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Revision as of 13:43, 28 August 2014
, 13:43, 28 August 2014sub \mathrm for \operatorname
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>
|}
|}
| valign="bottom" |
| valign="bottom" |
<math>\begin{matrix}
<math>\begin{matrix}
~x~ ~\operatorname{at}~ t
~x~ ~\mathrm{at}~ t
\\[4pt]
\\[4pt]
~x~ ~\operatorname{at}~ t
~x~ ~\mathrm{at}~ t
\\[4pt]
\\[4pt]
(x) ~\operatorname{at}~ t
(x) ~\mathrm{at}~ t
\\[4pt]
\\[4pt]
(x) ~\operatorname{at}~ t
(x) ~\mathrm{at}~ t
\end{matrix}</math>
\end{matrix}</math>
| valign="bottom" |
| valign="bottom" |
<math>\begin{matrix}
<math>\begin{matrix}
~\operatorname{d}x~ ~\operatorname{at}~ t
~\mathrm{d}x~ ~\mathrm{at}~ t
\\[4pt]
\\[4pt]
(\operatorname{d}x) ~\operatorname{at}~ t
(\mathrm{d}x) ~\mathrm{at}~ t
\\[4pt]
\\[4pt]
~\operatorname{d}x~ ~\operatorname{at}~ t
~\mathrm{d}x~ ~\mathrm{at}~ t
\\[4pt]
\\[4pt]
(\operatorname{d}x) ~\operatorname{at}~ t
(\mathrm{d}x) ~\mathrm{at}~ t
\end{matrix}</math>
\end{matrix}</math>
| valign="bottom" |
| valign="bottom" |
<math>\begin{matrix}
<math>\begin{matrix}
(x) ~\operatorname{at}~ t'
(x) ~\mathrm{at}~ t'
\\[4pt]
\\[4pt]
~x~ ~\operatorname{at}~ t'
~x~ ~\mathrm{at}~ t'
\\[4pt]
\\[4pt]
~x~ ~\operatorname{at}~ t'
~x~ ~\mathrm{at}~ t'
\\[4pt]
\\[4pt]
(x) ~\operatorname{at}~ t'
(x) ~\mathrm{at}~ t'
\end{matrix}</math>
\end{matrix}</math>
|}
|}
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.
It might be thought that a notion of real time <math>(t \in \mathbb{R})\!</math> is needed at this point to fund the account of sequential processes. From a logical point of view, however, I think it will be found that it is precisely out of such data that the notion of time has to be constructed.
The symbol <math>{}^{\backprime\backprime} \ominus\!\!- {}^{\prime\prime},</math> read ''thus'', ''then'', or ''yields'', can be used to mark sequential inferences, allowing for expressions like <math>x \land \operatorname{d}x \ominus\!\!-~ (x).\!</math> In each case, a suitable context of temporal moments <math>(t, t')\!</math> is understood to underlie the inference.
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 \operatorname{d}X \times X\!</math> is defined as follows.
