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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)

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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.

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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>

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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>

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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>

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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 LOC <math>*\!</math> on <math>X\!</math> is '''associative''' if and only if <math>(x*y)*z = x*(y*z)\!</math> for every <math>x, y, z \in X.\!</math>

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A '''monoid''' is a semigroup with a unit element. Formally, a monoid <math>\underline{X}\!</math> is an ordered triple <math>(X, *, e),\!</math> where <math>X\!</math> is a set, <math>*\!</math> is an associative LOC on the set <math>X,\!</math> and <math>e\!</math> is the unit element in the semigroup <math>(X, *).\!</math>

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An '''inverse''' of an element <math>x\!</math> in a monoid <math>\underline{X} = (X, *, e)\!</math> is an element <math>y \in X\!</math> such that <math>x*y = e = y*x.\!</math> An element that has an inverse in <math>\underline{X}\!</math> is said to be '''invertible''' (relative to <math>*\!</math> and <math>e\!</math>). If <math>x\!</math> has an inverse in <math>\underline{X},\!</math> then it is unique to <math>x.\!</math> To see this, suppose that <math>y'\!</math> is also an inverse of <math>x.\!</math> Then it follows that:

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An '''inverse''' of an element <math>x\!</math> in a monoid <math>\underline{X} = (X, *, e)\!</math> is an element <math>y \in X\!</math> such that <math>x*y = e = y*x.\!</math> An element that has an inverse in <math>\underline{X}\!</math> is said to be '''invertible''' (relative to <math>*\!</math> and <math>e\!</math>). If <math>x\!</math> has an inverse in <math>{\underline{X}},\!</math> then it is unique to <math>x.\!</math> To see this, suppose that <math>y'\!</math> is also an inverse of <math>x.\!</math> Then it follows that:

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It is customary to use a number of abbreviations and conventions in discussing semigroups, monoids, and groups. A system <math>\underline{X} = (X, *)\!</math> is given the adjective ''commutative'' if and only if <math>*\!</math> is commutative. Commutative groups, however, are traditionally called ''abelian groups''. By way of making comparisons with familiar systems and operations, the following usages are also common.

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One says that <math>\underline{X}\!</math> is '''written multiplicatively''' to mean that a raised dot <math>(\cdot)\!</math> or concatenation is used instead of a star for the LOC. In this case, the unit element is commonly written as an ordinary algebraic one, <math>1,\!</math> while the inverse of an element <math>x\!</math> is written as <math>x^{-1}.\!</math> The multiplicative manner of presentation is the one that is usually taken by default in the most general types of situations. In the multiplicative idiom, the following definitions of ''powers'', ''cyclic groups'', and ''generators'' are also common.

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One says that <math>\underline{X}\!</math> is '''written multiplicatively''' to mean that a raised dot <math>{(\cdot)}\!</math> or concatenation is used instead of a star for the LOC. In this case, the unit element is commonly written as an ordinary algebraic one, <math>1,\!</math> while the inverse of an element <math>x\!</math> is written as <math>x^{-1}.\!</math> The multiplicative manner of presentation is the one that is usually taken by default in the most general types of situations. In the multiplicative idiom, the following definitions of ''powers'', ''cyclic groups'', and ''generators'' are also common.

: In a semigroup, the <math>n^\text{th}\!</math> '''power''' of an element <math>x\!</math> is notated as <math>x^n\!</math> and defined for every positive integer <math>n\!</math> in the following manner. Proceeding recursively, let <math>x^1 = x\!</math> and let <math>x^n = x^{n-1} \cdot x\!</math> for all <math>n > 1.\!</math>

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|}

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To sum up the development so far in a general way: A ''homomorphism'' is a mapping from a system to a system that preserves an aspect of systematic structure, usually one that is relevant to an understood purpose or context. When the pertinent aspect of structure for both the source and the target system is a binary operation or a LOC, then the condition that the LOCs be preserved in passing from the pre-image to the image of the mapping is frequently expressed by stating that ''the image of the product is the product of the images''. That is, if <math>h : X_1 \to X_2\!</math> is a homomorphism from <math>\underline{X}_1 = (X_1, *_1)\!</math> to <math>\underline{X}_2 = (X_2, *_2),\!</math> then for every <math>x, y \in X_1\!</math> the following condition holds:

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To sum up the development so far in a general way: A ''homomorphism'' is a mapping from a system to a system that preserves an aspect of systematic structure, usually one that is relevant to an understood purpose or context. When the pertinent aspect of structure for both the source and the target system is a binary operation or a LOC, then the condition that the LOCs be preserved in passing from the pre-image to the image of the mapping is frequently expressed by stating that ''the image of the product is the product of the images''. That is, if <math>h : X_1 \to X_2\!</math> is a homomorphism from <math>{\underline{X}_1 = (X_1, *_1)}\!</math> to <math>{\underline{X}_2 = (X_2, *_2)},\!</math> then for every <math>x, y \in X_1\!</math> the following condition holds:

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Next, the concept of a homomorphism or ''structure-preserving map'' is specialized to the different kinds of structure of interest here.

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A '''semigroup homomorphism''' from a semigroup <math>\underline{X}_1 = (X_1, *_1)\!</math> to a semigroup <math>\underline{X}_2 = (X_2, *_2)\!</math> is a mapping between the underlying sets that preserves the structure appropriate to semigroups, namely, the LOCs. This makes it a map <math>h : X_1 \to X_2\!</math> whose induced action on the LOCs is such that it takes every element of <math>*_1\!</math> to an element of <math>*_2.\!</math> That is:

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A '''semigroup homomorphism''' from a semigroup <math>{\underline{X}_1 = (X_1, *_1)}\!</math> to a semigroup <math>{\underline{X}_2 = (X_2, *_2)}\!</math> is a mapping between the underlying sets that preserves the structure appropriate to semigroups, namely, the LOCs. This makes it a map <math>h : X_1 \to X_2\!</math> whose induced action on the LOCs is such that it takes every element of <math>*_1\!</math> to an element of <math>*_2.\!</math> That is:

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Finally, to introduce two pieces of language that are often useful: an '''endomorphism''' is a homomorphism from a system into itself, while an '''automorphism''' is an isomorphism from a system onto itself.

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If nothing more succinct is available, a group can be specified by means of its ''operation table'', usually styled either as a ''multiplication table'' or an ''addition table''. Table 32.1 illustrates the general scheme of a group operation table. In this case the group operation, treated as a “multiplication”, is formally symbolized by a star <math>(*),\!</math> as in <math>x * y = z.\!</math> In contexts where only algebraic operations are formalized it is common practice to omit the star, but when logical conjunctions (symbolized by a raised dot <math>(\cdot)\!</math> or by concatenation) appear in the same context, then the star is retained for the group operation.

