{Der}^L \upharpoonright (x, y)
| <math>=\!</math>
& = & \underset{o \in O}{\operatorname{Conj}} \\
| <math>((p) q)\!</math>
& & & \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>
|}
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>
|}
Compare this with the usual form of composition, typically notated <math>P \circ Q</math> and defined as follows:
{| 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>
{| 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{Logical 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>s ~\text{is a sentence about things in the universe X}</math>
|- style="height:48px"
|
| <math>\text{and}\!</math>
| <math>p ~\text{is a proposition} ~:~ X \to \underline\mathbb{B}</math>
|- style="height:48px"
|
| <math>\text{such that:}\!</math>
|
|- style="height:48px"
|
| <math>\text{L1a.}\!</math>
| <math>\downharpoonleft s \downharpoonright ~=~ 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{L1b}_{00}.\!</math>
| width="20%" style="border-top:1px solid black" |
<math>\downharpoonleft \operatorname{false} \downharpoonright</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{G2b}_{14}.\!</math>
| align="left" | <math>\text{L1b}_{01}.\!</math>
| <math>\upharpoonleft P ~\cup~ Q \upharpoonright</math>
| <math>\downharpoonleft \operatorname{not}~ s \downharpoonright</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>((\upharpoonleft P \upharpoonright)(\upharpoonleft Q \upharpoonright))</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{G2b}_{15}.\!</math>
| align="left" | <math>\text{L1b}_{10}.\!</math>
| <math>\upharpoonleft X \upharpoonright</math>
| <math>\downharpoonleft s \downharpoonright</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>((~))</math>
| <math>\downharpoonleft s \downharpoonright</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>((~))</math>
| <math>p ~:~ X \to \underline\mathbb{B}</math>
|}
|- style="height:52px"
|}
|
| align="left" | <math>\text{L1b}_{11}.\!</math>
<br>
| <math>\downharpoonleft \operatorname{true} \downharpoonright</math>
| <math>=\!</math>
==Document History==
| <math>((~))</math>
| <math>=\!</math>
| <math>\underline{1} ~:~ X \to \underline\mathbb{B}</math>
|}
|}
<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"
|
| <math>\text{such that:}\!</math>
|
|- style="height:48px"
|
| <math>\text{G1a.}\!</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"
|
| align="left" | <math>\text{G1b}_{01}.\!</math>
| <math>\upharpoonleft {}^{_\sim} Q \upharpoonright</math>
| <math>=\!</math>
| <math>(\upharpoonleft Q \upharpoonright)</math>
| <math>=\!</math>
| <math>(p) ~:~ X \to \underline\mathbb{B}</math>
|- style="height:52px"
|
| align="left" | <math>\text{G1b}_{10}.\!</math>
| <math>\upharpoonleft Q \upharpoonright</math>
| <math>=\!</math>
| <math>\upharpoonleft Q \upharpoonright</math>
| <math>=\!</math>
| <math>p ~:~ X \to \underline\mathbb{B}</math>
|- style="height:52px"
|
| align="left" | <math>\text{G1b}_{11}.\!</math>
| <math>\upharpoonleft X \upharpoonright</math>
| <math>=\!</math>
| <math>((~))</math>
| <math>=\!</math>
| <math>\underline{1} ~:~ X \to \underline\mathbb{B}</math>
|}
|}
<br>
===Logical Translation Rule 2===
<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{Logical Translation Rule 2}\!</math>
| width="2%" |
|}
|-
|
{| 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>
|
|- style="height:48px"
|
| <math>\text{L2a.}\!</math>
| <math>\downharpoonleft s \downharpoonright ~=~ p \quad \operatorname{and} \quad \downharpoonleft t \downharpoonright ~=~ q</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="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"
|
| align="left" | <math>\text{L2b}_{1}.\!</math>
| <math>\downharpoonleft \operatorname{neither}~ s ~\operatorname{nor}~ t \downharpoonright</math>
| <math>=\!</math>
| <math>(\downharpoonleft s \downharpoonright)(\downharpoonleft t \downharpoonright)</math>
| <math>=\!</math>
| <math>(p)(q)\!</math>
|- style="height:52px"
|
| align="left" | <math>\text{L2b}_{2}.\!</math>
| <math>\downharpoonleft \operatorname{not}~ s ~\operatorname{but}~ t \downharpoonright</math>
| <math>=\!</math>