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%"
{| align="center" cellspacing="8" width="90%"
| <math>\ominus ~=~ \{ (x, y, z) \in X \times \operatorname{d}X \times X : x \land y \ominus\!\!-~ z \}.\!</math>
| <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 \operatorname{d}X \times X\!</math> constituted by a sequential inference constraint can be interpreted as a proposition <math>\ominus : X \times \operatorname{d}X \times X \to \mathbb{B}\!</math> about sequential inference triples, and thus as a map <math>\ominus : \operatorname{d}X \to (X \times X \to \mathbb{B})\!</math> from the space <math>\operatorname{d}X\!</math> of differential states to the space of propositions about transitions in <math>X.\!</math>
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>
<br>
'''Question.''' Group Actions? <math>r : \operatorname{d}X \to (X \to X)\!</math>
'''Question.''' Group Actions? <math>r : \mathrm{d}X \to (X \to X)\!</math>
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" |
|+ style="height:30px" |
<math>\text{Table 70.1} ~~ \text{Group Representation} ~ \operatorname{Rep}^\text{A} (V_4)\!</math>
<math>\text{Table 70.1} ~~ \text{Group Representation} ~ \mathrm{Rep}^\text{A} (V_4)\!</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
| width="16%" | <math>\begin{matrix} \text{Abstract} \\ \text{Element} \end{matrix}</math>
| width="16%" | <math>\begin{matrix} \text{Abstract} \\ \text{Element} \end{matrix}</math>
| valign="bottom" |
| valign="bottom" |
<math>\begin{matrix}
<math>\begin{matrix}
(\operatorname{d}\underline{\underline{\text{a}}})
(\mathrm{d}\underline{\underline{\text{a}}})
(\operatorname{d}\underline{\underline{\text{b}}})
(\mathrm{d}\underline{\underline{\text{b}}})
(\operatorname{d}\underline{\underline{\text{i}}})
(\mathrm{d}\underline{\underline{\text{i}}})
(\operatorname{d}\underline{\underline{\text{u}}})
(\mathrm{d}\underline{\underline{\text{u}}})
\\[4pt]
\\[4pt]
~\operatorname{d}\underline{\underline{\text{a}}}~
~\mathrm{d}\underline{\underline{\text{a}}}~
(\operatorname{d}\underline{\underline{\text{b}}})
(\mathrm{d}\underline{\underline{\text{b}}})
~\operatorname{d}\underline{\underline{\text{i}}}~
~\mathrm{d}\underline{\underline{\text{i}}}~
(\operatorname{d}\underline{\underline{\text{u}}})
(\mathrm{d}\underline{\underline{\text{u}}})
\\[4pt]
\\[4pt]
(\operatorname{d}\underline{\underline{\text{a}}})
(\mathrm{d}\underline{\underline{\text{a}}})
~\operatorname{d}\underline{\underline{\text{b}}}~
~\mathrm{d}\underline{\underline{\text{b}}}~
(\operatorname{d}\underline{\underline{\text{i}}})
(\mathrm{d}\underline{\underline{\text{i}}})
~\operatorname{d}\underline{\underline{\text{u}}}~
~\mathrm{d}\underline{\underline{\text{u}}}~
\\[4pt]
\\[4pt]
~\operatorname{d}\underline{\underline{\text{a}}}~
~\mathrm{d}\underline{\underline{\text{a}}}~
~\operatorname{d}\underline{\underline{\text{b}}}~
~\mathrm{d}\underline{\underline{\text{b}}}~
~\operatorname{d}\underline{\underline{\text{i}}}~
~\mathrm{d}\underline{\underline{\text{i}}}~
~\operatorname{d}\underline{\underline{\text{u}}}~
~\mathrm{d}\underline{\underline{\text{u}}}~
\end{matrix}</math>
\end{matrix}</math>
| valign="bottom" |
| valign="bottom" |
<math>\begin{matrix}
<math>\begin{matrix}
\langle \operatorname{d}! \rangle
\langle \mathrm{d}! \rangle
\\[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
\rangle
\\[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
\rangle
\\[4pt]
\\[4pt]
\langle \operatorname{d}* \rangle
\langle \mathrm{d}* \rangle
\end{matrix}</math>
\end{matrix}</math>
| valign="bottom" |
| valign="bottom" |
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{d}!
\mathrm{d}!
\\[4pt]
\\[4pt]
\operatorname{d}\underline{\underline{\text{a}}} \cdot
\mathrm{d}\underline{\underline{\text{a}}} \cdot
\operatorname{d}\underline{\underline{\text{i}}} ~ !
\mathrm{d}\underline{\underline{\text{i}}} ~ !
\\[4pt]
\\[4pt]
\operatorname{d}\underline{\underline{\text{b}}} \cdot
\mathrm{d}\underline{\underline{\text{b}}} \cdot
\operatorname{d}\underline{\underline{\text{u}}} ~ !
\mathrm{d}\underline{\underline{\text{u}}} ~ !