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If nothing more succinct is available, a group can be specified by means of its ''operation table'', usually styled either as a ''multiplication table'' or an ''addition table''. Table 32.1 illustrates the general scheme of a group operation table. In this case the group operation, treated as a “multiplication”, is formally symbolized by a star <math>(*),\!</math> as in <math>x * y = z.\!</math> In contexts where only algebraic operations are formalized it is common practice to omit the star, but when logical conjunctions (symbolized by a raised dot <math>{(\cdot)}\!</math> or by concatenation) appear in the same context, then the star is retained for the group operation.

Another way of approaching the study or presenting the structure of a group is by means of a ''group representation'', in particular, one that represents the group in the special form of a ''transformation group''. This is a set of transformations acting on a concrete space of “points” or a designated set of “objects”. In providing an abstractly given group with a representation as a transformation group, one is seeking to know the group by its effects, that is, in terms of the action it induces, through the representation, on a concrete domain of objects. In the type of representation known as a ''regular representation'', one is seeking to know the group by its effects on itself.

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|+ <math>\text{Table 32.1}~~\text{Scheme of a Group Operation Table}</math>

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|+ style="height:30px" |

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<math>\text{Table 32.1} ~~ \text{Scheme of a Group Operation Table}\!</math>

|- style="height:50px"

| style="border-bottom:1px solid black; border-right:1px solid black" | <math>*\!</math>

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|+ <math>\text{Table 32.2}~~\text{Scheme of the Regular Ante-Representation}</math>

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|+ style="height:30px" |

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<math>\text{Table 32.2} ~~ \text{Scheme of the Regular Ante-Representation}\!</math>

|- style="height:50px"

| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>

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|+ <math>\text{Table 32.3}~~\text{Scheme of the Regular Post-Representation}</math>

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|+ style="height:30px" |

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<math>\text{Table 32.3} ~~ \text{Scheme of the Regular Post-Representation}\!</math>

|- style="height:50px"

| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>

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For the sake of comparison, I give a discussion of both these groups.

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The next series of Tables presents the group operations and regular representations for the groups <math>V_4\!</math> and <math>Z_4.\!</math> If a group is abelian, as both of these groups are, then its <math>h_1\!</math> and <math>h_2\!</math> representations are indistinguishable, and a single form of regular representation <math>h : G \to (G \to G)\!</math> will do for both.

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The next series of Tables presents the group operations and regular representations for the groups <math>V_4\!</math> and <math>Z_4.\!</math> If a group is abelian, as both of these groups are, then its <math>h_1\!</math> and <math>h_2\!</math> representations are indistinguishable, and a single form of regular representation <math>{h : G \to (G \to G)}\!</math> will do for both.

Table 33.1 shows the multiplication table of the group <math>V_4,\!</math> while Tables 33.2 and 33.3 present two versions of its regular representation. The first version, somewhat hastily, gives the functional representation of each group element as a set of ordered pairs of group elements. The second version, more circumspectly, gives the functional representative of each group element as a set of ordered pairs of element names, also referred to as ''objects'', ''points'', ''letters'', or ''symbols''.

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|+ <math>\text{Table 33.1}~~\text{Multiplication Operation of the Group}~V_4</math>

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|+ style="height:30px" |

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<math>\text{Table 33.1} ~~ \text{Multiplication Operation of the Group} ~ V_4\!</math>

|- style="height:50px"

| width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>

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| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{e}</math>

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| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{e}\!</math>

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| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{f}</math>

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| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{f}\!</math>

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| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{g}</math>

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| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{g}\!</math>

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| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{h}</math>

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| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{h}\!</math>

|- style="height:50px"

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| style="border-right:1px solid black" | <math>\operatorname{e}</math>

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| style="border-right:1px solid black" | <math>\operatorname{e}\!</math>

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| <math>\operatorname{e}</math>

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| <math>\operatorname{e}\!</math>

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| <math>\operatorname{f}</math>

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| <math>\operatorname{f}\!</math>

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| <math>\operatorname{g}</math>

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| <math>\operatorname{g}\!</math>

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| <math>\operatorname{h}</math>

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| <math>\operatorname{h}\!</math>

|- style="height:50px"

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| style="border-right:1px solid black" | <math>\operatorname{f}</math>

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| style="border-right:1px solid black" | <math>\operatorname{f}\!</math>

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| <math>\operatorname{f}</math>

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| <math>\operatorname{f}\!</math>

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| <math>\operatorname{e}</math>

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| <math>\operatorname{e}\!</math>

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| <math>\operatorname{h}</math>

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| <math>\operatorname{h}\!</math>

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| <math>\operatorname{g}</math>

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| <math>\operatorname{g}\!</math>

|- style="height:50px"

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| style="border-right:1px solid black" | <math>\operatorname{g}</math>

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| style="border-right:1px solid black" | <math>\operatorname{g}\!</math>

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| <math>\operatorname{g}</math>

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| <math>\operatorname{g}\!</math>

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| <math>\operatorname{h}</math>

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| <math>\operatorname{h}\!</math>

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| <math>\operatorname{e}</math>

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| <math>\operatorname{e}\!</math>

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| <math>\operatorname{f}</math>

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| <math>\operatorname{f}\!</math>

|- style="height:50px"

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| style="border-right:1px solid black" | <math>\operatorname{h}</math>

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| style="border-right:1px solid black" | <math>\operatorname{h}\!</math>

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| <math>\operatorname{h}</math>

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| <math>\operatorname{h}\!</math>

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| <math>\operatorname{g}</math>

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| <math>\operatorname{g}\!</math>

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| <math>\operatorname{f}</math>

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| <math>\operatorname{f}\!</math>

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| <math>\operatorname{e}</math>

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| <math>\operatorname{e}\!</math>

|}

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|+ <math>\text{Table 33.2}~~\text{Regular Representation of the Group}~V_4</math>

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|+ style="height:30px" |

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<math>\text{Table 33.2} ~~ \text{Regular Representation of the Group} ~ V_4\!</math>

|- style="height:50px"

| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>

| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>

|- style="height:50px"

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| width="20%" style="border-right:1px solid black" | <math>\operatorname{e}</math>

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| width="20%" style="border-right:1px solid black" | <math>\operatorname{e}\!</math>

| width="4%" | <math>\{\!</math>

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| width="16%" | <math>(\operatorname{e}, \operatorname{e}),</math>

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| width="16%" | <math>(\operatorname{e}, \operatorname{e}),\!</math>

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| width="20%" | <math>(\operatorname{f}, \operatorname{f}),</math>

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| width="20%" | <math>(\operatorname{f}, \operatorname{f}),\!</math>

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| width="20%" | <math>(\operatorname{g}, \operatorname{g}),</math>

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| width="20%" | <math>(\operatorname{g}, \operatorname{g}),\!</math>

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| width="16%" | <math>(\operatorname{h}, \operatorname{h})</math>

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| width="16%" | <math>(\operatorname{h}, \operatorname{h})\!</math>

| width="4%" | <math>\}\!</math>

|- style="height:50px"

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| style="border-right:1px solid black" | <math>\operatorname{f}</math>

