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First, <math>L\!</math> can be associated with a logical predicate or a proposition that says something about the space of effects, being true of certain effects and false of all others. In this way, <math>{}^{\backprime\backprime} L {}^{\prime\prime}</math> can be interpreted as naming a function from <math>\textstyle\prod_i X_i</math> to the domain of truth values <math>\mathbb{B} = \{ 0, 1 \}.</math> With the appropriate understanding, it is permissible to let the notation <math>{}^{\backprime\backprime} L : X_1 \times \ldots \times X_k \to \mathbb{B} {}^{\prime\prime}</math> indicate this interpretation.
First, <math>L\!</math> can be associated with a logical predicate or a proposition that says something about the space of effects, being true of certain effects and false of all others. In this way, <math>{}^{\backprime\backprime} L {}^{\prime\prime}</math> can be interpreted as naming a function from <math>\textstyle\prod_i X_i</math> to the domain of truth values <math>\mathbb{B} = \{ 0, 1 \}.</math> With the appropriate understanding, it is permissible to let the notation <math>{}^{\backprime\backprime} L : X_1 \times \ldots \times X_k \to \mathbb{B} {}^{\prime\prime}</math> indicate this interpretation.
Second, <math>L\!</math> can be associated with a piece of information that allows one to complete various sorts of partial data sets in the space of effects. In particular, if one is given a partial effect or an incomplete <math>k\!</math>-tuple, say, one that is missing a value in the <math>j^\text{th}\!</math> place, as indicated by the notation <math>{}^{\backprime\backprime} (x_1, \ldots, \hat{j}, \ldots, x_k) {}^{\prime\prime},</math> then <math>{}^{\backprime\backprime} L {}^{\prime\prime}</math> can be interpreted as naming a function from the cartesian product of the domains at the filled places to the power set of the domain at the missing place. With this in mind, it is permissible to let <math>{}^{\backprime\backprime} L : X_1 \times \ldots \times \hat{j} \times \ldots \times X_k \to \operatorname{Pow}(X_j) {}^{\prime\prime}</math> indicate this use of <math>{}^{\backprime\backprime} L {}^{\prime\prime}.</math> If the sets in the range of this function are all singletons, then it is permissible to let <math>{}^{\backprime\backprime} L : X_1 \times \ldots \times \hat{j} \times \ldots \times X_k \to X_j {}^{\prime\prime}</math> specify the corresponding use of <math>{}^{\backprime\backprime} L {}^{\prime\prime}.</math>
Second, <math>L\!</math> can be associated with a piece of information that allows one to complete various sorts of partial data sets in the space of effects. In particular, if one is given a partial effect or an incomplete <math>k\!</math>-tuple, say, one that is missing a value in the <math>j^\text{th}\!</math> place, as indicated by the notation <math>{}^{\backprime\backprime} (x_1, \ldots, \hat{j}, \ldots, x_k) {}^{\prime\prime},</math> then <math>{}^{\backprime\backprime} L {}^{\prime\prime}</math> can be interpreted as naming a function from the cartesian product of the domains at the filled places to the power set of the domain at the missing place. With this in mind, it is permissible to let <math>{}^{\backprime\backprime} L : X_1 \times \ldots \times \hat{j} \times \ldots \times X_k \to \mathrm{Pow}(X_j) {}^{\prime\prime}</math> indicate this use of <math>{}^{\backprime\backprime} L {}^{\prime\prime}.</math> If the sets in the range of this function are all singletons, then it is permissible to let <math>{}^{\backprime\backprime} L : X_1 \times \ldots \times \hat{j} \times \ldots \times X_k \to X_j {}^{\prime\prime}</math> specify the corresponding use of <math>{}^{\backprime\backprime} L {}^{\prime\prime}.</math>
In general, the indicated degrees of freedom in the interpretation of relation symbols can be exploited properly only if one understands the consequences of this interpretive liberality and is prepared to deal with the problems that arise from its “polymorphic” practices — from using the same sign in different contexts to refer to different types of objects. For example, one should consider what happens, and what sort of IF is demanded to deal with it, when the name <math>{}^{\backprime\backprime} L {}^{\prime\prime}</math> is used equivocally in a statement like <math>L = L^{-1}(1),\!</math> where a sensible reading requires it to denote the relational set <math>L \subseteq \textstyle\prod_i X_i</math> on the first appearance and the propositional function <math>L : \textstyle\prod_i X_i \to \mathbb{B}</math> on the second appearance.
In general, the indicated degrees of freedom in the interpretation of relation symbols can be exploited properly only if one understands the consequences of this interpretive liberality and is prepared to deal with the problems that arise from its “polymorphic” practices — from using the same sign in different contexts to refer to different types of objects. For example, one should consider what happens, and what sort of IF is demanded to deal with it, when the name <math>{}^{\backprime\backprime} L {}^{\prime\prime}</math> is used equivocally in a statement like <math>L = L^{-1}(1),\!</math> where a sensible reading requires it to denote the relational set <math>L \subseteq \textstyle\prod_i X_i</math> on the first appearance and the propositional function <math>L : \textstyle\prod_i X_i \to \mathbb{B}</math> on the second appearance.
A '''triadic relation''' is a relation on an ordered triple of nonempty sets. Thus, <math>L\!</math> is a triadic relation on <math>(X, Y, Z)\!</math> if and only if <math>L \subseteq X \times Y \times Z.\!</math> Exercising a proper degree of flexibility with notation, one can use the name of a triadic relation <math>L \subseteq X \times Y \times Z\!</math> to refer to a logical predicate or a propositional function, of the type <math>X \times Y \times Z \to \mathbb{B},\!</math> or any one of the derived binary operations, of the three types <math>X \times Y \to \operatorname{Pow}(Z),\!</math> <math>X \times Z \to \operatorname{Pow}(Y),\!</math> and <math>Y \times Z \to \operatorname{Pow}(X).\!</math>
A '''triadic relation''' is a relation on an ordered triple of nonempty sets. Thus, <math>L\!</math> is a triadic relation on <math>(X, Y, Z)\!</math> if and only if <math>L \subseteq X \times Y \times Z.\!</math> Exercising a proper degree of flexibility with notation, one can use the name of a triadic relation <math>L \subseteq X \times Y \times Z\!</math> to refer to a logical predicate or a propositional function, of the type <math>X \times Y \times Z \to \mathbb{B},\!</math> or any one of the derived binary operations, of the three types <math>X \times Y \to \mathrm{Pow}(Z),\!</math> <math>X \times Z \to \mathrm{Pow}(Y),\!</math> and <math>Y \times Z \to \mathrm{Pow}(X).\!</math>
A '''binary operation''' or '''law of composition''' (LOC) on a nonempty set <math>X\!</math> is a triadic relation <math>* \subseteq X \times X \times X\!</math> that is also a function <math>* : X \times X \to X.\!</math> The notation <math>{}^{\backprime\backprime} x * y {}^{\prime\prime}\!</math> is used to indicate the functional value <math>*(x, y) \in X,~\!</math> which is also referred to as the '''product''' of <math>x\!</math> and <math>y\!</math> under <math>*.\!</math>
A '''binary operation''' or '''law of composition''' (LOC) on a nonempty set <math>X\!</math> is a triadic relation <math>* \subseteq X \times X \times X\!</math> that is also a function <math>* : X \times X \to X.\!</math> The notation <math>{}^{\backprime\backprime} x * y {}^{\prime\prime}\!</math> is used to indicate the functional value <math>*(x, y) \in X,~\!</math> which is also referred to as the '''product''' of <math>x\!</math> and <math>y\!</math> under <math>*.\!</math>
|- style="height:50px"
|- style="height:50px"
| width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
| width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{e}\!</math>
| width="20%" style="border-bottom:1px solid black" | <math>\mathrm{e}\!</math>
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{f}\!</math>
| width="20%" style="border-bottom:1px solid black" | <math>\mathrm{f}\!</math>
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{g}\!</math>
| width="20%" style="border-bottom:1px solid black" | <math>\mathrm{g}\!</math>
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{h}\!</math>
| width="20%" style="border-bottom:1px solid black" | <math>\mathrm{h}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{e}\!</math>
| style="border-right:1px solid black" | <math>\mathrm{e}\!</math>
| <math>\operatorname{e}\!</math>
| <math>\mathrm{e}\!</math>
| <math>\operatorname{f}\!</math>
| <math>\mathrm{f}\!</math>
| <math>\operatorname{g}\!</math>
| <math>\mathrm{g}\!</math>
| <math>\operatorname{h}\!</math>
| <math>\mathrm{h}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{f}\!</math>
| style="border-right:1px solid black" | <math>\mathrm{f}\!</math>
| <math>\operatorname{f}\!</math>
| <math>\mathrm{f}\!</math>
| <math>\operatorname{e}\!</math>
| <math>\mathrm{e}\!</math>
| <math>\operatorname{h}\!</math>
| <math>\mathrm{h}\!</math>
| <math>\operatorname{g}\!</math>
| <math>\mathrm{g}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{g}\!</math>
| style="border-right:1px solid black" | <math>\mathrm{g}\!</math>
| <math>\operatorname{g}\!</math>
| <math>\mathrm{g}\!</math>
| <math>\operatorname{h}\!</math>
| <math>\mathrm{h}\!</math>
| <math>\operatorname{e}\!</math>
| <math>\mathrm{e}\!</math>
| <math>\operatorname{f}\!</math>
| <math>\mathrm{f}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{h}\!</math>
| style="border-right:1px solid black" | <math>\mathrm{h}\!</math>
| <math>\operatorname{h}\!</math>
| <math>\mathrm{h}\!</math>
| <math>\operatorname{g}\!</math>
| <math>\mathrm{g}\!</math>
| <math>\operatorname{f}\!</math>
| <math>\mathrm{f}\!</math>
| <math>\operatorname{e}\!</math>
| <math>\mathrm{e}\!</math>
|}
|}
| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
|- style="height:50px"
|- style="height:50px"
| width="20%" style="border-right:1px solid black" | <math>\operatorname{e}\!</math>
| width="20%" style="border-right:1px solid black" | <math>\mathrm{e}\!</math>
| width="4%" | <math>\{\!</math>
| width="4%" | <math>\{\!</math>
| width="16%" | <math>(\operatorname{e}, \operatorname{e}),\!</math>
| width="16%" | <math>(\mathrm{e}, \mathrm{e}),\!</math>
| width="20%" | <math>(\operatorname{f}, \operatorname{f}),\!</math>
| width="20%" | <math>(\mathrm{f}, \mathrm{f}),\!</math>
| width="20%" | <math>(\operatorname{g}, \operatorname{g}),\!</math>
| width="20%" | <math>(\mathrm{g}, \mathrm{g}),\!</math>
| width="16%" | <math>(\operatorname{h}, \operatorname{h})\!</math>
| width="16%" | <math>(\mathrm{h}, \mathrm{h})\!</math>
| width="4%" | <math>\}\!</math>
| width="4%" | <math>\}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{f}\!</math>
| style="border-right:1px solid black" | <math>\mathrm{f}\!</math>
| <math>\{\!</math>
| <math>\{\!</math>
| <math>(\operatorname{e}, \operatorname{f}),\!</math>
| <math>(\mathrm{e}, \mathrm{f}),\!</math>
| <math>(\operatorname{f}, \operatorname{e}),\!</math>
| <math>(\mathrm{f}, \mathrm{e}),\!</math>
| <math>(\operatorname{g}, \operatorname{h}),\!</math>
| <math>(\mathrm{g}, \mathrm{h}),\!</math>
| <math>(\operatorname{h}, \operatorname{g})\!</math>
| <math>(\mathrm{h}, \mathrm{g})\!</math>
| <math>\}\!</math>
| <math>\}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{g}\!</math>
| style="border-right:1px solid black" | <math>\mathrm{g}\!</math>
| <math>\{\!</math>
| <math>\{\!</math>
| <math>(\operatorname{e}, \operatorname{g}),\!</math>
| <math>(\mathrm{e}, \mathrm{g}),\!</math>
| <math>(\operatorname{f}, \operatorname{h}),\!</math>
| <math>(\mathrm{f}, \mathrm{h}),\!</math>
| <math>(\operatorname{g}, \operatorname{e}),\!</math>
| <math>(\mathrm{g}, \mathrm{e}),\!</math>
| <math>(\operatorname{h}, \operatorname{f})\!</math>
| <math>(\mathrm{h}, \mathrm{f})\!</math>
| <math>\}\!</math>
| <math>\}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{h}\!</math>
| style="border-right:1px solid black" | <math>\mathrm{h}\!</math>
| <math>\{\!</math>
| <math>\{\!</math>
| <math>(\operatorname{e}, \operatorname{h}),\!</math>
| <math>(\mathrm{e}, \mathrm{h}),\!</math>
| <math>(\operatorname{f}, \operatorname{g}),\!</math>
| <math>(\mathrm{f}, \mathrm{g}),\!</math>
| <math>(\operatorname{g}, \operatorname{f}),\!</math>
| <math>(\mathrm{g}, \mathrm{f}),\!</math>
| <math>(\operatorname{h}, \operatorname{e})\!</math>
| <math>(\mathrm{h}, \mathrm{e})\!</math>
| <math>\}\!</math>
| <math>\}\!</math>
|}
|}
| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Symbols}\!</math>
| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Symbols}\!</math>
|- style="height:50px"
|- style="height:50px"
| width="20%" style="border-right:1px solid black" | <math>\operatorname{e}\!</math>
| width="20%" style="border-right:1px solid black" | <math>\mathrm{e}\!</math>
| width="4%" | <math>\{\!</math>
| width="4%" | <math>\{\!</math>
| width="16%" | <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),\!</math>
| width="16%" | <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),\!</math>
| width="4%" | <math>\}\!</math>
| width="4%" | <math>\}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{f}\!</math>
| style="border-right:1px solid black" | <math>\mathrm{f}\!</math>
| <math>\{\!</math>
| <math>\{\!</math>
| <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),\!</math>
| <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),\!</math>
| <math>\}\!</math>
| <math>\}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{g}\!</math>
| style="border-right:1px solid black" | <math>\mathrm{g}\!</math>
| <math>\{\!</math>
| <math>\{\!</math>
| <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),\!</math>
| <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),\!</math>
| <math>\}\!</math>
| <math>\}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{h}\!</math>
| style="border-right:1px solid black" | <math>\mathrm{h}\!</math>
| <math>\{\!</math>
| <math>\{\!</math>
| <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),\!</math>
| <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),\!</math>
|- style="height:50px"
|- style="height:50px"
| width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
| width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{1}</math>
| width="20%" style="border-bottom:1px solid black" | <math>\mathrm{1}</math>
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{a}</math>
| width="20%" style="border-bottom:1px solid black" | <math>\mathrm{a}</math>
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{b}</math>
| width="20%" style="border-bottom:1px solid black" | <math>\mathrm{b}</math>
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{c}</math>
| width="20%" style="border-bottom:1px solid black" | <math>\mathrm{c}</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{1}</math>
| style="border-right:1px solid black" | <math>\mathrm{1}</math>
| <math>\operatorname{1}</math>
| <math>\mathrm{1}</math>
| <math>\operatorname{a}</math>
| <math>\mathrm{a}</math>
| <math>\operatorname{b}</math>
| <math>\mathrm{b}</math>
| <math>\operatorname{c}</math>
| <math>\mathrm{c}</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{a}</math>
| style="border-right:1px solid black" | <math>\mathrm{a}</math>
| <math>\operatorname{a}</math>
| <math>\mathrm{a}</math>
| <math>\operatorname{b}</math>
| <math>\mathrm{b}</math>
| <math>\operatorname{c}</math>
| <math>\mathrm{c}</math>
| <math>\operatorname{1}</math>
| <math>\mathrm{1}</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{b}</math>
| style="border-right:1px solid black" | <math>\mathrm{b}</math>
| <math>\operatorname{b}</math>
| <math>\mathrm{b}</math>
| <math>\operatorname{c}</math>
| <math>\mathrm{c}</math>
| <math>\operatorname{1}</math>
| <math>\mathrm{1}</math>
| <math>\operatorname{a}</math>
| <math>\mathrm{a}</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{c}</math>
| style="border-right:1px solid black" | <math>\mathrm{c}</math>
| <math>\operatorname{c}</math>
| <math>\mathrm{c}</math>
| <math>\operatorname{1}</math>
| <math>\mathrm{1}</math>
| <math>\operatorname{a}</math>
| <math>\mathrm{a}</math>
| <math>\operatorname{b}</math>
| <math>\mathrm{b}</math>
|}
|}
| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
|- style="height:50px"
|- style="height:50px"
| width="20%" style="border-right:1px solid black" | <math>\operatorname{1}\!</math>
| width="20%" style="border-right:1px solid black" | <math>\mathrm{1}\!</math>
| width="4%" | <math>\{\!</math>
| width="4%" | <math>\{\!</math>
| width="16%" | <math>(\operatorname{1}, \operatorname{1}),\!</math>
| width="16%" | <math>(\mathrm{1}, \mathrm{1}),\!</math>
| width="20%" | <math>(\operatorname{a}, \operatorname{a}),\!</math>
| width="20%" | <math>(\mathrm{a}, \mathrm{a}),\!</math>
| width="20%" | <math>(\operatorname{b}, \operatorname{b}),\!</math>
| width="20%" | <math>(\mathrm{b}, \mathrm{b}),\!</math>
| width="16%" | <math>(\operatorname{c}, \operatorname{c})\!</math>
| width="16%" | <math>(\mathrm{c}, \mathrm{c})\!</math>
| width="4%" | <math>\}\!</math>
| width="4%" | <math>\}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{a}\!</math>
| style="border-right:1px solid black" | <math>\mathrm{a}\!</math>
| <math>\{\!</math>
| <math>\{\!</math>
| <math>(\operatorname{1}, \operatorname{a}),\!</math>
| <math>(\mathrm{1}, \mathrm{a}),\!</math>
| <math>(\operatorname{a}, \operatorname{b}),\!</math>
| <math>(\mathrm{a}, \mathrm{b}),\!</math>
| <math>(\operatorname{b}, \operatorname{c}),\!</math>
| <math>(\mathrm{b}, \mathrm{c}),\!</math>
| <math>(\operatorname{c}, \operatorname{1})\!</math>
| <math>(\mathrm{c}, \mathrm{1})\!</math>
| <math>\}\!</math>
| <math>\}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{b}\!</math>
| style="border-right:1px solid black" | <math>\mathrm{b}\!</math>
| <math>\{\!</math>
| <math>\{\!</math>
| <math>(\operatorname{1}, \operatorname{b}),\!</math>
| <math>(\mathrm{1}, \mathrm{b}),\!</math>
| <math>(\operatorname{a}, \operatorname{c}),\!</math>
| <math>(\mathrm{a}, \mathrm{c}),\!</math>
| <math>(\operatorname{b}, \operatorname{1}),\!</math>
| <math>(\mathrm{b}, \mathrm{1}),\!</math>
| <math>(\operatorname{c}, \operatorname{a})\!</math>
| <math>(\mathrm{c}, \mathrm{a})\!</math>
| <math>\}\!</math>
| <math>\}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{c}\!</math>
| style="border-right:1px solid black" | <math>\mathrm{c}\!</math>
| <math>\{\!</math>
| <math>\{\!</math>
| <math>(\operatorname{1}, \operatorname{c}),\!</math>
| <math>(\mathrm{1}, \mathrm{c}),\!</math>
| <math>(\operatorname{a}, \operatorname{1}),\!</math>
| <math>(\mathrm{a}, \mathrm{1}),\!</math>
| <math>(\operatorname{b}, \operatorname{a}),\!</math>
| <math>(\mathrm{b}, \mathrm{a}),\!</math>
| <math>(\operatorname{c}, \operatorname{b})\!</math>
| <math>(\mathrm{c}, \mathrm{b})\!</math>
| <math>\}\!</math>
| <math>\}\!</math>
|}
|}
|- style="height:50px"
|- style="height:50px"
| width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>+\!</math>
| width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>+\!</math>
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{0}\!</math>
| width="20%" style="border-bottom:1px solid black" | <math>\mathrm{0}\!</math>
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{1}\!</math>
| width="20%" style="border-bottom:1px solid black" | <math>\mathrm{1}\!</math>
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{2}\!</math>
| width="20%" style="border-bottom:1px solid black" | <math>\mathrm{2}\!</math>
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{3}\!</math>
| width="20%" style="border-bottom:1px solid black" | <math>\mathrm{3}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{0}\!</math>
| style="border-right:1px solid black" | <math>\mathrm{0}\!</math>
| <math>\operatorname{0}\!</math>
| <math>\mathrm{0}\!</math>
| <math>\operatorname{1}\!</math>
| <math>\mathrm{1}\!</math>
| <math>\operatorname{2}\!</math>
| <math>\mathrm{2}\!</math>
| <math>\operatorname{3}\!</math>
| <math>\mathrm{3}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{1}\!</math>
| style="border-right:1px solid black" | <math>\mathrm{1}\!</math>
| <math>\operatorname{1}\!</math>
| <math>\mathrm{1}\!</math>
| <math>\operatorname{2}\!</math>
| <math>\mathrm{2}\!</math>
| <math>\operatorname{3}\!</math>
| <math>\mathrm{3}\!</math>
| <math>\operatorname{0}\!</math>
| <math>\mathrm{0}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{2}\!</math>
| style="border-right:1px solid black" | <math>\mathrm{2}\!</math>
| <math>\operatorname{2}\!</math>
| <math>\mathrm{2}\!</math>
| <math>\operatorname{3}\!</math>
| <math>\mathrm{3}\!</math>
| <math>\operatorname{0}\!</math>
| <math>\mathrm{0}\!</math>
| <math>\operatorname{1}\!</math>
| <math>\mathrm{1}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{3}\!</math>
| style="border-right:1px solid black" | <math>\mathrm{3}\!</math>
| <math>\operatorname{3}\!</math>
| <math>\mathrm{3}\!</math>
| <math>\operatorname{0}\!</math>
| <math>\mathrm{0}\!</math>
| <math>\operatorname{1}\!</math>
| <math>\mathrm{1}\!</math>
| <math>\operatorname{2}\!</math>
| <math>\mathrm{2}\!</math>
|}
|}
| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
|- style="height:50px"
|- style="height:50px"
| width="20%" style="border-right:1px solid black" | <math>\operatorname{0}\!</math>
| width="20%" style="border-right:1px solid black" | <math>\mathrm{0}\!</math>
| width="4%" | <math>\{\!</math>
| width="4%" | <math>\{\!</math>
| width="16%" | <math>(\operatorname{0}, \operatorname{0}),\!</math>
| width="16%" | <math>(\mathrm{0}, \mathrm{0}),\!</math>
| width="20%" | <math>(\operatorname{1}, \operatorname{1}),\!</math>
| width="20%" | <math>(\mathrm{1}, \mathrm{1}),\!</math>
| width="20%" | <math>(\operatorname{2}, \operatorname{2}),\!</math>
| width="20%" | <math>(\mathrm{2}, \mathrm{2}),\!</math>
| width="16%" | <math>(\operatorname{3}, \operatorname{3})~\!</math>
| width="16%" | <math>(\mathrm{3}, \mathrm{3})~\!</math>
| width="4%" | <math>\}\!</math>
| width="4%" | <math>\}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{1}\!</math>
| style="border-right:1px solid black" | <math>\mathrm{1}\!</math>
| <math>\{\!</math>
| <math>\{\!</math>
| <math>(\operatorname{0}, \operatorname{1}),\!</math>
| <math>(\mathrm{0}, \mathrm{1}),\!</math>
| <math>(\operatorname{1}, \operatorname{2}),\!</math>
| <math>(\mathrm{1}, \mathrm{2}),\!</math>
| <math>(\operatorname{2}, \operatorname{3}),\!</math>
| <math>(\mathrm{2}, \mathrm{3}),\!</math>
| <math>(\operatorname{3}, \operatorname{0})\!</math>
| <math>(\mathrm{3}, \mathrm{0})\!</math>
| <math>\}\!</math>
| <math>\}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{2}\!</math>
| style="border-right:1px solid black" | <math>\mathrm{2}\!</math>
| <math>\{\!</math>
| <math>\{\!</math>
| <math>(\operatorname{0}, \operatorname{2}),\!</math>
| <math>(\mathrm{0}, \mathrm{2}),\!</math>
| <math>(\operatorname{1}, \operatorname{3}),\!</math>
| <math>(\mathrm{1}, \mathrm{3}),\!</math>
| <math>(\operatorname{2}, \operatorname{0}),\!</math>
| <math>(\mathrm{2}, \mathrm{0}),\!</math>
| <math>(\operatorname{3}, \operatorname{1})\!</math>
| <math>(\mathrm{3}, \mathrm{1})\!</math>
| <math>\}\!</math>
| <math>\}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{3}\!</math>
| style="border-right:1px solid black" | <math>\mathrm{3}\!</math>
| <math>\{\!</math>
| <math>\{\!</math>
| <math>(\operatorname{0}, \operatorname{3}),\!</math>
| <math>(\mathrm{0}, \mathrm{3}),\!</math>
| <math>(\operatorname{1}, \operatorname{0}),\!</math>
| <math>(\mathrm{1}, \mathrm{0}),\!</math>
| <math>(\operatorname{2}, \operatorname{1}),\!</math>
| <math>(\mathrm{2}, \mathrm{1}),\!</math>
| <math>(\operatorname{3}, \operatorname{2})\!</math>
| <math>(\mathrm{3}, \mathrm{2})\!</math>
| <math>\}\!</math>
| <math>\}\!</math>
|}
|}
By way of definition, a sign <math>q\!</math> in a sign relation <math>L \subseteq O \times S \times I\!</math> is said to be, to constitute, or to make a '''plural indefinite reference''' ('''PIR''') to (every element in) a set of objects, <math>X \subseteq O,\!</math> if and only if <math>q\!</math> denotes every element of <math>X.\!</math> This relationship can be expressed in a succinct formula by making use of one additional definition.