\\[4pt]
\\[4pt]
\operatorname{d}*
\mathrm{d}*
\end{matrix}</math>
\end{matrix}</math>
| valign="bottom" |
| valign="bottom" |
1
1
\\[4pt]
\\[4pt]
\operatorname{d}_{\text{ai}}
\mathrm{d}_{\text{ai}}
\\[4pt]
\\[4pt]
\operatorname{d}_{\text{bu}}
\mathrm{d}_{\text{bu}}
\\[4pt]
\\[4pt]
\operatorname{d}_{\text{ai}} * \operatorname{d}_{\text{bu}}
\mathrm{d}_{\text{ai}} * \mathrm{d}_{\text{bu}}
\end{matrix}</math>
\end{matrix}</math>
|}
|}
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" |
|+ style="height:30px" |
<math>\text{Table 70.2} ~~ \text{Group Representation} ~ \operatorname{Rep}^\text{B} (V_4)\!</math>
<math>\text{Table 70.2} ~~ \text{Group Representation} ~ \mathrm{Rep}^\text{B} (V_4)\!</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
| width="16%" | <math>\begin{matrix} \text{Abstract} \\ \text{Element} \end{matrix}</math>
| width="16%" | <math>\begin{matrix} \text{Abstract} \\ \text{Element} \end{matrix}</math>
| valign="bottom" |
| valign="bottom" |
<math>\begin{matrix}
<math>\begin{matrix}
(\operatorname{d}\underline{\underline{\text{a}}})
(\mathrm{d}\underline{\underline{\text{a}}})
(\operatorname{d}\underline{\underline{\text{b}}})
(\mathrm{d}\underline{\underline{\text{b}}})
(\operatorname{d}\underline{\underline{\text{i}}})
(\mathrm{d}\underline{\underline{\text{i}}})
(\operatorname{d}\underline{\underline{\text{u}}})
(\mathrm{d}\underline{\underline{\text{u}}})
\\[4pt]
\\[4pt]
~\operatorname{d}\underline{\underline{\text{a}}}~
~\mathrm{d}\underline{\underline{\text{a}}}~
(\operatorname{d}\underline{\underline{\text{b}}})
(\mathrm{d}\underline{\underline{\text{b}}})
(\operatorname{d}\underline{\underline{\text{i}}})
(\mathrm{d}\underline{\underline{\text{i}}})
~\operatorname{d}\underline{\underline{\text{u}}}~
~\mathrm{d}\underline{\underline{\text{u}}}~
\\[4pt]
\\[4pt]
(\operatorname{d}\underline{\underline{\text{a}}})
(\mathrm{d}\underline{\underline{\text{a}}})
~\operatorname{d}\underline{\underline{\text{b}}}~
~\mathrm{d}\underline{\underline{\text{b}}}~
~\operatorname{d}\underline{\underline{\text{i}}}~
~\mathrm{d}\underline{\underline{\text{i}}}~
(\operatorname{d}\underline{\underline{\text{u}}})
(\mathrm{d}\underline{\underline{\text{u}}})
\\[4pt]
\\[4pt]
~\operatorname{d}\underline{\underline{\text{a}}}~
~\mathrm{d}\underline{\underline{\text{a}}}~
~\operatorname{d}\underline{\underline{\text{b}}}~
~\mathrm{d}\underline{\underline{\text{b}}}~
~\operatorname{d}\underline{\underline{\text{i}}}~
~\mathrm{d}\underline{\underline{\text{i}}}~
~\operatorname{d}\underline{\underline{\text{u}}}~
~\mathrm{d}\underline{\underline{\text{u}}}~
\end{matrix}</math>
\end{matrix}</math>
| valign="bottom" |
| valign="bottom" |
<math>\begin{matrix}
<math>\begin{matrix}
\langle \operatorname{d}! \rangle
\langle \mathrm{d}! \rangle
\\[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
\rangle
\\[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
\rangle
\\[4pt]
\\[4pt]
\langle \operatorname{d}* \rangle
\langle \mathrm{d}* \rangle
\end{matrix}</math>
\end{matrix}</math>
| valign="bottom" |
| valign="bottom" |
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{d}!
\mathrm{d}!
\\[4pt]
\\[4pt]
\operatorname{d}\underline{\underline{\text{a}}} \cdot
\mathrm{d}\underline{\underline{\text{a}}} \cdot
\operatorname{d}\underline{\underline{\text{u}}} ~ !
\mathrm{d}\underline{\underline{\text{u}}} ~ !