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| style="border-right:1px solid black" | <math>\operatorname{f}\!</math>

| <math>\{\!</math>

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| <math>(\operatorname{e}, \operatorname{f}),</math>

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| <math>(\operatorname{e}, \operatorname{f}),\!</math>

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| <math>(\operatorname{f}, \operatorname{e}),</math>

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| <math>(\operatorname{f}, \operatorname{e}),\!</math>

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| <math>(\operatorname{g}, \operatorname{h}),</math>

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| <math>(\operatorname{g}, \operatorname{h}),\!</math>

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| <math>(\operatorname{h}, \operatorname{g})</math>

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| <math>(\operatorname{h}, \operatorname{g})\!</math>

| <math>\}\!</math>

|- style="height:50px"

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| style="border-right:1px solid black" | <math>\operatorname{g}</math>

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| style="border-right:1px solid black" | <math>\operatorname{g}\!</math>

| <math>\{\!</math>

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| <math>(\operatorname{e}, \operatorname{g}),</math>

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| <math>(\operatorname{e}, \operatorname{g}),\!</math>

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| <math>(\operatorname{f}, \operatorname{h}),</math>

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| <math>(\operatorname{f}, \operatorname{h}),\!</math>

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| <math>(\operatorname{g}, \operatorname{e}),</math>

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| <math>(\operatorname{g}, \operatorname{e}),\!</math>

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| <math>(\operatorname{h}, \operatorname{f})</math>

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| <math>(\operatorname{h}, \operatorname{f})\!</math>

| <math>\}\!</math>

|- style="height:50px"

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| style="border-right:1px solid black" | <math>\operatorname{h}</math>

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| style="border-right:1px solid black" | <math>\operatorname{h}\!</math>

| <math>\{\!</math>

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| <math>(\operatorname{e}, \operatorname{h}),</math>

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| <math>(\operatorname{e}, \operatorname{h}),\!</math>

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| <math>(\operatorname{f}, \operatorname{g}),</math>

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| <math>(\operatorname{f}, \operatorname{g}),\!</math>

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| <math>(\operatorname{g}, \operatorname{f}),</math>

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| <math>(\operatorname{g}, \operatorname{f}),\!</math>

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| <math>(\operatorname{h}, \operatorname{e})</math>

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| <math>(\operatorname{h}, \operatorname{e})\!</math>

| <math>\}\!</math>

|}

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|+ <math>\text{Table 33.3}~~\text{Regular Representation of the Group}~V_4</math>

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|+ style="height:30px" |

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<math>\text{Table 33.3} ~~ \text{Regular Representation of the Group} ~ V_4\!</math>

|- style="height:50px"

| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>

| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Symbols}\!</math>

|- style="height:50px"

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| width="20%" style="border-right:1px solid black" | <math>\operatorname{e}</math>

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| width="20%" style="border-right:1px solid black" | <math>\operatorname{e}\!</math>

| width="4%" | <math>\{\!</math>

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| width="16%" | <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),</math>

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| width="16%" | <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),\!</math>

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| width="20%" | <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),</math>

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| width="20%" | <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),\!</math>

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| width="20%" | <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),</math>

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| width="20%" | <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),\!</math>

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| width="16%" | <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime})</math>

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| width="16%" | <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime})\!</math>

| width="4%" | <math>\}\!</math>

|- style="height:50px"

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| style="border-right:1px solid black" | <math>\operatorname{f}</math>

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| style="border-right:1px solid black" | <math>\operatorname{f}\!</math>

| <math>\{\!</math>

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| <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),</math>

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| <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),\!</math>

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| <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),</math>

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| <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),\!</math>

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| <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),</math>

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| <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),\!</math>

−

| <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime})</math>

+

| <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime})\!</math>

| <math>\}\!</math>

|- style="height:50px"

−

| style="border-right:1px solid black" | <math>\operatorname{g}</math>

+

| style="border-right:1px solid black" | <math>\operatorname{g}\!</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>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),</math>

+

| <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),\!</math>

−

| <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),</math>

+

| <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),\!</math>

−

| <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime})</math>

+

| <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime})\!</math>

| <math>\}\!</math>

|- style="height:50px"

−

| style="border-right:1px solid black" | <math>\operatorname{h}</math>

+

| style="border-right:1px solid black" | <math>\operatorname{h}\!</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>

−

| <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),</math>

+

| <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),\!</math>

−

| <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),</math>

+

| <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),\!</math>

−

| <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime})</math>

+

| <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime})\!</math>

| <math>\}\!</math>

|}

Line 1,196: Line 1,202:

{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"

−

|+ <math>\text{Table 34.1}~~\text{Multiplicative Presentation of the Group}~Z_4(\cdot)</math>

+

|+ style="height:30px" |

+

<math>\text{Table 34.1} ~~ \text{Multiplicative Presentation of the Group} ~ Z_4(\cdot)~\!</math>

|- style="height:50px"

| width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>

Line 1,232: Line 1,239:

{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"

−

|+ <math>\text{Table 34.2}~~\text{Regular Representation of the Group}~Z_4(\cdot)</math>

+

|+ style="height:30px" |

+

<math>\text{Table 34.2} ~~ \text{Regular Representation of the Group} ~ Z_4(\cdot)\!</math>

|- style="height:50px"

| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>

| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>

|- style="height:50px"

−

| width="20%" style="border-right:1px solid black" | <math>\operatorname{1}</math>

+

| width="20%" style="border-right:1px solid black" | <math>\operatorname{1}\!</math>

| width="4%" | <math>\{\!</math>

−

| width="16%" | <math>(\operatorname{1}, \operatorname{1}),</math>

+

| width="16%" | <math>(\operatorname{1}, \operatorname{1}),\!</math>

−

| width="20%" | <math>(\operatorname{a}, \operatorname{a}),</math>

+

| width="20%" | <math>(\operatorname{a}, \operatorname{a}),\!</math>

−

| width="20%" | <math>(\operatorname{b}, \operatorname{b}),</math>

+

| width="20%" | <math>(\operatorname{b}, \operatorname{b}),\!</math>

−

| width="16%" | <math>(\operatorname{c}, \operatorname{c})</math>

+

| width="16%" | <math>(\operatorname{c}, \operatorname{c})\!</math>

| width="4%" | <math>\}\!</math>

|- style="height:50px"

−

| style="border-right:1px solid black" | <math>\operatorname{a}</math>

+

| style="border-right:1px solid black" | <math>\operatorname{a}\!</math>

| <math>\{\!</math>

−

| <math>(\operatorname{1}, \operatorname{a}),</math>

+

| <math>(\operatorname{1}, \operatorname{a}),\!</math>

−

| <math>(\operatorname{a}, \operatorname{b}),</math>

+

| <math>(\operatorname{a}, \operatorname{b}),\!</math>

−

| <math>(\operatorname{b}, \operatorname{c}),</math>

+

| <math>(\operatorname{b}, \operatorname{c}),\!</math>

−

| <math>(\operatorname{c}, \operatorname{1})</math>

+

| <math>(\operatorname{c}, \operatorname{1})\!</math>

| <math>\}\!</math>

|- style="height:50px"