By way of definition, a sign <math>q\!</math> in a sign relation <math>L \subseteq O \times S \times I\!</math> is said to be, to constitute, or to make a '''plural indefinite reference''' ('''PIR''') to (every element in) a set of objects, <math>X \subseteq O,\!</math> if and only if <math>q\!</math> denotes every element of <math>X.\!</math> This relationship can be expressed in a succinct formula by making use of one additional definition.
The '''denotation''' of <math>q\!</math> in <math>L,\!</math> written <math>\operatorname{De}(q, L),\!</math> is defined as follows:
The '''denotation''' of <math>q\!</math> in <math>L,\!</math> written <math>\mathrm{De}(q, L),\!</math> is defined as follows:
{| align="center" cellspacing="8" width="90%"
{| align="center" cellspacing="8" width="90%"
| <math>\operatorname{De}(q, L) ~=~ \operatorname{Den}(L) \cdot q ~=~ L_{OS} \cdot q ~=~ \{ o \in O : (o, q, i) \in L, ~\text{for some}~ i \in I \}.</math>
| <math>\mathrm{De}(q, L) ~=~ \mathrm{Den}(L) \cdot q ~=~ L_{OS} \cdot q ~=~ \{ o \in O : (o, q, i) \in L, ~\text{for some}~ i \in I \}.</math>
|}
|}
Then <math>q\!</math> makes a PIR to <math>X\!</math> in <math>L\!</math> if and only if <math>X \subseteq \operatorname{De}(q, L).\!</math> Of course, this includes the limiting case where <math>X\!</math> is a singleton, say <math>X = \{ o \}.\!</math> In this case the reference is neither plural nor indefinite, properly speaking, but <math>q\!</math> denotes <math>o\!</math> uniquely.
Then <math>q\!</math> makes a PIR to <math>X\!</math> in <math>L\!</math> if and only if <math>X \subseteq \mathrm{De}(q, L).\!</math> Of course, this includes the limiting case where <math>X\!</math> is a singleton, say <math>X = \{ o \}.\!</math> In this case the reference is neither plural nor indefinite, properly speaking, but <math>q\!</math> denotes <math>o\!</math> uniquely.
The proper exploitation of PIRs in sign relations makes it possible to finesse the distinction between HI signs and HU signs, in other words, to provide a ready means of translating between the two kinds of signs that preserves all the relevant information, at least, for many purposes. This is accomplished by connecting the sides of the distinction in two directions. First, a HI sign that makes a PIR to many triples of the form <math>(o, s, i)\!</math> can be taken as tantamount to a HU sign that denotes the corresponding sign relation. Second, a HU sign that denotes a singleton sign relation can be taken as tantamount to a HI sign that denotes its single triple. The relation of one sign being “tantamount to” another is not exactly a full-fledged semantic equivalence, but it is a reasonable approximation to it, and one that serves a number of practical purposes.
The proper exploitation of PIRs in sign relations makes it possible to finesse the distinction between HI signs and HU signs, in other words, to provide a ready means of translating between the two kinds of signs that preserves all the relevant information, at least, for many purposes. This is accomplished by connecting the sides of the distinction in two directions. First, a HI sign that makes a PIR to many triples of the form <math>(o, s, i)\!</math> can be taken as tantamount to a HU sign that denotes the corresponding sign relation. Second, a HU sign that denotes a singleton sign relation can be taken as tantamount to a HI sign that denotes its single triple. The relation of one sign being “tantamount to” another is not exactly a full-fledged semantic equivalence, but it is a reasonable approximation to it, and one that serves a number of practical purposes.
In ordinary discourse HA signs are usually generated by means of linguistic devices for quoting pieces of text. In computational frameworks these quoting mechanisms are implemented as functions that map syntactic arguments into numerical or syntactic values. A quoting function, given a sign or expression as its single argument, needs to accomplish two things: first, to defer the reference of that sign, in other words, to inhibit, delay, or prevent the evaluation of its argument expression, and then, to exhibit or produce another sign whose object is precisely that argument expression.
In ordinary discourse HA signs are usually generated by means of linguistic devices for quoting pieces of text. In computational frameworks these quoting mechanisms are implemented as functions that map syntactic arguments into numerical or syntactic values. A quoting function, given a sign or expression as its single argument, needs to accomplish two things: first, to defer the reference of that sign, in other words, to inhibit, delay, or prevent the evaluation of its argument expression, and then, to exhibit or produce another sign whose object is precisely that argument expression.
The rest of this section considers the development of sign relations that have moderate capacities to reference their own signs as objects. In each case, these extensions are assumed to begin with sign relations like <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> that have disjoint sets of objects and signs and thus have no reflective capacity at the outset. The status of <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> as the reflective origins of the associated reflective developments is recalled by saying that <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> themselves are the ''zeroth order reflective extensions'' of <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> in symbols, <math>L(\text{A}) = \operatorname{Ref}^0 L(\text{A})\!</math> and <math>L(\text{B}) = \operatorname{Ref}^0 L(\text{B}).\!</math>
The rest of this section considers the development of sign relations that have moderate capacities to reference their own signs as objects. In each case, these extensions are assumed to begin with sign relations like <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> that have disjoint sets of objects and signs and thus have no reflective capacity at the outset. The status of <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> as the reflective origins of the associated reflective developments is recalled by saying that <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> themselves are the ''zeroth order reflective extensions'' of <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> in symbols, <math>L(\text{A}) = \mathrm{Ref}^0 L(\text{A})\!</math> and <math>L(\text{B}) = \mathrm{Ref}^0 L(\text{B}).\!</math>
The next set of Tables illustrates a few the most common ways that sign relations can begin to develop reflective extensions. For ease of reference, Tables 40 and 41 repeat the contents of Tables 1 and 2, respectively, merely replacing ordinary quotes with arch quotes.
The next set of Tables illustrates a few the most common ways that sign relations can begin to develop reflective extensions. For ease of reference, Tables 40 and 41 repeat the contents of Tables 1 and 2, respectively, merely replacing ordinary quotes with arch quotes.
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table 40.} ~~ \text{Reflective Origin} ~ \operatorname{Ref}^0 L(\text{A})\!</math>
|+ style="height:30px" | <math>\text{Table 40.} ~~ \text{Reflective Origin} ~ \mathrm{Ref}^0 L(\text{A})\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| <math>\text{Object}\!</math>
| <math>\text{Object}\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table 41.} ~~ \text{Reflective Origin} ~ \operatorname{Ref}^0 L(\text{B})\!</math>
|+ style="height:30px" | <math>\text{Table 41.} ~~ \text{Reflective Origin} ~ \mathrm{Ref}^0 L(\text{B})\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| <math>\text{Object}\!</math>
| <math>\text{Object}\!</math>
<br>
<br>
Tables 42 and 43 show one way that the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> can be extended in a reflective sense through the use of quotational devices, yielding the ''first order reflective extensions'', <math>\operatorname{Ref}^1 L(\text{A})\!</math> and <math>\operatorname{Ref}^1 L(\text{B}).\!</math> These extensions add one layer of HA signs and their objects to the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> respectively. The new triples specify that, for each <math>{}^{\langle} x {}^{\rangle}\!</math> in the set <math>\{ {}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{i} {}^{\rangle}, {}^{\langle} \text{u} {}^{\rangle} \},\!</math> the HA sign of the form <math>{}^{\langle\langle} x {}^{\rangle\rangle}\!</math> connotes itself while denoting <math>{}^{\langle} x {}^{\rangle}.\!</math>
Tables 42 and 43 show one way that the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> can be extended in a reflective sense through the use of quotational devices, yielding the ''first order reflective extensions'', <math>\mathrm{Ref}^1 L(\text{A})\!</math> and <math>\mathrm{Ref}^1 L(\text{B}).\!</math> These extensions add one layer of HA signs and their objects to the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> respectively. The new triples specify that, for each <math>{}^{\langle} x {}^{\rangle}\!</math> in the set <math>\{ {}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{i} {}^{\rangle}, {}^{\langle} \text{u} {}^{\rangle} \},\!</math> the HA sign of the form <math>{}^{\langle\langle} x {}^{\rangle\rangle}\!</math> connotes itself while denoting <math>{}^{\langle} x {}^{\rangle}.\!</math>
Notice that the semantic equivalences of nouns and pronouns referring to each interpreter do not extend to semantic equivalences of their higher order signs, exactly as demanded by the literal character of quotations. Also notice that the reflective extensions of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> coincide in their reflective parts, since exactly the same triples were added to each set.
Notice that the semantic equivalences of nouns and pronouns referring to each interpreter do not extend to semantic equivalences of their higher order signs, exactly as demanded by the literal character of quotations. Also notice that the reflective extensions of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> coincide in their reflective parts, since exactly the same triples were added to each set.
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table 42.} ~~ \text{Higher Ascent Sign Relation} ~ \operatorname{Ref}^1 L(\text{A})\!</math>
|+ style="height:30px" | <math>\text{Table 42.} ~~ \text{Higher Ascent Sign Relation} ~ \mathrm{Ref}^1 L(\text{A})\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| <math>\text{Object}\!</math>
| <math>\text{Object}\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table 43.} ~~ \text{Higher Ascent Sign Relation} ~ \operatorname{Ref}^1 L(\text{B})\!</math>
|+ style="height:30px" | <math>\text{Table 43.} ~~ \text{Higher Ascent Sign Relation} ~ \mathrm{Ref}^1 L(\text{B})\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| <math>\text{Object}\!</math>
| <math>\text{Object}\!</math>
<br>
<br>
There are many ways to extend sign relations in an effort to develop their reflective capacities. The implicit goal of a reflective project is to reach a condition of ''reflective closure'', a configuration satisfying the inclusion <math>S \subseteq O,\!</math> where every sign is an object. It is important to note that not every process of reflective extension can achieve a reflective closure in a finite sign relation. This can only happen if there are additional equivalence relations that keep the effective orders of signs within finite bounds. As long as there are higher order signs that remain distinct from all lower order signs, the sign relation driven by a reflective process is forced to keep expanding. In particular, the process that is ''freely'' suggested by the formation of <math>\operatorname{Ref}^1 L(\text{A})~\!</math> and <math>\operatorname{Ref}^1 L(\text{B})~\!</math> cannot reach closure if it continues as indicated, without further constraints.
There are many ways to extend sign relations in an effort to develop their reflective capacities. The implicit goal of a reflective project is to reach a condition of ''reflective closure'', a configuration satisfying the inclusion <math>S \subseteq O,\!</math> where every sign is an object. It is important to note that not every process of reflective extension can achieve a reflective closure in a finite sign relation. This can only happen if there are additional equivalence relations that keep the effective orders of signs within finite bounds. As long as there are higher order signs that remain distinct from all lower order signs, the sign relation driven by a reflective process is forced to keep expanding. In particular, the process that is ''freely'' suggested by the formation of <math>\mathrm{Ref}^1 L(\text{A})~\!</math> and <math>\mathrm{Ref}^1 L(\text{B})~\!</math> cannot reach closure if it continues as indicated, without further constraints.
Tables 44 and 45 present ''higher import extensions'' of <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> respectively. These are just higher order sign relations that add selections of higher import signs and their objects to the underlying set of triples in <math>L(\text{A})\!</math> and <math>L(\text{B}).\!</math> One way to understand these extensions is as follows. The interpreters <math>\text{A}\!</math> and <math>\text{B}\!</math> each use nouns and pronouns just as before, except that the nouns are given additional denotations that refer to the interpretive conduct of the interpreter named. In this form of development, using a noun as a canonical form that refers indifferently to all the <math>(o, s, i)\!</math> triples of a sign relation is a pragmatic way that a sign relation can refer to itself and to other sign relations.
Tables 44 and 45 present ''higher import extensions'' of <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> respectively. These are just higher order sign relations that add selections of higher import signs and their objects to the underlying set of triples in <math>L(\text{A})\!</math> and <math>L(\text{B}).\!</math> One way to understand these extensions is as follows. The interpreters <math>\text{A}\!</math> and <math>\text{B}\!</math> each use nouns and pronouns just as before, except that the nouns are given additional denotations that refer to the interpretive conduct of the interpreter named. In this form of development, using a noun as a canonical form that refers indifferently to all the <math>(o, s, i)\!</math> triples of a sign relation is a pragmatic way that a sign relation can refer to itself and to other sign relations.
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table 44.} ~~ \text{Higher Import Sign Relation} ~ \operatorname{HI}^1 L(\text{A})\!</math>
|+ style="height:30px" | <math>\text{Table 44.} ~~ \text{Higher Import Sign Relation} ~ \mathrm{HI}^1 L(\text{A})\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| <math>\text{Object}\!</math>
| <math>\text{Object}\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table 45.} ~~ \text{Higher Import Sign Relation} ~ \operatorname{HI}^1 L(\text{B})\!</math>
|+ style="height:30px" | <math>\text{Table 45.} ~~ \text{Higher Import Sign Relation} ~ \mathrm{HI}^1 L(\text{B})\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| <math>\text{Object}\!</math>
| <math>\text{Object}\!</math>
<br>
<br>
Several important facts about the class of higher order sign relations in general are illustrated by these examples. First, the notations appearing in the object columns of <math>\operatorname{HI}^1 L(\text{A})\!</math> and <math>\operatorname{HI}^1 L(\text{B})\!</math> are not the terms that these newly extended interpreters are depicted as using to describe their objects, but the kinds of language that you and I, or other external observers, would typically make available to distinguish them. The sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> as extended by the transactions of <math>\operatorname{HI}^1 L(\text{A})\!</math> and <math>\operatorname{HI}^1 L(\text{B}),\!</math> respectively, are still restricted to their original syntactic domain <math>\{ {}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{i} {}^{\rangle}, {}^{\langle} \text{u} {}^{\rangle} \}.\!</math> This means that there need be nothing especially articulate about a HI sign relation just because it qualifies as higher order. Indeed, the sign relations <math>\operatorname{HI}^1 L(\text{A})\!</math> and <math>\operatorname{HI}^1 L(\text{B})\!</math> are not very discriminating in their descriptions of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> referring to many different things under the very same signs that you and I and others would explicitly distinguish, especially in marking the distinction between an interpretive agent and any one of its individual transactions.
Several important facts about the class of higher order sign relations in general are illustrated by these examples. First, the notations appearing in the object columns of <math>\mathrm{HI}^1 L(\text{A})\!</math> and <math>\mathrm{HI}^1 L(\text{B})\!</math> are not the terms that these newly extended interpreters are depicted as using to describe their objects, but the kinds of language that you and I, or other external observers, would typically make available to distinguish them. The sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> as extended by the transactions of <math>\mathrm{HI}^1 L(\text{A})\!</math> and <math>\mathrm{HI}^1 L(\text{B}),\!</math> respectively, are still restricted to their original syntactic domain <math>\{ {}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{i} {}^{\rangle}, {}^{\langle} \text{u} {}^{\rangle} \}.\!</math> This means that there need be nothing especially articulate about a HI sign relation just because it qualifies as higher order. Indeed, the sign relations <math>\mathrm{HI}^1 L(\text{A})\!</math> and <math>\mathrm{HI}^1 L(\text{B})\!</math> are not very discriminating in their descriptions of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> referring to many different things under the very same signs that you and I and others would explicitly distinguish, especially in marking the distinction between an interpretive agent and any one of its individual transactions.
In practice, it does an interpreter little good to have the higher import signs for referring to triples of objects, signs, and interpretants if it does not also have the higher ascent signs for referring to each triple's syntactic portions. Consequently, the higher order sign relations that one is likely to observe in practice are typically a mixed bag, having both higher ascent and higher import sections. Moreover, the ambiguity involved in having signs that refer equivocally to simple world elements and also to complex structures formed from these ingredients would most likely be resolved by drawing additional information from context and fashioning more distinctive signs.
In practice, it does an interpreter little good to have the higher import signs for referring to triples of objects, signs, and interpretants if it does not also have the higher ascent signs for referring to each triple's syntactic portions. Consequently, the higher order sign relations that one is likely to observe in practice are typically a mixed bag, having both higher ascent and higher import sections. Moreover, the ambiguity involved in having signs that refer equivocally to simple world elements and also to complex structures formed from these ingredients would most likely be resolved by drawing additional information from context and fashioning more distinctive signs.
The technique illustrated here represents a general strategy, one that can be exploited to derive certain benefits of set theory without having to pay the overhead that is needed to maintain sets as abstract objects. Using an identified type of a sign as a canonical form that can refer indifferently to all the members of a set is a pragmatic way of making plural reference to the members of a set without invoking the set itself as an abstract object. Of course, it is not that one can get something for nothing by these means. One is merely banking on one's recurring investment in the setting of a certain sign relation, a particular set of elementary transactions that is taken for granted as already funded.
The technique illustrated here represents a general strategy, one that can be exploited to derive certain benefits of set theory without having to pay the overhead that is needed to maintain sets as abstract objects. Using an identified type of a sign as a canonical form that can refer indifferently to all the members of a set is a pragmatic way of making plural reference to the members of a set without invoking the set itself as an abstract object. Of course, it is not that one can get something for nothing by these means. One is merely banking on one's recurring investment in the setting of a certain sign relation, a particular set of elementary transactions that is taken for granted as already funded.
As a rule, it is desirable for the grammatical system that one uses to construct and interpret higher order signs, that is, signs for referring to signs as objects, to mesh in a comfortable fashion with the overall pragmatic system that one uses to assign syntactic codes to objects in general. For future reference, I call this requirement the problem of creating a ''conformally reflective extension'' (CRE) for a given sign relation. A good way to think about this task is to imagine oneself beginning with a sign relation <math>L \subseteq O \times S \times I,\!</math> and to consider its denotative component <math>\operatorname{Den}_L = L_{OS} \subseteq O \times S.\!</math> Typically one has a ''naming function'', say <math>\operatorname{Nom},\!</math> that maps objects into signs:
As a rule, it is desirable for the grammatical system that one uses to construct and interpret higher order signs, that is, signs for referring to signs as objects, to mesh in a comfortable fashion with the overall pragmatic system that one uses to assign syntactic codes to objects in general. For future reference, I call this requirement the problem of creating a ''conformally reflective extension'' (CRE) for a given sign relation. A good way to think about this task is to imagine oneself beginning with a sign relation <math>L \subseteq O \times S \times I,\!</math> and to consider its denotative component <math>\mathrm{Den}_L = L_{OS} \subseteq O \times S.\!</math> Typically one has a ''naming function'', say <math>\mathrm{Nom},\!</math> that maps objects into signs:
{| align="center" cellspacing="8" width="90%"
{| align="center" cellspacing="8" width="90%"
| <math>\operatorname{Nom} \subseteq \operatorname{Den}_L \subseteq O \times S ~\text{such that}~ \operatorname{Nom} : O \to S.\!</math>
| <math>\mathrm{Nom} \subseteq \mathrm{Den}_L \subseteq O \times S ~\text{such that}~ \mathrm{Nom} : O \to S.\!</math>
|}
|}
Part of the task of making a sign relation more reflective is to extend it in ways that turn more of its signs into objects. This is the reason for creating higher order signs, which are just signs for making objects out of signs. One effect of progressive reflection is to extend the initial naming function <math>\operatorname{Nom}\!</math> through a succession of new naming functions <math>\operatorname{Nom}',\!</math> <math>\operatorname{Nom}'',\!</math> and so on, assigning unique names to larger allotments of the original and subsequent signs. With respect to the difficulties of construction, the ''hard core'' or ''adamant part'' of creating extended naming functions resides in the initial portion <math>\operatorname{Nom}\!</math> that maps objects of the “external world” to signs in the “internal world”. The subsequent task of assigning conventional names to signs is supposed to be comparatively natural and ''easy'', perhaps on account of the ''nominal'' nature of signs themselves.
Part of the task of making a sign relation more reflective is to extend it in ways that turn more of its signs into objects. This is the reason for creating higher order signs, which are just signs for making objects out of signs. One effect of progressive reflection is to extend the initial naming function <math>\mathrm{Nom}\!</math> through a succession of new naming functions <math>\mathrm{Nom}',\!</math> <math>\mathrm{Nom}'',\!</math> and so on, assigning unique names to larger allotments of the original and subsequent signs. With respect to the difficulties of construction, the ''hard core'' or ''adamant part'' of creating extended naming functions resides in the initial portion <math>\mathrm{Nom}\!</math> that maps objects of the “external world” to signs in the “internal world”. The subsequent task of assigning conventional names to signs is supposed to be comparatively natural and ''easy'', perhaps on account of the ''nominal'' nature of signs themselves.