\\[4pt]
\\[4pt]
\operatorname{d}\underline{\underline{\text{b}}} \cdot
\mathrm{d}\underline{\underline{\text{b}}} \cdot
\operatorname{d}\underline{\underline{\text{i}}} ~ !
\mathrm{d}\underline{\underline{\text{i}}} ~ !
\\[4pt]
\\[4pt]
\operatorname{d}*
\mathrm{d}*
\end{matrix}</math>
\end{matrix}</math>
| valign="bottom" |
| valign="bottom" |
1
1
\\[4pt]
\\[4pt]
\operatorname{d}_{\text{au}}
\mathrm{d}_{\text{au}}
\\[4pt]
\\[4pt]
\operatorname{d}_{\text{bi}}
\mathrm{d}_{\text{bi}}
\\[4pt]
\\[4pt]
\operatorname{d}_{\text{au}} * \operatorname{d}_{\text{bi}}
\mathrm{d}_{\text{au}} * \mathrm{d}_{\text{bi}}
\end{matrix}</math>
\end{matrix}</math>
|}
|}
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" |
|+ style="height:30px" |
<math>{\text{Table 70.3} ~~ \text{Group Representation} ~ \operatorname{Rep}^\text{C} (V_4)}\!</math>
<math>{\text{Table 70.3} ~~ \text{Group Representation} ~ \mathrm{Rep}^\text{C} (V_4)}\!</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
| width="16%" | <math>\begin{matrix} \text{Abstract} \\ \text{Element} \end{matrix}</math>
| width="16%" | <math>\begin{matrix} \text{Abstract} \\ \text{Element} \end{matrix}</math>
| valign="bottom" |
| valign="bottom" |
<math>\begin{matrix}
<math>\begin{matrix}
(\operatorname{d}\text{m})
(\mathrm{d}\text{m})
(\operatorname{d}\text{n})
(\mathrm{d}\text{n})
\\[4pt]
\\[4pt]
~\operatorname{d}\text{m}~
~\mathrm{d}\text{m}~
(\operatorname{d}\text{n})
(\mathrm{d}\text{n})
\\[4pt]
\\[4pt]
(\operatorname{d}\text{m})
(\mathrm{d}\text{m})
~\operatorname{d}\text{n}~
~\mathrm{d}\text{n}~
\\[4pt]
\\[4pt]
~\operatorname{d}\text{m}~
~\mathrm{d}\text{m}~
~\operatorname{d}\text{n}~
~\mathrm{d}\text{n}~
\end{matrix}</math>
\end{matrix}</math>
| valign="bottom" |
| valign="bottom" |
<math>\begin{matrix}
<math>\begin{matrix}
\langle\operatorname{d}!\rangle
\langle\mathrm{d}!\rangle
\\[4pt]
\\[4pt]
\langle\operatorname{d}\text{m}\rangle
\langle\mathrm{d}\text{m}\rangle
\\[4pt]
\\[4pt]
\langle\operatorname{d}\text{n}\rangle
\langle\mathrm{d}\text{n}\rangle
\\[4pt]
\\[4pt]
\langle\operatorname{d}*\rangle
\langle\mathrm{d}*\rangle
\end{matrix}</math>
\end{matrix}</math>
| valign="bottom" |
| valign="bottom" |
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{d}!
\mathrm{d}!
\\[4pt]
\\[4pt]
\operatorname{d}\text{m}!
\mathrm{d}\text{m}!
\\[4pt]
\\[4pt]
\operatorname{d}\text{n}!
\mathrm{d}\text{n}!