−

| style="border-right:1px solid black" | <math>\operatorname{b}</math>

+

| style="border-right:1px solid black" | <math>\operatorname{b}\!</math>

| <math>\{\!</math>

−

| <math>(\operatorname{1}, \operatorname{b}),</math>

+

| <math>(\operatorname{1}, \operatorname{b}),\!</math>

−

| <math>(\operatorname{a}, \operatorname{c}),</math>

+

| <math>(\operatorname{a}, \operatorname{c}),\!</math>

−

| <math>(\operatorname{b}, \operatorname{1}),</math>

+

| <math>(\operatorname{b}, \operatorname{1}),\!</math>

−

| <math>(\operatorname{c}, \operatorname{a})</math>

+

| <math>(\operatorname{c}, \operatorname{a})\!</math>

| <math>\}\!</math>

|- style="height:50px"

−

| style="border-right:1px solid black" | <math>\operatorname{c}</math>

+

| style="border-right:1px solid black" | <math>\operatorname{c}\!</math>

| <math>\{\!</math>

−

| <math>(\operatorname{1}, \operatorname{c}),</math>

+

| <math>(\operatorname{1}, \operatorname{c}),\!</math>

−

| <math>(\operatorname{a}, \operatorname{1}),</math>

+

| <math>(\operatorname{a}, \operatorname{1}),\!</math>

−

| <math>(\operatorname{b}, \operatorname{a}),</math>

+

| <math>(\operatorname{b}, \operatorname{a}),\!</math>

−

| <math>(\operatorname{c}, \operatorname{b})</math>

+

| <math>(\operatorname{c}, \operatorname{b})\!</math>

| <math>\}\!</math>

|}

Line 1,273: Line 1,281:

{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"

−

|+ <math>\text{Table 35.1}~~\text{Additive Presentation of the Group}~Z_4(+)</math>

+

|+ style="height:30px" |

+

<math>\text{Table 35.1} ~~ \text{Additive Presentation of the Group} ~ Z_4(+)\!</math>

|- 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" | <math>\operatorname{0}</math>

+

| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{0}\!</math>

−

| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{1}</math>

+

| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{1}\!</math>

−

| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{2}</math>

+

| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{2}\!</math>

−

| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{3}</math>

+

| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{3}\!</math>

|- style="height:50px"

−

| style="border-right:1px solid black" | <math>\operatorname{0}</math>

+

| style="border-right:1px solid black" | <math>\operatorname{0}\!</math>

−

| <math>\operatorname{0}</math>

+

| <math>\operatorname{0}\!</math>

−

| <math>\operatorname{1}</math>

+

| <math>\operatorname{1}\!</math>

−

| <math>\operatorname{2}</math>

+

| <math>\operatorname{2}\!</math>

−

| <math>\operatorname{3}</math>

+

| <math>\operatorname{3}\!</math>

|- style="height:50px"

−

| style="border-right:1px solid black" | <math>\operatorname{1}</math>

+

| style="border-right:1px solid black" | <math>\operatorname{1}\!</math>

−

| <math>\operatorname{1}</math>

+

| <math>\operatorname{1}\!</math>

−

| <math>\operatorname{2}</math>

+

| <math>\operatorname{2}\!</math>

−

| <math>\operatorname{3}</math>

+

| <math>\operatorname{3}\!</math>

−

| <math>\operatorname{0}</math>

+

| <math>\operatorname{0}\!</math>

|- style="height:50px"

−

| style="border-right:1px solid black" | <math>\operatorname{2}</math>

+

| style="border-right:1px solid black" | <math>\operatorname{2}\!</math>

−

| <math>\operatorname{2}</math>

+

| <math>\operatorname{2}\!</math>

−

| <math>\operatorname{3}</math>

+

| <math>\operatorname{3}\!</math>

−

| <math>\operatorname{0}</math>

+

| <math>\operatorname{0}\!</math>

−

| <math>\operatorname{1}</math>

+

| <math>\operatorname{1}\!</math>

|- style="height:50px"

−

| style="border-right:1px solid black" | <math>\operatorname{3}</math>

+

| style="border-right:1px solid black" | <math>\operatorname{3}\!</math>

−

| <math>\operatorname{3}</math>

+

| <math>\operatorname{3}\!</math>

−

| <math>\operatorname{0}</math>

+

| <math>\operatorname{0}\!</math>

−

| <math>\operatorname{1}</math>

+

| <math>\operatorname{1}\!</math>

−

| <math>\operatorname{2}</math>

+

| <math>\operatorname{2}\!</math>

|}

Line 1,309: Line 1,318:

{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"

−

|+ <math>\text{Table 35.2}~~\text{Regular Representation of the Group}~Z_4(+)</math>

+

|+ style="height:30px" |

+

<math>\text{Table 35.2} ~~ \text{Regular Representation of the Group} ~ Z_4(+)\!</math>

|- style="height:50px"

| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>

| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>

|- style="height:50px"

−

| width="20%" style="border-right:1px solid black" | <math>\operatorname{0}</math>

+

| width="20%" style="border-right:1px solid black" | <math>\operatorname{0}\!</math>

| width="4%" | <math>\{\!</math>

−

| width="16%" | <math>(\operatorname{0}, \operatorname{0}),</math>

+

| width="16%" | <math>(\operatorname{0}, \operatorname{0}),\!</math>

−

| width="20%" | <math>(\operatorname{1}, \operatorname{1}),</math>

+

| width="20%" | <math>(\operatorname{1}, \operatorname{1}),\!</math>

−

| width="20%" | <math>(\operatorname{2}, \operatorname{2}),</math>

+

| width="20%" | <math>(\operatorname{2}, \operatorname{2}),\!</math>

−

| width="16%" | <math>(\operatorname{3}, \operatorname{3})</math>

+

| width="16%" | <math>(\operatorname{3}, \operatorname{3})~\!</math>

| width="4%" | <math>\}\!</math>

|- style="height:50px"

−

| style="border-right:1px solid black" | <math>\operatorname{1}</math>

+

| style="border-right:1px solid black" | <math>\operatorname{1}\!</math>

| <math>\{\!</math>

−

| <math>(\operatorname{0}, \operatorname{1}),</math>

+

| <math>(\operatorname{0}, \operatorname{1}),\!</math>

−

| <math>(\operatorname{1}, \operatorname{2}),</math>

+

| <math>(\operatorname{1}, \operatorname{2}),\!</math>

−

| <math>(\operatorname{2}, \operatorname{3}),</math>

+

| <math>(\operatorname{2}, \operatorname{3}),\!</math>

−

| <math>(\operatorname{3}, \operatorname{0})</math>

+

| <math>(\operatorname{3}, \operatorname{0})\!</math>

| <math>\}\!</math>

|- style="height:50px"