The effect of reflection on the original sign relation <math>L \subseteq O \times S \times I\!</math> can be analyzed as follows. Suppose that a step of reflection creates higher order signs for a subset of <math>S.\!</math> Then this step involves the construction of a newly extended sign relation:
The effect of reflection on the original sign relation <math>L \subseteq O \times S \times I\!</math> can be analyzed as follows. Suppose that a step of reflection creates higher order signs for a subset of <math>S.\!</math> Then this step involves the construction of a newly extended sign relation:
{| align="center" cellspacing="8" width="90%"
{| align="center" cellspacing="8" width="90%"
| <math>\operatorname{Nom}_1 : O_1 \to S_1 ~\text{such that}~ \operatorname{Nom}_1 : x \mapsto {}^{\langle} x {}^{\rangle}.\!</math>
| <math>\mathrm{Nom}_1 : O_1 \to S_1 ~\text{such that}~ \mathrm{Nom}_1 : x \mapsto {}^{\langle} x {}^{\rangle}.\!</math>
|}
|}
Finally, the reflectively extended naming function <math>\operatorname{Nom}' : O' \to S'\!</math> is defined as <math>\operatorname{Nom}' = \operatorname{Nom} \cup \operatorname{Nom}_1.\!</math>
Finally, the reflectively extended naming function <math>\mathrm{Nom}' : O' \to S'\!</math> is defined as <math>\mathrm{Nom}' = \mathrm{Nom} \cup \mathrm{Nom}_1.\!</math>
A few remarks are necessary to see how this way of defining a CRE can be regarded as legitimate.
A few remarks are necessary to see how this way of defining a CRE can be regarded as legitimate.
In the present context an application of the arch notation, for example, <math>{}^{\langle} x {}^{\rangle},\!</math> is read on analogy with the use of any other functional notation, for example, <math>f(x),\!</math> where <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> is the name of a function <math>f,\!</math> <math>{}^{\backprime\backprime} f(~) {}^{\prime\prime}\!</math> is the context of its application, <math>{}^{\backprime\backprime} x {}^{\prime\prime}\!</math> is the name of an argument <math>x,\!</math> and where the functional abstraction <math>{}^{\backprime\backprime} x \mapsto f(x) {}^{\prime\prime}\!</math> is just another name for the function <math>f.\!</math>
In the present context an application of the arch notation, for example, <math>{}^{\langle} x {}^{\rangle},\!</math> is read on analogy with the use of any other functional notation, for example, <math>f(x),\!</math> where <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> is the name of a function <math>f,\!</math> <math>{}^{\backprime\backprime} f(~) {}^{\prime\prime}\!</math> is the context of its application, <math>{}^{\backprime\backprime} x {}^{\prime\prime}\!</math> is the name of an argument <math>x,\!</math> and where the functional abstraction <math>{}^{\backprime\backprime} x \mapsto f(x) {}^{\prime\prime}\!</math> is just another name for the function <math>f.\!</math>
It is clear that some form of functional abstraction is being invoked in the above definition of <math>\operatorname{Nom}_1.\!</math> Otherwise, the expression <math>x \mapsto {}^{\langle} x {}^{\rangle}\!</math> would indicate a constant function, one that maps every <math>x\!</math> in its domain to the same code or sign for the letter <math>{}^{\backprime\backprime} x {}^{\prime\prime}.\!</math> But if this is allowed, then it appears to pose a dilemma, either to invoke a more powerful concept of functional abstraction than the concept being defined, or else to attempt an improper definition of the naming function in terms of itself.
It is clear that some form of functional abstraction is being invoked in the above definition of <math>\mathrm{Nom}_1.\!</math> Otherwise, the expression <math>x \mapsto {}^{\langle} x {}^{\rangle}\!</math> would indicate a constant function, one that maps every <math>x\!</math> in its domain to the same code or sign for the letter <math>{}^{\backprime\backprime} x {}^{\prime\prime}.\!</math> But if this is allowed, then it appears to pose a dilemma, either to invoke a more powerful concept of functional abstraction than the concept being defined, or else to attempt an improper definition of the naming function in terms of itself.
Although it appears that this form of functional abstraction is being used to define the CRE in terms of itself, trying to extend the definition of the naming function in terms of a definition that is already assumed to be available, in reality this only uses a finite function, a finite table look up, to define the naming function for an unlimited number of higher order signs.
Although it appears that this form of functional abstraction is being used to define the CRE in terms of itself, trying to extend the definition of the naming function in terms of a definition that is already assumed to be available, in reality this only uses a finite function, a finite table look up, to define the naming function for an unlimited number of higher order signs.
===6.11. Higher Order Sign Relations : Application===
===6.11. Higher Order Sign Relations : Application===
Given the language in which a notation like <math>{}^{\backprime\backprime} \operatorname{De}(q, L) {}^{\prime\prime}\!</math> makes sense, or in prospect of being given such a language, it is instructive to ask: “What must be assumed about the context of interpretation in which this language is supposed to make sense?” According to the theory of signs that is being examined here, the relevant formal aspects of that context are embodied in a particular sign relation, call it <math>{}^{\backprime\backprime} Q {}^{\prime\prime}.\!</math> With respect to the hypothetical sign relation <math>Q,\!</math> commonly personified as the prospective reader or the ideal interpreter of the intended language, the denotation of the expression <math>{}^{\backprime\backprime} \operatorname{De}(q, L) {}^{\prime\prime}\!</math> is given by:
Given the language in which a notation like <math>{}^{\backprime\backprime} \mathrm{De}(q, L) {}^{\prime\prime}\!</math> makes sense, or in prospect of being given such a language, it is instructive to ask: “What must be assumed about the context of interpretation in which this language is supposed to make sense?” According to the theory of signs that is being examined here, the relevant formal aspects of that context are embodied in a particular sign relation, call it <math>{}^{\backprime\backprime} Q {}^{\prime\prime}.\!</math> With respect to the hypothetical sign relation <math>Q,\!</math> commonly personified as the prospective reader or the ideal interpreter of the intended language, the denotation of the expression <math>{}^{\backprime\backprime} \mathrm{De}(q, L) {}^{\prime\prime}\!</math> is given by:
{| align="center" cellspacing="8" width="90%"
{| align="center" cellspacing="8" width="90%"
| <math>\operatorname{De}( {}^{\backprime\backprime} \operatorname{De}(q, L) {}^{\prime\prime}, Q ).\!</math>
| <math>\mathrm{De}( {}^{\backprime\backprime} \mathrm{De}(q, L) {}^{\prime\prime}, Q ).\!</math>
|}
|}
|
|
<math>\begin{array}{lccc}
<math>\begin{array}{lccc}
\operatorname{De}( & {}^{\backprime\backprime} \operatorname{De} {}^{\prime\prime} & , & Q)
\mathrm{De}( & {}^{\backprime\backprime} \mathrm{De} {}^{\prime\prime} & , & Q)
\\[6pt]
\\[6pt]
\operatorname{De}( & {}^{\backprime\backprime} q {}^{\prime\prime} & , & Q)
\mathrm{De}( & {}^{\backprime\backprime} q {}^{\prime\prime} & , & Q)
\\[6pt]
\\[6pt]
\operatorname{De}( & {}^{\backprime\backprime} L {}^{\prime\prime} & , & Q)
\mathrm{De}( & {}^{\backprime\backprime} L {}^{\prime\prime} & , & Q)
\end{array}</math>
\end{array}</math>
|}
|}
What are the roles of the signs <math>{}^{\backprime\backprime} \operatorname{De} {}^{\prime\prime},\!</math> <math>{}^{\backprime\backprime} q {}^{\prime\prime},\!</math> <math>{}^{\backprime\backprime} L {}^{\prime\prime}\!</math> and what are they supposed to mean to <math>Q\!</math>? Evidently, <math>{}^{\backprime\backprime} \operatorname{De} {}^{\prime\prime}\!</math> is a constant name that refers to a particular function, <math>{}^{\backprime\backprime} q {}^{\prime\prime}\!</math> is a variable name that makes a PIR to a collection of signs, and <math>{}^{\backprime\backprime} L {}^{\prime\prime}\!</math> is a variable name that makes a PIR to a collection of sign relations.
What are the roles of the signs <math>{}^{\backprime\backprime} \mathrm{De} {}^{\prime\prime},\!</math> <math>{}^{\backprime\backprime} q {}^{\prime\prime},\!</math> <math>{}^{\backprime\backprime} L {}^{\prime\prime}\!</math> and what are they supposed to mean to <math>Q\!</math>? Evidently, <math>{}^{\backprime\backprime} \mathrm{De} {}^{\prime\prime}\!</math> is a constant name that refers to a particular function, <math>{}^{\backprime\backprime} q {}^{\prime\prime}\!</math> is a variable name that makes a PIR to a collection of signs, and <math>{}^{\backprime\backprime} L {}^{\prime\prime}\!</math> is a variable name that makes a PIR to a collection of sign relations.
This is not the place to take up the possibility of an ideal, universal, or even a very comprehensive interpreter for the language indicated here, so I specialize the account to consider an interpreter <math>Q_{\text{AB}} = Q(\text{A}, \text{B})\!</math> that is competent to cover the initial level of reflections that arise from the dialogue of <math>\text{A}\!</math> and <math>\text{B}.\!</math>
This is not the place to take up the possibility of an ideal, universal, or even a very comprehensive interpreter for the language indicated here, so I specialize the account to consider an interpreter <math>Q_{\text{AB}} = Q(\text{A}, \text{B})\!</math> that is competent to cover the initial level of reflections that arise from the dialogue of <math>\text{A}\!</math> and <math>\text{B}.\!</math>
| valign="bottom" width="33%" |
| valign="bottom" width="33%" |
<math>\begin{matrix}
<math>\begin{matrix}
{}^{\langle} \operatorname{De} {}^{\rangle}
{}^{\langle} \mathrm{De} {}^{\rangle}
\\
\\
{}^{\langle} \operatorname{De} {}^{\rangle}
{}^{\langle} \mathrm{De} {}^{\rangle}
\\
\\
{}^{\langle} \operatorname{De} {}^{\rangle}
{}^{\langle} \mathrm{De} {}^{\rangle}
\\
\\
{}^{\langle} \operatorname{De} {}^{\rangle}
{}^{\langle} \mathrm{De} {}^{\rangle}
\\
\\
{}^{\langle} \operatorname{De} {}^{\rangle}
{}^{\langle} \mathrm{De} {}^{\rangle}
\\
\\
{}^{\langle} \operatorname{De} {}^{\rangle}
{}^{\langle} \mathrm{De} {}^{\rangle}
\\
\\
{}^{\langle} \operatorname{De} {}^{\rangle}
{}^{\langle} \mathrm{De} {}^{\rangle}
\\
\\
{}^{\langle} \operatorname{De} {}^{\rangle}
{}^{\langle} \mathrm{De} {}^{\rangle}
\end{matrix}\!</math>
\end{matrix}\!</math>
| valign="bottom" width="33%" |
| valign="bottom" width="33%" |
<math>\begin{matrix}
<math>\begin{matrix}
{}^{\langle} \operatorname{De} {}^{\rangle}
{}^{\langle} \mathrm{De} {}^{\rangle}
\\
\\
{}^{\langle} \operatorname{De} {}^{\rangle}
{}^{\langle} \mathrm{De} {}^{\rangle}
\\
\\
{}^{\langle} \operatorname{De} {}^{\rangle}
{}^{\langle} \mathrm{De} {}^{\rangle}
\\
\\
{}^{\langle} \operatorname{De} {}^{\rangle}
{}^{\langle} \mathrm{De} {}^{\rangle}
\\
\\
{}^{\langle} \operatorname{De} {}^{\rangle}
{}^{\langle} \mathrm{De} {}^{\rangle}
\\
\\
{}^{\langle} \operatorname{De} {}^{\rangle}
{}^{\langle} \mathrm{De} {}^{\rangle}
\\
\\
{}^{\langle} \operatorname{De} {}^{\rangle}
{}^{\langle} \mathrm{De} {}^{\rangle}
\\
\\
{}^{\langle} \operatorname{De} {}^{\rangle}
{}^{\langle} \mathrm{De} {}^{\rangle}
\end{matrix}\!</math>
\end{matrix}\!</math>
|}
|}
<br>
<br>
Following the manner of construction in this extremely reduced example, it is possible to see how answers to the above questions, concerning the meaning of <math>{}^{\backprime\backprime} \operatorname{De}(q, L) {}^{\prime\prime},\!</math> might be worked out. In the present instance:
Following the manner of construction in this extremely reduced example, it is possible to see how answers to the above questions, concerning the meaning of <math>{}^{\backprime\backprime} \mathrm{De}(q, L) {}^{\prime\prime},\!</math> might be worked out. In the present instance:
{| align="center" cellspacing="8" width="90%"
{| align="center" cellspacing="8" width="90%"
|
|
<math>\begin{array}{lll}
<math>\begin{array}{lll}
\operatorname{De} ({}^{\backprime\backprime} q {}^{\prime\prime}, Q_{\text{AB}})
\mathrm{De} ({}^{\backprime\backprime} q {}^{\prime\prime}, Q_{\text{AB}})
& = & S
& = & S
\\[6pt]
\\[6pt]
\operatorname{De} ({}^{\backprime\backprime} L {}^{\prime\prime}, Q_{\text{AB}})
\mathrm{De} ({}^{\backprime\backprime} L {}^{\prime\prime}, Q_{\text{AB}})
& = & \{ L(\text{A}), L(\text{B}) \}
& = & \{ L(\text{A}), L(\text{B}) \}
\end{array}</math>
\end{array}</math>
<p>The ''nominal resource'' (''nominal alphabet'' or ''nominal lexicon'') for <math>X\!</math> is a set of signs that is notated and defined as follows:</p>
<p>The ''nominal resource'' (''nominal alphabet'' or ''nominal lexicon'') for <math>X\!</math> is a set of signs that is notated and defined as follows:</p>
<p><math>X^{\backprime\backprime\prime\prime} = \operatorname{Nom}(X) = \{ {}^{\backprime\backprime} x_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} x_n {}^{\prime\prime} \}.</math></p>
<p><math>X^{\backprime\backprime\prime\prime} = \mathrm{Nom}(X) = \{ {}^{\backprime\backprime} x_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} x_n {}^{\prime\prime} \}.</math></p>
<p>This concept is intended to capture the ordinary usage of this set of signs in one familiar context or another.</p></li>
<p>This concept is intended to capture the ordinary usage of this set of signs in one familiar context or another.</p></li>
<p>The ''mediate resource'' (''mediate alphabet'' or ''mediate lexicon'') for <math>X\!</math> is a set of signs that is notated and defined as follows:</p>
<p>The ''mediate resource'' (''mediate alphabet'' or ''mediate lexicon'') for <math>X\!</math> is a set of signs that is notated and defined as follows:</p>
<p><math>X^{\langle\rangle} = \operatorname{Med}(X) = \{ {}^{\langle} x_1 {}^{\rangle}, \ldots, {}^{\langle} x_n {}^{\rangle} \}.</math></p>
<p><math>X^{\langle\rangle} = \mathrm{Med}(X) = \{ {}^{\langle} x_1 {}^{\rangle}, \ldots, {}^{\langle} x_n {}^{\rangle} \}.</math></p>
<p>This concept provides a middle ground between the nominal resource above and the literal resource described next.</p></li>
<p>This concept provides a middle ground between the nominal resource above and the literal resource described next.</p></li>
<p>The ''literal resource'' (''literal alphabet'' or ''literal lexicon'') for <math>X\!</math> is a set of signs that is notated and defined as follows:</p>
<p>The ''literal resource'' (''literal alphabet'' or ''literal lexicon'') for <math>X\!</math> is a set of signs that is notated and defined as follows:</p>
<p><math>X = \operatorname{Lit}(X) = \{ x_1, \ldots, x_n \}.</math></p>
<p><math>X = \mathrm{Lit}(X) = \{ x_1, \ldots, x_n \}.</math></p>
<p>This concept is intended to supply a set of signs that can be used in ways analogous to familiar usages, but which are more subject to free variation and thematic control.</p></li></ol>
<p>This concept is intended to supply a set of signs that can be used in ways analogous to familiar usages, but which are more subject to free variation and thematic control.</p></li></ol>
{| align="center" cellspacing="8" width="90%"
{| align="center" cellspacing="8" width="90%"
| <math>\underline{\underline{X}} = \operatorname{Lit}(X) = \{ \underline{\underline{x_1}}, \ldots, \underline{\underline{x_n}} \}.\!</math>
| <math>\underline{\underline{X}} = \mathrm{Lit}(X) = \{ \underline{\underline{x_1}}, \ldots, \underline{\underline{x_n}} \}.\!</math>
|}
|}
In this approach to propositional logic, with a view toward computational realization, one begins with a space <math>X,\!</math> called a ''universe of discourse'', whose points can be reasonably well described by means of a finite set of logical features. Since the points of the space <math>X\!</math> are effectively known only in terms of their computable features, one can assume that there is a finite set of computable coordinate projections <math>x_i : X \to \mathbb{B},\!</math> for <math>{i = 1 ~\text{to}~ n,}\!</math> for some <math>n,\!</math> that can serve to describe the points of <math>X.\!</math> This means that there is a computable coordinate representation for <math>X,\!</math> in other words, a computable map <math>T : X \to \mathbb{B}^n\!</math> that describes the points of <math>X\!</math> insofar as they are known. Thus, each proposition <math>F : X \to \mathbb{B}\!</math> can be factored through the coordinate representation <math>T : X \to \mathbb{B}^n\!</math> to yield a related proposition <math>f : \mathbb{B}^n \to \mathbb{B},\!</math> one that speaks directly about coordinate <math>n\!</math>-tuples but indirectly about points of <math>X.\!</math> Composing maps on the right, the mapping <math>f\!</math> is defined by the equation <math>F = T \circ f.\!</math> For all practical purposes served by the representation <math>T,\!</math> the proposition <math>f\!</math> can be taken as a proxy for the proposition <math>F,\!</math> saying things about the points of <math>X\!</math> by means of <math>X\!</math>'s encoding to <math>\mathbb{B}^n.\!</math>
In this approach to propositional logic, with a view toward computational realization, one begins with a space <math>X,\!</math> called a ''universe of discourse'', whose points can be reasonably well described by means of a finite set of logical features. Since the points of the space <math>X\!</math> are effectively known only in terms of their computable features, one can assume that there is a finite set of computable coordinate projections <math>x_i : X \to \mathbb{B},\!</math> for <math>{i = 1 ~\text{to}~ n,}\!</math> for some <math>n,\!</math> that can serve to describe the points of <math>X.\!</math> This means that there is a computable coordinate representation for <math>X,\!</math> in other words, a computable map <math>T : X \to \mathbb{B}^n\!</math> that describes the points of <math>X\!</math> insofar as they are known. Thus, each proposition <math>F : X \to \mathbb{B}\!</math> can be factored through the coordinate representation <math>T : X \to \mathbb{B}^n\!</math> to yield a related proposition <math>f : \mathbb{B}^n \to \mathbb{B},\!</math> one that speaks directly about coordinate <math>n\!</math>-tuples but indirectly about points of <math>X.\!</math> Composing maps on the right, the mapping <math>f\!</math> is defined by the equation <math>F = T \circ f.\!</math> For all practical purposes served by the representation <math>T,\!</math> the proposition <math>f\!</math> can be taken as a proxy for the proposition <math>F,\!</math> saying things about the points of <math>X\!</math> by means of <math>X\!</math>'s encoding to <math>\mathbb{B}^n.\!</math>
Working under the functional perspective, the formal system known as ''propositional calculus'' is introduced as a general system of notations for referring to boolean functions. Typically, one takes a space <math>X\!</math> and a coordinate representation <math>T : X \to \mathbb{B}^n\!</math> as parameters of a particular system and speaks of the propositional calculus on a finite set of variables <math>\{ \underline{\underline{x_i}} \}.\!</math> In objective terms, this constitutes the ''domain of propositions'' on the basis <math>\{ \underline{\underline{x_i}} \},\!</math> notated as <math>\operatorname{DOP}\{ \underline{\underline{x_i}} \}.\!</math> Ideally, one does not want to become too fixed on a particular set of logical features or to let the momentary dimensions of the space be cast in stone. In practice, this means that the formalism and its computational implementation should allow for the automatic embedding of <math>\operatorname{DOP}(\underline{\underline{X}})\!</math> into <math>\operatorname{DOP}(\underline{\underline{Y}})\!</math> whenever <math>\underline{\underline{X}} \subseteq \underline{\underline{Y}}.\!</math>
Working under the functional perspective, the formal system known as ''propositional calculus'' is introduced as a general system of notations for referring to boolean functions. Typically, one takes a space <math>X\!</math> and a coordinate representation <math>T : X \to \mathbb{B}^n\!</math> as parameters of a particular system and speaks of the propositional calculus on a finite set of variables <math>\{ \underline{\underline{x_i}} \}.\!</math> In objective terms, this constitutes the ''domain of propositions'' on the basis <math>\{ \underline{\underline{x_i}} \},\!</math> notated as <math>\mathrm{DOP}\{ \underline{\underline{x_i}} \}.\!</math> Ideally, one does not want to become too fixed on a particular set of logical features or to let the momentary dimensions of the space be cast in stone. In practice, this means that the formalism and its computational implementation should allow for the automatic embedding of <math>\mathrm{DOP}(\underline{\underline{X}})\!</math> into <math>\mathrm{DOP}(\underline{\underline{Y}})\!</math> whenever <math>\underline{\underline{X}} \subseteq \underline{\underline{Y}}.\!</math>
The rest of this section presents the elements of a particular calculus for propositional logic. First, I establish the basic notations and summarize the axiomatic presentation of the calculus, and then I give special attention to its functional and geometric interpretations.
The rest of this section presents the elements of a particular calculus for propositional logic. First, I establish the basic notations and summarize the axiomatic presentation of the calculus, and then I give special attention to its functional and geometric interpretations.
A sign relation is a complex object and its representations, insofar as they faithfully preserve its structure, are complex signs. Accordingly, the problems of translating between ERs and IRs of sign relations, of detecting when representations alleged to be of sign relations do indeed represent objects of the specified character, and of recognizing whether different representations do or do not represent the same sign relation as their common object — these are the familiar questions that would be asked of the signs and interpretants in a simple sign relation, but this time asked at a higher level, in regard to the complex signs and complex interpretants that are posed by the different stripes of representation. At the same time, it should be obvious that these are also the natural questions to be faced in building a bridge between representations.
A sign relation is a complex object and its representations, insofar as they faithfully preserve its structure, are complex signs. Accordingly, the problems of translating between ERs and IRs of sign relations, of detecting when representations alleged to be of sign relations do indeed represent objects of the specified character, and of recognizing whether different representations do or do not represent the same sign relation as their common object — these are the familiar questions that would be asked of the signs and interpretants in a simple sign relation, but this time asked at a higher level, in regard to the complex signs and complex interpretants that are posed by the different stripes of representation. At the same time, it should be obvious that these are also the natural questions to be faced in building a bridge between representations.
How many different sorts of entities are conceivably involved in translating between ERs and IRs of sign relations? To address this question it helps to introduce a system of type notations that can be used to keep track of the various sorts of things, or the varieties of objects of thought, that are generated in the process of answering it. Table 47.1 summarizes the basic types of things that are needed in this pursuit, while the rest can be derived by constructions of the form <math>X ~\operatorname{of}~ Y,\!</math> notated <math>X(Y)\!</math> or just <math>XY,\!</math> for any basic types <math>X\!</math> and <math>Y.\!</math> The constructed types of things involved in the ERs and IRs of sign relations are listed in Tables 47.2 and 47.3, respectively.