\\[4pt]
\\[4pt]
\operatorname{d}*
\mathrm{d}*
\end{matrix}</math>
\end{matrix}</math>
| valign="bottom" |
| valign="bottom" |
1
1
\\[4pt]
\\[4pt]
\operatorname{d}_{\text{m}}
\mathrm{d}_{\text{m}}
\\[4pt]
\\[4pt]
\operatorname{d}_{\text{n}}
\mathrm{d}_{\text{n}}
\\[4pt]
\\[4pt]
\operatorname{d}_{\text{m}} * \operatorname{d}_{\text{n}}
\mathrm{d}_{\text{m}} * \mathrm{d}_{\text{n}}
\end{matrix}</math>
\end{matrix}</math>
|}
|}
| valign="bottom" |
| valign="bottom" |
<math>\begin{matrix}
<math>\begin{matrix}
(\operatorname{d}\text{m})
(\mathrm{d}\text{m})
(\operatorname{d}\text{n})
(\mathrm{d}\text{n})
\\[4pt]
\\[4pt]
~\operatorname{d}\text{m}~
~\mathrm{d}\text{m}~
(\operatorname{d}\text{n})
(\mathrm{d}\text{n})
\\[4pt]
\\[4pt]
(\operatorname{d}\text{m})
(\mathrm{d}\text{m})
~\operatorname{d}\text{n}~
~\mathrm{d}\text{n}~
\\[4pt]
\\[4pt]
~\operatorname{d}\text{m}~
~\mathrm{d}\text{m}~
~\operatorname{d}\text{n}~
~\mathrm{d}\text{n}~
\end{matrix}</math>
\end{matrix}</math>
| valign="bottom" |
| valign="bottom" |
<math>\begin{matrix}
<math>\begin{matrix}
\langle\operatorname{d}!\rangle
\langle\mathrm{d}!\rangle
\\[4pt]
\\[4pt]
\langle\operatorname{d}\text{m}\rangle
\langle\mathrm{d}\text{m}\rangle
\\[4pt]
\\[4pt]
\langle\operatorname{d}\text{n}\rangle
\langle\mathrm{d}\text{n}\rangle
\\[4pt]
\\[4pt]
\langle\operatorname{d}*\rangle
\langle\mathrm{d}*\rangle
\end{matrix}</math>
\end{matrix}</math>
| valign="bottom" |
| valign="bottom" |
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{d}!
\mathrm{d}!
\\[4pt]
\\[4pt]
\operatorname{d}\text{m}!
\mathrm{d}\text{m}!
\\[4pt]
\\[4pt]
\operatorname{d}\text{n}!
\mathrm{d}\text{n}!
\\[4pt]
\\[4pt]
\operatorname{d}*
\mathrm{d}*
\end{matrix}</math>
\end{matrix}</math>
| valign="bottom" |
| valign="bottom" |
1
1
\\[4pt]
\\[4pt]
\operatorname{d}_{\text{m}}
\mathrm{d}_{\text{m}}
\\[4pt]
\\[4pt]
\operatorname{d}_{\text{n}}
\mathrm{d}_{\text{n}}
\\[4pt]
\\[4pt]
\operatorname{d}_{\text{m}} * \operatorname{d}_{\text{n}}
\mathrm{d}_{\text{m}} * \mathrm{d}_{\text{n}}
\end{matrix}</math>
\end{matrix}</math>
|}
|}
|-
|-
| <math>G_\text{m}\!</math>
| <math>G_\text{m}\!</math>
| <math>(\operatorname{d}\text{m})\!</math>
| <math>(\mathrm{d}\text{m})\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
(\operatorname{d}\text{m})(\operatorname{d}\text{n})
(\mathrm{d}\text{m})(\mathrm{d}\text{n})
\\[4pt]
\\[4pt]
(\operatorname{d}\text{m})~\operatorname{d}\text{n}~
(\mathrm{d}\text{m})~\mathrm{d}\text{n}~
\end{matrix}</math>
\end{matrix}</math>
|
|
1
1
\\[4pt]
\\[4pt]
\operatorname{d}_\text{n}
\mathrm{d}_\text{n}
\end{matrix}</math>
\end{matrix}</math>
|-
|-
| <math>G_\text{m} * \operatorname{d}_\text{m}\!</math>
| <math>G_\text{m} * \mathrm{d}_\text{m}\!</math>
| <math>\operatorname{d}\text{m}\!</math>
| <math>\mathrm{d}\text{m}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
~\operatorname{d}\text{m}~(\operatorname{d}\text{n})
~\mathrm{d}\text{m}~(\mathrm{d}\text{n})
\\[4pt]
\\[4pt]
~\operatorname{d}\text{m}~~\operatorname{d}\text{n}~