−

| style="border-right:1px solid black" | <math>\operatorname{2}</math>

+

| style="border-right:1px solid black" | <math>\operatorname{2}\!</math>

| <math>\{\!</math>

−

| <math>(\operatorname{0}, \operatorname{2}),</math>

+

| <math>(\operatorname{0}, \operatorname{2}),\!</math>

−

| <math>(\operatorname{1}, \operatorname{3}),</math>

+

| <math>(\operatorname{1}, \operatorname{3}),\!</math>

−

| <math>(\operatorname{2}, \operatorname{0}),</math>

+

| <math>(\operatorname{2}, \operatorname{0}),\!</math>

−

| <math>(\operatorname{3}, \operatorname{1})</math>

+

| <math>(\operatorname{3}, \operatorname{1})\!</math>

| <math>\}\!</math>

|- style="height:50px"

−

| style="border-right:1px solid black" | <math>\operatorname{3}</math>

+

| style="border-right:1px solid black" | <math>\operatorname{3}\!</math>

| <math>\{\!</math>

−

| <math>(\operatorname{0}, \operatorname{3}),</math>

+

| <math>(\operatorname{0}, \operatorname{3}),\!</math>

−

| <math>(\operatorname{1}, \operatorname{0}),</math>

+

| <math>(\operatorname{1}, \operatorname{0}),\!</math>

−

| <math>(\operatorname{2}, \operatorname{1}),</math>

+

| <math>(\operatorname{2}, \operatorname{1}),\!</math>

−

| <math>(\operatorname{3}, \operatorname{2})</math>

+

| <math>(\operatorname{3}, \operatorname{2})\!</math>

| <math>\}\!</math>

|}

Line 1,393: Line 1,403:

|}

−

By convention for the case where <math>k = 0,\!</math> this gives <math>\underline{\underline{X}}^0 = \{ () \},</math> that is, the singleton set consisting of the empty sequence. Depending on the setting, the empty sequence is referred to as the ''empty word'' or the ''empty sentence'', and is commonly denoted by an epsilon <math>{}^{\backprime\backprime} \varepsilon {}^{\prime\prime}</math> or a lambda <math>{}^{\backprime\backprime} \lambda {}^{\prime\prime}.</math> In this text a variant epsilon symbol will be used for the empty sequence, <math>\varepsilon = ().\!</math> In addition, a singly underlined epsilon will be used for the language that consists of a single empty sequence, <math>\underline\varepsilon = \{ \varepsilon \} = \{ () \}.</math>

+

By convention for the case where <math>k = 0,\!</math> this gives <math>\underline{\underline{X}}^0 = \{ () \},</math> that is, the singleton set consisting of the empty sequence. Depending on the setting, the empty sequence is referred to as the ''empty word'' or the ''empty sentence'', and is commonly denoted by an epsilon <math>{}^{\backprime\backprime} \varepsilon {}^{\prime\prime}</math> or a lambda <math>{}^{\backprime\backprime} \lambda {}^{\prime\prime}.</math> In this text a variant epsilon symbol will be used for the empty sequence, <math>{\varepsilon = ()}.\!</math> In addition, a singly underlined epsilon will be used for the language that consists of a single empty sequence, <math>\underline\varepsilon = \{ \varepsilon \} = \{ () \}.</math>

It is probably worth remarking at this point that all empty sequences are indistinguishable (in a one-level formal language, that is), and thus all sets that consist of a single empty sequence are identical. Consequently, <math>\underline{\underline{X}}^0 = \{ () \} = \underline{\varepsilon} = \underline{\underline{Y}}^0,</math> for all resources <math>\underline{\underline{X}}</math> and <math>\underline{\underline{Y}}.</math> However, the empty language <math>\varnothing = \{ \}</math> and the language that consists of a single empty sequence <math>\underline\varepsilon = \{ \varepsilon \} = \{ () \}</math> need to be distinguished from each other.

Line 1,605: Line 1,615:

|}

−

The intent of this succession, as interpreted in FL environments, is that <math>{}^{\langle\langle} x {}^{\rangle\rangle}</math> denotes or refers to <math>{}^{\langle} x {}^{\rangle},</math> which denotes or refers to <math>x.\!</math> Moreover, its computational realization, as implemented in CL environments, is that <math>{}^{\langle\langle} x {}^{\rangle\rangle}</math> addresses or evaluates to <math>{}^{\langle} x {}^{\rangle},</math> which addresses or evaluates to <math>x.\!</math>

+

The intent of this succession, as interpreted in FL environments, is that <math>{}^{\langle\langle} x {}^{\rangle\rangle}\!</math> denotes or refers to <math>{}^{\langle} x {}^{\rangle},\!</math> which denotes or refers to <math>x.\!</math> Moreover, its computational realization, as implemented in CL environments, is that <math>{}^{\langle\langle} x {}^{\rangle\rangle}\!</math> addresses or evaluates to <math>{}^{\langle} x {}^{\rangle},\!</math> which addresses or evaluates to <math>x.\!</math>

The designations ''higher order'' and ''lower order'' are attributed to signs in a casual, local, and transitory way. At this point they signify nothing beyond the occurrence in a sign relation of a pair of triples having the form shown in Table 37.

Line 2,106: Line 2,116:

−

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>\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.

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.

Line 2,585: Line 2,595:

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>

−

For the interpreter <math>Q_{\text{AB}},\!</math> the sign variable <math>q\!</math> need only range over the syntactic domain <math>S = \{ {}^{\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> and the relation variable <math>L\!</math> need only range over the set of sign relations <math>\{ L(\text{A}), L(\text{B}) \}.\!</math> These requirements can be accomplished as follows:

+

For the interpreter <math>Q_\text{AB},\!</math> the sign variable <math>q\!</math> need only range over the syntactic domain <math>S = \{ {}^{\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> and the relation variable <math>L\!</math> need only range over the set of sign relations <math>\{ L(\text{A}), L(\text{B}) \}.\!</math> These requirements can be accomplished as follows:

−

# The variable name <math>{}^{\backprime\backprime} q {}^{\prime\prime}</math> is a HA sign that makes a PIR to the elements of <math>S.\!</math>

+

# The variable name <math>{}^{\backprime\backprime} q {}^{\prime\prime}</math> is a HA sign that makes a PIR to the elements of <math>S.~\!</math>

−

# The variable name <math>{}^{\backprime\backprime} L {}^{\prime\prime}</math> is a HU sign that makes a PIR to the elements of <math>\{ L(\text{A}), L(\text{B}) \}.\!</math>

+

# The variable name <math>{}^{\backprime\backprime} L {}^{\prime\prime}</math> is a HU sign that makes a PIR to the elements of <math>\{ L(\text{A}), L(\text{B}) \}.~\!</math>

−

# The constant name <math>{}^{\backprime\backprime} L(\text{A}) {}^{\prime\prime}</math> is a HI sign that makes a PIR to the elements of <math>L(\text{A}).\!</math>

+

# The constant name <math>{}^{\backprime\backprime} L(\text{A}) {}^{\prime\prime}</math> is a HI sign that makes a PIR to the elements of <math>L(\text{A}).~\!</math>

−

# The constant name <math>{}^{\backprime\backprime} L(\text{B}) {}^{\prime\prime}</math> is a HI sign that makes a PIR to the elements of <math>L(\text{B}).\!</math>

+

# The constant name <math>{}^{\backprime\backprime} L(\text{B}) {}^{\prime\prime}</math> is a HI sign that makes a PIR to the elements of <math>L(\text{B}).~\!</math>

−

This results in a higher order sign relation for <math>Q_{\text{AB}},\!</math> that is shown in Table 46.