How many different sorts of entities are conceivably involved in translating between ERs and IRs of sign relations? To address this question it helps to introduce a system of type notations that can be used to keep track of the various sorts of things, or the varieties of objects of thought, that are generated in the process of answering it. Table 47.1 summarizes the basic types of things that are needed in this pursuit, while the rest can be derived by constructions of the form <math>X ~\mathrm{of}~ Y,\!</math> notated <math>X(Y)\!</math> or just <math>XY,\!</math> for any basic types <math>X\!</math> and <math>Y.\!</math> The constructed types of things involved in the ERs and IRs of sign relations are listed in Tables 47.2 and 47.3, respectively.
<br>
<br>
Starting from a standpoint in concrete constructions, the easiest way to begin developing an explicit treatment of ERs is to gather the relevant materials in the forms already presented, to fill out the missing details and expand the abbreviated contents of these forms, and to review their full structures in a more formal light. Consequently, this section inaugurates the formal discussion of ERs by taking a second look at the interpreters <math>\text{A}\!</math> and <math>\text{B},\!</math> recollecting the Tables of their sign relations and finishing up the Tables of their dyadic components. Since the form of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> no longer presents any novelty, I can exploit their second presentation as a first opportunity to examine a selection of finer points, previously overlooked. Also, in the process of reviewing this material it is useful to anticipate a number of incidental issues that are reaching the point of becoming critical within this discussion and to begin introducing the generic types of technical devices that are needed to deal with them.
Starting from a standpoint in concrete constructions, the easiest way to begin developing an explicit treatment of ERs is to gather the relevant materials in the forms already presented, to fill out the missing details and expand the abbreviated contents of these forms, and to review their full structures in a more formal light. Consequently, this section inaugurates the formal discussion of ERs by taking a second look at the interpreters <math>\text{A}\!</math> and <math>\text{B},\!</math> recollecting the Tables of their sign relations and finishing up the Tables of their dyadic components. Since the form of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> no longer presents any novelty, I can exploit their second presentation as a first opportunity to examine a selection of finer points, previously overlooked. Also, in the process of reviewing this material it is useful to anticipate a number of incidental issues that are reaching the point of becoming critical within this discussion and to begin introducing the generic types of technical devices that are needed to deal with them.
The next set of Tables summarizes the ERs of <math>L(\text{A})\!</math> and <math>L(\text{B}).\!</math> For ease of reference, Tables 48.1 and 49.1 repeat the contents of Tables 1 and 2, respectively, the only difference being that appearances of ordinary quotation marks <math>({}^{\backprime\backprime} \ldots {}^{\prime\prime})\!</math> are transcribed as invocations of the ''arch operator'' <math>({}^{\langle} \ldots {}^{\rangle}).\!</math> The reason for this slight change of notation will be explained shortly. The denotative components <math>\operatorname{Den}(\text{A})\!</math> and <math>\operatorname{Den}(\text{B})\!</math> are shown in the first two columns of Tables 48.2 and 49.2, respectively, while the third column gives the transition from sign to object as an ordered pair <math>(s, o).\!</math> The connotative components <math>\operatorname{Con}(\text{A})\!</math> and <math>\operatorname{Con}(\text{B})\!</math> are shown in the first two columns of Tables 48.3 and 49.3, respectively, while the third column gives the transition from sign to interpretant as an ordered pair <math>(s, i).\!</math>
The next set of Tables summarizes the ERs of <math>L(\text{A})\!</math> and <math>L(\text{B}).\!</math> For ease of reference, Tables 48.1 and 49.1 repeat the contents of Tables 1 and 2, respectively, the only difference being that appearances of ordinary quotation marks <math>({}^{\backprime\backprime} \ldots {}^{\prime\prime})\!</math> are transcribed as invocations of the ''arch operator'' <math>({}^{\langle} \ldots {}^{\rangle}).\!</math> The reason for this slight change of notation will be explained shortly. The denotative components <math>\mathrm{Den}(\text{A})\!</math> and <math>\mathrm{Den}(\text{B})\!</math> are shown in the first two columns of Tables 48.2 and 49.2, respectively, while the third column gives the transition from sign to object as an ordered pair <math>(s, o).\!</math> The connotative components <math>\mathrm{Con}(\text{A})\!</math> and <math>\mathrm{Con}(\text{B})\!</math> are shown in the first two columns of Tables 48.3 and 49.3, respectively, while the third column gives the transition from sign to interpretant as an ordered pair <math>(s, i).\!</math>
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<br>
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
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<math>\text{Table 48.1} ~~ \operatorname{ER}(L_\text{A}) : \text{Extensional Representation of} ~ L_\text{A}\!</math>
<math>\text{Table 48.1} ~~ \mathrm{ER}(L_\text{A}) : \text{Extensional Representation of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| <math>\text{Object}\!</math>
| <math>\text{Object}\!</math>
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
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<math>\text{Table 48.2} ~~ \operatorname{ER}(\operatorname{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!</math>
<math>\text{Table 48.2} ~~ \mathrm{ER}(\mathrm{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| <math>\text{Object}\!</math>
| <math>\text{Object}\!</math>
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
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<math>\text{Table 48.3} ~~ \operatorname{ER}(\operatorname{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!</math>
<math>\text{Table 48.3} ~~ \mathrm{ER}(\mathrm{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| <math>\text{Sign}\!</math>
| <math>\text{Sign}\!</math>
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
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<math>\text{Table 49.1} ~~ \operatorname{ER}(L_\text{B}) : \text{Extensional Representation of} ~ L_\text{B}\!</math>
<math>\text{Table 49.1} ~~ \mathrm{ER}(L_\text{B}) : \text{Extensional Representation of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| <math>\text{Object}\!</math>
| <math>\text{Object}\!</math>
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
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<math>\text{Table 49.2} ~~ \operatorname{ER}(\operatorname{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}~\!</math>
<math>\text{Table 49.2} ~~ \mathrm{ER}(\mathrm{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}~\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| <math>\text{Object}\!</math>
| <math>\text{Object}\!</math>
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
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<math>\text{Table 49.3} ~~ \operatorname{ER}(\operatorname{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!</math>
<math>\text{Table 49.3} ~~ \mathrm{ER}(\mathrm{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| <math>\text{Sign}\!</math>
| <math>\text{Sign}\!</math>
For the sake of maximum clarity and reusability of results, I begin by articulating the abstract skeleton of the paradigm structure, treating the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> as sundry aspects of a single, unitary, but still uninterpreted object. Then I return at various successive stages to differentiate and individualize the two interpreters, to arrange more functional flesh on the basis provided by their structural bones, and to illustrate how their bare forms can be arrayed in many different styles of qualitative detail.
For the sake of maximum clarity and reusability of results, I begin by articulating the abstract skeleton of the paradigm structure, treating the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> as sundry aspects of a single, unitary, but still uninterpreted object. Then I return at various successive stages to differentiate and individualize the two interpreters, to arrange more functional flesh on the basis provided by their structural bones, and to illustrate how their bare forms can be arrayed in many different styles of qualitative detail.
In building connections between ERs and IRs of sign relations the discussion turns on two types of partially ordered sets, or ''posets''. Suppose that <math>L\!</math> is one of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> and let <math>\operatorname{ER}(L)\!</math> be an ER of <math>L.\!</math>
In building connections between ERs and IRs of sign relations the discussion turns on two types of partially ordered sets, or ''posets''. Suppose that <math>L\!</math> is one of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> and let <math>\mathrm{ER}(L)\!</math> be an ER of <math>L.\!</math>
In the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> both of their ERs are based on a common world set:
In the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> both of their ERs are based on a common world set:
To devise an IR of any relation <math>L\!</math> one needs to describe <math>L\!</math> in terms of properties of its ingredients. Broadly speaking, the ingredients of a relation include its elementary relations or <math>n\!</math>-tuples and the elementary components of these <math>n\!</math>-tuples that reside in the relational domains.
To devise an IR of any relation <math>L\!</math> one needs to describe <math>L\!</math> in terms of properties of its ingredients. Broadly speaking, the ingredients of a relation include its elementary relations or <math>n\!</math>-tuples and the elementary components of these <math>n\!</math>-tuples that reside in the relational domains.
The poset <math>\operatorname{Pos}(W)\!</math> of interest here is the power set <math>\mathcal{P}(W) = \operatorname{Pow}(W).\!</math>
The poset <math>\mathrm{Pos}(W)\!</math> of interest here is the power set <math>\mathcal{P}(W) = \mathrm{Pow}(W).\!</math>
The elements of these posets are abstractly regarded as ''properties'' or ''propositions'' that apply to the elements of <math>W.\!</math> These properties and propositions are independently given entities. In other words, they are primitive elements in their own right, and cannot in general be defined in terms of points, but they exist in relation to these points, and their extensions can be represented as sets of points.
The elements of these posets are abstractly regarded as ''properties'' or ''propositions'' that apply to the elements of <math>W.\!</math> These properties and propositions are independently given entities. In other words, they are primitive elements in their own right, and cannot in general be defined in terms of points, but they exist in relation to these points, and their extensions can be represented as sets of points.
'''[Variant]''' There is a foundational issue arising in this context that I do not pretend to fully understand and cannot attempt to finally dispatch. What I do understand I will try to express in terms of an aesthetic principle: On balance, it seems best to regard extensional elements and intensional features as independently given entities. This involves treating points and properties as fundamental realities in their own rights, placing them on an equal basis with each other, and seeking their relation to each other, but not trying to reduce one to the other.
'''[Variant]''' There is a foundational issue arising in this context that I do not pretend to fully understand and cannot attempt to finally dispatch. What I do understand I will try to express in terms of an aesthetic principle: On balance, it seems best to regard extensional elements and intensional features as independently given entities. This involves treating points and properties as fundamental realities in their own rights, placing them on an equal basis with each other, and seeking their relation to each other, but not trying to reduce one to the other.
The discussion is now specialized to consider the IRs of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> their denotative projections as the digraphs <math>\operatorname{Den}(L_\text{A})\!</math> and <math>\operatorname{Den}(L_\text{B}),\!</math> and their connotative projections as the digraphs <math>\operatorname{Con}(L_\text{A})\!</math> and <math>\operatorname{Con}(L_\text{B}).\!</math> In doing this I take up two different strategies of representation:
The discussion is now specialized to consider the IRs of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> their denotative projections as the digraphs <math>\mathrm{Den}(L_\text{A})\!</math> and <math>\mathrm{Den}(L_\text{B}),\!</math> and their connotative projections as the digraphs <math>\mathrm{Con}(L_\text{A})\!</math> and <math>\mathrm{Con}(L_\text{B}).\!</math> In doing this I take up two different strategies of representation:
# The first strategy is called the ''literal coding'', because it sticks to obvious features of each syntactic element to contrive its code, or the ''<math>{\mathcal{O}(n)}\!</math> coding'', because it uses a number on the order of <math>n\!</math> logical features to represent a domain of <math>n\!</math> elements.
# The first strategy is called the ''literal coding'', because it sticks to obvious features of each syntactic element to contrive its code, or the ''<math>{\mathcal{O}(n)}\!</math> coding'', because it uses a number on the order of <math>n\!</math> logical features to represent a domain of <math>n\!</math> elements.
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
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<math>\text{Table 53.1} ~~ \text{Elements of} ~ \operatorname{ER}(W)\!</math>
<math>\text{Table 53.1} ~~ \text{Elements of} ~ \mathrm{ER}(W)\!</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
| <math>\text{Mnemonic Element}\!</math> <br><br> <math>w \in W\!</math>
| <math>\text{Mnemonic Element}\!</math> <br><br> <math>w \in W\!</math>
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
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<math>\text{Table 53.2} ~~ \text{Features of} ~ \operatorname{LIR}(W)\!</math>
<math>\text{Table 53.2} ~~ \text{Features of} ~ \mathrm{LIR}(W)\!</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
|
|
<br>
<br>
If the world of <math>\text{A}\!</math> and <math>\text{B},\!</math> the set <math>W = \{ \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> is viewed abstractly, as an arbitrary set of six atomic points, then there are exactly <math>2^6 = 64\!</math> ''abstract properties'' or ''potential attributes'' that might be applied to or recognized in these points. The elements of <math>W\!</math> that possess a given property form a subset of <math>W\!</math> called the ''extension'' of that property. Thus the extensions of abstract properties are exactly the subsets of <math>W.\!</math> The set of all subsets of <math>W\!</math> is called the ''power set'' of <math>W,\!</math> notated as <math>\operatorname{Pow}(W)\!</math> or <math>\mathcal{P}(W).\!</math> In order to make this way of talking about properties consistent with the previous definition of reality, it is necessary to say that one potential property is never realized, since no point has it, and its extension is the empty set <math>\varnothing = \{ \}.\!</math> All the ''natural'' properties of points that one observes in a concrete situation, properties whose extensions are known as ''natural kinds'', can be recognized among the ''abstract'', ''arbitrary'', or ''set-theoretic'' properties that are systematically generated in this way. Typically, however, many of these abstract properties will not be recognized as falling among the more natural kinds.
If the world of <math>\text{A}\!</math> and <math>\text{B},\!</math> the set <math>W = \{ \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> is viewed abstractly, as an arbitrary set of six atomic points, then there are exactly <math>2^6 = 64\!</math> ''abstract properties'' or ''potential attributes'' that might be applied to or recognized in these points. The elements of <math>W\!</math> that possess a given property form a subset of <math>W\!</math> called the ''extension'' of that property. Thus the extensions of abstract properties are exactly the subsets of <math>W.\!</math> The set of all subsets of <math>W\!</math> is called the ''power set'' of <math>W,\!</math> notated as <math>\mathrm{Pow}(W)\!</math> or <math>\mathcal{P}(W).\!</math> In order to make this way of talking about properties consistent with the previous definition of reality, it is necessary to say that one potential property is never realized, since no point has it, and its extension is the empty set <math>\varnothing = \{ \}.\!</math> All the ''natural'' properties of points that one observes in a concrete situation, properties whose extensions are known as ''natural kinds'', can be recognized among the ''abstract'', ''arbitrary'', or ''set-theoretic'' properties that are systematically generated in this way. Typically, however, many of these abstract properties will not be recognized as falling among the more natural kinds.
Tables 54.1, 54.2, and 54.3 show three different ways of representing the elements of the world set <math>W\!</math> as vectors in the coordinate space <math>\underline{W}\!</math> and as singular propositions in the universe of discourse <math>W^\Box.\!</math> Altogether, these Tables present the ''literal'' codes for the elements of <math>\underline{W}\!</math> and <math>W^\circ\!</math> in their ''mnemonic'', ''pragmatic'', and ''abstract'' versions, respectively. In each Table, Column 1 lists the element <math>w \in W,\!</math> while Column 2 gives the corresponding coordinate vector <math>\underline{w} \in \underline{W}\!</math> in the form of a bit string. The next two Columns represent each <math>w \in W\!</math> as a proposition in <math>W^\circ\!,</math> in effect, reconstituting it as a function <math>w : \underline{W} \to \mathbb{B}.</math> Column 3 shows the propositional expression of each element in the form of a conjunct term, in other words, as a logical product of positive and negative features. Column 4 gives the compact code for each element, using a conjunction of positive features in subscripted angle brackets to represent the singular proposition corresponding to each element.
Tables 54.1, 54.2, and 54.3 show three different ways of representing the elements of the world set <math>W\!</math> as vectors in the coordinate space <math>\underline{W}\!</math> and as singular propositions in the universe of discourse <math>W^\Box.\!</math> Altogether, these Tables present the ''literal'' codes for the elements of <math>\underline{W}\!</math> and <math>W^\circ\!</math> in their ''mnemonic'', ''pragmatic'', and ''abstract'' versions, respectively. In each Table, Column 1 lists the element <math>w \in W,\!</math> while Column 2 gives the corresponding coordinate vector <math>\underline{w} \in \underline{W}\!</math> in the form of a bit string. The next two Columns represent each <math>w \in W\!</math> as a proposition in <math>W^\circ\!,</math> in effect, reconstituting it as a function <math>w : \underline{W} \to \mathbb{B}.</math> Column 3 shows the propositional expression of each element in the form of a conjunct term, in other words, as a logical product of positive and negative features. Column 4 gives the compact code for each element, using a conjunction of positive features in subscripted angle brackets to represent the singular proposition corresponding to each element.
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
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<math>\text{Table 55.1} ~~ \operatorname{LIR}_1 (L_\text{A}) : \text{Literal Representation of} ~ L_\text{A}\!</math>
<math>\text{Table 55.1} ~~ \mathrm{LIR}_1 (L_\text{A}) : \text{Literal Representation of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Object}\!</math>
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
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<math>\text{Table 55.2} ~~ \operatorname{LIR}_1 (\operatorname{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!</math>
<math>\text{Table 55.2} ~~ \mathrm{LIR}_1 (\mathrm{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Object}\!</math>
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
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<math>\text{Table 55.3} ~~ \operatorname{LIR}_1 (\operatorname{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!</math>
<math>\text{Table 55.3} ~~ \mathrm{LIR}_1 (\mathrm{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
| valign="bottom" |
| valign="bottom" |
<math>\begin{matrix}
<math>\begin{matrix}
0_{\operatorname{d}W}
0_{\mathrm{d}W}
\\[4pt]
\\[4pt]
{\langle
{\langle
\operatorname{d}\underline{\underline{\text{a}}}
\mathrm{d}\underline{\underline{\text{a}}}
~
~
\operatorname{d}\underline{\underline{\text{i}}}
\mathrm{d}\underline{\underline{\text{i}}}
\rangle}_{\operatorname{d}W}
\rangle}_{\mathrm{d}W}
\\[4pt]
\\[4pt]
{\langle
{\langle
\operatorname{d}\underline{\underline{\text{a}}}
\mathrm{d}\underline{\underline{\text{a}}}
~
~
\operatorname{d}\underline{\underline{\text{i}}}
\mathrm{d}\underline{\underline{\text{i}}}
\rangle}_{\operatorname{d}W}
\rangle}_{\mathrm{d}W}
\\[4pt]
\\[4pt]
0_{\operatorname{d}W}
0_{\mathrm{d}W}
\end{matrix}</math>
\end{matrix}</math>
|-
|-
| valign="bottom" |
| valign="bottom" |
<math>\begin{matrix}
<math>\begin{matrix}
0_{\operatorname{d}W}
0_{\mathrm{d}W}
\\[4pt]
\\[4pt]
{\langle
{\langle
\operatorname{d}\underline{\underline{\text{b}}}
\mathrm{d}\underline{\underline{\text{b}}}
~
~
\operatorname{d}\underline{\underline{\text{u}}}
\mathrm{d}\underline{\underline{\text{u}}}
\rangle}_{\operatorname{d}W}
\rangle}_{\mathrm{d}W}
\\[4pt]
\\[4pt]
{\langle
{\langle
\operatorname{d}\underline{\underline{\text{b}}}
\mathrm{d}\underline{\underline{\text{b}}}
~
~
\operatorname{d}\underline{\underline{\text{u}}}
\mathrm{d}\underline{\underline{\text{u}}}
\rangle}_{\operatorname{d}W}
\rangle}_{\mathrm{d}W}
\\[4pt]
\\[4pt]
0_{\operatorname{d}W}
0_{\mathrm{d}W}
\end{matrix}</math>
\end{matrix}</math>
|}
|}
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
|+ style="height:30px" |
<math>\text{Table 56.1} ~~ \operatorname{LIR}_1 (L_\text{B}) : \text{Literal Representation of} ~ L_\text{B}\!</math>
