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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
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===6.26. Differential Logic and Directed Graphs===
This section extracts the graph-theoretic content of the previous series of Tables, using it to illustrate the logical description, or ''intensional representation'' (IR), of graphs and digraphs. Where the points of graphs and digraphs are described by conjunctions of logical features, the edges and arcs are described by differential features, possibly in conjunction with the ordinary features that depict their points of origin and destination.
Because of the formal confound that I mentioned earlier, anchored in the essentially accidential circumstance that <math>\text{A}\!</math> and <math>\text{B}\!</math> and I all use the same proper names for <math>\text{A}\!</math> and <math>\text{B},\!</math> I cannot analyze the denotative aspects of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> without analyzing the corresponding parts of my own denotative conduct, namely, those actions of mine that involve parallel uses of the signs <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \text{B} {}^{\prime\prime}.\!</math> However, it will soon become obvious that I have not prepared the discussion at this point with the technical means it needs to carry out this task in any meaningful way. In order to do this, it would be necessary to consider the common world <math>W =\!</math> <math>\{ \text{A}, \text{B}, {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \},\!</math> of the two sign relations as an initially homogeneous set and then to provide explicit logical features that mark the distinction between objects and signs. For the sake of simplicity, I am putting these considerations off to a subsequent round of analysis. On this pass, the denotative sections of each analytic scheme are filled in with what amount to inert proxies for the actual analyses to be carried out later.
===6.27. Differential Logic and Group Operations===
This section isolates the group-theoretic content of the previous series of Tables, using it to illustrate the following principle: When a geometric object, like a graph or digraph, is given an intensional representation (IR) in terms of a set of logical properties or propositional features, then many of the transformational aspects of that object can be represented in the ''differential extension'' of that IR.
One approach to the study of a temporal system is through the paradigm or principle of ''sequential inference''.
Principle of ''sequential inference''. A sequential inference rule is operative in any setting where the following list of ingredients can be identified.
# There is a frame of observation that affords, arranges for, or determines a sequence of observations on a system.
# There is an observable property or a logical feature <math>x\!</math> that can be true or false of the system at any given moment <math>t\!</math> of observation.
# There is a pair <math>(t, t')\!</math> of succeeding moments of observation.
Relative to a setting of this kind, the rules of sequential inference are exemplified by the schematism shown in Table 41.
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>\text{Table 69.} ~~ \text{Schematism of Sequential Inference}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Initial Premiss}\!</math>
| width="33%" | <math>\text{Differential Premiss}\!</math>
| width="33%" | <math>\text{Inferred Sequel}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
~x~ ~\operatorname{at}~ t
\\[4pt]
~x~ ~\operatorname{at}~ t
\\[4pt]
(x) ~\operatorname{at}~ t
\\[4pt]
(x) ~\operatorname{at}~ t
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
~\operatorname{d}x~ ~\operatorname{at}~ t
\\[4pt]
(\operatorname{d}x) ~\operatorname{at}~ t
\\[4pt]
~\operatorname{d}x~ ~\operatorname{at}~ t
\\[4pt]
(\operatorname{d}x) ~\operatorname{at}~ t
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(x) ~\operatorname{at}~ t'
\\[4pt]
~x~ ~\operatorname{at}~ t'
\\[4pt]
~x~ ~\operatorname{at}~ t'
\\[4pt]
(x) ~\operatorname{at}~ t'
\end{matrix}</math>
|}
<br>
It might be thought that a notion of real time <math>(t \in \mathbb{R})\!</math> is needed at this point to fund the account of sequential processes. From a logical point of view, however, I think it will be found that it is precisely out of such data that the notion of time has to be constructed.
The symbol <math>{}^{\backprime\backprime} \ominus\!\!- {}^{\prime\prime},</math> read ''thus'', ''then'', or ''yields'', can be used to mark sequential inferences, allowing for expressions like <math>x \land \operatorname{d}x \ominus\!\!-~ (x).\!</math> In each case, a suitable context of temporal moments <math>(t, t')\!</math> is understood to underlie the inference.
A ''sequential inference constraint'' is a logical condition that applies to a temporal system, providing information about the kinds of sequential inference that apply to the system in a hopefully large number of situations. Typically, a sequential inference constraint is formulated in intensional terms and expressed by means of a collection of sequential inference rules or schemata that tell what sequential inferences apply to the system in particular situations. Since it has the status of logical theory about an empirical system, a sequential inference constraint is subject to being reformulated in terms of its set-theoretic extension, and it can be established as existing in the customary sort of dual relationship with this extension. Logically, it determines, and, empirically, it is determined by the corresponding set of ''sequential inference triples'', the <math>(x, y, z)\!</math> such that <math>x \land y \ominus\!\!-~ z.\!</math> The set-theoretic extension of a sequential inference constraint is thus a triadic relation, generically notated as <math>\ominus,\!</math> where <math>\ominus \subseteq X \times \operatorname{d}X \times X\!</math> is defined as follows.
{| align="center" cellspacing="8" width="90%"
| <math>\ominus ~=~ \{ (x, y, z) \in X \times \operatorname{d}X \times X : x \land y \ominus\!\!-~ z \}.\!</math>
|}
Using the appropriate isomorphisms, or recognizing how, in terms of the information given, that each of several descriptions is tantamount to the same object, the triadic relation <math>\ominus \subseteq X \times \operatorname{d}X \times X\!</math> constituted by a sequential inference constraint can be interpreted as a proposition <math>\ominus : X \times \operatorname{d}X \times X \to \mathbb{B}\!</math> about sequential inference triples, and thus as a map <math>\ominus : \operatorname{d}X \to (X \times X \to \mathbb{B})\!</math> from the space <math>\operatorname{d}X\!</math> of differential states to the space of propositions about transitions in <math>X.\!</math>
<br>
'''Question.''' Group Actions? <math>r : \operatorname{d}X \to (X \to X)\!</math>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" |
<math>\text{Table 70.1} ~~ \text{Group Representation} ~ \operatorname{Rep}^\text{A} (V_4)\!</math>
|- style="background:#f0f0ff"
| width="16%" | <math>\begin{matrix} \text{Abstract} \\ \text{Element} \end{matrix}</math>
| width="36%" | <math>\begin{matrix} \text{Logical} \\ \text{Element} \end{matrix}</math>
| width="16%" | <math>\begin{matrix} \text{Active} \\ \text{List} \end{matrix}</math>
| width="16%" | <math>\begin{matrix} \text{Active} \\ \text{Term} \end{matrix}</math>
| width="16%" | <math>\begin{matrix} \text{Genetic} \\ \text{Element} \end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
1
\\[4pt]
r
\\[4pt]
s
\\[4pt]
t
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(\operatorname{d}\underline{\underline{\text{a}}})
(\operatorname{d}\underline{\underline{\text{b}}})
(\operatorname{d}\underline{\underline{\text{i}}})
(\operatorname{d}\underline{\underline{\text{u}}})
\\[4pt]
~\operatorname{d}\underline{\underline{\text{a}}}~
(\operatorname{d}\underline{\underline{\text{b}}})
~\operatorname{d}\underline{\underline{\text{i}}}~
(\operatorname{d}\underline{\underline{\text{u}}})
\\[4pt]
(\operatorname{d}\underline{\underline{\text{a}}})
~\operatorname{d}\underline{\underline{\text{b}}}~
(\operatorname{d}\underline{\underline{\text{i}}})
~\operatorname{d}\underline{\underline{\text{u}}}~
\\[4pt]
~\operatorname{d}\underline{\underline{\text{a}}}~
~\operatorname{d}\underline{\underline{\text{b}}}~
~\operatorname{d}\underline{\underline{\text{i}}}~
~\operatorname{d}\underline{\underline{\text{u}}}~
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\langle \operatorname{d}! \rangle
\\[4pt]
\langle
\operatorname{d}\underline{\underline{\text{a}}} ~
\operatorname{d}\underline{\underline{\text{i}}}
\rangle
\\[4pt]
\langle
\operatorname{d}\underline{\underline{\text{b}}} ~
\operatorname{d}\underline{\underline{\text{u}}}
\rangle
\\[4pt]
\langle \operatorname{d}* \rangle
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\operatorname{d}!
\\[4pt]
\operatorname{d}\underline{\underline{\text{a}}} \cdot
\operatorname{d}\underline{\underline{\text{i}}} ~ !
\\[4pt]
\operatorname{d}\underline{\underline{\text{b}}} \cdot
\operatorname{d}\underline{\underline{\text{u}}} ~ !
\\[4pt]
\operatorname{d}*
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
1
\\[4pt]
\operatorname{d}_{\text{ai}}
\\[4pt]
\operatorname{d}_{\text{bu}}
\\[4pt]
\operatorname{d}_{\text{ai}} * \operatorname{d}_{\text{bu}}
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" |
<math>\text{Table 70.2} ~~ \text{Group Representation} ~ \operatorname{Rep}^\text{B} (V_4)\!</math>
|- style="background:#f0f0ff"
| width="16%" | <math>\begin{matrix} \text{Abstract} \\ \text{Element} \end{matrix}</math>
| width="36%" | <math>\begin{matrix} \text{Logical} \\ \text{Element} \end{matrix}</math>
| width="16%" | <math>\begin{matrix} \text{Active} \\ \text{List} \end{matrix}</math>
| width="16%" | <math>\begin{matrix} \text{Active} \\ \text{Term} \end{matrix}</math>
| width="16%" | <math>\begin{matrix} \text{Genetic} \\ \text{Element} \end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
1
\\[4pt]
r
\\[4pt]
s
\\[4pt]
t
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(\operatorname{d}\underline{\underline{\text{a}}})
(\operatorname{d}\underline{\underline{\text{b}}})
(\operatorname{d}\underline{\underline{\text{i}}})
(\operatorname{d}\underline{\underline{\text{u}}})
\\[4pt]
~\operatorname{d}\underline{\underline{\text{a}}}~
(\operatorname{d}\underline{\underline{\text{b}}})
(\operatorname{d}\underline{\underline{\text{i}}})
~\operatorname{d}\underline{\underline{\text{u}}}~
\\[4pt]
(\operatorname{d}\underline{\underline{\text{a}}})
~\operatorname{d}\underline{\underline{\text{b}}}~
~\operatorname{d}\underline{\underline{\text{i}}}~
(\operatorname{d}\underline{\underline{\text{u}}})
\\[4pt]
~\operatorname{d}\underline{\underline{\text{a}}}~
~\operatorname{d}\underline{\underline{\text{b}}}~
~\operatorname{d}\underline{\underline{\text{i}}}~
~\operatorname{d}\underline{\underline{\text{u}}}~
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\langle \operatorname{d}! \rangle
\\[4pt]
\langle
\operatorname{d}\underline{\underline{\text{a}}} ~
\operatorname{d}\underline{\underline{\text{u}}}
\rangle
\\[4pt]
\langle
\operatorname{d}\underline{\underline{\text{b}}} ~
\operatorname{d}\underline{\underline{\text{i}}}
\rangle
\\[4pt]
\langle \operatorname{d}* \rangle
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\operatorname{d}!
\\[4pt]
\operatorname{d}\underline{\underline{\text{a}}} \cdot
\operatorname{d}\underline{\underline{\text{u}}} ~ !
\\[4pt]
\operatorname{d}\underline{\underline{\text{b}}} \cdot
\operatorname{d}\underline{\underline{\text{i}}} ~ !