+

This results in a higher order sign relation for <math>Q_\text{AB},\!</math> that is shown in Table 46.

Line 2,608: Line 2,618:

\\

\text{B}

−

\end{matrix}</math>

+

\end{matrix}\!</math>

| valign="bottom" width="33%" |

<math>\begin{matrix}

Line 2,614: Line 2,624:

\\

{}^{\langle} L {}^{\rangle}

−

\end{matrix}</math>

+

\end{matrix}\!</math>

| valign="bottom" width="33%" |

<math>\begin{matrix}

Line 2,620: Line 2,630:

\\

{}^{\langle} L {}^{\rangle}

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|-

| valign="bottom" width="33%" |

Line 2,631: Line 2,641:

\\

{}^{\langle} \text{u} {}^{\rangle}

−

\end{matrix}</math>

+

\end{matrix}\!</math>

| valign="bottom" width="33%" |

<math>\begin{matrix}

Line 2,641: Line 2,651:

\\

{}^{\langle} q {}^{\rangle}

−

\end{matrix}</math>

+

\end{matrix}\!</math>

| valign="bottom" width="33%" |

<math>\begin{matrix}

Line 2,651: Line 2,661:

\\

{}^{\langle} q {}^{\rangle}

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\end{matrix}</math>

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\end{matrix}\!</math>

|-

| valign="bottom" width="33%" |

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( & \text{B} & , & {}^{\langle} \text{u} {}^{\rangle} & , & {}^{\langle} \text{u} {}^{\rangle} & )

−

\end{matrix}</math>

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\end{matrix}\!</math>

| valign="bottom" width="33%" |

<math>\begin{matrix}

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{}^{\langle} \text{A} {}^{\rangle}

−

\end{matrix}</math>

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\end{matrix}\!</math>

| valign="bottom" width="33%" |

<math>\begin{matrix}

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{}^{\langle} \text{A} {}^{\rangle}

−

\end{matrix}</math>

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\end{matrix}\!</math>

|-

| valign="bottom" width="33%" |

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( & \text{B} & , & {}^{\langle} \text{i} {}^{\rangle} & , & {}^{\langle} \text{i} {}^{\rangle} & )

−

\end{matrix}</math>

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\end{matrix}\!</math>

| valign="bottom" width="33%" |

<math>\begin{matrix}

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{}^{\langle} \text{B} {}^{\rangle}

−

\end{matrix}</math>

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\end{matrix}\!</math>

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<math>\begin{matrix}

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{}^{\langle} \text{B} {}^{\rangle}

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\end{matrix}</math>

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\end{matrix}\!</math>

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(( & {}^{\langle} \text{u} {}^{\rangle} & , & \text{B} & ), & \text{A} & )

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\end{matrix}</math>

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\end{matrix}\!</math>

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<math>\begin{matrix}

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{}^{\langle} \operatorname{De} {}^{\rangle}

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\end{matrix}</math>

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\end{matrix}\!</math>

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<math>\begin{matrix}

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{}^{\langle} \operatorname{De} {}^{\rangle}

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\end{matrix}</math>

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\end{matrix}\!</math>

|}

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In the ''elemental construal'' of variables, a variable <math>x\!</math> is just an existing object <math>x\!</math> that is an element of a set <math>X,\!</math> the catch being “which element?” In spite of this lack of information, one is still permitted to write <math>{}^{\backprime\backprime} x \in X {}^{\prime\prime}\!</math> as a syntactically well-formed expression and otherwise treat the variable name <math>{}^{\backprime\backprime} x {}^{\prime\prime}\!</math> as a pronoun on a grammatical par with a noun. Given enough information about the contexts of usage and interpretation, this explanation of the variable <math>x\!</math> as an unknown object would complete itself in a determinate indication of the element intended, just as if a constant object had always been named by <math>{}^{\backprime\backprime} x {}^{\prime\prime}.\!</math>

−

In the ''functional construal'' of variables, a variable is a function of unknown circumstances that results in a known range of definite values. This tactic pushes the ostensible location of the uncertainty back a bit, into the domain of a named function, but it cannot eliminate it entirely. Thus, a variable is a function <math>x : X \to Y\!</math> that maps a domain of unknown circumstances, or a ''sample space'' <math>X,\!</math> into a range <math>Y\!</math> of outcome values. Typically, variables of this sort come in sets of the form <math>\{ x_i : X \to Y \},\!</math> collectively called ''coordinate projections'' and together constituting a basis for a whole class of functions <math>x : X \to Y\!</math> sharing a similar type. This construal succeeds in giving each variable name <math>{}^{\backprime\backprime} x_i {}^{\prime\prime}\!</math> an objective referent, namely, the coordinate projection <math>x_i,\!</math> but the explanation is partial to the extent that the domain of unknown circumstances remains to be explained. Completing this explanation of variables, to the extent that it can be accomplished, requires an account of how these unknown circumstances can be known exactly to the extent that they are in fact described, that is, in terms of their effects under the given projections.

+

In the ''functional construal'' of variables, a variable is a function of unknown circumstances that results in a known range of definite values. This tactic pushes the ostensible location of the uncertainty back a bit, into the domain of a named function, but it cannot eliminate it entirely. Thus, a variable is a function <math>x : X \to Y\!</math> that maps a domain of unknown circumstances, or a ''sample space'' <math>X,\!</math> into a range <math>Y\!</math> of outcome values. Typically, variables of this sort come in sets of the form <math>\{ x_i : X \to Y \},\!</math> collectively called ''coordinate projections'' and together constituting a basis for a whole class of functions <math>x : X \to Y\!</math> sharing a similar type. This construal succeeds in giving each variable name <math>{}^{\backprime\backprime} x_i {}^{\prime\prime}\!</math> an objective referent, namely, the coordinate projection <math>{x_i},\!</math> but the explanation is partial to the extent that the domain of unknown circumstances remains to be explained. Completing this explanation of variables, to the extent that it can be accomplished, requires an account of how these unknown circumstances can be known exactly to the extent that they are in fact described, that is, in terms of their effects under the given projections.

As suggested by the whole direction of the present work, the ultimate explanation of variables is to be given by the pragmatic theory of signs, where variables are treated as a special class of signs called ''indices''.