| <math>(\downharpoonleft s \downharpoonright) \downharpoonleft t \downharpoonright</math>
| <math>=\!</math>
| <math>(p) q\!</math>
|- style="height:52px"
|
| align="left" | <math>\text{L2b}_{3}.\!</math>
| <math>\downharpoonleft \operatorname{not}~ s \downharpoonright</math>
| <math>=\!</math>
| <math>(\downharpoonleft s \downharpoonright)</math>
| <math>=\!</math>
| <math>(p)\!</math>
|- style="height:52px"
|
| 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"
|
| 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"
|
| align="left" | <math>\text{L2b}_{6}.\!</math>
| <math>\downharpoonleft s ~\operatorname{or}~ t, ~\operatorname{not~both} \downharpoonright</math>
| <math>=\!</math>
| <math>(\downharpoonleft s \downharpoonright ~,~ \downharpoonleft t \downharpoonright)</math>
| <math>=\!</math>
| <math>(p, q)\!</math>
|- style="height:52px"
|
| 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>
|- style="height:52px"
|
| align="left" | <math>\text{L2b}_{8}.\!</math>
| <math>\downharpoonleft s ~\operatorname{and}~ t \downharpoonright</math>
| <math>=\!</math>
| <math>\downharpoonleft s \downharpoonright ~ \downharpoonleft t \downharpoonright</math>
| <math>=\!</math>
| <math>p q\!</math>
|- style="height:52px"
|
| align="left" | <math>\text{L2b}_{9}.\!</math>
| <math>\downharpoonleft s ~\operatorname{is~equivalent~to}~ t \downharpoonright</math>
| <math>=\!</math>
| <math>((\downharpoonleft s \downharpoonright ~,~ \downharpoonleft t \downharpoonright))</math>
| <math>=\!</math>
| <math>((p, q))\!</math>
|- style="height:52px"
|
| align="left" | <math>\text{L2b}_{10}.\!</math>
| <math>\downharpoonleft t \downharpoonright</math>
| <math>=\!</math>
| <math>\downharpoonleft t \downharpoonright</math>
| <math>=\!</math>
| <math>q\!</math>
|- style="height:52px"
|
| align="left" | <math>\text{L2b}_{11}.\!</math>
| <math>\downharpoonleft s ~\operatorname{implies}~ t \downharpoonright</math>
| <math>=\!</math>
| <math>(\downharpoonleft s \downharpoonright (\downharpoonleft t \downharpoonright))</math>
| <math>=\!</math>
| <math>(p (q))\!</math>
|- style="height:52px"
|
| align="left" | <math>\text{L2b}_{12}.\!</math>
| <math>\downharpoonleft s \downharpoonright</math>
| <math>=\!</math>
| <math>\downharpoonleft s \downharpoonright</math>
| <math>=\!</math>
| <math>p\!</math>
|- style="height:52px"
|
| align="left" | <math>\text{L2b}_{13}.\!</math>
| <math>\downharpoonleft s ~\operatorname{is~implied~by}~ t \downharpoonright</math>
| <math>=\!</math>
| <math>((\downharpoonleft s \downharpoonright) \downharpoonleft t \downharpoonright)</math>
| <math>=\!</math>
| <math>((p) q)\!</math>
|- style="height:52px"
|
| align="left" | <math>\text{L2b}_{14}.\!</math>
| <math>\downharpoonleft s ~\operatorname{or}~ t \downharpoonright</math>
| <math>=\!</math>
| <math>((\downharpoonleft s \downharpoonright)(\downharpoonleft t \downharpoonright))</math>
| <math>=\!</math>
| <math>((p)(q))\!</math>
|- style="height:52px"
|
| align="left" | <math>\text{L2b}_{15}.\!</math>
| <math>\downharpoonleft \operatorname{true} \downharpoonright</math>
| <math>=\!</math>
| <math>((~))</math>
| <math>=\!</math>
| <math>((~))</math>
|}
|}
<br>
===Geometric Translation Rule 2===
<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 2}\!</math>
| width="2%" |
|}
|-
|
{| 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"
|
| <math>\text{G2a.}\!</math>
| <math>\upharpoonleft P \upharpoonright ~=~ p \quad \operatorname{and} \quad \upharpoonleft Q \upharpoonright ~=~ q</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="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>(p)(q)\!</math>
|- style="height:52px"
|
| align="left" | <math>\text{G2b}_{2}.\!</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>((\upharpoonleft P \upharpoonright) \upharpoonleft Q \upharpoonright)</math>
| <math>=\!</math>
| <math>((p) q)\!</math>
|- style="height:52px"
|
| align="left" | <math>\text{G2b}_{14}.\!</math>
| <math>\upharpoonleft P ~\cup~ 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}_{15}.\!</math>
| <math>\upharpoonleft X \upharpoonright</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>
***