\\[4pt]
\operatorname{d}*
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
1
\\[4pt]
\operatorname{d}_{\text{au}}
\\[4pt]
\operatorname{d}_{\text{bi}}
\\[4pt]
\operatorname{d}_{\text{au}} * \operatorname{d}_{\text{bi}}
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" |
<math>{\text{Table 70.3} ~~ \text{Group Representation} ~ \operatorname{Rep}^\text{C} (V_4)}\!</math>
|- style="background:#f0f0ff"
| width="16%" | <math>\begin{matrix} \text{Abstract} \\ \text{Element} \end{matrix}</math>
| width="36%" | <math>\begin{matrix} \text{Logical} \\ \text{Element} \end{matrix}</math>
| width="16%" | <math>\begin{matrix} \text{Active} \\ \text{List} \end{matrix}</math>
| width="16%" | <math>\begin{matrix} \text{Active} \\ \text{Term} \end{matrix}</math>
| width="16%" | <math>\begin{matrix} \text{Genetic} \\ \text{Element} \end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
1
\\[4pt]
r
\\[4pt]
s
\\[4pt]
t
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(\operatorname{d}\text{m})
(\operatorname{d}\text{n})
\\[4pt]
~\operatorname{d}\text{m}~
(\operatorname{d}\text{n})
\\[4pt]
(\operatorname{d}\text{m})
~\operatorname{d}\text{n}~
\\[4pt]
~\operatorname{d}\text{m}~
~\operatorname{d}\text{n}~
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\langle\operatorname{d}!\rangle
\\[4pt]
\langle\operatorname{d}\text{m}\rangle
\\[4pt]
\langle\operatorname{d}\text{n}\rangle
\\[4pt]
\langle\operatorname{d}*\rangle
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\operatorname{d}!
\\[4pt]
\operatorname{d}\text{m}!
\\[4pt]
\operatorname{d}\text{n}!
\\[4pt]
\operatorname{d}*
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
1
\\[4pt]
\operatorname{d}_{\text{m}}
\\[4pt]
\operatorname{d}_{\text{n}}
\\[4pt]
\operatorname{d}_{\text{m}} * \operatorname{d}_{\text{n}}
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" |
<math>\text{Table 71.1} ~~ \text{The Differential Group} ~ G = V_4\!</math>
|- style="background:#f0f0ff"
| width="16%" | <math>\begin{matrix} \text{Abstract} \\ \text{Element} \end{matrix}</math>
| width="36%" | <math>\begin{matrix} \text{Logical} \\ \text{Element} \end{matrix}</math>
| width="16%" | <math>\begin{matrix} \text{Active} \\ \text{List} \end{matrix}</math>
| width="16%" | <math>\begin{matrix} \text{Active} \\ \text{Term} \end{matrix}</math>
| width="16%" | <math>\begin{matrix} \text{Genetic} \\ \text{Element} \end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
1
\\[4pt]
r
\\[4pt]
s
\\[4pt]
t
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(\operatorname{d}\text{m})
(\operatorname{d}\text{n})
\\[4pt]
~\operatorname{d}\text{m}~
(\operatorname{d}\text{n})
\\[4pt]
(\operatorname{d}\text{m})
~\operatorname{d}\text{n}~
\\[4pt]
~\operatorname{d}\text{m}~
~\operatorname{d}\text{n}~
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\langle\operatorname{d}!\rangle
\\[4pt]
\langle\operatorname{d}\text{m}\rangle
\\[4pt]
\langle\operatorname{d}\text{n}\rangle
\\[4pt]
\langle\operatorname{d}*\rangle
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\operatorname{d}!
\\[4pt]
\operatorname{d}\text{m}!
\\[4pt]
\operatorname{d}\text{n}!
\\[4pt]
\operatorname{d}*
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
1
\\[4pt]
\operatorname{d}_{\text{m}}
\\[4pt]
\operatorname{d}_{\text{n}}
\\[4pt]
\operatorname{d}_{\text{m}} * \operatorname{d}_{\text{n}}
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" |
<math>\text{Table 71.2} ~~ \text{Cosets of} ~ G_\text{m} ~ \text{in} ~ G\!</math>
|- style="background:#f0f0ff"
| width="25%" | <math>\text{Group Coset}\!</math>
| width="25%" | <math>\text{Logical Coset}~\!</math>
| width="25%" | <math>\text{Logical Element}\!</math>
| width="25%" | <math>\text{Group Element}\!</math>
|-
| <math>G_\text{m}\!</math>
| <math>(\operatorname{d}\text{m})\!</math>
|
<math>\begin{matrix}
(\operatorname{d}\text{m})(\operatorname{d}\text{n})
\\[4pt]
(\operatorname{d}\text{m})~\operatorname{d}\text{n}~
\end{matrix}</math>
|
<math>\begin{matrix}
1
\\[4pt]
\operatorname{d}_\text{n}
\end{matrix}</math>
|-
| <math>G_\text{m} * \operatorname{d}_\text{m}\!</math>
| <math>\operatorname{d}\text{m}\!</math>
|
<math>\begin{matrix}
~\operatorname{d}\text{m}~(\operatorname{d}\text{n})
\\[4pt]
~\operatorname{d}\text{m}~~\operatorname{d}\text{n}~
\end{matrix}</math>
|
<math>\begin{matrix}
\operatorname{d}_\text{m}
\\[4pt]
\operatorname{d}_\text{n} * \operatorname{d}_\text{m}
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" |
<math>\text{Table 71.3} ~~ \text{Cosets of} ~ G_\text{n} ~ \text{in} ~ G\!</math>
|- style="background:#f0f0ff"
| width="25%" | <math>\text{Group Coset}\!</math>
| width="25%" | <math>\text{Logical Coset}~\!</math>
| width="25%" | <math>\text{Logical Element}\!</math>
| width="25%" | <math>\text{Group Element}\!</math>
|-
| <math>G_\text{n}\!</math>
| <math>({\operatorname{d}\text{n})}\!</math>
|
<math>\begin{matrix}
(\operatorname{d}\text{m})(\operatorname{d}\text{n})
\\[4pt]
~\operatorname{d}\text{m}~(\operatorname{d}\text{n})
\end{matrix}</math>
|
<math>\begin{matrix}
1
\\[4pt]
\operatorname{d}_\text{m}
\end{matrix}</math>
|-
| <math>G_\text{n} * \operatorname{d}_\text{n}\!</math>
| <math>\operatorname{d}\text{n}\!</math>
|
<math>\begin{matrix}
(\operatorname{d}\text{m})~\operatorname{d}\text{n}~
\\[4pt]
~\operatorname{d}\text{m}~~\operatorname{d}\text{n}~
\end{matrix}</math>
|
<math>\begin{matrix}
\operatorname{d}_\text{n}
\\[4pt]
\operatorname{d}_\text{m} * \operatorname{d}_\text{n}
\end{matrix}</math>
|}
<br>
===6.28. The Bridge : From Obstruction to Opportunity===
There are many reasons for using intensional representations to describe formal objects, especially as the size and complexity of these objects grows beyond the bounds of finite information capacities to represent them in practical terms. This is extremely pertinent to the progress of the present discussion. As often happens, when a top-down investigation of complex families of formal objects actually succeeds in arriving at examples that are simple enough to contemplate in extensional terms, it can be difficult to see the relation of such impoverished examples to the cases of original interest, all of them typically having infinite cardinality and indefinite complexity. In short, once a discussion is brought down to the level of its smallest cases it can be nearly impossible to bring it back up to the level of its intended application. Without invoking intensional representations of sign relations there is little hope that this discussion can rise far beyond its present level, eternally elaborating the subtleties of cases as elementary as <math>L(\text{A})\!</math> and <math>L(\text{B}).\!</math>
There are many obstacles to building this bridge, but if these forms of obstruction are understood in the proper fashion, it is possible to use them as stepping stones, to capitalize on their redoubtable structures, and to convert their recalcitrant materials into a formal calculus that can serve the aims and means of instruction.
This approach requires me to consider a chain of relationships that connects signs, names, concepts, properties, sets, and objects, along with various ways that these classes of entities have been viewed at different periods in the development of mathematical logic.
I would like to begin by giving an “impressionistic capsule history” of the relevant developments in mathematical logic, admittedly as viewed from a certain perspective, but hoping to allow room for alternative perspectives to have their way and present themselves in their own best light.
Variant 1. The human mind, boggling at the many to many relation between objects and signs that it finds in the world as soon as it begins to reflect on its own reasoning process, hits upon the strategy of interposing a realm of intermediate nodes between objects and signs, and looking through this medium for ways to factor the original relation into simpler components.
Variant 2. At the beginning of logic, the human mind, as soon as it begins to reflect on its own reasoning process, boggles at the many to many relation between objects and signs that it finds itself conducting through the world.
There are two methods for attempting to disentangle this confusion that are generally tried, the first more rarely, the second quite frequently, though apparently in opposite proportion to their respective chances of actual success. In order to describe the rationales of these methods I need to introduce a number of technical concepts.
Suppose <math>P\!</math> and <math>Q\!</math> are dyadic relations, with <math>P \subseteq X \times Y\!</math> and <math>Q \subseteq Y \times Z.\!</math> Then the ''contension'' of <math>P\!</math> and <math>Q\!</math> is a triadic relation <math>R \subseteq X \times Y \times Z\!</math> that is notated as <math>R = P\!\!\And\!\!Q</math> and defined as follows.
{| align="center" cellspacing="8" width="90%"
| <math>P\!\!\And\!\!Q ~=~ \{ (x, y, z) \in X \times Y \times Z : (x, y) \in P ~\text{and}~ (y, z) \in Q \}.</math>
|}
In other words, <math>P\!\!\And\!\!Q</math> is the intersection of the ''inverse projections'' <math>P' = \operatorname{Pr}_{12}^{-1}(P)\!</math> and <math>Q' = \operatorname{Pr}_{23}^{-1}(Q),\!</math> which are defined as follows:
{| align="center" cellspacing="8" width="90%"
|
<math>\begin{matrix}
\operatorname{Pr}_{12}^{-1}(P) & = & P \times Z & = & \{ (x, y, z) \in X \times Y \times Z : (x, y) \in P \}.
\\[4pt]
\operatorname{Pr}_{23}^{-1}(Q) & = & X \times Q & = & \{ (x, y, z) \in X \times Y \times Z : (y, z) \in Q \}.
\end{matrix}</math>
|}
Inverse projections are often referred to as ''extensions'', in spite of the conflict this creates with the ''extensions'' of concepts and terms.
One of the standard turns of phrase that finds use in this setting, not only for translating between extensional representations and intensional representations, but for converting both into computational forms, is to associate any set <math>S\!</math> contained in a space <math>X\!</math> with two other types of formal objects: (1) a logical proposition <math>p_S\!</math> known as the characteristic, indicative, or selective proposition of <math>S,\!</math> and (2) a boolean-valued function <math>f_S : X \to \mathbb{B}\!</math> known as the characteristic, indicative, or selective function of <math>S.\!</math>
Strictly speaking, the logical entity <math>p_S\!</math> is the intensional representation of the tribe, presiding at the highest level of abstraction, while <math>f_S\!</math> and <math>S\!</math> are its more concrete extensional representations, rendering its concept in functional and geometric materials, respectively. Whenever it is possible to do so without confusion, I try to use identical or similar names for the corresponding objects and species of each type, and I generally ignore the distinctions that otherwise set them apart. For instance, in moving toward computational settings, <math>f_S\!</math> makes the best computational proxy for <math>p_S,\!</math> so I commonly refer to the mapping <math>f_S : X \to \mathbb{B}\!</math> as a proposition on <math>X.\!</math>
Regarded as logical models, the elements of the contension <math>P\!\!\And\!\!Q</math> satisfy the proposition referred to as the ''conjunction of extensions'' <math>P^\prime\!</math> and <math>Q^\prime.\!</math>
Next, the ''composition'' of <math>P\!</math> and <math>Q\!</math> is a dyadic relation <math>R' \subseteq X \times Z\!</math> that is notated as <math>R' = P \circ Q\!</math> and defined as follows.