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# The reflective (or critical) acceptation is to see the list before all else as a list of signs, each of which may or may not have a EU-object. This is the attitude that must be taken in formal language theory and in any setting where computational constraints on interpretation are being contemplated. In these contexts it cannot be assumed without question that every sign, whose participation in a denotation relation would have to be indicated by a recursive function and implemented by an effective program, does in fact have an existential denotation, much less a unique object. The entire body of implicit assumptions that go to make up this acceptation, although they operate more like interpretive suspicions than automatic dispositions, will be referred to as the ''sign convention''.

−

In the present context, I can answer questions about the ontology of a “variable” by saying that each variable <math>x_i\!</math> is a kind of a sign, in the boolean case capable of denoting an element of <math>\mathbb{B} = \{ 0, 1 \}\!</math> as its object, with the actual value depending on the interpretation of the moment. Note that <math>x_i\!</math> is a sign, and that <math>{}^{\backprime\backprime} x_i {}^{\prime\prime}\!</math> is another sign that denotes it. This acceptation of the list <math>X = \{ x_i \}\!</math> corresponds to what was just called the ''sign convention''.

+

In the present context, I can answer questions about the ontology of a “variable” by saying that each variable <math>x_i\!</math> is a kind of a sign, in the boolean case capable of denoting an element of <math>{\mathbb{B} = \{ 0, 1 \}}\!</math> as its object, with the actual value depending on the interpretation of the moment. Note that <math>x_i\!</math> is a sign, and that <math>{}^{\backprime\backprime} x_i {}^{\prime\prime}\!</math> is another sign that denotes it. This acceptation of the list <math>X = \{ x_i \}\!</math> corresponds to what was just called the ''sign convention''.

In a context where all the signs that ought to have EU-objects are in fact safely assured to do so, then it is usually less bothersome to assume the object convention. Otherwise, discussion must resort to the less natural but more careful sign convention. This convention is only “artificial” in the sense that it recalls the artifactual nature and the instrumental purpose of signs, and does nothing more out of the way than to call an implement “an implement”.

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<li>

−

The sign <math>{}^{\backprime\backprime} x_i {}^{\prime\prime},\!</math> appearing in the contextual frame <math>{}^{\backprime\backprime} \underline{~~~} : \mathbb{B}^n \to \mathbb{B} {}^{\prime\prime},\!</math> or interpreted as belonging to that frame, denotes the <math>i^\text{th}\!</math> coordinate function <math>\underline{\underline{x_i}} : \mathbb{B}^n \to \mathbb{B}.</math> The entire collection of coordinate maps in <math>\underline{\underline{X}} = \{ \underline{\underline{x_i}} \}\!</math> contributes to the definition of the ''coordinate space'' or ''vector space'' <math>\underline{X} : \mathbb{B}^n,\!</math> notated as follows:

+

The sign <math>{}^{\backprime\backprime} x_i {}^{\prime\prime},\!</math> appearing in the contextual frame <math>{}^{\backprime\backprime} \underline{~~~} : \mathbb{B}^n \to \mathbb{B} {}^{\prime\prime},\!</math> or interpreted as belonging to that frame, denotes the <math>i^\text{th}\!</math> coordinate function <math>\underline{\underline{x_i}} : \mathbb{B}^n \to \mathbb{B}.</math> The entire collection of coordinate maps in <math>{\underline{\underline{X}} = \{ \underline{\underline{x_i}} \}}\!</math> contributes to the definition of the ''coordinate space'' or ''vector space'' <math>\underline{X} : \mathbb{B}^n,\!</math> notated as follows:

<math>\underline{X} = \langle \underline{\underline{X}} \rangle = \langle \underline{\underline{x_1}}, \ldots, \underline{\underline{x_n}} \rangle = \{ (\underline{\underline{x_1}}, \ldots, \underline{\underline{x_n}}) \} : \mathbb{B}^n.\!</math>

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Next, it is necessary to consider the stylistic differences among the logical, functional, and geometric conceptions of propositional logic. Logically, a domain of properties or propositions is known by the axioms it is subject to. Concretely, one thinks of a particular property or proposition as applying to the things or situations it is true of. With the synthesis just indicated, this can be expressed in a unified form: In abstract logical terms, a DOP is known by the axioms to which it is subject. In concrete functional or geometric terms, a particular element of a DOP is known by the things of which it is true.

−

With the appropriate correspondences between these three domains in mind, the general term ''proposition'' can be interpreted in a flexible manner to cover logical, functional, and geometric types of objects. Thus, a locution like <math>{}^{\backprime\backprime} \text{the proposition}~ F {}^{\prime\prime}\!</math> can be interpreted in three ways: (1) literally, to denote a logical proposition, (2) functionally, to denote a mapping from a space <math>X\!</math> of propertied or proposed objects to the domain <math>\mathbb{B} = \{ 0, 1 \}\!</math> of truth values, and (3) geometrically, to denote the so-called ''fiber of truth'' <math>F^{-1}(1)\!</math> as a region or a subset of <math>X.\!</math> For all of these reasons, it is desirable to set up a suitably flexible interpretive framework for propositional logic, where an object introduced as a logical proposition <math>F\!</math> can be recast as a boolean function <math>F : X \to \mathbb{B},\!</math> and understood to indicate the region of the space <math>X\!</math> that is ruled by <math>F.\!</math>

+

With the appropriate correspondences between these three domains in mind, the general term ''proposition'' can be interpreted in a flexible manner to cover logical, functional, and geometric types of objects. Thus, a locution like <math>{}^{\backprime\backprime} \text{the proposition}~ F {}^{\prime\prime}\!</math> can be interpreted in three ways: (1) literally, to denote a logical proposition, (2) functionally, to denote a mapping from a space <math>X\!</math> of propertied or proposed objects to the domain <math>{\mathbb{B} = \{ 0, 1 \}}\!</math> of truth values, and (3) geometrically, to denote the so-called ''fiber of truth'' <math>F^{-1}(1)\!</math> as a region or a subset of <math>X.\!</math> For all of these reasons, it is desirable to set up a suitably flexible interpretive framework for propositional logic, where an object introduced as a logical proposition <math>F\!</math> can be recast as a boolean function <math>F : X \to \mathbb{B},\!</math> and understood to indicate the region of the space <math>X\!</math> that is ruled by <math>F.\!</math>

Generally speaking, it does not seem possible to disentangle these three domains from each other or to determine which one is more fundamental. In practice, due to its concern with the computational implementations of every concept it uses, the present work is biased toward the functional interpretation of propositions. From this point of view, the abstract intention of a logical proposition <math>F\!</math> is regarded as being realized only when a program is found that computes the function <math>F : X \to \mathbb{B}.\!</math>

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One of the reasons for pursuing a pragmatic hybrid of semantic and syntactic approaches, rather than keeping to the purely syntactic ways of manipulating meaningless tokens according to abstract rules of proof, is that the model theoretic strategy preserves the form of connection that exists between an agent's concrete particular experiences and the abstract propositions and general properties that it uses to describe its experience. This makes it more likely that a hybrid approach will serve in the realistic pursuits of inquiry, since these efforts involve the integration of deductive, inductive, and abductive sources of knowledge.