~\mathrm{d}\text{m}~~\mathrm{d}\text{n}~
\end{matrix}</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{d}_\text{m}
\mathrm{d}_\text{m}
\\[4pt]
\\[4pt]
\operatorname{d}_\text{n} * \operatorname{d}_\text{m}
\mathrm{d}_\text{n} * \mathrm{d}_\text{m}
\end{matrix}</math>
\end{matrix}</math>
|}
|}
|-
|-
| <math>G_\text{n}\!</math>
| <math>G_\text{n}\!</math>
| <math>({\operatorname{d}\text{n})}\!</math>
| <math>({\mathrm{d}\text{n})}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
(\operatorname{d}\text{m})(\operatorname{d}\text{n})
(\mathrm{d}\text{m})(\mathrm{d}\text{n})
\\[4pt]
\\[4pt]
~\operatorname{d}\text{m}~(\operatorname{d}\text{n})
~\mathrm{d}\text{m}~(\mathrm{d}\text{n})
\end{matrix}</math>
\end{matrix}</math>
|
|
1
1
\\[4pt]
\\[4pt]
\operatorname{d}_\text{m}
\mathrm{d}_\text{m}
\end{matrix}</math>
\end{matrix}</math>
|-
|-
| <math>G_\text{n} * \operatorname{d}_\text{n}\!</math>
| <math>G_\text{n} * \mathrm{d}_\text{n}\!</math>
| <math>\operatorname{d}\text{n}\!</math>
| <math>\mathrm{d}\text{n}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
(\operatorname{d}\text{m})~\operatorname{d}\text{n}~
(\mathrm{d}\text{m})~\mathrm{d}\text{n}~
\\[4pt]
\\[4pt]
~\operatorname{d}\text{m}~~\operatorname{d}\text{n}~
~\mathrm{d}\text{m}~~\mathrm{d}\text{n}~
\end{matrix}</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{d}_\text{n}
\mathrm{d}_\text{n}
\\[4pt]
\\[4pt]
\operatorname{d}_\text{m} * \operatorname{d}_\text{n}
\mathrm{d}_\text{m} * \mathrm{d}_\text{n}
\end{matrix}</math>
\end{matrix}</math>
|}
|}
|}
|}
In other words, <math>P\!\!\And\!\!Q</math> is the intersection of the ''inverse projections'' <math>P' = \operatorname{Pr}_{12}^{-1}(P)\!</math> and <math>Q' = \operatorname{Pr}_{23}^{-1}(Q),\!</math> which are defined as follows:
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%"
{| align="center" cellspacing="8" width="90%"
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{Pr}_{12}^{-1}(P) & = & P \times Z & = & \{ (x, y, z) \in X \times Y \times Z : (x, y) \in P \}.
\mathrm{Pr}_{12}^{-1}(P) & = & P \times Z & = & \{ (x, y, z) \in X \times Y \times Z : (x, y) \in P \}.
\\[4pt]
\\[4pt]
\operatorname{Pr}_{23}^{-1}(Q) & = & X \times Q & = & \{ (x, y, z) \in X \times Y \times Z : (y, z) \in Q \}.
\mathrm{Pr}_{23}^{-1}(Q) & = & X \times Q & = & \{ (x, y, z) \in X \times Y \times Z : (y, z) \in Q \}.
\end{matrix}</math>
\end{matrix}</math>
|}
|}
{| align="center" cellspacing="8" width="90%"
{| align="center" cellspacing="8" width="90%"
| <math>P \circ Q ~=~ \operatorname{Pr}_{13} (P\!\!\And\!\!Q) ~=~ \{ (x, z) \in X \times Z : (x, y, z) \in P\!\!\And\!\!Q \}.</math>
| <math>P \circ Q ~=~ \mathrm{Pr}_{13} (P\!\!\And\!\!Q) ~=~ \{ (x, z) \in X \times Z : (x, y, z) \in P\!\!\And\!\!Q \}.</math>
|}
|}
For a <math>k\!</math>-place relation <math>L \subseteq X_1 \times \ldots \times X_k,\!</math> we have the following usages.
For a <math>k\!</math>-place relation <math>L \subseteq X_1 \times \ldots \times X_k,\!</math> we have the following usages.