{| align="center" cellspacing="8" width="90%"
| <math>P \circ Q ~=~ \operatorname{Pr}_{13} (P\!\!\And\!\!Q) ~=~ \{ (x, z) \in X \times Z : (x, y, z) \in P\!\!\And\!\!Q \}.</math>
|}
In other words:
{| align="center" cellspacing="8" width="90%"
| <math>P \circ Q ~=~ \{ (x, z) \in X \times Z : (x, y) \in P ~\text{and}~ (y, z) \in Q \}.\!</math>
|}
'''Begin Fragment.''' I will have to find my notes on this.
Using these notions, the customary methods for disentangling a many-to-many relation can be explained as follows:
#
#
In the logic of the ancients, the many-to-one relation of things to general names ...
'''End Fragment.'''
In early approaches to mathematical logic, from Leibniz to Peirce and Frege, one ordinarily spoke of the extensions and intensions of concepts.
Typically, one starts a work of bridge-building by casting a thin line across the intervening gap, using this expediency to conduct a slightly more substantial linkage over the rift, and then proceeding through a train of successors to draw increasingly stronger connections between the opposing shores until a load-bearing framework can be established. There is an analogue of this operation that fits the current situation, and this is something I can do this by taking up the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> already introduced in extensional terms, and re-describing the abstract features of their structures in intensional terms.
This would be the ideal plan. But bridging the “tensions”, “ex-” and “in-”, that subsist within the forms of representation is not as easy as that. In order to convey the importance of the task and provide a motivation for carrying it out, I will plot a chain of relationships that stretches from signs, names, and concepts to properties, sets, and objects.
As a way of resolving the discerned “tensions”, posed here to fall into “ex-” and “in-” kinds, the strategy just described affords a way of approaching the problem that is less like a bridge than a pole vault, taking its pivot on a fixed set of narrowly circumscribed sign relations to make a transit from extensional to intensional outlooks on their form. With time and reflection, the logical depth of the supposed distinction, the “pretension” of maintaining a couple of separate but equal tensions in isolation from each other, does not withstand a persistent probing. Accordingly, the gulf between the two realms can always be fathomed by a finitely informed creature, in fact, by the very form of interpreter that created the fault in the first place. Consequently, converting the form of a transient vault into the substance of a usable bridge requires in adjunction only that initially pliable and ultimately tensile sorts of connecting lines be conducted along the tracery of the vault until the work of castling the gap can begin.
In the pragmatic theory of signs, the word ''representation'' is a technical term that is synonymous with the word ''sign'', in other words, it applies to an entity in the most general category of things that can enter into sign relations in the roles of signs and interpretants. In this usage the scope of the term ''representation'' includes all sorts of syntactic, descriptive, and conceptual entities, a range of options I will frequently find it convenient to suggest by drawing on a pair of stock phrases: ''terms and concepts'' (TACs) in a conjunctive context versus ''terms or concepts'' (TOCs) in a disjunctive context.
In mathematics, the word ''representation'' is commonly reserved for referring to a ''homomorphism'', that is, a linear transformation or a structure-preserving mapping <math>h : X \to Y\!</math> between mathematical objects, that is, structure-bearing spaces in a category of comparable domains.
In keeping with the spirit of the current discussion, I will first present a set of examples that are designed to illustrate what I mean by an intensional representation. In general, an intensional representation of any object is a sign, description, or concept that denotes, describes, or conceives its object in terms of its properties, that is, in terms of the logical attributes that the object possesses or the propositional features that the object is supposed to have. If the object to be represented is a complex formal object like a sign relation, then there needs to be an intensional representation of each elementary sign relation and an intensional representation of the sign relation as a whole.
But first, before I try to tackle this project, it is advisable to seek a measure of theoretical advantage that I can bring to bear on the task. This I can do by anchoring my focal outlook on sign relations within a more global consideration of <math>n\!</math>-place relations. Not only will this help with the conceptual recasting of <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> but it will also support later stages of the present work, especially in the effort to build a collection of readily accessible linkages between the extensions and the intensions of each construct that I try to use, and ultimately of each concept and term that might conceivably find a use in inquiry.
===6.29. Projects of Representation===
'''Note.''' This section is very rough and will need to be rewritten.
There are numerous modalities of description and representation that are involved in linking the extensions and intensions of terms and concepts. To facilitate the building of a suitable analytic and synthetic framework for this task, and to abbreviate future references to the categories of modalities that come into play, I will employ a set of technical notions, along with their aliases and acronyms, to be indicated next.
First of all, I want to call attention to a ''mode of description'' or a ''category of representation'' that logically precedes all forms of extensional representation and intensional representation. To do this, I introduce the notion of a ''project of representation'' and the mode or category of ''pre-tensional representation'' that contains the elements for attempting its actualization.
A ''project of representation'', in the interval before its completion or its viability can be assured with any measurable degree of certainty, initiates a mode of description called a ''prospective representation'' or a ''pretended representation''. Taken over time, the project of representation issues in a series of prospective representation notices, a sequence of explicit expressions that constitute its ''prospectus'' or ''pretension''. If the project of representation turns out to be feasible, and if all goes well with its realization, then the promises tendered by these notes can be redeemed in full. Regarded from a retrospective vantage point, and ever contingent on the eventual success of the project of representation, what amounts to little more than a species of potentially attractive prospective representation bulletins can be valued ultimately as a set of successive approximations to the intended concept and, further, to the objective reality intended. It is only at this point that a piece of prospective representation achieves the virtues of an ''actual representation'', embodying in its purported description a modicum of ''actually resolved tension'', and becoming amenable to formalization as a non-degenerate example of a sign relation. This tells how a project of representation develops in time under conditions ideal enough to achieve its aim. Next, I describe the features of prospective representation as a form of existence ''sub specie aeternitatis''.
Under the banner of prospective representation parades an uncontrolled and wide-ranging variety of pretended signs and prospective signs, entities with the alleged or apparent significance of signs, of likely stories that are potentially meaningful and useful as descriptions and representations, but the character of whose participation in a sign relation is yet to be judged and warranted. A certain quality of PR marks the brand of significance that a candidate sign possesses before it is known for certain whether it will be genuine or spurious in its role as a sign, the meagre benefit of the doubt that can be granted any reputed sign before its character as an authentic sign has been tested in the performance of its assigned functions.
The prospects and pretensions of prospective representation are associated with a quality of tension that prevails on the scene of representation even before it is definite that objects have signs and signs have objects, an uncertain character that supervenes on the stage before one is sure that a project of representation will succeed in its intentions — far enough for its extension to stretch over many determinate instances, or well enough for its intension to hold forth any distinctive properties. This whole modality of allegation and aspiration, given the self described significance and uncertified self-advertisement that signs in all their immature phases and developmental crises cannot help but affect, taken along with the valid ambitions of potential signs to become signs in fact, is a class of pretended and prospective meaning that it seems fitting to label as a category of ''pretensional representation''.
The assortment of prospective representation mechanisms that goes into a run-of-the-mill project of representation is a rather odd lot, drawing into its curious train every style of potential, preliminary, prospective, provisional, purported, putative, and otherwise ''pretensional representation''. The motley array of artifice and device that is permitted under this gangly heading seems ill-suited to becoming recognized as a natural category, and perhaps it is destined to persist for all time pretty much as one presently finds it, too quixotic to regiment fully and too recalcitrant to organize under compact terms. For the purposes of the current project, the point of distinguishing the category of prospective representation as a mode of description is twofold:
# The category of prospective representation is drawn up to encompass the categories of extensional representation and intensional representation and to allow the creation or recognition of a collection of continuities and correspondences between them.
# The prospective representation mode of description draws attention to several important facts about the more problematic phases of interpretation and inquiry processes, especially including the inchoate actions of their initial stages and the moments when global paradigm shifts are manifestly in progress. All that interpreters have to go on for much of the intermediate time that they spend involved in the sign formation processes of inquiry is a type of prospective representation.
Variant. The purpose of these prospective representation labels is to draw attention to the indirect character and the allusory nature of the interpretive processes that contribute to the initial stages and non-routine phases of inquiry.
Variant. The point of pointing out the indirect character and the allusory nature of all prospective representation is to draw attention to the circumstance that intelligent agents of interpretation and inquiry have to be capable of waiting, trading, and acting on purported and potential representations. To proceed at all from the conditions prevailing at the outset of inquiry, they have to lead off from the slightest inklings that a problematic phenomenon may disclose an objective reality that is a key to its resolution, to take their initial direction from uncertain signs that an object of value might be in the offing, and to open the bidding on a brand of improper symbols whose very qualifications as representations are required by the nature of interpretive inquiry to remain in question for much of the mean time taken up by the pursuit of the alleged objects. It is part of the task that prospective representation is all these agents have to go on, …
Extensional descriptions of <math>n\!</math>-place relations are the kinds that can be presented in relational data tables, or at least initiated and partially illustrated in this form. If an <math>n\!</math>-place relation is constituted as a ''finite information construction'', where the relation as a whole plus each of its elements is specified in discrete and finite terms, then the prospective tabulation can be carried out to completion, at least in principle, explicitly enumerating the ''elementary relations'' or the <math>n\!</math>-tuples of relational domain elements that enter into the relation.
Extensional descriptions are so close to what one casually and commonly regards as “immediate experience” that the knowledge of a relation one gains by means of their indications is often not thought to lie in the medium of description at all. The acquaintance with the character of an objective relation that extensional descriptions so successfully manage to record and convey is a type of impression that one often fails to reflect on as arriving through signs at all, and thus it can leave an impression of knowing its object that is susceptible to being confounded with a direct experience of the relation itself.
This does not have to be a bad thing in practice. Indeed, it is a factor contributing to the success of extensional descriptions that an agent can usually afford to remain oblivious to the more indirect aspects of their interpretation, to proceed with impunity to take them at face value, and unless there is some obvious trouble to work on the assumption that they are in fact nothing more than what they seem to denote. Thus, one often finds extensional presentations treated as though they arose from radically empirical sources of data, instituting fundamentally pure modes of “knowledge by acquaintance” and reputed to lie in meaningful contradistinction to every other type of representation, the rest falling into categories that grade them as mere “knowledge by description”. However, when it comes to the ends of deliberate design and analysis, the usual assumptions can no longer be relied on to justify their own usage, and they need to make themselves available for examination whenever the limits of their usefulness are called into question.
One of the purposes of introducing an explicit theory of sign relations into the present study of inquiry is to examine the status of this idea, namely, that it makes sense to posit an absolute distinction between knowledge by acquaintance and knowledge by description. Pragmatic thinking begins with a certain amount of skepticism toward this notion, on account of the many illusions that appear to trace their origins to it, but overall it seeks to arrive at a language in which to examine the question thoroughly, and to devise a means for individual interpreters to make a clear choice for themselves, with respect to the possibly of their purposes being divergent in the mean time, one way or the other.