<math>\text{Table 56.1} ~~ \mathrm{LIR}_1 (L_\text{B}) : \text{Literal Representation of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Object}\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
|+ style="height:30px" |
<math>\text{Table 56.2} ~~ \operatorname{LIR}_1 (\operatorname{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!</math>
<math>\text{Table 56.2} ~~ \mathrm{LIR}_1 (\mathrm{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Object}\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
|+ style="height:30px" |
<math>\text{Table 56.3} ~~ \operatorname{LIR}_1 (\operatorname{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!</math>
<math>\text{Table 56.3} ~~ \mathrm{LIR}_1 (\mathrm{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
| valign="bottom" |
| valign="bottom" |
<math>\begin{matrix}
<math>\begin{matrix}
0_{\operatorname{d}W}
0_{\mathrm{d}W}
\\[4pt]
\\[4pt]
{\langle
{\langle
\operatorname{d}\underline{\underline{\text{a}}}
\mathrm{d}\underline{\underline{\text{a}}}
~
~
\operatorname{d}\underline{\underline{\text{u}}}
\mathrm{d}\underline{\underline{\text{u}}}
\rangle}_{\operatorname{d}W}
\rangle}_{\mathrm{d}W}
\\[4pt]
\\[4pt]
{\langle
{\langle
\operatorname{d}\underline{\underline{\text{a}}}
\mathrm{d}\underline{\underline{\text{a}}}
~
~
\operatorname{d}\underline{\underline{\text{u}}}
\mathrm{d}\underline{\underline{\text{u}}}
\rangle}_{\operatorname{d}W}
\rangle}_{\mathrm{d}W}
\\[4pt]
\\[4pt]
0_{\operatorname{d}W}
0_{\mathrm{d}W}
\end{matrix}</math>
\end{matrix}</math>
|-
|-
| valign="bottom" |
| valign="bottom" |
<math>\begin{matrix}
<math>\begin{matrix}
0_{\operatorname{d}W}
0_{\mathrm{d}W}
\\[4pt]
\\[4pt]
{\langle
{\langle
\operatorname{d}\underline{\underline{\text{b}}}
\mathrm{d}\underline{\underline{\text{b}}}
~
~
\operatorname{d}\underline{\underline{\text{i}}}
\mathrm{d}\underline{\underline{\text{i}}}
\rangle}_{\operatorname{d}W}
\rangle}_{\mathrm{d}W}
\\[4pt]
\\[4pt]
{\langle
{\langle
\operatorname{d}\underline{\underline{\text{b}}}
\mathrm{d}\underline{\underline{\text{b}}}
~
~
\operatorname{d}\underline{\underline{\text{i}}}
\mathrm{d}\underline{\underline{\text{i}}}
\rangle}_{\operatorname{d}W}
\rangle}_{\mathrm{d}W}
\\[4pt]
\\[4pt]
0_{\operatorname{d}W}
0_{\mathrm{d}W}
\end{matrix}</math>
\end{matrix}</math>
|}
|}
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
|+ style="height:30px" |
<math>\text{Table 58.1} ~~ \operatorname{LIR}_2 (L_\text{A}) : \text{Lateral Representation of} ~ L_\text{A}\!</math>
<math>\text{Table 58.1} ~~ \mathrm{LIR}_2 (L_\text{A}) : \text{Lateral Representation of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Object}\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
|+ style="height:30px" |
<math>\text{Table 58.2} ~~ \operatorname{LIR}_2 (\operatorname{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!</math>
<math>\text{Table 58.2} ~~ \mathrm{LIR}_2 (\mathrm{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Object}\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
|+ style="height:30px" |
<math>\text{Table 58.3} ~~ \operatorname{LIR}_2 (\operatorname{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!</math>
<math>\text{Table 58.3} ~~ \mathrm{LIR}_2 (\mathrm{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
|+ style="height:30px" |
<math>\text{Table 59.1} ~~ \operatorname{LIR}_2 (L_\text{B}) : \text{Lateral Representation of} ~ L_\text{B}\!</math>
<math>\text{Table 59.1} ~~ \mathrm{LIR}_2 (L_\text{B}) : \text{Lateral Representation of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Object}\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
|+ style="height:30px" |
<math>\text{Table 59.2} ~~ \operatorname{LIR}_2 (\operatorname{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!</math>
<math>\text{Table 59.2} ~~ \mathrm{LIR}_2 (\mathrm{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Object}\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
|+ style="height:30px" |
<math>\text{Table 59.3} ~~ \operatorname{LIR}_2 (\operatorname{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!</math>
<math>\text{Table 59.3} ~~ \mathrm{LIR}_2 (\mathrm{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
|+ style="height:30px" |
<math>\text{Table 60.1} ~~ \operatorname{LIR}_3 (L_\text{A}) : \text{Lateral Representation of} ~ L_\text{A}\!</math>
<math>\text{Table 60.1} ~~ \mathrm{LIR}_3 (L_\text{A}) : \text{Lateral Representation of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Object}\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
|+ style="height:30px" |
<math>\text{Table 60.2} ~~ \operatorname{LIR}_3 (\operatorname{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!</math>
<math>\text{Table 60.2} ~~ \mathrm{LIR}_3 (\mathrm{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Object}\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
|+ style="height:30px" |
<math>\text{Table 60.3} ~~ \operatorname{LIR}_3 (\operatorname{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!</math>
<math>\text{Table 60.3} ~~ \mathrm{LIR}_3 (\mathrm{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
| valign="bottom" |
| valign="bottom" |
<math>\begin{matrix}
<math>\begin{matrix}
0_{\operatorname{d}Y}
0_{\mathrm{d}Y}
\\[4pt]
\\[4pt]
{\langle
{\langle
\operatorname{d}\underline{\underline{\text{a}}}
\mathrm{d}\underline{\underline{\text{a}}}
~
~
\operatorname{d}\underline{\underline{\text{i}}}
\mathrm{d}\underline{\underline{\text{i}}}
\rangle}_{\operatorname{d}Y}
\rangle}_{\mathrm{d}Y}
\\[4pt]
\\[4pt]
{\langle
{\langle
\operatorname{d}\underline{\underline{\text{a}}}
\mathrm{d}\underline{\underline{\text{a}}}
~
~
\operatorname{d}\underline{\underline{\text{i}}}
\mathrm{d}\underline{\underline{\text{i}}}
\rangle}_{\operatorname{d}Y}
\rangle}_{\mathrm{d}Y}
\\[4pt]
\\[4pt]
0_{\operatorname{d}Y}
0_{\mathrm{d}Y}
\end{matrix}</math>
\end{matrix}</math>
|-
|-
| valign="bottom" |
| valign="bottom" |
<math>\begin{matrix}
<math>\begin{matrix}
0_{\operatorname{d}Y}
0_{\mathrm{d}Y}
\\[4pt]
\\[4pt]
{\langle
{\langle
\operatorname{d}\underline{\underline{\text{b}}}
\mathrm{d}\underline{\underline{\text{b}}}
~
~
\operatorname{d}\underline{\underline{\text{u}}}
\mathrm{d}\underline{\underline{\text{u}}}
\rangle}_{\operatorname{d}Y}
\rangle}_{\mathrm{d}Y}
\\[4pt]
\\[4pt]
{\langle
{\langle
\operatorname{d}\underline{\underline{\text{b}}}
\mathrm{d}\underline{\underline{\text{b}}}
~
~
\operatorname{d}\underline{\underline{\text{u}}}
\mathrm{d}\underline{\underline{\text{u}}}
\rangle}_{\operatorname{d}Y}
\rangle}_{\mathrm{d}Y}
\\[4pt]
\\[4pt]
0_{\operatorname{d}Y}
0_{\mathrm{d}Y}
\end{matrix}</math>
\end{matrix}</math>
|}
|}
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
|+ style="height:30px" |
<math>\text{Table 61.1} ~~ \operatorname{LIR}_3 (L_\text{B}) : \text{Lateral Representation of} ~ L_\text{B}\!</math>
<math>\text{Table 61.1} ~~ \mathrm{LIR}_3 (L_\text{B}) : \text{Lateral Representation of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Object}\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
|+ style="height:30px" |
<math>\text{Table 61.2} ~~ \operatorname{LIR}_3 (\operatorname{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!</math>
<math>\text{Table 61.2} ~~ \mathrm{LIR}_3 (\mathrm{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Object}\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
|+ style="height:30px" |
<math>\text{Table 61.3} ~~ \operatorname{LIR}_3 (\operatorname{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!</math>
<math>\text{Table 61.3} ~~ \mathrm{LIR}_3 (\mathrm{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
| valign="bottom" |
| valign="bottom" |
<math>\begin{matrix}
<math>\begin{matrix}
0_{\operatorname{d}Y}
0_{\mathrm{d}Y}
\\[4pt]
\\[4pt]
{\langle
{\langle
\operatorname{d}\underline{\underline{\text{a}}}
\mathrm{d}\underline{\underline{\text{a}}}
~
~
\operatorname{d}\underline{\underline{\text{u}}}
\mathrm{d}\underline{\underline{\text{u}}}
\rangle}_{\operatorname{d}Y}
\rangle}_{\mathrm{d}Y}
\\[4pt]
\\[4pt]
{\langle
{\langle
\operatorname{d}\underline{\underline{\text{a}}}
\mathrm{d}\underline{\underline{\text{a}}}
~
~
\operatorname{d}\underline{\underline{\text{u}}}
\mathrm{d}\underline{\underline{\text{u}}}
\rangle}_{\operatorname{d}Y}
\rangle}_{\mathrm{d}Y}
\\[4pt]
\\[4pt]
0_{\operatorname{d}Y}
0_{\mathrm{d}Y}
\end{matrix}</math>
\end{matrix}</math>
|-
|-
| valign="bottom" |
| valign="bottom" |
<math>\begin{matrix}
<math>\begin{matrix}
0_{\operatorname{d}Y}
0_{\mathrm{d}Y}
\\[4pt]
\\[4pt]
{\langle
{\langle
\operatorname{d}\underline{\underline{\text{b}}}
\mathrm{d}\underline{\underline{\text{b}}}
~
~
\operatorname{d}\underline{\underline{\text{i}}}
\mathrm{d}\underline{\underline{\text{i}}}
\rangle}_{\operatorname{d}Y}
\rangle}_{\mathrm{d}Y}
\\[4pt]
\\[4pt]
{\langle
{\langle
\operatorname{d}\underline{\underline{\text{b}}}
\mathrm{d}\underline{\underline{\text{b}}}
~
~
\operatorname{d}\underline{\underline{\text{i}}}
\mathrm{d}\underline{\underline{\text{i}}}
\rangle}_{\operatorname{d}Y}
\rangle}_{\mathrm{d}Y}
\\[4pt]
\\[4pt]
0_{\operatorname{d}Y}
0_{\mathrm{d}Y}
\end{matrix}</math>
\end{matrix}</math>
|}
|}
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
|+ style="height:30px" |
<math>\text{Table 65.1} ~~ \operatorname{AIR}_1 (L_\text{A}) : \text{Analytic Representation of} ~ L_\text{A}\!</math>
<math>\text{Table 65.1} ~~ \mathrm{AIR}_1 (L_\text{A}) : \text{Analytic Representation of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Object}\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
|+ style="height:30px" |
<math>\text{Table 65.2} ~~ \operatorname{AIR}_1 (\operatorname{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!</math>
<math>\text{Table 65.2} ~~ \mathrm{AIR}_1 (\mathrm{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Object}\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
|+ style="height:30px" |
<math>\text{Table 65.3} ~~ \operatorname{AIR}_1 (\operatorname{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!</math>
<math>\text{Table 65.3} ~~ \mathrm{AIR}_1 (\mathrm{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
|+ style="height:30px" |
<math>\text{Table 66.1} ~~ \operatorname{AIR}_1 (L_\text{B}) : \text{Analytic Representation of} ~ L_\text{B}\!</math>
<math>\text{Table 66.1} ~~ \mathrm{AIR}_1 (L_\text{B}) : \text{Analytic Representation of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Object}\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
|+ style="height:30px" |
<math>\text{Table 66.2} ~~ \operatorname{AIR}_1 (\operatorname{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!</math>
<math>\text{Table 66.2} ~~ \mathrm{AIR}_1 (\mathrm{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Object}\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
|+ style="height:30px" |
<math>\text{Table 66.3} ~~ \operatorname{AIR}_1 (\operatorname{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!</math>
<math>\text{Table 66.3} ~~ \mathrm{AIR}_1 (\mathrm{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
|+ style="height:30px" |
<math>\text{Table 67.1} ~~ \operatorname{AIR}_2 (L_\text{A}) : \text{Analytic Representation of} ~ L_\text{A}\!</math>
<math>\text{Table 67.1} ~~ \mathrm{AIR}_2 (L_\text{A}) : \text{Analytic Representation of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Object}\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
|+ style="height:30px" |
<math>\text{Table 67.2} ~~ \operatorname{AIR}_2 (\operatorname{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!</math>
<math>\text{Table 67.2} ~~ \mathrm{AIR}_2 (\mathrm{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Object}\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
|+ style="height:30px" |
<math>\text{Table 67.3} ~~ \operatorname{AIR}_2 (\operatorname{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!</math>
<math>\text{Table 67.3} ~~ \mathrm{AIR}_2 (\mathrm{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
| valign="bottom" |
| valign="bottom" |
<math>\begin{matrix}
<math>\begin{matrix}
{\langle\operatorname{d}!\rangle}_{\operatorname{d}Y}
{\langle\mathrm{d}!\rangle}_{\mathrm{d}Y}
\\[4pt]
\\[4pt]
{\langle\operatorname{d}\text{n}\rangle}_{\operatorname{d}Y}
{\langle\mathrm{d}\text{n}\rangle}_{\mathrm{d}Y}
\\[4pt]
\\[4pt]
{\langle\operatorname{d}\text{n}\rangle}_{\operatorname{d}Y}
{\langle\mathrm{d}\text{n}\rangle}_{\mathrm{d}Y}
\\[4pt]
\\[4pt]
{\langle\operatorname{d}!\rangle}_{\operatorname{d}Y}
{\langle\mathrm{d}!\rangle}_{\mathrm{d}Y}
\end{matrix}</math>
\end{matrix}</math>
|-
|-
| valign="bottom" |
| valign="bottom" |
<math>\begin{matrix}
<math>\begin{matrix}
{\langle\operatorname{d}!\rangle}_{\operatorname{d}Y}
{\langle\mathrm{d}!\rangle}_{\mathrm{d}Y}
\\[4pt]
\\[4pt]
{\langle\operatorname{d}\text{n}\rangle}_{\operatorname{d}Y}
{\langle\mathrm{d}\text{n}\rangle}_{\mathrm{d}Y}
\\[4pt]
\\[4pt]
{\langle\operatorname{d}\text{n}\rangle}_{\operatorname{d}Y}
{\langle\mathrm{d}\text{n}\rangle}_{\mathrm{d}Y}
\\[4pt]
\\[4pt]
{\langle\operatorname{d}!\rangle}_{\operatorname{d}Y}
{\langle\mathrm{d}!\rangle}_{\mathrm{d}Y}
\end{matrix}</math>
\end{matrix}</math>
|}
|}
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
|+ style="height:30px" |
<math>\text{Table 68.1} ~~ \operatorname{AIR}_2 (L_\text{B}) : \text{Analytic Representation of} ~ L_\text{B}\!</math>
<math>\text{Table 68.1} ~~ \mathrm{AIR}_2 (L_\text{B}) : \text{Analytic Representation of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Object}\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
|+ style="height:30px" |
<math>\text{Table 68.2} ~~ \operatorname{AIR}_2 (\operatorname{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!</math>
<math>\text{Table 68.2} ~~ \mathrm{AIR}_2 (\mathrm{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Object}\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
|+ style="height:30px" |
<math>\text{Table 68.3} ~~ \operatorname{AIR}_2 (\operatorname{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!</math>
<math>\text{Table 68.3} ~~ \mathrm{AIR}_2 (\mathrm{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
| valign="bottom" |
| valign="bottom" |
<math>\begin{matrix}
<math>\begin{matrix}
{\langle\operatorname{d}!\rangle}_{\operatorname{d}Y}
{\langle\mathrm{d}!\rangle}_{\mathrm{d}Y}
\\[4pt]
\\[4pt]
{\langle\operatorname{d}\text{n}\rangle}_{\operatorname{d}Y}
{\langle\mathrm{d}\text{n}\rangle}_{\mathrm{d}Y}
\\[4pt]
\\[4pt]
{\langle\operatorname{d}\text{n}\rangle}_{\operatorname{d}Y}
{\langle\mathrm{d}\text{n}\rangle}_{\mathrm{d}Y}
\\[4pt]
\\[4pt]
{\langle\operatorname{d}!\rangle}_{\operatorname{d}Y}
{\langle\mathrm{d}!\rangle}_{\mathrm{d}Y}
\end{matrix}</math>
\end{matrix}</math>
|-
|-
| valign="bottom" |
| valign="bottom" |
<math>\begin{matrix}
<math>\begin{matrix}
{\langle\operatorname{d}!\rangle}_{\operatorname{d}Y}
{\langle\mathrm{d}!\rangle}_{\mathrm{d}Y}
\\[4pt]
\\[4pt]
{\langle\operatorname{d}\text{n}\rangle}_{\operatorname{d}Y}
{\langle\mathrm{d}\text{n}\rangle}_{\mathrm{d}Y}
\\[4pt]
\\[4pt]
{\langle\operatorname{d}\text{n}\rangle}_{\operatorname{d}Y}
{\langle\mathrm{d}\text{n}\rangle}_{\mathrm{d}Y}
\\[4pt]
\\[4pt]
{\langle\operatorname{d}!\rangle}_{\operatorname{d}Y}
{\langle\mathrm{d}!\rangle}_{\mathrm{d}Y}
\end{matrix}</math>
\end{matrix}</math>
|}
|}
<br>
<br>
===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~ ~\mathrm{at}~ t
\\[4pt]
~x~ ~\mathrm{at}~ t
\\[4pt]
(x) ~\mathrm{at}~ t
\\[4pt]
(x) ~\mathrm{at}~ t
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
~\mathrm{d}x~ ~\mathrm{at}~ t
\\[4pt]
(\mathrm{d}x) ~\mathrm{at}~ t
\\[4pt]
~\mathrm{d}x~ ~\mathrm{at}~ t
\\[4pt]
(\mathrm{d}x) ~\mathrm{at}~ t
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
(x) ~\mathrm{at}~ t'
\\[4pt]
~x~ ~\mathrm{at}~ t'
\\[4pt]
~x~ ~\mathrm{at}~ t'
\\[4pt]
(x) ~\mathrm{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 \mathrm{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 \mathrm{d}X \times X\!</math> is defined as follows.
{| align="center" cellspacing="8" width="90%"
| <math>\ominus ~=~ \{ (x, y, z) \in X \times \mathrm{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 \mathrm{d}X \times X\!</math> constituted by a sequential inference constraint can be interpreted as a proposition <math>\ominus : X \times \mathrm{d}X \times X \to \mathbb{B}\!</math> about sequential inference triples, and thus as a map <math>\ominus : \mathrm{d}X \to (X \times X \to \mathbb{B})\!</math> from the space <math>\mathrm{d}X\!</math> of differential states to the space of propositions about transitions in <math>X.\!</math>
<br>
'''Question.''' Group Actions? <math>r : \mathrm{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} ~ \mathrm{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}
(\mathrm{d}\underline{\underline{\text{a}}})
(\mathrm{d}\underline{\underline{\text{b}}})
(\mathrm{d}\underline{\underline{\text{i}}})
(\mathrm{d}\underline{\underline{\text{u}}})
\\[4pt]
~\mathrm{d}\underline{\underline{\text{a}}}~
(\mathrm{d}\underline{\underline{\text{b}}})
~\mathrm{d}\underline{\underline{\text{i}}}~
(\mathrm{d}\underline{\underline{\text{u}}})
\\[4pt]
(\mathrm{d}\underline{\underline{\text{a}}})
~\mathrm{d}\underline{\underline{\text{b}}}~
(\mathrm{d}\underline{\underline{\text{i}}})
~\mathrm{d}\underline{\underline{\text{u}}}~
\\[4pt]
~\mathrm{d}\underline{\underline{\text{a}}}~
~\mathrm{d}\underline{\underline{\text{b}}}~
~\mathrm{d}\underline{\underline{\text{i}}}~
~\mathrm{d}\underline{\underline{\text{u}}}~
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\langle \mathrm{d}! \rangle
\\[4pt]
\langle
\mathrm{d}\underline{\underline{\text{a}}} ~
\mathrm{d}\underline{\underline{\text{i}}}
\rangle
\\[4pt]
\langle
\mathrm{d}\underline{\underline{\text{b}}} ~
\mathrm{d}\underline{\underline{\text{u}}}
\rangle
\\[4pt]
\langle \mathrm{d}* \rangle
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\mathrm{d}!
\\[4pt]
\mathrm{d}\underline{\underline{\text{a}}} \cdot
\mathrm{d}\underline{\underline{\text{i}}} ~ !
\\[4pt]
\mathrm{d}\underline{\underline{\text{b}}} \cdot
\mathrm{d}\underline{\underline{\text{u}}} ~ !
\\[4pt]
\mathrm{d}*
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
1
\\[4pt]
\mathrm{d}_{\text{ai}}
\\[4pt]
\mathrm{d}_{\text{bu}}
\\[4pt]
\mathrm{d}_{\text{ai}} * \mathrm{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} ~ \mathrm{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}
(\mathrm{d}\underline{\underline{\text{a}}})
(\mathrm{d}\underline{\underline{\text{b}}})
(\mathrm{d}\underline{\underline{\text{i}}})
(\mathrm{d}\underline{\underline{\text{u}}})
\\[4pt]
~\mathrm{d}\underline{\underline{\text{a}}}~
(\mathrm{d}\underline{\underline{\text{b}}})
(\mathrm{d}\underline{\underline{\text{i}}})
~\mathrm{d}\underline{\underline{\text{u}}}~
\\[4pt]
(\mathrm{d}\underline{\underline{\text{a}}})
~\mathrm{d}\underline{\underline{\text{b}}}~
~\mathrm{d}\underline{\underline{\text{i}}}~
(\mathrm{d}\underline{\underline{\text{u}}})
\\[4pt]
~\mathrm{d}\underline{\underline{\text{a}}}~
~\mathrm{d}\underline{\underline{\text{b}}}~
~\mathrm{d}\underline{\underline{\text{i}}}~
~\mathrm{d}\underline{\underline{\text{u}}}~
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\langle \mathrm{d}! \rangle
\\[4pt]
\langle
\mathrm{d}\underline{\underline{\text{a}}} ~
\mathrm{d}\underline{\underline{\text{u}}}
\rangle
\\[4pt]
\langle
\mathrm{d}\underline{\underline{\text{b}}} ~
\mathrm{d}\underline{\underline{\text{i}}}
\rangle
\\[4pt]
\langle \mathrm{d}* \rangle
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\mathrm{d}!
\\[4pt]
\mathrm{d}\underline{\underline{\text{a}}} \cdot
\mathrm{d}\underline{\underline{\text{u}}} ~ !
\\[4pt]
\mathrm{d}\underline{\underline{\text{b}}} \cdot
\mathrm{d}\underline{\underline{\text{i}}} ~ !
\\[4pt]
\mathrm{d}*
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
1
\\[4pt]
\mathrm{d}_{\text{au}}
\\[4pt]
\mathrm{d}_{\text{bi}}
\\[4pt]
\mathrm{d}_{\text{au}} * \mathrm{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} ~ \mathrm{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}
(\mathrm{d}\text{m})
(\mathrm{d}\text{n})
\\[4pt]
~\mathrm{d}\text{m}~
(\mathrm{d}\text{n})
\\[4pt]
(\mathrm{d}\text{m})
~\mathrm{d}\text{n}~
\\[4pt]
~\mathrm{d}\text{m}~
~\mathrm{d}\text{n}~
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\langle\mathrm{d}!\rangle
\\[4pt]
\langle\mathrm{d}\text{m}\rangle
\\[4pt]
\langle\mathrm{d}\text{n}\rangle
\\[4pt]
\langle\mathrm{d}*\rangle
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\mathrm{d}!
\\[4pt]
\mathrm{d}\text{m}!
\\[4pt]
\mathrm{d}\text{n}!
\\[4pt]
\mathrm{d}*
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
1
\\[4pt]
\mathrm{d}_{\text{m}}
\\[4pt]
\mathrm{d}_{\text{n}}
\\[4pt]
\mathrm{d}_{\text{m}} * \mathrm{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}
(\mathrm{d}\text{m})
(\mathrm{d}\text{n})
\\[4pt]
~\mathrm{d}\text{m}~
(\mathrm{d}\text{n})
\\[4pt]
(\mathrm{d}\text{m})
~\mathrm{d}\text{n}~
\\[4pt]
~\mathrm{d}\text{m}~
~\mathrm{d}\text{n}~
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\langle\mathrm{d}!\rangle
\\[4pt]
\langle\mathrm{d}\text{m}\rangle
\\[4pt]
\langle\mathrm{d}\text{n}\rangle
\\[4pt]
\langle\mathrm{d}*\rangle
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\mathrm{d}!
\\[4pt]
\mathrm{d}\text{m}!
\\[4pt]
\mathrm{d}\text{n}!
\\[4pt]
\mathrm{d}*
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
1
\\[4pt]
\mathrm{d}_{\text{m}}
\\[4pt]
\mathrm{d}_{\text{n}}
\\[4pt]
\mathrm{d}_{\text{m}} * \mathrm{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>(\mathrm{d}\text{m})\!</math>
|
<math>\begin{matrix}
(\mathrm{d}\text{m})(\mathrm{d}\text{n})
\\[4pt]
(\mathrm{d}\text{m})~\mathrm{d}\text{n}~
\end{matrix}</math>
|
<math>\begin{matrix}
1
\\[4pt]
\mathrm{d}_\text{n}
\end{matrix}</math>
|-
| <math>G_\text{m} * \mathrm{d}_\text{m}\!</math>
| <math>\mathrm{d}\text{m}\!</math>
|
<math>\begin{matrix}
~\mathrm{d}\text{m}~(\mathrm{d}\text{n})
\\[4pt]
~\mathrm{d}\text{m}~~\mathrm{d}\text{n}~
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}_\text{m}
\\[4pt]
\mathrm{d}_\text{n} * \mathrm{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>({\mathrm{d}\text{n})}\!</math>
|
<math>\begin{matrix}
(\mathrm{d}\text{m})(\mathrm{d}\text{n})
\\[4pt]
~\mathrm{d}\text{m}~(\mathrm{d}\text{n})
\end{matrix}</math>
|
<math>\begin{matrix}
1
\\[4pt]
\mathrm{d}_\text{m}
\end{matrix}</math>
|-
| <math>G_\text{n} * \mathrm{d}_\text{n}\!</math>
| <math>\mathrm{d}\text{n}\!</math>
|
<math>\begin{matrix}
(\mathrm{d}\text{m})~\mathrm{d}\text{n}~
\\[4pt]
~\mathrm{d}\text{m}~~\mathrm{d}\text{n}~
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}_\text{n}
\\[4pt]
\mathrm{d}_\text{m} * \mathrm{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' = \mathrm{Pr}_{12}^{-1}(P)\!</math> and <math>Q' = \mathrm{Pr}_{23}^{-1}(Q),\!</math> which are defined as follows:
{| align="center" cellspacing="8" width="90%"
|
<math>\begin{matrix}
\mathrm{Pr}_{12}^{-1}(P) & = & P \times Z & = & \{ (x, y, z) \in X \times Y \times Z : (x, y) \in P \}.