# The notation <math>{}^{\backprime\backprime} \operatorname{Dom}_j (L) {}^{\prime\prime}\!</math> denotes the set <math>X_j,\!</math> called the ''domain of <math>L\!</math> at <math>j\!</math>'' or the ''<math>j^\text{th}\!</math> domain of <math>L.\!</math>''.
# The notation <math>{}^{\backprime\backprime} \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} \operatorname{Quo}_j (L) {}^{\prime\prime}\!</math> denotes a subset of <math>{X_j}\!</math> called the ''quorum of <math>L\!</math> at <math>j\!</math>'' or the ''<math>j^\text{th}\!</math> quorum of <math>L,\!</math>'' defined as follows.
# 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%"
{| align="center" cellspacing="8" width="90%"
|
|
<math>\begin{array}{lll}
<math>\begin{array}{lll}
\operatorname{Quo}_j (L)
\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,
\text{the largest}~ Q \subseteq X_j ~\text{such that}~ ~L_{Q \,\text{at}\, j}~ ~\text{is}~ (> 1)\text{-regular at}~ j,
# 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 arbitrarily designated domains <math>X_1 = X\!</math> and <math>X_2 = Y\!</math> that form the widest sets admitted to the dyadic relation are referred to as the ''domain'' or ''source'' and the ''codomain'' or ''target'', respectively, of the relation in question.
# The terms ''quota'' and ''range'' are reserved for those uniquely defined sets whose elements actually appear as the first and second members, respectively, of the ordered pairs in that relation. Thus, for a dyadic relation <math>L \subseteq X \times Y,\!</math> we identify <math>\operatorname{Quo} (L) = \operatorname{Quo}_1 (L) \subseteq X\!</math> with what is usually called the ''domain of definition'' of <math>L\!</math> and we identify <math>\operatorname{Ran} (L) = \operatorname{Quo}_2 (L) \subseteq Y\!</math> with the usual ''range'' of <math>L.\!</math>
# 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>\operatorname{Quo} (L) \subseteq X.\!</math> In this situation, <math>[x]_L\!</math> is empty for each <math>x\!</math> in <math>X\!</math> that is not in <math>\operatorname{Quo} (L).\!</math> Another way of reaching the same concept is to call a PER a dyadic relation that is symmetric and transitive, but not necessarily reflexive. Like the “self-identical elements” of old that epitomized the very definition of self-consistent existence in classical logic, the property of being a self-related or self-equivalent element in the purview of a PER on <math>X\!</math> singles out the members of <math>\operatorname{Quo} (L)\!</math> as those for which a properly meaningful existence can be contemplated.
A ''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.
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.
{| align="center" cellspacing="8" width="90%"
{| align="center" cellspacing="8" width="90%"
| <math>\operatorname{Proj}^{(2)} L ~=~ (\operatorname{proj}_{12} L, ~ \operatorname{proj}_{13} L, ~ \operatorname{proj}_{23} L).\!</math>
| <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>\operatorname{Proj}^{(2)} L\!</math> can be visualized as the arrangement or ordered collection of shadows it throws on the <math>XY, ~ XZ, ~ YZ\!</math> planes, respectively.
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>\operatorname{Proj}^{(2)}.\!</math>
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>\operatorname{Pow}(S)\!</math> or <math>2^S\!</math> and defined as follows:
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%"
{| align="center" cellspacing="8" width="90%"
| <math>\operatorname{Pow}(S) ~=~ 2^S ~=~ \{ T : T \subseteq S \}.\!</math>
| <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>\operatorname{Pow}(S),\!</math> in other words, as <math>L \in \operatorname{Pow}(X_1 \times \ldots \times X_n).\!</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>\operatorname{Explo}(X, Y, Z ~|~ 2),\!</math> read as the ''explosion of <math>X \times Y \times Z\!</math> by twos'', or more simply as <math>X, Y, Z ~\operatorname{choose}~ 2,\!</math> and defined as follows:
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%"
{| align="center" cellspacing="8" width="90%"
| <math>\operatorname{Explo}(X, Y, Z ~|~ 2) ~=~ \operatorname{Pow}(X \times Y) \times \operatorname{Pow}(X \times Z) \times \operatorname{Pow}(Y \times Z).\!</math>
| <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.