Whatever the outcome of these individual decisions, independent in principle no matter how they turn out in practice, and without forcing the preliminary acknowledgement of any unavoidable gulf or unbridgeable abyss that might be imagined to separate the modalities of acquaintance and description, one nevertheless wants to preserve the practical uses of the comparative, interpretive, relative, and otherwise sufficiently qualified and circumspect orderings of data and descriptions along these lines that can be organized by individual interpreters and communities of interpretation. It is for these reasons that I have taken the trouble to conduct this discussion in a way that diverts some attention toward its own casual context, and I hope that this strategy is able to reflect, or at least scatter, enough light back on the enfolding context of informal sign relations that it can dispel any inkling of an automatic distinction of this form, or at least to cast doubts on the remaining traces of its illusion.
For the rest of this section I restrict the discussion to sign relations of the type <math>L \subseteq O \times S \times I\!</math> and elementary sign relations of the form <math>(o, s, i) \in L.\!</math>
A ''discussion of concrete examples'', intended to serve as a preparatory treatment for approaching a significantly more complex area, is necessarily limited in its focus to isolated cases, in effect, to those that remain simple enough to be instructive in a preliminary approach to the topic. This means that the observable properties of the initial examples, with respect to the class they are aimed to exemplify, will sort themselves into two kinds: (1) their essential, generic, or genuine properties, and (2) their accidental, factitious, or spurious properties.
But the present discussion of sign relations cannot illustrate the properties of even these elementary examples in an adequate way without considering extended multitudes of other relations, both those that share the same properties and those that do not. Consequently, by way of getting the comparative study of sign relations started on a casual basis, an end that is served in addition by placing sign relations within the broader setting of <math>n\!</math>-place relations, I will exploit a few devices of taxonomic nomenclature, intending them to be applied for the moment in a purely informal way.
An ''order'', ''genus'', or ''species'' of relations is a class or set of relations that obey a particular collection of axioms, or that satisfy a certain combination of operational constraints and axiomatic properties. These respective terms, given in order of increasing specificity, are not intended to be applied too systematically, but only roughly to indicate how many axioms are listed in the specification of a class of relations and thus how narrowly the indicated class is pinned down relative to other classes within the context of a particular discussion.
For example, this terminology allows me to indicate a ''general order'' <math>G\!</math> of sign relations <math>L,\!</math> each of whose connotative components <math>L_{SI}\!</math> is an equivalence relation, and then it allows me to extend this investigation by pursuing the prospective existence of a generalized order <math>G'\!</math> of sign relations <math>L,\!</math> each of which has many properties analogous to the sign relations in <math>G,\!</math> with the exception or extension that <math>G'\!</math> is more broadly formulated in certain designated respects.
The purpose of these informal taxonomic distinctions is not to specify absolute levels of generality, something that could not be achieved in a global manner without splitting hierarchies of hereditary properties and the whole host of their successive heirs down to the ultimate pedigrees, but merely to organize properties relative to each other in comparative terms, in which case three levels of generality are usually enough to orient oneself locally in any ontology, no matter how wide or deep. Thus, the main interest that the terms ''order'', ''genus'', and ''species'' will subserve in this connection is to indicate the taxonomic directions of generalization and specialization that a particular investigation is trying to achieve among classes of relations: ''generalizing'' a class by abstracting features or removing constraints from its original definition, and ''specializing'' a class by concretizing features or adding constraints to its initial characterization.
In order to talk and think about any sign relation at all, not to mention addressing the topic of a ''generic order'' of sign relations, one has to use signs to do it, and this requires one's taking part in what can be called a ''higher order'' of sign relations. By way of definition, a sign relation is a ''higher order sign relation'' if some of its signs refer to objects that are themselves sign relations or classes of sign relations.
So long as one expects to deal with only a few sign relations at a time, managing to use only a few conventional names to denote each of them, then one's participation in a higher order sign relation hardly ever becomes too problematic, and it rarely needs to be formalized in order for one to cope with the duties of serving as its unofficial interpreter. Once a reflective involvement with higher order sign relations gets started, however, there will be difficulties that continue to grow and lurk just beneath the apparently conversant surface of their all too facile fluency.
By way of example, a singular sign that denotes an entire sign relation refers by extension to a class of elementary sign relations, or a set of ''transaction triples'' <math>(o, s, i).\!</math> So far, this is still not too much of a problem. But when one begins to develop large numbers of conventional symbols and complicated formulas for referring to the same classes of sign transactions, then considerations of effective and efficient interpretation will demand that these symbols and formulas be organized into semantic equivalence classes with recognizable characters. That is, one is forced to find computable types of similarity relations defined on pairs of symbols and pairs of formulas that tell whether they refer to the same class of sign transactions or not. It is almost inevitable in such a situation that canonical representatives of these equivalence classes will have to be developed, and a means for transforming arbitrarily complex and obscure expressions into optimally simple and clear equivalents will also become necessary.
At this stage one is brought face to face with the task of implementing a full fledged interpreter for a particular higher order sign relation, summarized as follows:
# The objects of <math>Q\!</math> are the abstract classes of transactions that constitute the sign relations in question.
# The signs of <math>Q\!</math> are the collection of symbols and formulas used as conventional names and analytic expressions for the sign relations in question.
# The interpretants of <math>Q\!</math> …
But a generic name intended to reference a whole class of sign relations is another matter altogether, especially when it comes into play in a comparative study of many different orders of relations.
===6.30. Connected, Integrated, Reflective Symbols===
Triadic relations need to be recognized as the minimal subsistents or staple elements of continuity that are capable of keeping the symbols for generalized objects or “hypostatic abstractions” viable in practice. In order to remain fully functioning in all the ways that initially make them useful, abstract terms have to stay connected in each of the many directions of relationship that make their use both flexible and stable, namely, (1) attached to the substantive particulars of their denotations, (2) dedicated to the associational and definitional connotations that constitute their law-abiding participation in a commonwealth of other abstract terms, (3) relevant to the ongoing understanding of inquiring agents and interpretive communities. Anything less, any attempt to use staple structural elements of lower arity than triadic bonds is bound to corrupt in time the dimensional solidity of these symbolic amalgamations.
In order for its knowledge to be reflective, an intelligent system must have the ability to reason about sign relations, not only the ones in which it operates but also the ones in which it might participate. A natural way of approaching this task is to consider the domain of sign relations set within the embedding framework of <math>n\!</math>-place relations, since resourcefulness with relations in general is something that a reasonably competent knowledge-based system will need anyway.
Now here is a class of mathematical objects, <math>n\!</math>-place relations, that are worthy of some thought, no matter what application might be intended, and given the levels of combinatorial complexity that their study raises, it is likely that suitable software will need to play a role in their investigation.
One of the ways that the design principles declared above bear on the application to <math>n\!</math>-place relations is as follows. In order to support reasoning about general classes of relations, and sign relations in particular, a computational system (or implemented formal system) must have signs or names that are available to refer to the subject matter of particular relations and symbols or formulas that are able to represent predicates of relations. If these references and representations are to avoid all the various ways of becoming logically empty and effectively vacuous — something they can do (1) by failing to have sufficient denotation from the very outset or (2) by exceeding the conceptual and computational bounds needed to maintain consistency and tractability at any subsequent stage of processing their indications — then …
===6.31. Relations in General===
In a realistic computational framework, where incomplete and inconsistent information is the rule, it is necessary to work with genera of relations that are increasingly relaxed in their constraining characters but still preserve a measure of analogy with the fundamental species of relations that are found to be prevalent in perfect information contexts.
In the present application the kinds of relations of primary interest are functions, equivalence relations, and other species of relations defined by axiomatic properties. Thus, the information-theoretic generalizations of these structures lead to partially defined functions and partially constrained versions of these specially defined classes of relations.
The purpose of this Section is to outline the kinds of generalized functions and other families of relations that are needed to extend the discussion of the present example. In this connection, to frame the problem in concrete terms, I need to adapt the square bracket notation for two generalizations of equivalence relations, to be defined below. But first, a number of broader issues need to be treated.
Generally speaking, one is free to interpret references to generalized objects either as indications of partially formed analogues or as partially informed descriptions of their corresponding species. I refer to these alternatives as the ''object-theoretic'' and the ''sign-theoretic'' options, respectively. The first interpretation assumes that vague and general references still have denotations, merely to vague and general objects. The second interpretation ascribes the partialities of information to the characters of the signs and expressions that are doing the denoting. In most cases that arise in casual discussion the choice between these conventions is purely stylistic. However, in many of the more intricate situations that arise in formal discussion the object choice often fails utterly, and whenever the utmost care is required it will usually be the attention to signs that saves the day.
In order to speak of generalized orders of relations I need to outline the dimensions of variation along which I intend the characters of already familiar orders of relations to be broadened. Generally speaking, the taxonomic features of <math>n\!</math>-place relations that I wish to liberalize can be read off from their ''local incidence properties'' (LIPs).
'''Definition.''' A ''local incidence property'' of a <math>k\!</math>-place relation <math>L \subseteq X_1 \times \ldots \times X_k\!</math> is one that is based on the following type of data. Pick an element <math>x\!</math> in one of the domains <math>{X_j}\!</math> of <math>L.\!</math> Let <math>L_{x \,\text{at}\, j}\!</math> be a subset of <math>L\!</math> called the ''flag of <math>L\!</math> with <math>x\!</math> at <math>{j},\!</math>'' or the ''<math>x \,\text{at}\, j\!</math> flag of <math>L.\!</math>'' The ''local flag'' <math>L_{x \,\text{at}\, j} \subseteq L\!</math> is defined as follows.
{| align="center" cellspacing="8" width="90%"
| <math>L_{x \,\text{at}\, j} = \{ (x_1, \ldots, x_j, \ldots, x_k) \in L : x_j = x \}.\!</math>
|}
Any property <math>P\!</math> of <math>L_{x \,\text{at}\, j}\!</math> constitutes a ''local incidence property'' of <math>L\!</math> with reference to the locus <math>x \,\text{at}\, j.\!</math>
'''Definition.''' A <math>k\!</math>-place relation <math>L \subseteq X_1 \times \ldots \times X_k\!</math> is ''<math>P\!</math>-regular at <math>j\!</math>'' if and only if every flag of <math>L\!</math> with <math>x\!</math> at <math>j\!</math> is <math>P,\!</math> letting <math>x\!</math> range over the domain <math>X_j,\!</math> in symbols, if and only if <math>P(L_{x \,\text{at}\, j})\!</math> is true for all <math>{x \in X_j}.\!</math>
Of particular interest are the local incidence properties of relations that can be calculated from the cardinalities of their local flags, and these are naturally called ''numerical incidence properties'' (NIPs).
For example, <math>L\!</math> is <math>c\text{-regular at}~ j\!</math> if and only if the cardinality of the local flag <math>L_{x \,\text{at}\, j}\!</math> is equal to <math>c\!</math> for all <math>x \in X_j,\!</math> coded in symbols, if and only if <math>|L_{x \,\text{at}\, j}| = c\!</math> for all <math>{x \in X_j}.\!</math>
In a similar fashion, it is possible to define the numerical incidence properties <math>(< c)\text{-regular at}~ j,\!</math> <math>(> c)\text{-regular at}~ j,\!</math> and so on. For ease of reference, a few of these definitions are recorded below.
{| align="center" cellspacing="8" width="90%"
|
<math>\begin{array}{lll}
L ~\text{is}~ c\text{-regular at}~ j
& \iff &
|L_{x \,\text{at}\, j}| = c ~\text{for all}~ x \in X_j.
\\[6pt]
L ~\text{is}~ (< c)\text{-regular at}~ j
& \iff &
|L_{x \,\text{at}\, j}| < c ~\text{for all}~ x \in X_j.