\\[4pt]
\mathrm{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 ~=~ \mathrm{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} \mathrm{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} \mathrm{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}
\mathrm{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>\mathrm{Quo} (L) = \mathrm{Quo}_1 (L) \subseteq X\!</math> with what is usually called the ''domain of definition'' of <math>L\!</math> and we identify <math>\mathrm{Ran} (L) = \mathrm{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>\mathrm{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>\mathrm{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>\mathrm{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>\mathrm{Proj}^{(2)} L ~=~ (\mathrm{proj}_{12} L, ~ \mathrm{proj}_{13} L, ~ \mathrm{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>\mathrm{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>\mathrm{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>\mathrm{Pow}(S)\!</math> or <math>2^S\!</math> and defined as follows:
{| align="center" cellspacing="8" width="90%"
| <math>\mathrm{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>\mathrm{Pow}(S),\!</math> in other words, as <math>L \in \mathrm{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>\mathrm{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 ~\mathrm{choose}~ 2,\!</math> and defined as follows:
{| align="center" cellspacing="8" width="90%"
| <math>\mathrm{Explo}(X, Y, Z ~|~ 2) ~=~ \mathrm{Pow}(X \times Y) \times \mathrm{Pow}(X \times Z) \times \mathrm{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>\mathrm{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>\mathrm{Proj}^{(2)} : \mathrm{Pow}(X \times Y \times Z) \to \mathrm{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>\mathrm{Proj}^{(2)}\!</math> is injective. The mapping <math>\mathrm{Proj}^{(2)}\!</math> is said to ''preserve information'' about the triadic relations <math>L \in \mathrm{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 \mathrm{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>(\mathrm{Proj}^{(2)})^{-1}(\mathrm{Proj}^{(2)}L)\!</math> is the singleton set <math>\{ L \}.\!</math> Otherwise, there exists an <math>L'\!</math> such that <math>\mathrm{Proj}^{(2)}L = \mathrm{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>\mathrm{Proj}^{(2)}.\!</math>
The next series of Tables illustrates the operation of <math>\mathrm{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>\mathrm{Proj}^{(2)}L_\text{A}\!</math> and <math>\mathrm{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>\mathrm{Proj}^{(2)} L(\text{A})\!</math> and <math>\mathrm{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>\mathrm{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 \mathrm{Pow} (O \times S \times I)\!</math> is reducible in this sense it is necessary to show that no distinct <math>L' \in \mathrm{Pow} (O \times S \times I)\!</math> exists such that <math>\mathrm{Proj}^{(2)} L = \mathrm{Proj}^{(2)} L',\!</math> and this can take a rather more exhaustive or comprehensive investigation of the space <math>\mathrm{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>\mathrm{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 \mathrm{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''.
===6.36. Irreducibly Triadic Relations===
Most likely, any triadic relation <math>L \subseteq X \times Y \times Z\!</math> imposed on arbitrary domains <math>X, Y, Z\!</math> could find use as a sign relation, provided it embodies any constraint at all, in other words, so long as it forms a proper subset of its total space, a relationship symbolized by writing <math>L \subset X \times Y \times Z.\!</math> However, triadic relations of this sort are not guaranteed to form the most natural examples of sign relations.
In order to show what an irreducibly triadic relation looks like, this Section presents a pair of triadic relations that have the same dyadic projections, and thus cannot be distinguished from each other on this basis alone. As it happens, these examples of triadic relations can be discussed independently of sign relational concerns, but structures of their general ilk are frequently found arising in signal-theoretic applications, and they are undoubtedly closely associated with problems of reliable coding and communication.
Tables 74.1 and 75.1 show a pair of irreducibly triadic relations <math>L_0\!</math> and <math>L_1,\!</math> respectively. Tables 74.2 to 74.4 and Tables 75.2 to 75.4 show the dyadic relations comprising <math>\mathrm{Proj}^{(2)} L_0\!</math> and <math>\mathrm{Proj}^{(2)} L_1,\!</math> respectively.
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 74.1} ~~ \text{Relation} ~ L_0 =\{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>x\!</math>
| width="33%" | <math>y\!</math>
| width="33%" | <math>z\!</math>
|-
| valign="bottom" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| valign="bottom" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| valign="bottom" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"
|+ style="height:30px" | <math>\text{Table 74.2} ~~ \text{Dyadic Projection} ~ (L_0)_{12}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>x\!</math>
| width="33%" | <math>y\!</math>
|-
| valign="bottom" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| valign="bottom" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"
|+ style="height:30px" | <math>\text{Table 74.3} ~~ \text{Dyadic Projection} ~ (L_0)_{13}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>x\!</math>
| width="33%" | <math>z\!</math>
|-
| valign="bottom" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| valign="bottom" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"
|+ style="height:30px" | <math>\text{Table 74.4} ~~ \text{Dyadic Projection} ~ (L_0)_{23}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>y\!</math>
| width="33%" | <math>z\!</math>
|-
| valign="bottom" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| valign="bottom" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 75.1} ~~ \text{Relation} ~ L_1 =\{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>x\!</math>
| width="33%" | <math>y\!</math>
| width="33%" | <math>z\!</math>
|-
| valign="bottom" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| valign="bottom" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| valign="bottom" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"
|+ style="height:30px" | <math>\text{Table 75.2} ~~ \text{Dyadic Projection} ~ (L_1)_{12}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>x\!</math>
| width="33%" | <math>y\!</math>
|-
| valign="bottom" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| valign="bottom" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"
|+ style="height:30px" | <math>\text{Table 75.3} ~~ \text{Dyadic Projection} ~ (L_1)_{13}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>x\!</math>
| width="33%" | <math>z\!</math>
|-
| valign="bottom" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
| valign="bottom" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"
|+ style="height:30px" | <math>\text{Table 75.4} ~~ \text{Dyadic Projection} ~ (L_1)_{23}\!</math>
|- style="height:40px; background:#f0f0ff"
| width="33%" | <math>y\!</math>
| width="33%" | <math>z\!</math>
|-
| valign="bottom" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
| valign="bottom" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}</math>
|}
<br>
The relations <math>L_0, L_1 \subseteq \mathbb{B}^3\!</math> are defined by the following equations, with algebraic operations taking place as in <math>\text{GF}(2),\!</math> that is, with <math>1 + 1 = 0.\!</math>
# The triple <math>(x, y, z)\!</math> in <math>\mathbb{B}^3\!</math> belongs to <math>L_0\!</math> if and only if <math>{x + y + z = 0}.\!</math> Thus, <math>L_0\!</math> is the set of even-parity bit vectors, with <math>x + y = z.\!</math>
# The triple <math>(x, y, z)\!</math> in <math>\mathbb{B}^3\!</math> belongs to <math>L_1\!</math> if and only if <math>{x + y + z = 1}.\!</math> Thus, <math>L_1\!</math> is the set of odd-parity bit vectors, with <math>x + y = z + 1.\!</math>
The corresponding projections of <math>\mathrm{Proj}^{(2)} L_0\!</math> and <math>\mathrm{Proj}^{(2)} L_1\!</math> are identical. In fact, all six projections, taken at the level of logical abstraction, constitute precisely the same dyadic relation, isomorphic to the whole of <math>\mathbb{B} \times \mathbb{B}\!</math> and expressed by the universal constant proposition <math>1 : \mathbb{B} \times \mathbb{B} \to \mathbb{B}.\!</math> In summary:
{| align="center" cellspacing="8" width="90%"
|
<math>\begin{array}{lllll}
(L_0)_{12} & = & (L_1)_{12} & \cong & \mathbb{B}^2
\\[4pt]
(L_0)_{13} & = & (L_1)_{13} & \cong & \mathbb{B}^2
\\[4pt]
(L_0)_{23} & = & (L_1)_{23} & \cong & \mathbb{B}^2
\end{array}</math>
|}
Thus, <math>L_0\!</math> and <math>L_1\!</math> are both examples of irreducibly triadic relations.
===6.37. Propositional Types===
This Section describes a formal system of ''type expressions'' that are analogous to formulas of propositional logic and discusses their use as a calculus of predicates for classifying, analyzing, and drawing typical inferences about <math>k\!</math>-place relations, in particular, for reasoning about the results of operations on relations and about the properties of their transformations and combinations.
'''Definition.''' Given a cartesian product <math>X \times Y,\!</math> an ordered pair <math>(x, y) \in X \times Y\!</math> has the type <math>S \cdot T,\!</math> written <math>(x, y) : S \cdot T,\!</math> if and only if <math>x \in S \subseteq X\!</math> and <math>y \in T \subseteq Y.\!</math> Notice that an ordered pair may have many types.
'''Definition.''' A relation <math>L \subseteq X \times Y\!</math> has type <math>S \cdot T,\!</math> written <math>L : S \cdot T,\!</math> if and only if every <math>(x, y) \in L\!</math> has type <math>S \cdot T,\!</math> that is, if and only if <math>L \subseteq S \times T\!</math> for some <math>S \subseteq X\!</math> and <math>T \subseteq Y.\!</math>
'''Notation.''' Parentheses in the Courier or Teletype font, <math>\texttt{( ... )},\!</math> are used to indicate the negations of propositions and the complements of sets. When a <math>k\!</math>-place relation <math>L\!</math> is initially given relative to the domains <math>X_1, \ldots, X_k\!</math> and a set <math>S\!</math> is mentioned as a subset of one of them, say <math>S \subseteq X_j,\!</math> then the ''relevant complement'' of <math>S\!</math> in such a context is the one taken relative to <math>X_j.\!</math> Thus we have the following equivalents.
{| align="center" cellspacing="8" width="90%"
| <math>\texttt{(} S \texttt{)} ~=~ -\!S ~=~ X_j - S\!</math>
|}
In case of ambiguities that are not resolved by context, indices may be used as follows.
{| align="center" cellspacing="8" width="90%"
| <math>\texttt{(} S \texttt{)}_j ~=~ X_j - S\!</math>
|}
In any case, the intended term can always be written out in full, as <math>X_j - S.\!</math>
<br>
<center>'''Fragments'''</center>
Consider a relation <math>L\!</math> of the following type.
{| align="center" cellspacing="8" width="90%"
| <math>L : \texttt{(} S \texttt{(} T \texttt{))}\!</math>
|}
[The following piece occurs in § 6.35.]
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>\mathrm{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 ~\mathrm{choose}~ 2,\!</math> and defined as follows:
{| align="center" cellspacing="8" width="90%"
| <math>\mathrm{Explo}(X, Y, Z ~|~ 2) ~=~ \mathrm{Pow}(X \times Y) \times \mathrm{Pow}(X \times Z) \times \mathrm{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.
[Maybe the following piece belongs there, too.]
Just to provide a hint of what's at stake, consider the following suggestive identity:
{| align="center" cellspacing="8" width="90%"
| <math>2^{XY} \times 2^{XZ} \times 2^{YZ} ~=~ 2^{(XY + XY + YZ)}\!</math>
|}
What sense would have to be found for the sums on the right in order to interpret this equation as a set theoretic isomorphism? Answering this question requires the concept of a ''co-product'', roughly speaking, a “disjointed union” of sets. By the time this discussion has detailed the forms of indexing necessary to maintain these constructions, it should have become patently obvious that the forms of analysis and synthesis that are called on to achieve the putative reductions to and reconstructions from dyadic relations in actual fact never really leave the realm of genuinely triadic relations, but merely reshuffle its contents in various convenient fashions.
===6.38. Considering the Source===
There are several ways to contemplate the supplementation of signs, the sorts of augmentation that are crucial to meaning in the case of indices. Some approaches are analytic, in the sense that they regard signs as derivative compounds and try to break up the unitary concept of an individual sign into a congeries of seemingly more real, more actual, or more determinate sign instances. Other approaches are synthetic, in the sense that they accept a given collection of signs at face value and try to reconstruct more objective realities through the formation of abstract categories on this basis.
====6.38.1. Attributed Signs====
One type of analytic method takes it as a maxim for the logic of context that “Every sign or text is indexed by the context in which it occurs”. This means that all signs, including indices, are themselves indexed, though initially only tacitly, by the objective situation, the syntactic context, and the actual interpreter that makes use of them.
To begin formalizing this brand of supplementation, it is necessary to mark salient aspects of the situational, contextual, and inclusively interpretive features of sign usage that were previously held tacit. In effect, signs once regarded as primitive objects need to be newly analyzed as categorical abstractions that cover multitudes of existential sign instances or ''signs in use''.
One way to develop these dimensions of the <math>\text{A}\!</math> and <math>\text{B}\!</math> example is to articulate the interpretive parameters of signs by means of subscripts or superscripts attached to the signs or their quotations, in this way forming a corresponding set of ''situated signs'' or ''attributed remarks''.
The attribution of signs to their interpreters preserves the original object domain but produces an expanded syntactic domain, a corresponding set of ''attributed signs''. In our <math>\text{A}\!</math> and <math>\text{B}\!</math> example this gives the following domains.
{| align="center" cellspacing="6" width="90%"
|
<math>\begin{array}{ccl}
O & = &
\{ \text{A}, \text{B} \}
\\[6pt]
S & = &
\{
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}},
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{A}},
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}},
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{A}},
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}},
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{B}},
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{B}},
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}}
\}
\\[6pt]
I & = &
\{
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}},
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{A}},
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}},
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{A}},
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}},
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{B}},
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{B}},
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}}
\}
\end{array}</math>
|}
Table 76 displays the results of indexing every sign of the <math>\text{A}\!</math> and <math>\text{B}\!</math> example with a superscript indicating its source or ''exponent'', namely, the interpreter who actively communicates or transmits the sign. The operation of attribution produces two new sign relations, but it turns out that both sign relations have the same form and content, so a single Table will do. The new sign relation generated by this operation will be denoted <math>\mathrm{At} (\text{A}, \text{B})\!</math> and called the ''attributed sign relation'' for the <math>\text{A}\!</math> and <math>\text{B}\!</math> example.
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table 76.} ~~ \text{Attributed Sign Relation for Interpreters A and 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\text{A}}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}}
\end{matrix}</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\text{B}}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}}
\end{matrix}</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{i} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}}
\end{matrix}</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{u} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}}
\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\text{A}}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{A}}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{A}}
\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\text{B}}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{B}}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{A}}
\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{i} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{B}}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{A}}
\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{u} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{A}}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{A}}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{B}}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{A}}
\end{matrix}</math>
|}
<br>
Thus informed, the semiotic equivalence relation for interpreter <math>\text{A}\!</math> yields the following semiotic equations.
{| cellpadding="10"
| width="10%" |
| <math>[{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}}]_\text{A}\!</math>
| <math>=\!</math>
| <math>[{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}}]_\text{A}\!</math>
| <math>=\!</math>
| <math>[{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}}]_\text{A}\!</math>
| <math>=\!</math>
| <math>[{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}}]_\text{A}\!</math>
|-
| width="10%" | or
| <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}}\!</math>
| valign="bottom" | <math>=_\text{A}\!</math>
| <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}}\!</math>
| valign="bottom" | <math>=_\text{A}\!</math>
| <math>{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}}\!</math>
| valign="bottom" | <math>=_\text{A}\!</math>
| <math>{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}}\!</math>
|}
In comparison, the semiotic equivalence relation for interpreter <math>\text{B}\!</math> yields the following semiotic equations.
{| cellpadding="10"
| width="10%" |
| <math>[{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}}]_\text{B}\!</math>
| <math>=\!</math>
| <math>[{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}}]_\text{B}\!</math>
| <math>=\!</math>
| <math>[{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}}]_\text{B}\!</math>
| <math>=\!</math>
| <math>[{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}}]_\text{B}\!</math>
|-
| width="10%" | or
| <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}}\!</math>
| valign="bottom" | <math>=_\text{B}\!</math>
| <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}}\!</math>
| valign="bottom" | <math>=_\text{B}\!</math>
| <math>{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}}\!</math>
| valign="bottom" | <math>=_\text{B}\!</math>
| <math>{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}}\!</math>
|}
Consequently, the semiotic equivalence relations for <math>\text{A}\!</math> and <math>\text{B}\!</math> both induce the same semiotic partition on <math>S,\!</math> namely, the following.
{| align="center" cellspacing="6" width="90%"
|
<math>
\{ \{
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{A}},
{}^{\backprime\backprime} \text{A} {}^{\prime\prime\text{B}},
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{A}},
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{B}}
\}~,~\{
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{A}},
{}^{\backprime\backprime} \text{B} {}^{\prime\prime\text{B}},
{}^{\backprime\backprime} \text{i} {}^{\prime\prime\text{B}},
{}^{\backprime\backprime} \text{u} {}^{\prime\prime\text{A}}
\} \}.\!
</math>
|}
By means of a simple attribution step a certain level of congruity has been reached in the community of interpretation comprised of <math>\text{A}\!</math> and <math>\text{B}.\!</math> This new-found agreement on what is abstractly a single semiotic equivalence relation means that its equivalence classes reconstruct the structure of the object domain within the parts of the corresponding semiotic partition. This allows a measure of objectivity or inter-subjectivity to be predicated of the sign relation's representation.
An instance of <math>\text{Y}\!</math> using <math>{}^{\backprime\backprime} \text{X} {}^{\prime\prime}\!</math> is considered to be an objective event, the kind of happening to which all suitably placed observers can point, and adverting to an occurrence of <math>{}^{\backprime\backprime} \text{X} {}^{\prime\prime\text{Y}}\!</math> is more specific and less vague than resorting to instances of <math>{}^{\backprime\backprime} \text{X} {}^{\prime\prime}\!</math> as if being issued by anonymous sources. The situated sign <math>{}^{\backprime\backprime} \text{X} {}^{\prime\prime\text{Y}}\!</math> is a ''wider sign'' than <math>{}^{\backprime\backprime} \text{X} {}^{\prime\prime}\!</math> in the sense that it takes in a broader field of view on the interpretive situation and provides more information about the context of use. As to the reception of attributed remarks, the interpreter that can recognize signs of the form <math>{}^{\backprime\backprime} \text{X} {}^{\prime\prime\text{Y}}\!</math> is one that knows what it means to ''consider the source''.
It is best to read the superscripts on attributed signs as accentuations and integral parts of the quotation marks, taking <math>{}^{\backprime\backprime} \ldots {}^{\prime\prime\text{A}}\!</math> and <math>{}^{\backprime\backprime} \ldots {}^{\prime\prime\text{B}}\!</math> as variant inflections of <math>{}^{\backprime\backprime} \ldots {}^{\prime\prime}.\!</math> Thus, I can refer to the sign <math>{}^{\backprime\backprime} \text{X} {}^{\prime\prime\text{Y}}\!</math> just as I would refer to the sign <math>{}^{\backprime\backprime} \text{X} {}^{\prime\prime}\!</math> in the present informal context, without any additional marks of quotation.
Taking a cue from this usage, the ordinary quotes that I use to mark salient relationships of signs and expressions with respect to the informal context can now be regarded as quotes that I myself, operating as a casual interpreter, tacitly index. Even without knowing the complete sign relation that I have in mind, the one that I presumably use to conduct this discussion, the sign relation that <math>{}^{\backprime\backprime} \text{I} {}^{\prime\prime}\!</math> represents can nevertheless be partially formalized by means of a certain functional equation, namely, the following equation between semantic functions:
{| align="center" cellspacing="8" width="90%"
| <math>{}^{\backprime\backprime} \ldots {}^{\prime\prime} ~=~ {}^{\backprime\backprime} \ldots {}^{\prime\prime\text{I}}\!</math>
|}
By way of vocal expression, the attributed sign <math>{}^{\backprime\backprime} \text{X} {}^{\prime\prime\text{Y}}\!</math> can be pronounced in any of the following ways.
{| align="center" cellspacing="8" width="90%"
|
<math>\begin{array}{l}
{}^{\backprime\backprime} \text{X} {}^{\prime\prime} ~\text{quoth}~ \text{Y}
\\[4pt]
{}^{\backprime\backprime} \text{X} {}^{\prime\prime} ~\text{said by}~ \text{Y}
\\[4pt]
{}^{\backprime\backprime} \text{X} {}^{\prime\prime} ~\text{used by}~ \text{Y}
\end{array}</math>
|}
To facilitate visual imagery, each token of the type <math>{}^{\backprime\backprime} \text{X} {}^{\prime\prime\text{Y}}\!</math> can be pictured as a specific occasion where the sign <math>{}^{\backprime\backprime} \text{X} {}^{\prime\prime}\!</math> is being used or issued by the interpreter <math>\text{Y}.\!</math>
The construal of objects as classes of attributed signs leads to a measure of inter-subjective agreement between the interpreters <math>\text{A}\!</math> and <math>\text{B}.\!</math> Something like this must be the goal of any system of communication, and analogous forms of congruity and gregarity are likely to be found in any system for establishing mutually intelligible responses and maintaining socially coordinated practices.
Nevertheless, the particular types of “analytic” solutions that were proposed for resolving the conflict of interpretations between <math>\text{A}\!</math> and <math>\text{B}\!</math> are conceptually unsatisfactory in several ways. The constructions instituted retain the quality of hypotheses, especially due to the level of speculation about fundamental objects that is required to support them. There remains something fictional and imaginary about the nature of the object instances that are posited to form the ontological infrastructure, the supposedly more determinate strata of being that are presumed to anchor the initial objects of discussion.
Founding objects on a particular selection of object instances is always initially an arbitrary choice, a meet response to a judgment call and a responsibility that cannot be avoided, but still a bit of guesswork that needs to be tested for its reality in practice.
This means that the postulated objects of objects cannot have their reality probed and proved in detail but evaluated only in terms of their conceivable practical effects.
====6.38.2. Augmented Signs====
One synthetic method …
Suppose now that each of the agents <math>\text{A}\!</math> and <math>\text{B}\!</math> reflects on the situational context of their discussion and observes on every occasion of utterance exactly who is saying what. By this critically reflective operation of ''considering the source'' each interpreter is empowered to create, in effect, an ''extended token'' or ''situated sign'' out of each utterance by indexing it with the proper name of its utterer. Though it arises by reflection, the augmented sign is not a higher order of abstraction so much as a restoration or reconstitution of what was lost by abstracting the sign from the signer in the first instance.
In order to continue the development of this example, I need to employ a more precise system of marking quotations in order to keep track of who says what and in what kinds of context. To help with this, I use raised angle brackets <math>{}^\langle \ldots {}^\rangle\!</math> on a par with ordinary quotation marks <math>{}^{\backprime\backprime} \ldots {}^{\prime\prime}\!</math> to call attention to pieces of text as signs or expressions. The angle quotes are especially useful for embedded quotations and for text regarded as used or mentioned by interpreters other than myself, for instance, by the fictional characters <math>\text{A}\!</math> and <math>\text{B}.\!</math> Whenever possible, I save ordinary quotes for the outermost level, the one that interfaces with the context of informal discussion.