This domain is defined well enough to serve the immediate purposes of this section, but later it will become necessary to examine its construction more closely.
By means of these constructions the operation that forms <math>\operatorname{Proj}^{(2)} L\!</math> for each triadic relation <math>L \subseteq X \times Y \times Z\!</math> can be expressed as a function:
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%"
{| align="center" cellspacing="8" width="90%"
| <math>\operatorname{Proj}^{(2)} : \operatorname{Pow}(X \times Y \times Z) \to \operatorname{Explo}(X, Y, Z ~|~ 2).\!</math>
| <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>\operatorname{Proj}^{(2)}\!</math> is injective. The mapping <math>\operatorname{Proj}^{(2)}\!</math> is said to ''preserve information'' about the triadic relations <math>L \in \operatorname{Pow}(X \times Y \times Z)\!</math> if and only if it is injective, otherwise one says that some loss of information has occurred in taking the projections. Given a specific instance of a triadic relation <math>L \in \operatorname{Pow}(X \times Y \times Z),\!</math> it can be said that <math>L\!</math> is ''determined by'' (''reducible to'' or ''reconstructible from'') its dyadic projections if and only if <math>(\operatorname{Proj}^{(2)})^{-1}(\operatorname{Proj}^{(2)}L)\!</math> is the singleton set <math>\{ L \}.\!</math> Otherwise, there exists an <math>L'\!</math> such that <math>\operatorname{Proj}^{(2)}L = \operatorname{Proj}^{(2)}L',\!</math> and in this case <math>L\!</math> is said to be ''irreducibly triadic'' or ''genuinely triadic''. Notice that irreducible or genuine triadic relations, when they exist, naturally occur in sets of two or more, the whole collection of them being equated or confounded with one another under <math>\operatorname{Proj}^{(2)}.\!</math>
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>\operatorname{Proj}^{(2)}\!</math> by means of its actions on the sign relations <math>L_\text{A}\!</math> and <math>L_\text{B}.\!</math> For ease of reference, Tables 72.1 and 73.1 repeat the contents of Tables 1 and 2, respectively, while the dyadic relations comprising <math>\operatorname{Proj}^{(2)}L_\text{A}\!</math> and <math>\operatorname{Proj}^{(2)}L_\text{B}\!</math> are shown in Tables 72.2 to 72.4 and Tables 73.2 to 73.4, respectively.
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>
<br>
<br>
<br>
A comparison of the corresponding projections in <math>\operatorname{Proj}^{(2)} L(\text{A})\!</math> and <math>\operatorname{Proj}^{(2)} L(\text{B})\!</math> shows that the distinction between the triadic relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> is preserved by <math>\operatorname{Proj}^{(2)},\!</math> and this circumstance allows one to say that this much information, at least, can be derived from the dyadic projections. However, to say that a triadic relation <math>L \in \operatorname{Pow} (O \times S \times I)\!</math> is reducible in this sense it is necessary to show that no distinct <math>L' \in \operatorname{Pow} (O \times S \times I)\!</math> exists such that <math>\operatorname{Proj}^{(2)} L = \operatorname{Proj}^{(2)} L',\!</math> and this can take a rather more exhaustive or comprehensive investigation of the space <math>\operatorname{Pow} (O \times S \times I).\!</math>
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>\operatorname{Proj}^{(2)} L.\!</math> This can be seen as follows.
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 \operatorname{proj}_{SI}^{-1}((s, i))?\!</math> The fact that <math>L_{OS}\!</math> has exactly one element <math>(o, s)\!</math> for each coordinate <math>s \in S\!</math> and that <math>L_{OI}\!</math> has exactly one element <math>(o, i)\!</math> for each coordinate <math>i \in I,\!</math> plus the “coincidence” of it being the same <math>o\!</math> at any one choice for <math>(s, i),\!</math> tells us that <math>L\!</math> has just the one element <math>(o, s, i)\!</math> over each point of <math>S \times I.\!</math> This proves that both <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> are reducible in an informational sense to triples of dyadic relations, that is, they are ''dyadically reducible''.
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''.