\\[6pt]
L ~\text{is}~ (> c)\text{-regular at}~ j
& \iff &
|L_{x \,\text{at}\, j}| > c ~\text{for all}~ x \in X_j.
\\[6pt]
L ~\text{is}~ (\le c)\text{-regular at}~ j
& \iff &
|L_{x \,\text{at}\, j}| \le c ~\text{for all}~ x \in X_j.
\\[6pt]
L ~\text{is}~ (\ge c)\text{-regular at}~ j
& \iff &
|L_{x \,\text{at}\, j}| \ge c ~\text{for all}~ x \in X_j.
\end{array}\!</math>
|}
The definition of local flags can be broadened to give a definition of ''regional flags''. Suppose <math>L \subseteq X_1 \times \ldots \times X_k\!</math> and choose a subset <math>M \subseteq X_j.\!</math> Let <math>L_{M \,\text{at}\, j}\!</math> be a subset of <math>L\!</math> called the ''flag of <math>L\!</math> with <math>M\!</math> at <math>{j},\!</math>'' or the ''<math>M \,\text{at}\, j\!</math> flag of <math>L,\!</math>'' defined as follows.
{| align="center" cellspacing="8" width="90%"
| <math>L_{M \,\text{at}\, j} = \{ (x_1, \ldots, x_j, \ldots, x_k) \in L : x_j \in M \}.\!</math>
|}
Returning to dyadic relations, it is useful to describe some familiar classes of objects in terms of their local and numerical incidence properties. Let <math>L \subseteq X \times Y\!</math> be an arbitrary dyadic relation. The following properties of <math>L\!</math> can then be defined.
{| align="center" cellspacing="8" width="90%"
|
<math>\begin{array}{lll}
L ~\text{is total at}~ X
& \iff &
L ~\text{is}~ (\ge 1)\text{-regular}~ \text{at}~ X.
\\[6pt]
L ~\text{is total at}~ Y
& \iff &
L ~\text{is}~ (\ge 1)\text{-regular}~ \text{at}~ Y.
\\[6pt]
L ~\text{is tubular at}~ X
& \iff &
L ~\text{is}~ (\le 1)\text{-regular}~ \text{at}~ X.
\\[6pt]
L ~\text{is tubular at}~ Y
& \iff &
L ~\text{is}~ (\le 1)\text{-regular}~ \text{at}~ Y.
\end{array}</math>
|}
If <math>L\!</math> is tubular at <math>X,\!</math> then <math>L\!</math> is known as a ''partial function'' or a ''prefunction'' from <math>X\!</math> to <math>Y,\!</math> indicated by writing <math>L : X \rightharpoonup Y.\!</math> We have the following definitions and notations.
{| align="center" cellspacing="8" width="90%"
|
<math>\begin{array}{lll}
L ~\text{is a prefunction}~ L : X \rightharpoonup Y
& \iff &
L ~\text{is tubular at}~ X.
\\[6pt]
L ~\text{is a prefunction}~ L : X \leftharpoonup Y
& \iff &
L ~\text{is tubular at}~ Y.
\end{array}</math>
|}
If <math>L\!</math> is a prefunction <math>L : X \rightharpoonup Y\!</math> that happens to be total at <math>X,\!</math> then <math>L\!</math> is known as a ''function'' from <math>X\!</math> to <math>Y,\!</math> indicated by writing <math>L : X \to Y.\!</math> To say that a relation <math>L \subseteq X \times Y\!</math> is ''totally tubular'' at <math>X\!</math> is to say that <math>L\!</math> is 1-regular at <math>X.\!</math> Thus, we may formalize the following definitions.
{| align="center" cellspacing="8" width="90%"
|
<math>\begin{array}{lll}
L ~\text{is a function}~ L : X \to Y
& \iff &
L ~\text{is}~ 1\text{-regular at}~ X.
\\[6pt]
L ~\text{is a function}~ L : X \leftarrow Y
& \iff &
L ~\text{is}~ 1\text{-regular at}~ Y.
\end{array}\!</math>
|}
In the case of a 2-adic relation <math>L \subseteq X \times Y\!</math> that has the qualifications of a function <math>f : X \to Y,\!</math> there are a number of further differentia that arise.
{| align="center" cellspacing="8" width="90%"
|
<math>\begin{array}{lll}
f ~\text{is surjective}
& \iff &
f ~\text{is total at}~ Y.
\\[6pt]
f ~\text{is injective}
& \iff &
f ~\text{is tubular at}~ Y.
\\[6pt]
f ~\text{is bijective}
& \iff &
f ~\text{is}~ 1\text{-regular at}~ Y.
\end{array}</math>
|}
A few more comments on terminology are needed in further preparation. One of the constant practical demands encountered in this project is to have available a language and a calculus for relations that can permit discussion and calculation to range over functions, dyadic relations, and <math>n\!</math>-place relations with a minimum amount of trouble in making transitions from subject to subject and in drawing the appropriate generalizations.
Up to this point in the discussion, the analysis of the <math>\text{A}\!</math> and <math>\text{B}\!</math> dialogue has concerned itself almost exclusively with the relationship of triadic sign relations to the dyadic relations obtained from them by taking their projections onto various relational planes. In particular, a major focus of interest was the extent to which salient properties of sign relations can be gleaned from a study of their dyadic projections.
Two important topics for later discussion will be concerned with: (1) the sense in which every <math>n\!</math>-place relation can be decomposed in terms of triadic relations, and (2) the fact that not every triadic relation can be further reduced to conjunctions of dyadic relations.
'''Variant.''' It is one of the constant technical needs of this project to maintain a flexible language for talking about relations, one that permits discussion to shift from functional to relational emphases and from dyadic relations to <math>n\!</math>-place relations with a maximum of ease. It is not possible to do this without violating the favored conventions of one technical linguistic community or another. I have chosen a strategy of use that respects as many different usages as possible, but in the end it cannot help but to reflect a few personal choices. To some extent my choices are guided by an interest in developing the information, computation, and decision-theoretic aspects of the mathematical language used. Eventually, this requires one to render every distinction, even that of appearing or not in a particular category, as being relative to an interpretive framework.
While operating in this context, it is necessary to distinguish ''domains'' in the broad sense from ''domains of definition'' in the narrow sense. For <math>k\!</math>-place relations it is convenient to use the terms ''domain'' and ''quorum'' as references to the wider and narrower sets, respectively.
For a <math>k\!</math>-place relation <math>L \subseteq X_1 \times \ldots \times X_k,\!</math> we have the following usages.
# The notation <math>{}^{\backprime\backprime} \operatorname{Dom}_j (L) {}^{\prime\prime}\!</math> denotes the set <math>X_j,\!</math> called the ''domain of <math>L\!</math> at <math>j\!</math>'' or the ''<math>j^\text{th}\!</math> domain of <math>L.\!</math>''.
# The notation <math>{}^{\backprime\backprime} \operatorname{Quo}_j (L) {}^{\prime\prime}\!</math> denotes a subset of <math>{X_j}\!</math> called the ''quorum of <math>L\!</math> at <math>j\!</math>'' or the ''<math>j^\text{th}\!</math> quorum of <math>L,\!</math>'' defined as follows.
{| align="center" cellspacing="8" width="90%"
|
<math>\begin{array}{lll}
\operatorname{Quo}_j (L)
& = &
\text{the largest}~ Q \subseteq X_j ~\text{such that}~ ~L_{Q \,\text{at}\, j}~ ~\text{is}~ (> 1)\text{-regular at}~ j,
\\[6pt]
& = &
\text{the largest}~ Q \subseteq X_j ~\text{such that}~ |L_{Q \,\text{at}\, j}| > 1 ~\text{for all}~ x \in Q \subseteq X_j.
\end{array}</math>
|}
In the special case of a dyadic relation <math>L \subseteq X_1 \times X_2 = X \times Y,\!</math> including the case of a partial function <math>p : X \rightharpoonup Y\!</math> or a total function <math>f : X \to Y,\!</math> we have the following conventions.
# The arbitrarily designated domains <math>X_1 = X\!</math> and <math>X_2 = Y\!</math> that form the widest sets admitted to the dyadic relation are referred to as the ''domain'' or ''source'' and the ''codomain'' or ''target'', respectively, of the relation in question.
# The terms ''quota'' and ''range'' are reserved for those uniquely defined sets whose elements actually appear as the first and second members, respectively, of the ordered pairs in that relation. Thus, for a dyadic relation <math>L \subseteq X \times Y,\!</math> we identify <math>\operatorname{Quo} (L) = \operatorname{Quo}_1 (L) \subseteq X\!</math> with what is usually called the ''domain of definition'' of <math>L\!</math> and we identify <math>\operatorname{Ran} (L) = \operatorname{Quo}_2 (L) \subseteq Y\!</math> with the usual ''range'' of <math>L.\!</math>
A ''partial equivalence relation'' (PER) on a set <math>X\!</math> is a relation <math>L \subseteq X \times X\!</math> that is an equivalence relation on its domain of definition <math>\operatorname{Quo} (L) \subseteq X.\!</math> In this situation, <math>[x]_L\!</math> is empty for each <math>x\!</math> in <math>X\!</math> that is not in <math>\operatorname{Quo} (L).\!</math> Another way of reaching the same concept is to call a PER a dyadic relation that is symmetric and transitive, but not necessarily reflexive. Like the “self-identical elements” of old that epitomized the very definition of self-consistent existence in classical logic, the property of being a self-related or self-equivalent element in the purview of a PER on <math>X\!</math> singles out the members of <math>\operatorname{Quo} (L)\!</math> as those for which a properly meaningful existence can be contemplated.
A ''moderate equivalence relation'' (MER) on the ''modus'' <math>M \subseteq X\!</math> is a relation on <math>X\!</math> whose restriction to <math>M\!</math> is an equivalence relation on <math>M.\!</math> In symbols, <math>L \subseteq X \times X\!</math> such that <math>L|M \subseteq M \times M\!</math> is an equivalence relation. Notice that the subset of restriction, or modus <math>M,\!</math> is a part of the definition, so the same relation <math>L\!</math> on <math>X\!</math> could be a MER or not depending on the choice of <math>M.\!</math> In spite of how it sounds, a moderate equivalence relation can have more ordered pairs in it than the ordinary sort of equivalence relation on the same set.
In applying the equivalence class notation to a sign relation <math>L,\!</math> the definitions and examples considered so far cover only the case where the connotative component <math>L_{SI}\!</math> is a total equivalence relation on the whole syntactic domain <math>S.\!</math> The next job is to adapt this usage to PERs.