A notation like <math>{}^{\backprime\backprime ~ \langle\langle} \text{A} {}^\rangle, \text{B}, \text{C} {}^{\rangle ~ \prime\prime}\!</math> is intended to indicate the construction of an extended (attributed, indexed, or situated) sign, in this case, by enclosing an initial sign <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math> in a contextual envelope <math>{}^{\backprime\backprime ~ \langle\langle} ~\underline{~}~ {}^\rangle, ~\underline{~}~, ~\underline{~}~ {}^{\rangle ~ \prime\prime}\!</math> and inscribing it with relevant items of situational data, as represented by the signs <math>{}^{\backprime\backprime} \text{B} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \text{C} {}^{\prime\prime}.\!</math>
# When a salient component of the situational data represents an observation of the agent <math>\text{B}\!</math> communicating the sign <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime},\!</math> then the compressed form <math>{}^{\backprime\backprime ~ \langle\langle} \text{A} {}^\rangle \text{B}, \text{C} {}^{\rangle ~ \prime\prime}\!</math> can be used to mark that fact.
# When there is no additional contextual information beyond the marking of a sign's source, the form <math>{}^{\backprime\backprime ~ \langle\langle} \text{A} {}^\rangle \text{B} {}^{\rangle ~ \prime\prime}\!</math> suffices to say that <math>\text{B}\!</math> said <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}.\!</math>
With this last modification, angle quotes become like ascribed quotes or attributed remarks, indexed with the name of the interpretive agent that issued the message in question. In sum, the notation <math>{}^{\backprime\backprime ~ \langle\langle} \text{A} {}^\rangle \text{B} {}^{\rangle ~ \prime\prime}\!</math> is intended to situate the sign <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math> in the context of its contemplated use and to index the sign <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math> with the name of the interpreter that is considered to be using it on a given occasion.
The notation <math>{}^{\backprime\backprime ~ \langle\langle} \text{A} {}^\rangle \text{B} {}^{\rangle ~ \prime\prime},~\!</math> read <math>{}^{\backprime\backprime ~ \langle} \text{A} {}^\rangle ~\text{quoth}~ \text{B} {}^{\prime\prime}\!</math> or <math>{}^{\backprime\backprime ~ \langle} \text{A} {}^\rangle ~\text{used by}~ \text{B} {}^{\prime\prime},\!</math> is an expression that indicates the use of the sign <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math> by the interpreter <math>\text{B}.\!</math> The expression inside the outer quotes is referred to as an ''indexed quotation'', since it is indexed by the name of the interpreter to which it is referred.
Since angle quotes with a blank index are equivalent to ordinary quotes, we have the following equivalence. [Not sure about this.]
{| align="center" cellspacing="6" width="90%"
|
<math>{}^{\backprime\backprime} ~ {}^\langle \text{A} {}^\rangle \text{B} ~ {}^{\prime\prime} ~=~ {}^{\langle\langle} \text{A} {}^\rangle \text{B} {}^\rangle\!</math>
|}
Enclosing a piece of text with raised angle brackets and following it with the name of an interpreter is intended to call to mind …
The augmentation of signs by the names of their interpreters preserves the original object domain but produces an extended syntactic domain. In our <math>\text{A}\!</math> and <math>\text{B}\!</math> example this gives the following domains.
{| align="center" cellspacing="8" width="90%"
|
<math>\begin{array}{lll}
O & = & \{ \text{A}, \text{B} \}
\end{array}</math>
|}
{| align="center" cellspacing="8" width="90%"
|
<math>\begin{array}{lllllll}
S
& = &
\{ &
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{A} {}^{\prime\prime},
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{A} {}^{\prime\prime},
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{A} {}^{\prime\prime},
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{A} {}^{\prime\prime},
&
\\[4pt]
& & &
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime},
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{B} {}^{\prime\prime},
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{B} {}^{\prime\prime},
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{B} {}^{\prime\prime}
& \}
\\[10pt]
I
& = &
\{ &
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{A} {}^{\prime\prime},
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{A} {}^{\prime\prime},
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{A} {}^{\prime\prime},
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{A} {}^{\prime\prime},
&
\\[4pt]
& & &
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime},
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{B} {}^{\prime\prime},
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{B} {}^{\prime\prime},
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{B} {}^{\prime\prime}
& \}
\end{array}</math>
|}
The situated sign or indexed expression <math>{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime}\!</math> presents the sign or expression <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math> as used by the interpreter <math>\text{B}.\!</math> In other words, the sign is indexed by the name of an interpreter to indicate a use of that sign by that interpreter. Thus, <math>{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime}\!</math> augments <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math> to form a new and more complete sign by including additional information about the context of its transmission, in particular, by the consideration of its source.
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table 77.} ~~ \text{Augmented Sign Relation for Interpreters A and 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} [ {}^\langle \text{A} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{A} {}^{\prime\prime}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{B} {}^{\prime\prime}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{B} {}^{\prime\prime}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{A} {}^{\prime\prime}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{B} {}^{\prime\prime}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{B} {}^{\prime\prime}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{A} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{B} {}^{\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} [ {}^\langle \text{B} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{A} {}^{\prime\prime}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{A} {}^{\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} [ {}^\langle \text{B} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{B} {}^{\prime\prime}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{A} {}^{\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} [ {}^\langle \text{i} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{B} {}^{\prime\prime}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{A} {}^{\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} [ {}^\langle \text{u} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{A} {}^{\prime\prime}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{B} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{i} {}^\rangle ]_\text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} [ {}^\langle \text{u} {}^\rangle ]_\text{A} {}^{\prime\prime}
\end{matrix}</math>
|}
<br>
===6.39. Prospective Indices : Pointers to Future Work===
In the effort to unify dynamical, connectionist, and symbolic approaches to intelligent systems, indices supply important stepping stones between the sorts of signs that remain bound to circumscribed theaters of action and the kinds of signs that can function globally as generic symbols. Current technology presents an array of largely accidental discoveries that have been brought into being for implementing indexical systems. Bringing systematic study to bear on this variety of accessory devices and trying to discern within the wealth of incidental features their essential principles and effective ingredients could help to improve the traction this form of bridge affords.
In the points where this project addresses work on the indexical front, a primary task is to show how the ''actual connections'' promised by the definition of indexical signs can be translated into system-theoretic terms and implemented by means of the class of ''dynamic connections'' that can persist in realistic systems.
An offshoot of this investigation would be to explore how indices like pointer variables could be realized within “connectionist” systems. There is no reason in principle why this cannot be done, but I think that pragmatic reasons and practical success will force the contemplation of higher orders of connectivity than those currently fashioned in two-dimensional arrays of connections. To be specific, further advances will require the generative power of genuinely triadic relations to be exploited to the fullest possible degree.
To avert one potential misunderstanding of what this entails, computing with triadic relations is not really a live option unless the algebraic tools and logical calculi needed to do so are developed to greater levels of facility than they are at present. Merely officiating over the storage of “dead letters” in higher dimensional arrays will not do the trick. Turning static sign relations into the orders of dynamic sign processes that can support live inquiries will demand new means of representation and new methods of computation.
To fulfill their intended roles, a formal calculus for sign relations and the associated implementation must be able to address and restore the full dimensionalities of the existential and social matrices in which inquiry takes place. Informational constraints that define objective situations of interest need to be freed from the locally linear confines of the “dia-matrix” and reposted within the realm of the “tri-matrix”, that is, reconstituted in a manner that allows critical reflection on their form and content.
The descriptive and conceptual architectures needed to frame this task must allow space for interlacing forms of “open work”, projects that anticipate the desirability of higher order relations and build in the capability for higher order reflections at the very beginning, and do not merely hope against hope to arrange these capacities as afterthoughts.
===6.40. Dynamic and Evaluative Frameworks===
The sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> are lacking in several dimensions of realistic properties that would ordinarily be more fully developed in the kinds of sign relations that are found to be involved in inquiry. This section initiates a discussion of two such dimensions, the ''dynamic'' and the ''evaluative'' aspects of sign relations, and it treats the materials that are organized along these lines at two broad levels, either ''within'' or ''between'' particular examples of sign relations.
# The ''dynamic dimension'' deals with change. Thus, it details the forms of diversity that sign relations distribute in a temporal process. It is concerned with the transitions that take place from element to element within a sign relation and also with the changes that take place from one whole sign relation to another, thereby generating various types and levels of ''sign process''.
# The ''evaluative dimension'' deals with goals. Thus, it details the forms of diversity that sign relations contribute to a definite purpose. It is concerned with the comparisons that can be made on a scale of values between the elements within a sign relation and also between whole sign relations themselves, with a view toward deciding which is better for a ''designated purpose''.
At the primary level of analysis, one is concerned with the application of these two dimensions ''within'' particular sign relations. At every subsequent level of analysis, one deals with the dynamic transitions and evaluative comparisons that can be contemplated ''between'' particular sign relations. In order to cover all these dimensions, types, and levels of diversity in a unified way, there is need for a substantive term that can allow one to indicate any of the above objects of discussion and thought — including elements of sign relations, particular sign relations, and states of systems — and to regard it as an “object, sign, or state in a certain stage of construction”. I will use the word ''station'' for this purpose.
In order to organize the discussion of these two dimensions, both within and between particular sign relations, and to coordinate their ordinary relation to each other in practical situations, it pays to develop a combined form of ''dynamic evaluative framework'' (DEF), similar in design and utility to the objective frameworks set up earlier.
A ''dynamic evaluative framework'' (DEF) encompasses two dimensions of comparison between stations:
<ol style="list-style-type:decimal">
<li>
<p>A dynamic dimension, as swept out by a process of changing stations, permits comparison between stations in terms of before and after on a scale of temporal order.</p>
<p>A terminal station on a dynamic dimension is called a ''stable station''.</p></li>
<li>
<p>An evaluative dimension permits comparison between stations on a scale of values.</p>
<p>A terminal station on an evaluative dimension is called a ''canonical station'' or a ''standard station''.</p></li></ol>
A station that is both stable and standard is called a ''normal station''.
Consider the following analogies or correspondences that exist between different orders of sign relational structure:
# Just as a sign represents its object and becomes associated with more or less equivalent signs in the minds of interpretive agents, the corpus of signs that embodies a SOI represents in a collective way its own proper object, intended objective, or ''try at objectivity'' (TAO).
# Just as the relationship of a sign to its semantic objects and interpretive associates can be formalized within a single sign relation, the relation of a dynamically changing SOI to its reference environment, developmental goals, and desired characteristics of interpretive performance can be formalized by means of a higher order sign relation, one that further establishes a grounds of comparison for relating the growing SOI, not only to its former and future selves, but to a diverse company of other SOIs.
From an outside perspective the distinction between a sign and its object is usually regarded as obvious, though agents operating in the thick of a SOI often act as though they cannot see the difference. Nevertheless, as a rule in practice, a sign is not a good thing to be confused with its object. Even in the rare and usually controversial cases where an identity of substance is contemplated, usually only for the sake of argument, there is still a distinction of roles to be maintained between the sign and its object. Just so, …
Although there are aspects of inquiry processes that operate within the single sign relation, the characteristic features of inquiry do not come into full bloom until one considers the whole diversity of dynamically developing sign relations. Because it will be some time before this discussion acquires the formal power it needs to deal with higher order sign relations, these issues will need to be treated on an informal basis as they arise, and often in cursory and ''ad hoc'' manner.
===6.41. Elective and Motive Forces===
The <math>\text{A}\!</math> and <math>\text{B}\!</math> example, in the fragmentary aspects of its sign relations presented so far, is unrealistic in its simplification of semantic issues, lacking a full development of many kinds of attributes that almost always become significant in situations of practical interest. Just to mention two related features of importance to inquiry that are missing from this example, there is no sense of directional process and no dimension of differential value defined either within or between the semantic equivalence classes.
When there is a clear sense of dynamic tendency or purposeful direction driving the passage from signs to interpretants in the connotative project of a sign relation, then the study moves from sign relations, statically viewed, to genuine sign processes. In the pragmatic theory of signs, such processes are usually dignified with the name ''semiosis'' and their systematic investigation is called ''semiotics''.
Further, when this dynamism or purpose is consistent and confluent with a differential value system defined on the syntactic domain, then the sign process in question becomes a candidate for the kind of clarity-gaining, canon-seeking process, capable of supporting learning and reasoning, that I classify as an ''inquiry driven system''.
There is a mathematical turn of thought that I will often take in discussing these kinds of issues. Instead of saying that a system has no attribute of a particular type, I will say that it has the attribute, but in a degenerate or trivial sense. This is merely a strategy of classification that allows one to include null cases in a taxonomy and to make use of continuity arguments in passing from case to case in a class of examples. Viewed in this way, each of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> can be taken to exhibit a trivial dynamic process and a trivial standard of value defined on the syntactic domain.
===6.42. Sign Processes : A Start===
To articulate the dynamic aspects of a sign relation, one can interpret it as determining a discrete or finite state transition system. In the usual ways of doing this, the states of the system are given by the elements of the syntactic domain, while the elements of the object domain correspond to input data or control parameters that affect transitions from signs to interpretant signs in the syntactic state space.
Working from these principles alone, there are numerous ways that a plausible dynamics can be invented for a given sign relation. I will concentrate on two principal forms of dynamic realization, or two ways of interpreting and augmenting sign relations as sign processes.
One form of realization lets each element of the object domain <math>O\!</math> correspond to the observed presence of an object in the environment of the systematic agent. In this interpretation, the object <math>x\!</math> acts as an input datum that causes the system <math>Y\!</math> to shift from whatever sign state it happens to occupy at a given moment to a random sign state in <math>[x]_Y.\!</math> Expressed in a cognitive vein, <math>{}^{\backprime\backprime} Y ~\mathrm{notes}~ x {}^{\prime\prime}.</math>
Another form of realization lets each element of the object domain <math>O\!</math> correspond to the autonomous intention of the systematic agent to denote an object, achieve an objective, or broadly speaking to accomplish any other purpose with respect to an object in its domain. In this interpretation, the object <math>x\!</math> is a control parameter that brings the system <math>Y\!</math> into line with realizing a target set <math>[x]_Y.\!</math>
Tables 78 and 79 show how the sign relations for <math>\text{A}\!</math> and <math>\text{B}\!</math> can be filled out as finite state processes in conformity with the interpretive principles just described. Rather than letting the actions go undefined for some combinations of inputs in <math>O\!</math> and states in <math>S,\!</math> transitions have been added that take the interpreters from whatever else they might have been thinking about to the semantic equivalence classes of their objects. In either modality of realization, cognitive-oriented or control-oriented, the abstract structure of the resulting sign process is exactly the same.
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table 78.} ~~ \text{Sign Process 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}
\\
\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{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\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{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\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}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\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}
\\
{}^{\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 79.} ~~ \text{Sign Process 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}
\\
\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{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\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}
\\
{}^{\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}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\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{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\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>
Treated in accord with these interpretations, the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> constitute partially degenerate cases of dynamic processes, in which the transitions are totally non-deterministic up to semantic equivalence classes but still manage to preserve those classes. Whether construed as present observation or projective speculation, the most significant feature to note about a sign process is how the contemplation of an object or objective leads the system from a less determined to a more determined condition.
On reflection, one observes that these processes are not completely trivial since they preserve the structure of their semantic partitions. In fact, each sign process preserves the entire topology — the family of sets closed under finite intersections and arbitrary unions — that is generated by its semantic equivalence classes. These topologies, <math>\mathrm{Top}(\text{A})\!</math> and <math>\mathrm{Top}(\text{B}),\!</math> can be viewed as partially ordered sets, <math>\mathrm{Poset}(\text{A})\!</math> and <math>\mathrm{Poset}(\text{B}),\!</math> by taking the inclusion ordering <math>(\subseteq)\!</math> as <math>(\le).\!</math> For each of the interpreters <math>\text{A}\!</math> and <math>\text{B},\!</math> as things stand in their respective orderings <math>\mathrm{Poset}(\text{A})\!</math> and <math>\mathrm{Poset}(\text{B}),\!</math> the semantic equivalence classes of <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \text{B} {}^{\prime\prime}\!</math> are situated as intermediate elements that are incomparable to each other.
{| align="center" cellspacing="6" width="90%"
|
<math>\begin{array}{lllll}
\mathrm{Top}(\text{A})
& = &
\mathrm{Poset}(\text{A})
& = &
\{
\varnothing,
\{
{}^{\backprime\backprime} \text{A} {}^{\prime\prime},
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\},
\{
{}^{\backprime\backprime} \text{B} {}^{\prime\prime},
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\},
S
\}.
\\[6pt]
\mathrm{Top}(\text{B})
& = &
\mathrm{Poset}(\text{B})
& = &
\{ \varnothing,
\{
{}^{\backprime\backprime} \text{A} {}^{\prime\prime},
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\},
\{
{}^{\backprime\backprime} \text{B} {}^{\prime\prime},
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\},
S
\}.
\end{array}</math>
|}
In anticipation of things to come, these orderings are germinal versions of the kinds of semantic hierarchies that will be used in this project to define the ''ontologies'', ''perspectives'', or ''world views'' corresponding to individual interpreters.
When it comes to discussing the stability properties of dynamic systems, the sets that remain invariant under iterated applications of a process are called its ''attractors'' or ''basins of attraction''.
'''Note.''' More care needed here. Strongly and weakly connected components of digraphs?
The dynamic realizations of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> augment their semantic equivalence relations in an “attractive” way. To describe this additional structure, I introduce a set of graph-theoretical concepts and notations.
The ''attractor'' of <math>x\!</math> in <math>Y.\!</math>
{| align="center" cellspacing="6" width="90%"
|
<math>Y ~\text{at}~ x ~=~ \mathrm{At}[x]_Y ~=~ [x]_Y \cup \{ \text{arcs into}~ [x]_Y \}.</math>
|}
In effect, this discussion of dynamic realizations of sign relations has advanced from considering semiotic partitions as partitioning the set of points in <math>S\!</math> to considering attractors as partitioning the set of arcs in <math>S \times I = S \times S.\!</math>
===6.43. Reflective Extensions===
This section takes up the topic of reflective extensions in a more systematic fashion, starting from the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> once again and keeping its focus within their vicinity, but exploring the space of nearby extensions in greater detail.
Tables 80 and 81 show one way that the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> can be extended in a reflective sense through the use of quotational devices, yielding the ''first order reflective extensions'', <math>\mathrm{Ref}^1 (\text{A})\!</math> and <math>\mathrm{Ref}^1 (\text{B}).\!</math>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>{\text{Table 80.} ~~ \text{Reflective Extension} ~ \mathrm{Ref}^1 (\text{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}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle\langle} \text{A} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{B} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{i} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{u} {}^{\rangle\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle\langle} \text{A} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{B} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{i} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{u} {}^{\rangle\rangle}
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" |
<math>{\text{Table 81.} ~~ \text{Reflective Extension} ~ \mathrm{Ref}^1 (\text{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}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle\langle} \text{A} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{B} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{i} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{u} {}^{\rangle\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle\langle} \text{A} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{B} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{i} {}^{\rangle\rangle}
\\
{}^{\langle\langle} \text{u} {}^{\rangle\rangle}
\end{matrix}</math>
|}
<br>
The common ''world'' <math>W\!</math> of the reflective extensions <math>\mathrm{Ref}^1 (\text{A})\!</math> and <math>\mathrm{Ref}^1 (\text{B})\!</math> is the totality of objects and signs they contain, namely, the following set of 10 elements.
{| align="center" cellspacing="8" width="90%"
| <math>W = \{ \text{A}, \text{B}, {}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{i} {}^{\rangle}, {}^{\langle} \text{u} {}^{\rangle}, {}^{\langle\langle} \text{A} {}^{\rangle\rangle}, {}^{\langle\langle} \text{B} {}^{\rangle\rangle}, {}^{\langle\langle} \text{i} {}^{\rangle\rangle}, {}^{\langle\langle} \text{u} {}^{\rangle\rangle} \}.</math>
|}
Raised angle brackets or ''supercilia'' <math>({}^{\langle} \ldots {}^{\rangle})\!</math> are here being used on a par with ordinary quotation marks <math>({}^{\backprime\backprime} \ldots {}^{\prime\prime})\!</math> to construct a new sign whose object is precisely the sign they enclose.
Regarded as sign relations in their own right, <math>\mathrm{Ref}^1 (\text{A})\!</math> and <math>\mathrm{Ref}^1 (\text{B})\!</math> are formed on the following relational domains.
{| align="center" cellspacing="6" width="90%"
|
<math>\begin{array}{ccccl}
O & = & O^{(1)} \cup O^{(2)} & = &
\{ \text{A}, \text{B} \}
~ \cup ~
\{
{}^{\langle} \text{A} {}^{\rangle},
{}^{\langle} \text{B} {}^{\rangle},
{}^{\langle} \text{i} {}^{\rangle},
{}^{\langle} \text{u} {}^{\rangle}
\}
\\[8pt]
S & = & S^{(1)} \cup S^{(2)} & = &
\{
{}^{\langle} \text{A} {}^{\rangle},
{}^{\langle} \text{B} {}^{\rangle},
{}^{\langle} \text{i} {}^{\rangle},
{}^{\langle} \text{u} {}^{\rangle}
\}
~ \cup ~
\{
{}^{\langle\langle} \text{A} {}^{\rangle\rangle},
{}^{\langle\langle} \text{B} {}^{\rangle\rangle},
{}^{\langle\langle} \text{i} {}^{\rangle\rangle},
{}^{\langle\langle} \text{u} {}^{\rangle\rangle}
\}
\\[8pt]
I & = & I^{(1)} \cup I^{(2)} & = &
\{
{}^{\langle} \text{A} {}^{\rangle},
{}^{\langle} \text{B} {}^{\rangle},
{}^{\langle} \text{i} {}^{\rangle},
{}^{\langle} \text{u} {}^{\rangle}
\}
~ \cup ~
\{
{}^{\langle\langle} \text{A} {}^{\rangle\rangle},
{}^{\langle\langle} \text{B} {}^{\rangle\rangle},
{}^{\langle\langle} \text{i} {}^{\rangle\rangle},
{}^{\langle\langle} \text{u} {}^{\rangle\rangle}
\}
\end{array}</math>
|}
It may be observed that <math>S\!</math> overlaps with <math>O\!</math> in the set of first-order signs or second-order objects, <math>S^{(1)} = O^{(2)},\!</math> exemplifying the extent to which signs have become objects in the new sign relations.
To discuss how the denotative and connotative aspects of a sign related are affected by its reflective extension it is useful to introduce a few abbreviations. For each sign relation <math>L\!</math> in <math>\{ L_\text{A}, L_\text{B} \}\!</math> the following operations may be defined.
{| align="center" cellspacing="6" width="90%"
|
<math>\begin{array}{lllll}
\mathrm{Den}^1 (L)
& = &
(\mathrm{Ref}^1 (L))_{SO}
& = &
\mathrm{proj}_{OS} (\mathrm{Ref}^1 (L))
\\[6pt]
\mathrm{Con}^1 (L)
& = &
(\mathrm{Ref}^1 (L))_{SI}
& = &
\mathrm{proj}_{SI} (\mathrm{Ref}^1 (L))
\end{array}\!</math>
|}
The dyadic components of sign relations can be given graph-theoretic representations, namely, as ''digraphs'' (directed graphs), that provide concise pictures of their structural and potential dynamic properties. By way of terminology, a directed edge <math>(x, y)\!</math> is called an ''arc'' from point <math>x\!</math> to point <math>y,\!</math> and a self-loop <math>(x, x)\!</math> is called a ''sling'' at <math>x.\!</math>
The denotative components <math>\mathrm{Den}^1 (L_\text{A})\!</math> and <math>\mathrm{Den}^1 (L_\text{B})\!</math> can be viewed as digraphs on the 10 points of the world set <math>W.\!</math> The arcs of these digraphs are given as follows.