If <math>L\!</math> is a sign relation whose syntactic projection <math>L_{SI}\!</math> is a PER on <math>S\!</math> then we may still write <math>{}^{\backprime\backprime} [s]_L {}^{\prime\prime}\!</math> for the “equivalence class of <math>s\!</math> under <math>L_{SI}\!</math>”. But now, <math>[s]_L\!</math> can be empty if <math>s\!</math> has no interpretant, that is, if <math>s\!</math> lies outside the “adequately meaningful” subset of the syntactic domain, where synonymy and equivalence of meaning are defined. Otherwise, if <math>s\!</math> has an <math>i\!</math> then it also has an <math>o,\!</math> by the definition of <math>L_{SI}.\!</math> In this case, there is a triple <math>{(o, s, i) \in L},\!</math> and it is permissible to let <math>[o]_L = [s]_L.\!</math>
===6.32. Partiality : Selective Operations===
One of the main subtasks of this project is to develop a computational framework for carrying out set-theoretic operations on abstractly represented classes and for reasoning about their indicated results. This effort has the general aim of enabling one to articulate the structures of <math>n\!</math>-place relations and the special aim of allowing one to reflect theoretically on the properties and projections of sign relations. A prototype system that makes a beginning in this direction has already been implemented, to which the current work contributes a major part of the design philosophy and technical documentation. This section presents the rudiments of set-theoretic notation in a way that conforms to these goals, taking the development only so far as needed for immediate application to sign relations like <math>L(\text{A})\!</math> and <math>L(\text{B}).\!</math>
One of the most important design considerations that goes into building the requisite software system is how well it furthers certain lines of abstraction and generalization. One of these dimensions of abstraction or directions of generalization is discussed in this section, where I attempt to unify its many appearances under the theme of ''partiality''. This name is chosen to suggest the desired sense of abstract intention since the extensions of concepts that it favors and for which it leaves room are outgrowths of the limitation that finite signs and expressions can never provide more than partial information about the richness of individual detail that is always involved in any real object. All in all, this modicum of tolerance for uncertainty is the very play in the wheels of determinism that provides a significant chance for luck to play a part in the finer steps toward finishing every real objective.
If a slogan is needed to charge this form of propagation, it is only that “Necessity is the mother of invention.” In other words, it is precisely this lack of perfect information that yields the opportunity for novel forms of speciation to develop among finitely informed creatures (FICs), and just this need of perfect information that drives the evolving forms of independent determination and spontaneous creation in any area, no matter how well the arena is circumscribed by the restrictions of signs.
In tracing the echoes of this theme, it is necessary to reflect on the circumstance that degenerate sign relations happen to be perfectly possible in practice, and it is desirable to provide a critical method that can address the facts of their flaws in theoretically insightful terms. Relative to particular environments of interpretation, nothing proscribes the occurrence of sign relations that are defective in any of their various facets, namely: (1) with signs that fail to denote or connote, (2) with interpretants that lack of being faithfully represented or reliably objectified, and (3) with objects that make no impression or remain ineffable in the preferred medium.
A cursory examination of the topic of ''partiality'', as just surveyed, reveals two strains fixing how this “quality of murky” in general reigns. This division depends on the disposition of <math>n\!</math>-tuples as the individual elements that inhabit an <math>n\!</math>-place relation.
<ol style="list-style-type:decimal">
<li>If the integrity of elementary relations as n-tuples is maintained, then the predicate of ''partiality'' characterizes only the state of information that one has, either about elementary relations or about entire relations, or both. Thus, this strain of partiality affects the determination of relations at two distinct levels of their formation:</li>
<ol style="list-style-type:lower-alpha">
<li>At the level of elementary relations, it frees up the point to which <math>n\!</math>-tuples are pinned down by signs or expressions of relations by modifying the name that indicates or the formula that specifies a relation.</li>
<li>At the level of entire relations, it relaxes the grip that axioms and constraints have on the character of a relation by modifying the strictness or generalizing the form of their application.</li></ol>
<li>If ''partial <math>n\!</math>-tuples'' are admitted, and not permitted to be confused with ''<math>(< n)\!</math>-tuples'', then one arrives at the concept of an ''<math>n\!</math>-place relational complex''.</li></ol>
'''Relational Complex?'''
{| align="center" cellspacing="8" width="90%"
| <math>L ~=~ L^{(1)} \cup \ldots \cup L^{(k)}\!</math>
|}
'''Sign Relational Complex?'''
{| align="center" cellspacing="8" width="90%"
| <math>L ~=~ L^{(1)} \cup L^{(2)} \cup L^{(3)}\!</math>
|}
It is possible to see two directions of remove that signs and concepts can take in departing from complete specifications of individual objects, and thus to see two dimensions of variation in the requisite varieties of partiality, each of which leads off into its own distinctive realm of abstraction.
# In a direction of ''generality'', with ''general'' signs and concepts, one loses an amount of certainty as to exactly what object the sign or concept applies at any given moment, and thus this can be recognized as an extensional type of abstraction.
# In a direction of ''vagueness'', with ''vague'' signs and concepts, one loses a degree of security as to exactly what property the sign or concept implies in the current context, and thus this can be classified as an intensional mode of abstraction.
The first order of business is to draw some distinctions, and at the same time to note some continuities, between the varieties of partiality that remain to be sufficiently clarified and the more mundane brands of partiality that are already familiar enough for present purposes, but lack perhaps only the formality of being recognized under that heading.
The most familiar illustrations of information-theoretic partiality, partial indication, or “signs bearing partial information about objects” occur every time one uses a general name, for example, the name of a class, genus, or set. Almost as commonly, the formula that expresses a logical proposition can be regarded as a partial specification of its logical models or satisfying interpretations. Just as the name of a class or genus can be taken as a ''partially informed reference'' or a ''plural indefinite reference'' (PIR) to one of its elements or species, so the name of an <math>n\!</math>-place relation can be viewed as a PIR to one of its elementary relations or <math>n\!</math>-tuples, and the formula or expression of a proposition can be understood as a PIR to one its models or satisfying interpretations. For brevity, this variety of referential indetermination can be called the ''generic partiality'' of signs as information bearers.
'''Note.''' In this discussion I will not systematically distinguish between the logical entity typically called a ''proposition'' or a ''statement'' and the syntactic entity usually called an ''expression'', ''formula'', or ''sentence''. Instead, I work on the assumption that both types of entity are always involved in everything one proposes and also on the hope that context will determine which aspect of proposing is most apt. For precision, the abstract category of propositions proper will have to be reconstituted as logical equivalence classes of syntactically diverse expressions. For the present, I will use the phrase ''propositional expression'' whenever it is necessary to call particular attention to the syntactic entity. Likewise, I will not always separate ''higher order propositions'', that is, propositions about propositions, from their corresponding formulations in the guise of ''higher order propositional expressions''.
Even though partial information is the usual case of information (as rendered by signs about objects) I will continue to use this phrase, for all its informative redundancy, to emphasize the issues of partial definition, determination, and specification that arise under the pervasive theme of partiality.
In speaking of properties and classes of relations, one would like to allude to ''all relations'' as the implicit domain of discussion, setting each particular topic against this optimally generous and neutral background. But even before discussion is restricted to a computational framework the notion of ''all'' (of almost anything) proves to be problematic in its very conception, not always amenable to assuming a consistent concept. So the connotation of ''all relations'' — really just a passing phrase that pops up in casual and careless discussions — must be relegated to the status of an informal concept, one that takes on definite meaning only when related to a context of constructive examples and formal models.
Thus, in talking sensibly about properties and classes of relations, one is always invoking, explicitly or implicitly, a preconceived domain of discussion or an established universe of discourse <math>X,\!</math> and in relation to this <math>X\!</math> one is always talking, expressly or otherwise, about a selected subset <math>A \subset X\!</math> that exhibits the property in question and a binary-valued selector function <math>f_A : X \to \mathbb{B}\!</math> that picks out the class in question.
When the subject matter of discussion is bounded by a universal set <math>X,\!</math> out of which all objects referred to must come, then every PIR to an object can be identified with the name or formula (sign or expression) of a subset <math>A \subseteq X\!</math> or else with that of its selector function <math>f_A : X \to \mathbb{B}.\!</math> Conceptually, one imagines generating all the objects in <math>X\!</math> and then selecting the ones that satisfy a definitive test for membership in <math>A.\!</math>
In a realistic computational framework, however, when the domain of interest is given generatively in a genuine sense of the word, that is, defined solely in terms of the primitive elements and operations that are needed to generate it, and when the resource limitations in actual effect make it impractical to enumerate all the possibilities in advance of selecting the adumbrated subset, then the implementation of PIRs becomes a genuine computational problem.
Considered in its application to <math>n\!</math>-place relations, the generic brand of partial specification constitutes a rather limited type of partiality, in that every element conceived as falling under the specified relation, no matter how indistinctly indicated, is still envisioned to maintain its full arity and to remain every bit a complete, though unknown, <math>n\!</math>-tuple. Still, there is a simple way to extend the concept of generic partiality in a significant fashion, achieving a form of PIRs to relations by making use of ''higher order propositions''.
Extending the concept of generic partiality, by iterating the principle on which it is based, leads to higher order propositions about elementary relations, or propositions about relations, as one way to achieve partial specifications of relations, or PIRs to relations.
This direction of generalization expands the scope of PIRs by means of an analogical extension, and can be charted in the following manner. If the sign or expression (name or formula) of an <math>n\!</math>-place relation can be interpreted as a proposition about <math>n\!</math>-tuples and thus as a PIR to an elementary relation, then a higher order proposition about <math>n\!</math>-tuples is a proposition about <math>n\!</math>-place relations that can be used to formulate a PIR to an <math>n\!</math>-place relation.
In order to formalize these ideas, it is helpful to have notational devices for switching back and forth among different ways of exemplifying what is abstractly the same contents of information, in particular, for translating among sets, their logical expressions, and their functional indications.
Given a set <math>X\!</math> and a subset <math>A \subseteq X,\!</math> let the ''selector function of <math>A\!</math> in <math>X\!</math>'' be notated as <math>A^\sharp\!</math> and defined as follows.
{| align="center" cellspacing="8" width="90%"
|
<math>\begin{array}{lll}
A^\sharp : X \to \mathbb{B} & \text{such that} & A^\sharp (x) = 1 \iff x \in A.
\end{array}</math>
|}
Other names for the same concept, appearing under various notations, are the ''characteristic function'' or the ''indicator function'' of <math>A\!</math> in <math>X.\!</math>
Conversely, given a boolean-valued function <math>f : X \to \mathbb{B},\!</math> let the ''selected set of <math>f\!</math> in <math>X\!</math>'' be notated as <math>f_\flat\!</math> and defined as follows.
{| align="center" cellspacing="8" width="90%"
|
<math>\begin{array}{lll}
f_\flat \subseteq X & \text{such that} & f_\flat = f^{-1}(1) = \{ x \in X : f(x) = 1 \}.
\end{array}</math>
|}
Other names for the same concept are the ''fiber'', ''level set'', or ''pre-image'' of 1 under the mapping <math>f : X \to \mathbb{B}.\!</math>
Obviously, the relation between these operations is such that the following equations hold.
{| align="center" cellspacing="8" width="90%"
|
<math>\begin{array}{lll}
(A^\sharp)_\flat = A & \text{and} & (f_\flat)^\sharp = f.
\end{array}</math>
|}
It will facilitate future discussions to go through the details of applying these selective operations to the case of <math>n\!</math>-place relations. If <math>L \subseteq X_1 \times \ldots \times X_n\!</math> is an <math>n\!</math>-place relation, then <math>L^\sharp : X_1 \times \ldots \times X_n \to \mathbb{B}\!</math> is the selector of <math>L\!</math> defined as follows.
{| align="center" cellspacing="8" width="90%"
|
<math>\begin{array}{lll}
L^\sharp (x_1, \ldots, x_n) = 1 & \iff & (x_1, \ldots, x_n) \in L.
\end{array}</math>
|}
===6.33. Sign Relational Complexes===
In a computational framework, indeed, in any constructively analytic and practically applied setting, the problem of working with insufficient information to fully determine one's object is a constant feature that goes with the territory of ''finite information constructions'' (FICs). The fineness of detail that is able to be specified by formal symbols is continually bedeviled by the frustrating truncations of every signal to a finite code and by the resistive constrictions of every intention to the restrictive confines of what can actually be conducted. Of course, one tries to get around the more finessible limitations, but the figurative extensions that one hopes to achieve by recourse to quasi-circular definitions and by reversion to parable and hyperbole — all of these tactics appeal to a pre-established aptness of reception on the part of interpreters that begs the very question of a determinate understanding and that risks falling short of the exact attitude needed for success. At any rate, the indirect strategy of approach relies on such large reserves of enthymeme to fuel its course that the grasp of a period to set bounds on its argument and fix a term to its conclusion is often found diverging in ways that both underreach and overreach its object, and well-founded or not the search for a generic method of definition typically ends so completely dumbfounded that it often trails off into the inescapable vacuity of a quasi terminal ellipsis …
This section treats the problems of insufficient information and indeterminate objects under the heading of ''partializations'', using this as a briefer term for the information-theoretic generalizations of the relevant object domains that take the use of indeterminate denotations, or partial determinations of objects, explicitly into account.
In working with ''partializations'' or information-theoretic generalizations of any subject matter, one has a choice between two options:
# Under the ''object-theoretic'' alternative one views the partiality as something attaching to the objects of discussion. Consequently, one operates as if the problems distinctive of the extended subject matter were questions of managing ordinary information about a strange new breed of partial objects.
# Under the ''sign-theoretic'' alternative one takes the partiality as something affecting only the signs used in discussion. Accordingly, one approaches the task as a matter of handling partial information about ordinary objects, namely, the same domains of objects initially given at the outset of discussion.
But a working maxim of information theory says that “Partial information is your ordinary information.” Applied to the principle regulating the sign-theoretic convention this means that the adjective ''partial'' is swallowed up by the substantive ''information'', so that the ostensibly more general case is always already subsumed within the ordinary case. Because partiality is part and parcel to the usual nature of information, it is a perfectly typical feature of the signs and expressions bearing it to provide normally only partial information about ordinary objects.
The only time when a finite sign or expression can give the appearance of determining a perfectly precise content or a post-finite amount of information, for example, when the symbol <math>{}^{\backprime\backprime} e {}^{\prime\prime}\!</math> is used to denote the number also known as “the unique base of the natural logarithms” — this can only happen when interpreters are prepared, by dint of the information embodied in their prior design and preliminary training, to accept as meaningful and be terminally satisfied with what is still only a finite content, syntactically speaking. Every remaining impression that a perfectly determinate object, an ''individual'' in the original sense of the word, has nevertheless been successfully specified — this can only be the aftermath of some prestidigitation, that is, the effect of some pre-arranged consensus, for example, of accepting a finite system of definitions and axioms that are supposed to define the space <math>\mathbb{R}\!</math> and the element <math>e\!</math> within it, and of remembering or imagining that an effective proof system has once been able or will yet be able to convince one of its demonstrations.
Ultimately, one must be prepared to work with probability distributions that are defined on entire spaces <math>O\!</math> of the relevant objects or outcomes. But probability distributions are just a special class of functions <math>f : O \to [0, 1] \subseteq \mathbb{R},\!</math> where <math>\mathbb{R}\!</math> is the real line, and this means that the corresponding theory of partializations involves the dual aspect of the domain <math>O,\!</math> dealing with the ''functionals'' defined on it, or the functions that map it into ''coefficient'' spaces. And since it is unavoidable in a computational framework, one way or another every type of coefficient information, real or otherwise, must be approached bit by bit. That is, all information is defined in terms of the either-or decisions that must be made to determine it. So, to make a long story short, one might as well approach this dual aspect by starting with the functions <math>f : O \to \{ 0, 1 \} = \mathbb{B},\!</math> in effect, with the logic of propositions.
I turn now to the question of ''partially specified relations'', or ''partially informed relations'' (PIRs), in other words, to the explicit treatment of relations in terms of the information that is logically possessed or actually expressed about them. There seem to be several ways to approach the concept of an <math>n\!</math>-place PIR and the supporting notion of a partially specified <math>n\!</math>-tuple. Since the term ''partial relation'' is already implicitly in use for the general class of relations that are not necessarily total on any of their domains, I will coin the term ''pro-relation'', on analogy with ''pronoun'' and ''proposition'', to denote an expression of information about a relation, a contingent indication that, if and when completed, conceivably points to a particular relation.
One way to deal with ''partially informed categories'' of <math>n\!</math>-place relations is to contemplate incomplete relational forms or schemata. Regarded over the years chiefly in logical and intensional terms, constructs of roughly this type have been variously referred to as ''rhemes'' or ''rhemata'' (Peirce), ''unsaturated relations'' (Frege), or ''frames'' (in current AI parlance). Expressed in extensional terms, talking about partially informed categories of <math>n\!</math>-place relations is tantamount to admitting elementary relations with missing elements. The question is not just syntactic — How to represent an <math>n\!</math>-tuple with empty places? — but also semantic — How to make sense of an <math>n\!</math>-tuple with less than <math>n\!</math> elements?
In order to deal with PIRs in a thoroughly consistent fashion, it appears necessary to contemplate elementary relations that present themselves as being ''unsaturated'' (in Frege's sense of that term), in other words, to consider elements of a presumptive product space that in some sense ''wanna be'' <math>n\!</math>-tuples or ''would be'' sequences of a certain length, but are currently missing components in some of their places.
To the extent that the issues of partialization become obvious at the level of symbols and can be dealt with by elementary syntactic means, they initially make their appearance in terms of the various ways that data can go missing.
The alternate notation <math>a \widehat{~} b\!</math> is provided for the ordered pair <math>(a, b).\!</math> This choice of representation for ordered pairs is especially apt in the case of ''concrete indices'' and ''localized addresses'', where one wants the lead item to serve as a pointed reminder of the itemized content, as in <math>j \widehat{~} X_j = (j, X_j),\!</math> and it helps to stress the individuality of each member in the indexed family, as in the following set of equivalent notations.
{| align="center" cellspacing="8" width="90%"
|
<math>\begin{matrix}
G & = & \{ G_j \} & = & \{ j \widehat{~} G_j \} & = & \{ (j, G_j ) \}.
\end{matrix}</math>
|}
The ''link'' device <math>(\,\widehat{~}~)\!</math> works well in any situation where one desires to accentuate the fact that a formal subscript is being reclaimed and elevated to the status of an actual parameter. By way of the operation indicated by the link symbol the index bound to an object term can be rehabilitated as a full-fledged component of an elementary relation, thereby schematically embedding the indicated object in the experiential space of a typical agent.
The form of the link notation is intended to suggest the use of ''pointers'' and ''views'' in computational frameworks, letting one interpret <math>j \widehat{~} x\!</math> in several different ways, for example, any one of the following.
{| align="center" cellspacing="8" width="90%"
|
<math>\begin{array}{lllll}
j \widehat{~} x
& =
& j^\texttt{,}\text{s access to}~ x,
& j^\texttt{,}\text{s allusion to}~ x,
& j^\texttt{,}\text{s copy of}~ x,
\\[4pt]
&
& j^\texttt{,}\text{s indication of}~ x,
& j^\texttt{,}\text{s information on}~ x,
& j^\texttt{,}\text{s view of}~ x.
\end{array}</math>
|}
Presently, the distinction between indirect pointers and direct pointers, that is, between virtual copies and actual views of an objective domain, is not yet relevant here, being a dimension of variation that the discussion is currently abstracting over.
===6.34. Set-Theoretic Constructions===
The next few sections deal with the informational relationships that exist between <math>n\!</math>-place relations and the relations of fewer dimensions that arise as their projections. A number of set-theoretic constructions of constant use in this investigation are brought together and described in the present section. Because their intended application is mainly to sign relations and other triadic relations, and since the current focus is restricted to discrete examples of these types, no attempt is made to present these constructions in their most general and elegant fashions, but only to deck them out in the forms that are most readily pressed into immediate service.
An initial set of operations, required to establish the subsequent constructions, all have in common the property that they do exactly the opposite of what is normally done in abstracting sets from situations. These operations reconstitute, though still in a generic, schematic, or stereotypical manner, some of the details of concrete context and interpretive nuance that are commonly suppressed in forming sets. Stretching points back along the direction of their initial pointing out, these extensions return to the mix a well-chosen selection of features, putting back in those dimensions from which ordinary sets are forced to abstract and in their ordination to treat as distractions.
In setting up these constructions, one typically makes use of two kinds of index sets, in colloquial terms, ''clipboards'' and ''scrapbooks''.
<ol style="list-style-type:decimal">
<li>
<p>The smaller and shorter-term index sets, typically having the form <math>I = \{ 1, \ldots, n \},\!</math> are used to keep tabs on the terms of finite sets and sequences, unions and intersections, sums and products.</p>
<p>In this context and elsewhere, the notation <math>{[n] = \{ 1, \ldots, n \}}\!</math> will be used to refer to a ''standard segment'' (finite initial subset) of the natural numbers <math>\mathbb{N} = \{ 1, 2, 3, \ldots \}.\!</math></p></li>
<li>
<p>The larger and longer-term index sets, typically having the form <math>J \subseteq \mathbb{N} = \{ 1, 2, 3, \ldots \},\!</math> are used to enumerate families of objects that enjoy a more abiding reference throughout the course of a discussion.</p></li></ol>
'''Definition.''' An ''indicated set'' <math>j \widehat{~} S\!</math> is an ordered pair <math>j \widehat{~} S = (j, S),\!</math> where <math>j \in J\!</math> is the ''indicator'' of the set and <math>S\!</math> is the set indicated.
'''Definition.''' An ''indited set'' <math>j \widehat{~} S\!</math> extends the incidental and extraneous indication of a set into a constant indictment of its entire membership.