<ol>
<li><math>\mathrm{Den}^1 (L_\text{A})\!</math> has an arc from each point of <math>[\text{A}]_\text{A} = \{ {}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{i}{}^{\rangle} \}\!</math> to <math>\text{A}\!</math> and from each point of <math>[\text{B}]_\text{A} = \{ {}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{u} {}^{\rangle} \}\!</math> to <math>\text{B}.\!</math></li>
<li><math>\mathrm{Den}^1 (L_\text{B})\!</math> has an arc from each point of <math>[\text{A}]_\text{B} = \{ {}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{u}{}^{\rangle} \}\!</math> to <math>\text{A}\!</math> and from each point of <math>[\text{B}]_\text{B} = \{ {}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{i} {}^{\rangle} \}\!</math> to <math>\text{B}.\!</math></li>
<li>In the parts added by reflective extension <math>\mathrm{Den}^1 (L_\text{A})\!</math> and <math>\mathrm{Den}^1 (L_\text{B})\!</math> both have arcs from <math>{}^{\langle} s {}^{\rangle}\!</math> to <math>s,\!</math> for each <math>s \in S^{(1)}.\!</math></li>
</ol>
Taken as transition digraphs, <math>\mathrm{Den}^1 (L_\text{A})\!</math> and <math>\mathrm{Den}^1 (L_\text{B})\!</math> summarize the upshots, end results, or effective steps of computation that are involved in the respective evaluations of signs in <math>S\!</math> by <math>\mathrm{Ref}^1 (\text{A})\!</math> and <math>\mathrm{Ref}^1 (\text{B}).\!</math>
The connotative components <math>\mathrm{Con}^1 (L_\text{A})~\!</math> and <math>\mathrm{Con}^1 (L_\text{B})~\!</math> can be viewed as digraphs on the eight points of the syntactic domain <math>S.\!</math> The arcs of these digraphs are given as follows.
<ol>
<li><math>\mathrm{Con}^1 (L_\text{A})\!</math> inherits from <math>L_\text{A}\!</math> the structure of a semiotic equivalence relation on <math>S^{(1)},\!</math> having a sling on each point of <math>S^{(1)},\!</math> arcs in both directions between <math>{}^{\langle} \text{A} {}^{\rangle}\!</math> and <math>{}^{\langle} \text{i}{}^{\rangle},\!</math> and arcs in both directions between <math>{}^{\langle} \text{B} {}^{\rangle}~\!</math> and <math>{}^{\langle} \text{u}{}^{\rangle}.~\!</math> The reflective extension <math>\mathrm{Ref}^1 (L_\text{A})\!</math> adds a sling on each point of <math>S^{(2)},\!</math> creating a semiotic equivalence relation on <math>S.\!</math></li>
<li><math>\mathrm{Con}^1 (L_\text{B})~\!</math> inherits from <math>L_\text{B}\!</math> the structure of a semiotic equivalence relation on <math>S^{(1)},\!</math> having a sling on each point of <math>S^{(1)},\!</math> arcs in both directions between <math>{}^{\langle} \text{A} {}^{\rangle}\!</math> and <math>{}^{\langle} \text{u}{}^{\rangle},\!</math> and arcs in both directions between <math>{}^{\langle} \text{B} {}^{\rangle}~\!</math> and <math>{}^{\langle} \text{i}{}^{\rangle}.~\!</math> The reflective extension <math>\mathrm{Ref}^1 (L_\text{B})\!</math> adds a sling on each point of <math>S^{(2)},\!</math> creating a semiotic equivalence relation on <math>S.\!</math></li>
</ol>
Taken as transition digraphs, <math>\mathrm{Con}^1 (L_\text{A})~\!</math> and <math>\mathrm{Con}^1 (L_\text{B})~\!</math> highlight the associations between signs in <math>\mathrm{Ref}^1 (L_\text{A})\!</math> and <math>\mathrm{Ref}^1 (L_\text{B}),\!</math> respectively.
The semiotic equivalence relation given by <math>\mathrm{Con}^1 (L_\text{A})\!</math> for interpreter <math>\text{A}\!</math> has the following semiotic equations.
{| cellpadding="10"
| width="10%" |
| <math>[ {}^{\langle} \text{A} {}^{\rangle} ]_\text{A}\!</math>
| <math>=\!</math>
| <math>[ {}^{\langle} \text{i} {}^{\rangle} ]_\text{A}\!</math>
| width="20%" |
| <math>[ {}^{\langle} \text{B} {}^{\rangle} ]_\text{A}\!</math>
| <math>=\!</math>
| <math>[ {}^{\langle} \text{u} {}^{\rangle} ]_\text{A}\!</math>
|-
| width="10%" | or
| <math>{}^{\langle} \text{A} {}^{\rangle}~\!</math>
| <math>=_\text{A}\!</math>
| <math>{}^{\langle} \text{i} {}^{\rangle}~\!</math>
| width="20%" |
| <math>{}^{\langle} \text{B} {}^{\rangle}~\!</math>
| <math>=_\text{A}\!</math>
| <math>{}^{\langle} \text{u} {}^{\rangle}~\!</math>
|}
These equations induce the following semiotic partition.
{| align="center" cellspacing="6" width="90%"
|
<math>
\{
\{ {}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{i} {}^{\rangle} \},
\{ {}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{u} {}^{\rangle} \},
\{ {}^{\langle\langle} \text{A} {}^{\rangle\rangle} \},
\{ {}^{\langle\langle} \text{i} {}^{\rangle\rangle} \},
\{ {}^{\langle\langle} \text{B} {}^{\rangle\rangle} \},
\{ {}^{\langle\langle} \text{u} {}^{\rangle\rangle} \}
\}.\!
</math>
|}
The semiotic equivalence relation given by <math>\mathrm{Con}^1 (L_\text{B})~\!</math> for interpreter <math>\text{B}\!</math> has the following semiotic equations.
{| cellpadding="10"
| width="10%" |
| <math>[ {}^{\langle} \text{A} {}^{\rangle} ]_\text{B}\!</math>
| <math>=\!</math>
| <math>[ {}^{\langle} \text{u} {}^{\rangle} ]_\text{B}\!</math>
| width="20%" |
| <math>[ {}^{\langle} \text{B} {}^{\rangle} ]_\text{B}\!</math>
| <math>=\!</math>
| <math>[ {}^{\langle} \text{i} {}^{\rangle} ]_\text{B}\!</math>
|-
| width="10%" | or
| <math>{}^{\langle} \text{A} {}^{\rangle}~\!</math>
| <math>=_\text{B}\!</math>
| <math>{}^{\langle} \text{u} {}^{\rangle}~\!</math>
| width="20%" |
| <math>{}^{\langle} \text{B} {}^{\rangle}~\!</math>
| <math>=_\text{B}\!</math>
| <math>{}^{\langle} \text{i} {}^{\rangle}~\!</math>
|}
These equations induce the following semiotic partition.
{| align="center" cellspacing="6" width="90%"
|
<math>
\{
\{ {}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{u} {}^{\rangle} \},
\{ {}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{i} {}^{\rangle} \},
\{ {}^{\langle\langle} \text{A} {}^{\rangle\rangle} \},
\{ {}^{\langle\langle} \text{i} {}^{\rangle\rangle} \},
\{ {}^{\langle\langle} \text{B} {}^{\rangle\rangle} \},
\{ {}^{\langle\langle} \text{u} {}^{\rangle\rangle} \}
\}.\!
</math>
|}
Notice that the semiotic equivalences of nouns and pronouns for each interpreter do not extend to equivalences of their second-order signs, exactly as demanded by the literal character of quotations. Moreover, the new sign relations for interpreters <math>\text{A}\!</math> and <math>\text{B}\!</math> coincide in their reflective parts, since exactly the same triples are added to each set.
There are many ways to extend sign relations in an effort to increase their reflective capacities. The implicit goal of a reflective project is to achieve ''reflective closure'', <math>S \subseteq O,\!</math> where every sign is an object.
Considered as reflective extensions, there is nothing unique about the constructions of <math>\mathrm{Ref}^1 (\text{A})\!</math> and <math>\mathrm{Ref}^1 (\text{B})\!</math> but their common pattern of development illustrates a typical approach toward reflective closure. In a sense it epitomizes the project of ''free'', ''naive'', or ''uncritical'' reflection, since continuing this mode of production to its closure would generate an infinite sign relation, passing through infinitely many higher orders of signs, but without examining critically to what purpose the effort is directed or evaluating alternative constraints that might be imposed on the initial generators toward this end.
At first sight it seems as though the imposition of reflective closure has multiplied a finite sign relation into an infinite profusion of highly distracting and largely redundant signs, all by itself and all in one step. But this explosion of orders happens only with the complicity of another requirement, that of deterministic interpretation.
There are two types of non-determinism, denotative and connotative, that can affect a sign relation.
<ol>
<li>A sign relation <math>L\!</math> has a non-deterministic denotation if its dyadic component <math>{L_{SO}}\!</math> is not a function <math>L_{SO} : S \to O,\!</math> in other words, if there are signs in <math>S\!</math> with missing or multiple objects in <math>O.\!</math></li>
<li>A sign relation <math>L\!</math> has a non-deterministic connotation if its dyadic component <math>L_{SI}\!</math> is not a function <math>L_{SI} : S \to I,\!</math> in other words, if there are signs in <math>S\!</math> with missing or multiple interpretants in <math>I.\!</math> As a rule, sign relations are rife with this variety of non-determinism, but it is usually felt to be under control so long as <math>L_{SI}\!</math> remains close to being an equivalence relation.</li>
</ol>
Thus, it is really the denotative type of indeterminacy that is felt to be a problem in this context.
The next two pairs of reflective extensions demonstrate that there are ways of achieving reflective closure that do not generate infinite sign relations.
As a flexible and fairly general strategy for describing reflective extensions, it is convenient to take the following tack. Given a syntactic domain <math>S,\!</math> there is an independent formal language <math>F = F(S) = S \langle {}^{\langle\rangle} \rangle,\!</math> called the ''free quotational extension of <math>S,\!</math>'' that can be generated from <math>S\!</math> by embedding each of its signs to any depth of quotation marks. Within <math>F,\!</math> the quoting operation can be regarded as a syntactic generator that is inherently free of constraining relations. In other words, for every <math>s \in S,\!</math> the sequence <math>s, {}^{\langle} s {}^{\rangle}, {}^{\langle\langle} s {}^{\rangle\rangle}, \ldots\!</math> contains nothing but pairwise distinct elements in <math>F\!</math> no matter how far it is produced. The set <math>F(s) = s \langle {}^{\langle\rangle} \rangle \subseteq F\!</math> that collects the elements of this sequence is called the ''subset of <math>F\!</math> generated from <math>s\!</math> by quotation''.
Against this background, other varieties of reflective extension can be specified by means of semantic equations that are considered to be imposed on the elements of <math>F.\!</math> Taking the reflective extensions <math>\mathrm{Ref}^1 (\text{A})\!</math> and <math>\mathrm{Ref}^1 (\text{B})\!</math> as the first orders of a “free” project toward reflective closure, variant extensions can be described by relating their entries with those of comparable members in the standard sequences <math>\mathrm{Ref}^n (\text{A})\!</math> and <math>\mathrm{Ref}^n (\text{B}).\!</math>
A variant pair of reflective extensions, <math>\mathrm{Ref}^1 (\text{A} | E_1)\!</math> and <math>\mathrm{Ref}^1 (\text{B} | E_1),\!</math> is presented in Tables 82 and 83, respectively. These are identical to the corresponding free variants, <math>\mathrm{Ref}^1 (\text{A})~\!</math> and <math>\mathrm{Ref}^1 (\text{B}),~\!</math> with the exception of those entries that are constrained by the following system of semantic equations.
{| align="center" cellspacing="8" width="90%"
|
<math>\begin{matrix}
E_1 :
&
{}^{\langle\langle} \text{A} {}^{\rangle\rangle} = {}^{\langle} \text{A} {}^{\rangle},
&
{}^{\langle\langle} \text{B} {}^{\rangle\rangle} = {}^{\langle} \text{B} {}^{\rangle},
&
{}^{\langle\langle} \text{i} {}^{\rangle\rangle} = {}^{\langle} \text{i} {}^{\rangle},
&
{}^{\langle\langle} \text{u} {}^{\rangle\rangle} = {}^{\langle} \text{u} {}^{\rangle}.
\end{matrix}</math>
|}
This has the effect of making all levels of quotation equivalent.
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table 82.} ~~ \text{Reflective Extension} ~ \mathrm{Ref}^1 (\text{A} | E_1)\!</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}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table 83.} ~~ \text{Reflective Extension} ~ \mathrm{Ref}^1 (\text{B} | E_1)\!</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}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
|}
<br>
Another pair of reflective extensions, <math>\mathrm{Ref}^1 (\text{A} | E_2)\!</math> and <math>\mathrm{Ref}^1 (\text{B} | E_2),\!</math> is presented in Tables 84 and 85, respectively. These are identical to the corresponding free variants, <math>\mathrm{Ref}^1 (\text{A})~\!</math> and <math>\mathrm{Ref}^1 (\text{B}),~\!</math> except for the entries constrained by the following semantic equations.
{| align="center" cellspacing="8" width="90%"
|
<math>\begin{matrix}
E_2 :
&
{}^{\langle\langle} \text{A} {}^{\rangle\rangle} = \text{A},
&
{}^{\langle\langle} \text{B} {}^{\rangle\rangle} = \text{B},
&
{}^{\langle\langle} \text{i} {}^{\rangle\rangle} = \text{i},
&
{}^{\langle\langle} \text{u} {}^{\rangle\rangle} = \text{u}.
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table 84.} ~~ \text{Reflective Extension} ~ \mathrm{Ref}^1 (\text{A} | E_2)\!</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}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\text{A}
\\
\text{B}
\\
\text{A}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\text{A}
\\
\text{B}
\\
\text{A}
\\
\text{B}
\end{matrix}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table 85.} ~~ \text{Reflective Extension} ~ \mathrm{Ref}^1 (\text{B} | E_2)\!</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}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\\
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
{}^{\langle} \text{A} {}^{\rangle}
\\
{}^{\langle} \text{B} {}^{\rangle}
\\
{}^{\langle} \text{i} {}^{\rangle}
\\
{}^{\langle} \text{u} {}^{\rangle}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\text{A}
\\
\text{B}
\\
\text{B}
\\
\text{A}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\text{A}
\\
\text{B}
\\
\text{B}
\\
\text{A}
\end{matrix}</math>
|}
<br>
By calling attention to their intended status as ''semantic'' equations, meaning that signs are being set equal in the semantic equivalence classes they inhabit or the objects they denote, I hope to emphasize that these equations are able to say something significant about objects.
'''Question.''' Redo <math>F(S)\!</math> over <math>W\!</math>? Use <math>W_F = O \cup F\!</math>?
===6.44. Reflections on Closure===
The previous section dealt with a formal operation that was dubbed ''reflection'' and found that it was closely associated with the device of ''quotation'' that makes it possible to treat signs as objects by making or finding other signs that refer to them. Clearly, an ability to take signs as objects is one component of a cognitive capacity for reflection. But a genuine and less superficial species of reflection can do more than grasp just the isolated signs and the separate interpretants of the thinking process as objects — it can pause the fleeting procession of signs upon signs and seize their generic patterns of transition as valid objects of discussion. This involves the conception and composition of not just ''higher order'' signs but also ''higher type'' signs, orders of signs that aspire to catch whole sign relations up in one breath.
…
===6.45. Intelligence ⇒ Critical Reflection===
It is just at this point that the discussion of sign relations is forced to contemplate the prospects of intelligent interpretation. For starters, I consider an intelligent interpreter to be one that can pursue alternative interpretations of a sign or text and pick one that makes sense. If an interpreter can find all of the most sensible interpretations and order them according to a scale of meaningfulness, but without losing the time required to act on their import, then so much the better.
Intelligent interpreters are a centrally important species of intelligent agents in general, since hardly any intelligent action at all can be taken without the ability to interpret signs and texts, even if read only in the sense of “the text of nature”. In other words, making sense of dubious signs is a central component of all sensible action.
Thus, I regard the determining trait of intelligent agency to be its response to non-deterministic situations. Agents that find themselves at junctures of unavoidable uncertainty are required by objective features of the situation to gather together the available options and select among the multitude of possibilities a few choices that further their active purposes.
Reflection enables an interpreter to stand back from signs and view them as objects, that is, as objective possibilities for choice to be followed up in a critical and experimental fashion rather than pursued as automatic reactions whose habitual connections cannot be questioned.
The mark of an intelligent interpreter that is relevant in this context is the ability to face (encounter, countenance) a non-deterministic juncture of choices in a sign relation and to respond to it as such with actions appropriate to the uncertain nature of the situation.
'''[Variants]'''
An intelligent interpreter is one that can follow up several different interpretations at once, experimenting with the denotations and connotations that are available in a non-deterministic sign relation, …
An intelligent interpreter is one that can face a situation of non deterministic choice and choose an interpretation (denotation or connotation) that fits the objective and syntactic context.
An intelligent interpreter is one that can deal with non-deterministic situations, that is, one that can follow up several lines of possible meaning for signs and read between the lines to pick out meanings that are sensitive to both the objective situation and the syntactic context of interpretation.
An intelligent interpreter is one that can reflect critically on the process of interpretation. This involves a capacity for standing back from signs and interpretants and viewing them as objects, seeing their connections as objective possibilities for choice, to be compared with each other and tested against the objective and syntactic contexts, rather than taking the usual paths responding in a reflexive manner with the …
To do this it is necessary to interrupt the customary connections and favored associations of signs and interpretants in a sign relation and to consider a plurality of interpretations, not merely to pursue many lines of meaning in a parallel or experimental fashion, but to question seriously whether anything at all is meant by a sign.
… follow up alternatives in an experimental fashion, evaluate choices with a sensitivity to both the objective and syntactic contexts.
The mark of intelligence that is relevant to this context is the ability to comprehend a non deterministic situation of choice precisely as it is, …
If a species of determinism is nevertheless expected, then the extra measure of determination must be attributed to a worldly context of objects and signs extending beyond those taken into account by the sign relation in question, or else to powers of choice as yet unformalized in the character of interpreters.
This means that the recursions involved in the process of interpretation, besides having recourse to the inner resources of interpreters, will also recur to interfaces with objective situations and syntactic contexts. Interpretation, to be intelligent, must have the capacity to address the full scope of objects and signs and must be given the room to operate interactively with everything up to and including the undetermined horizons of the external world.
===6.46. Looking Ahead===
On the whole throughout this project, the “meta” issue that has been raised here will be treated at three different levels of sophistication.
<ol>
<li>The way I have chosen to deal with this issue in the present case is not by injecting more features of the informal discussion into the dialogue of <math>\text{A}\!</math> and <math>\text{B},\!</math> but by trying to imagine how agents like <math>\text{A}\!</math> and <math>\text{B}\!</math> might be enabled to reflect on these aspects of their own discussion.</li>
<li>
<p>In the series of examples that I will use to develop further aspects of the <math>\text{A}\!</math> and <math>\text{B}\!</math> dialogue, several different ways of extending the sign relations for <math>\text{A}\!</math> and <math>\text{B}\!</math> will be explored. The most pressing task is to capture facts of the following sort.</p>
{| align="center" cellspacing="8" width="90%"
| <math>\text{A}\!</math> knows that <math>\text{B}\!</math> uses <math>{}^{\backprime\backprime} \text{i} {}^{\prime\prime}\!</math> to denote <math>\text{B}\!</math> and <math>{}^{\backprime\backprime} \text{u} {}^{\prime\prime}\!</math> to denote <math>\text{A}.\!</math>
|-
| <math>\text{B}\!</math> knows that <math>\text{A}\!</math> uses <math>{}^{\backprime\backprime} \text{i} {}^{\prime\prime}\!</math> to denote <math>\text{A}\!</math> and <math>{}^{\backprime\backprime} \text{u} {}^{\prime\prime}\!</math> to denote <math>\text{B}.\!</math>
|}
<p>Toward this aim, I will present a variety of constructions for motivating ''extended'', ''indexed'', or ''situated'' sign relations, all designed to meet the following requirements.</p>
<ol style="list-style-type:lower-alpha">
<li>To incorporate higher components of “meta-knowledge” about language use as it works in a community of interpreters, in reality the most basic ingredients of pragmatic competence.</li>
<li>To amalgamate the fragmentary sign relations of individual interpreters into “broader-minded” sign relations, in the use and understanding of which a plurality of agents can share.</li>
</ol>
<p>Work at this level of concrete investigation will proceed in an incremental fashion, augmenting the discussion of A and B with features of increasing interest and relevance to inquiry. The plan for this series of developments is as follows.</p>
<ol style="list-style-type:lower-alpha" start="3">
<li>I start by gathering materials and staking out intermediate goals for investigation. This involves making a tentative foray into ways that dimensions of directed change and motivated value can be added to the sign relations initially given for <math>\text{A}\!</math> and <math>\text{B}.\!</math></li>
<li>With this preparation, I return to the dialogue of <math>\text{A}\!</math> and <math>\text{B}\!</math> and pursue ways of integrating their independent selections of information into a unified system of interpretation.
<ol style="list-style-type:lower-roman">
<li>First, I employ the sign relations <math>L_\text{A}\!</math> and <math>L_\text{B}\!</math> to illustrate two basic kinds of set theoretic merges, the ordinary or ''simple'' union and the indexed or ''situated'' union of extensional relations. On review, both forms of combination are observed to fall short of what is needed to constitute the desired characteristics of a shared sign relation.</li>
<li>Next, I present two other ways of extending the sign relations <math>L_\text{A}\!</math> and <math>L_\text{B}\!</math> into a common system of interpretation. These extensions succeed in capturing further aspects of what interpreters know about their shared language use. Although motivated on different grounds, the alternative constructions that develop coincide in exactly the same abstract structure.</li>
</ol>
</li>
</ol>
</li>
<li>As this project begins to take on sign relations that are complex enough to convey the impression of genuine inquiry processes, a fuller explication of this issue will become mandatory. Eventually, this will demand a concept of ''higher-order sign relations'', whose objects, signs, and interpretants can all be complete sign relations in their own rights.</li>
</ol>
In principle, the successive grades of complexity enumerated above could be ascended in a straightforward way, if only the steps did not go straight up the cliffs of abstraction. As always, the kinds of intentional objects that are the toughest to face are those whose realization is so distant that even the gear needed to approach their construction is not yet in existence.
===6.50. Revisiting the Source===
'''…'''
----
<div align="center">
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems|Contents]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 1|Part 1]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 2|Part 2]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 3|Part 3]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 4|Part 4]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 5|Part 5]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6|Part 6]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 7|Part 7]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 8|Part 8]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Appendices|Appendices]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : References|References]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Document History|Document History]]
•
</div>
----
[[Category:Artificial Intelligence]]
[[Category:Critical Thinking]]
[[Category:Cybernetics]]
[[Category:Education]]
[[Category:Hermeneutics]]
[[Category:Information Systems]]
[[Category:Inquiry]]
[[Category:Intelligence Amplification]]
[[Category:Learning Organizations]]
[[Category:Knowledge Representation]]
[[Category:Logic]]
[[Category:Philosophy]]
[[Category:Pragmatics]]
[[Category:Semantics]]
[[Category:Semiotics]]
[[Category:Systems Science]]