{| align="center" cellspacing="8" width="90%"
|
<math>\begin{array}{lll}
j \widehat{~} S
& = & j \widehat{~} \{ j \widehat{~} s : s \in S \}
\\[4pt]
& = & j \widehat{~} \{ (j, s) : s \in S \}
\\[4pt]
& = & (j, \{ j \} \times S)
\end{array}</math>
|}
Notice the difference between these notions and the more familiar concepts of an ''indexed set'', ''numbered set'', and ''enumerated set''. In each of these cases the construct that results is one where each element has a distinctive index attached to it. In contrast, the above indications and indictments attach to the set <math>S\!</math> as a whole, and respectively to each element of it, the same index number <math>j.\!</math>
'''Definition.''' An ''indexed set'' <math>(S, L)\!</math> is constructed from two components: its ''underlying set'' <math>S\!</math> and its ''indexing relation'' <math>L : S \to \mathbb{N},\!</math> where <math>L\!</math> is total at <math>S\!</math> and tubular at <math>\mathbb{N}.\!</math> It is defined as follows:
{| align="center" cellspacing="8" width="90%"
| <math>(S, L) = \{ \{ s \} \times L(s) : s \in S \} = \{ (s, j) : s \in S, j \in L(s) \}.\!</math>
|}
<math>L\!</math> assigns a unique set of “local habitations” <math>L(s)\!</math> to each element <math>s\!</math> in the underlying set <math>S.\!</math>
'''Definition.''' A ''numbered set'' <math>(S, f),\!</math> based on the set <math>S\!</math> and the injective function <math>{f : S \to \mathbb{N}},</math> is defined as follows. …
'''Definition.''' An ''enumerated set'' <math>(S, f)\!</math> is a numbered set with a bijective <math>f.\!</math> …
'''Definition.''' The ''<math>n\!</math>-fold sum'' (''co-product'', ''disjoint union'') of the sets <math>X_1, \ldots, X_n\!</math> is notated and defined as follows:
{| align="center" cellspacing="8" width="90%"
| <math>\coprod_{i=1}^n X_i ~=~ X_1 + \ldots + X_n ~=~ 1 \widehat{~} X_1 \cup \ldots \cup n \widehat{~} X_n.\!</math>
|}
'''Definition.''' The ''<math>n\!</math>-fold product'' (''cartesian product'') of the sets <math>X_1, \ldots, X_n\!</math> is notated and defined as follows:
{| align="center" cellspacing="8" width="90%"
| <math>\prod_{i=1}^n X_i ~=~ X_1 \times \ldots \times X_n ~=~ \{ (x_1, \ldots, x_n) : x_i \in X_i \}.\!</math>
|}
As an alternative definition, the <math>n\!</math>-tuples of <math>\prod_{i=1}^n X_i\!</math> can be regarded as sequences of elements from the successive <math>X_i\!</math> and thus as functions that map <math>[n]\!</math> into the sum of the <math>X_i,\!</math> namely, as the functions <math>f : [n] \to \coprod_{i=1}^n X_i\!</math> that obey the condition <math>f(i) \in i \widehat{~} X_i.\!</math>
{| align="center" cellspacing="8" width="90%"
| <math>\prod_{i=1}^n X_i ~=~ X_1 \times \ldots \times X_n ~=~ \{ f : [n] \to \coprod_{i=1}^n X_i ~|~ f(i) \in X_i \}.\!</math>
|}
Viewing these functions as relations <math>f \subseteq J \times J \times X,\!</math> where <math>X = \bigcup_{i=1}^n X_i\!</math> …
Another way to view these elements is as triples <math>i \widehat{~} j \widehat{~} x\!</math> such that <math>i = j\!</math> …
===6.35. Reducibility of Sign Relations===
This Section introduces a topic of fundamental importance to the whole theory of sign relations, namely, the question of whether triadic relations are ''determined by'', ''reducible to'', or ''reconstructible from'' their dyadic projections.
Suppose <math>L \subseteq X \times Y \times Z\!</math> is an arbitrary triadic relation and consider the information about <math>L\!</math> that is provided by collecting its dyadic projections. To formalize this information define the ''projective triple'' of <math>L\!</math> as follows:
{| align="center" cellspacing="8" width="90%"
| <math>\operatorname{Proj}^{(2)} L ~=~ (\operatorname{proj}_{12} L, ~ \operatorname{proj}_{13} L, ~ \operatorname{proj}_{23} L).\!</math>
|}
If <math>L\!</math> is visualized as a solid body in the 3-dimensional space <math>X \times Y \times Z,\!</math> then <math>\operatorname{Proj}^{(2)} L\!</math> can be visualized as the arrangement or ordered collection of shadows it throws on the <math>XY, ~ XZ, ~ YZ\!</math> planes, respectively.
Two more set-theoretic constructions are worth introducing at this point, in particular for describing the source and target domains of the projection operator <math>\operatorname{Proj}^{(2)}.\!</math>
The set of subsets of a set <math>S\!</math> is called the ''power set'' of <math>S.\!</math> This object is denoted by either of the forms <math>\operatorname{Pow}(S)\!</math> or <math>2^S\!</math> and defined as follows:
{| align="center" cellspacing="8" width="90%"
| <math>\operatorname{Pow}(S) ~=~ 2^S ~=~ \{ T : T \subseteq S \}.\!</math>
|}
The power set notation can be used to provide an alternative description of relations. In the case where <math>S\!</math> is a cartesian product, say <math>{S = X_1 \times \ldots \times X_n},\!</math> then each <math>n\!</math>-place relation <math>L\!</math> described as a subset of <math>S,\!</math> say <math>L \subseteq X_1 \times \ldots \times X_n,\!</math> is equally well described as an element of <math>\operatorname{Pow}(S),\!</math> in other words, as <math>L \in \operatorname{Pow}(X_1 \times \ldots \times X_n).\!</math>
The set of triples of dyadic relations, with pairwise cartesian products chosen in a pre-arranged order from a triple of three sets <math>(X, Y, Z),\!</math> is called the ''dyadic explosion'' of <math>X \times Y \times Z.\!</math> This object is denoted <math>\operatorname{Explo}(X, Y, Z ~|~ 2),\!</math> read as the ''explosion of <math>X \times Y \times Z\!</math> by twos'', or more simply as <math>X, Y, Z ~\operatorname{choose}~ 2,\!</math> and defined as follows:
{| align="center" cellspacing="8" width="90%"
| <math>\operatorname{Explo}(X, Y, Z ~|~ 2) ~=~ \operatorname{Pow}(X \times Y) \times \operatorname{Pow}(X \times Z) \times \operatorname{Pow}(Y \times Z).\!</math>
|}
This domain is defined well enough to serve the immediate purposes of this section, but later it will become necessary to examine its construction more closely.
By means of these constructions the operation that forms <math>\operatorname{Proj}^{(2)} L\!</math> for each triadic relation <math>L \subseteq X \times Y \times Z\!</math> can be expressed as a function:
{| align="center" cellspacing="8" width="90%"
| <math>\operatorname{Proj}^{(2)} : \operatorname{Pow}(X \times Y \times Z) \to \operatorname{Explo}(X, Y, Z ~|~ 2).\!</math>
|}
In this setting the issue of whether triadic relations are ''reducible to'' or ''reconstructible from'' their dyadic projections, both in general and in specific cases, can be identified with the question of whether <math>\operatorname{Proj}^{(2)}\!</math> is injective. The mapping <math>\operatorname{Proj}^{(2)}\!</math> is said to ''preserve information'' about the triadic relations <math>L \in \operatorname{Pow}(X \times Y \times Z)\!</math> if and only if it is injective, otherwise one says that some loss of information has occurred in taking the projections. Given a specific instance of a triadic relation <math>L \in \operatorname{Pow}(X \times Y \times Z),\!</math> it can be said that <math>L\!</math> is ''determined by'' (''reducible to'' or ''reconstructible from'') its dyadic projections if and only if <math>(\operatorname{Proj}^{(2)})^{-1}(\operatorname{Proj}^{(2)}L)\!</math> is the singleton set <math>\{ L \}.\!</math> Otherwise, there exists an <math>L'\!</math> such that <math>\operatorname{Proj}^{(2)}L = \operatorname{Proj}^{(2)}L',\!</math> and in this case <math>L\!</math> is said to be ''irreducibly triadic'' or ''genuinely triadic''. Notice that irreducible or genuine triadic relations, when they exist, naturally occur in sets of two or more, the whole collection of them being equated or confounded with one another under <math>\operatorname{Proj}^{(2)}.\!</math>
The next series of Tables illustrates the operation of <math>\operatorname{Proj}^{(2)}\!</math> by means of its actions on the sign relations <math>L_\text{A}\!</math> and <math>L_\text{B}.\!</math> For ease of reference, Tables 72.1 and 73.1 repeat the contents of Tables 1 and 2, respectively, while the dyadic relations comprising <math>\operatorname{Proj}^{(2)}L_\text{A}\!</math> and <math>\operatorname{Proj}^{(2)}L_\text{B}\!</math> are shown in Tables 72.2 to 72.4 and Tables 73.2 to 73.4, respectively.
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table 72.1} ~~ \text{Sign Relation of Interpreter A}~\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Interpretant}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"
|+ style="height:30px" | <math>\text{Table 72.2} ~~ \text{Dyadic Projection} ~ L(\text{A})_{OS}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="50%" | <math>\text{Object}\!</math>
| width="50%" | <math>\text{Sign}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{A}
\\
\text{A}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{B}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"
|+ style="height:30px" | <math>\text{Table 72.3} ~~ \text{Dyadic Projection} ~ L(\text{A})_{OI}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="50%" | <math>\text{Object}\!</math>
| width="50%" | <math>\text{Interpretant}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{A}
\\
\text{A}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{B}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"
|+ style="height:30px" | <math>\text{Table 72.4} ~~ \text{Dyadic Projection} ~ L(\text{A})_{SI}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="50%" | <math>\text{Sign}\!</math>
| width="50%" | <math>\text{Interpretant}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table 73.1} ~~ \text{Sign Relation of Interpreter B}~\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Interpretant}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"
|+ style="height:30px" | <math>\text{Table 73.2} ~~ \text{Dyadic Projection} ~ L(\text{B})_{OS}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="50%" | <math>\text{Object}\!</math>
| width="50%" | <math>\text{Sign}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{A}
\\
\text{A}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{B}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"
|+ style="height:30px" | <math>\text{Table 73.3} ~~ \text{Dyadic Projection} ~ L(\text{B})_{OI}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="50%" | <math>\text{Object}\!</math>
| width="50%" | <math>\text{Interpretant}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{A}
\\
\text{A}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{B}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"
|+ style="height:30px" | <math>\text{Table 73.4} ~~ \text{Dyadic Projection} ~ L(\text{B})_{SI}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="50%" | <math>\text{Sign}\!</math>
| width="50%" | <math>\text{Interpretant}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\end{matrix}</math>
|}
<br>
A comparison of the corresponding projections in <math>\operatorname{Proj}^{(2)} L(\text{A})\!</math> and <math>\operatorname{Proj}^{(2)} L(\text{B})\!</math> shows that the distinction between the triadic relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> is preserved by <math>\operatorname{Proj}^{(2)},\!</math> and this circumstance allows one to say that this much information, at least, can be derived from the dyadic projections. However, to say that a triadic relation <math>L \in \operatorname{Pow} (O \times S \times I)\!</math> is reducible in this sense it is necessary to show that no distinct <math>L' \in \operatorname{Pow} (O \times S \times I)\!</math> exists such that <math>\operatorname{Proj}^{(2)} L = \operatorname{Proj}^{(2)} L',\!</math> and this can take a rather more exhaustive or comprehensive investigation of the space <math>\operatorname{Pow} (O \times S \times I).\!</math>
As it happens, each of the relations <math>L = L(\text{A})\!</math> or <math>L = L(\text{B})\!</math> is uniquely determined by its projective triple <math>\operatorname{Proj}^{(2)} L.\!</math> This can be seen as follows.
Consider any coordinate position <math>(s, i)\!</math> in the plane <math>S \times I.\!</math> If <math>(s, i)\!</math> is not in <math>L_{SI}\!</math> then there can be no element <math>(o, s, i)\!</math> in <math>L,\!</math> therefore we may restrict our attention to positions <math>(s, i)\!</math> in <math>L_{SI},\!</math> knowing that there exist at least <math>|L_{SI}| = 8\!</math> elements in <math>L,\!</math> and seeking only to determine what objects <math>o\!</math> exist such that <math>(o, s, i)\!</math> is an element in the objective ''fiber'' of <math>(s, i).\!</math> In other words, for what <math>{o \in O}\!</math> is <math>(o, s, i) \in \operatorname{proj}_{SI}^{-1}((s, i))?\!</math> The fact that <math>L_{OS}\!</math> has exactly one element <math>(o, s)\!</math> for each coordinate <math>s \in S\!</math> and that <math>L_{OI}\!</math> has exactly one element <math>(o, i)\!</math> for each coordinate <math>i \in I,\!</math> plus the “coincidence” of it being the same <math>o\!</math> at any one choice for <math>(s, i),\!</math> tells us that <math>L\!</math> has just the one element <math>(o, s, i)\!</math> over each point of <math>S \times I.\!</math> This proves that both <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> are reducible in an informational sense to triples of dyadic relations, that is, they are ''dyadically reducible''.