−

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>

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In order to render this MON instructive for the development of a RIF, something intended to be a deliberately ''self-conscious'' construction, it is important to remedy the excessive lucidity of this MONs reflections, the confusing mix of opacity and transparency that comes in proportion to one's very familiarity with an object and that is compounded by one's very fluency in a language. To do this, it is incumbent on a proper analysis of the situation to slow the MON down, to interrupt one's own comprehension of its developing intent, and to articulate the details of the sign process that mediates it much more carefully than is customary.

−

These goals can be achieved by singling out the formal language that is used by this MON to denote its set theoretic objects. This involves separating the object domain <math>O = O_\text{MON}\!</math> from the sign domain <math>S = S_\text{MON},\!</math> paying closer attention to the naive level of set notation that is actually used by this MON, and treating its primitive set theoretic expressions as a formal language all its own.

+

These goals can be achieved by singling out the formal language that is used by this MON to denote its set theoretic objects. This involves separating the object domain <math>{O = O_\text{MON}}\!</math> from the sign domain <math>{S = S_\text{MON}},\!</math> paying closer attention to the naive level of set notation that is actually used by this MON, and treating its primitive set theoretic expressions as a formal language all its own.

Thus, I need to discuss a variety of formal languages on the following alphabet:

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I close this section by discussing the relationship among the three views of systems that are relevant to the example of <math>\text{A}\!</math> and <math>\text{B}.\!</math>

−

[Variant] How do these three perspectives bear on the example of <math>\text{A}\!</math> and <math>\text{B}\!</math>?

+

'''[Variant]''' How do these three perspectives bear on the example of <math>\text{A}\!</math> and <math>\text{B}\!</math>?

−

[Variant] In order to show how these three perspectives bear on the present inquiry, I will now discuss the relationship they exhibit in the example of <math>\text{A}\!</math> and <math>\text{B}.\!</math>

+

'''[Variant]''' In order to show how these three perspectives bear on the present inquiry, I will now discuss the relationship they exhibit in the example of <math>\text{A}\!</math> and <math>\text{B}.\!</math>

In the present example, concerned with the form of communication that takes place between the interpreters <math>\text{A}\!</math> and <math>\text{B},\!</math> the topic of interest is not the type of dynamics that would change one of the original objects, <math>\text{A}\!</math> or <math>\text{B},\!</math> into the other. Thus, the object system is nothing more than the object domain <math>O = \{ \text{A}, \text{B} \}\!</math> shared between the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}).\!</math> In this case, where the object system reduces to an abstract set, falling under the action of a trivial dynamics, one says that the object system is ''stable'' or ''static''. In more developed examples, when the dynamics at the level of the object system becomes more interesting, the ''objects'' in the object system are usually referred to as ''objective configurations'' or ''object states''. Later examples will take on object systems that enjoy significant variations in the sequences of their objective states.

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|-

| <math>\text{Relation}\!</math>

−

| <math>R\!</math>

+

| <math>{R}\!</math>

−

| <math>S(T(U))\!</math>

+

| <math>{S(T(U))}\!</math>

|}

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Let <math>\underline{S}\!</math> be the type of signs, <math>S\!</math> the type of sets, <math>T\!</math> the type of triples, and <math>U\!</math> the type of underlying objects. Now consider the various sorts of things, or the varieties of objects of thought, that are invoked on each side, annotating each type as it is mentioned:

−

ERs of sign relations describe them as sets <math>(Ss)\!</math> of triples <math>(Ts)\!</math> of underlying elements <math>(Us).\!</math> This makes for three levels of objective structure that must be put in coordination with each other, a task that is projected to be carried out in the appropriate OF of sign relations. Corresponding to this aspect of structure in the OF, there is a parallel aspect of structure in the IF of sign relations. Namely, the accessory sign relations that are used to discuss a targeted sign relation need to have signs for sets <math>(\underline{S}Ss),\!</math> signs for triples <math>(\underline{S}Ts),\!</math> and signs for the underlying elements <math>(\underline{S}Us).\!</math> This accounts for three levels of syntactic structure in the IF of sign relations that must be coordinated with each other and also with the targeted levels of objective structure.

+

ERs of sign relations describe them as sets <math>(Ss)\!</math> of triples <math>(Ts)\!</math> of underlying elements <math>(Us).\!</math> This makes for three levels of objective structure that must be put in coordination with each other, a task that is projected to be carried out in the appropriate OF of sign relations. Corresponding to this aspect of structure in the OF, there is a parallel aspect of structure in the IF of sign relations. Namely, the accessory sign relations that are used to discuss a targeted sign relation need to have signs for sets <math>{(\underline{S}Ss)},\!</math> signs for triples <math>{(\underline{S}Ts)},\!</math> and signs for the underlying elements <math>{(\underline{S}Us)}.\!</math> This accounts for three levels of syntactic structure in the IF of sign relations that must be coordinated with each other and also with the targeted levels of objective structure.

−

[Variant] IRs of sign relations describe them in terms of properties <math>(Ps)\!</math> that are taken as primitive entities in their own right. /// refer to properties <math>(Ps)\!</math> of transactions <math>(Ts)\!</math> of underlying elements <math>(Us).\!</math>

+

'''[Variant]''' IRs of sign relations describe them in terms of properties <math>(Ps)\!</math> that are taken as primitive entities in their own right. /// refer to properties <math>(Ps)\!</math> of transactions <math>(Ts)\!</math> of underlying elements <math>(Us).\!</math>

−

[Variant] IRs of sign relations refer to properties of sets <math>(PSs),\!</math> properties of triples <math>(PTs),\!</math> and properties of underlying elements <math>(PUs).\!</math> This amounts to three more levels of objective structure in the OF of the IR that need to be coordinated with each other and interlaced with the OF of the ER if the two are to be brought into the same discussion, possibly for the purpose of translating either into the other. Accordingly, the accessory sign relations that are used to discuss an IR of a targeted sign relation need to have <math>\underline{S}PSs,\!</math> <math>\underline{S}PTs,\!</math> and <math>\underline{S}PUs.\!</math>

+

'''[Variant]''' IRs of sign relations refer to properties of sets <math>(PSs),\!</math> properties of triples <math>(PTs),\!</math> and properties of underlying elements <math>(PUs).\!</math> This amounts to three more levels of objective structure in the OF of the IR that need to be coordinated with each other and interlaced with the OF of the ER if the two are to be brought into the same discussion, possibly for the purpose of translating either into the other. Accordingly, the accessory sign relations that are used to discuss an IR of a targeted sign relation need to have <math>\underline{S}PSs,\!</math> <math>\underline{S}PTs,\!</math> and <math>\underline{S}PUs.\!</math>

===6.22. Extensional Representations of Sign Relations===

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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"

|+ style="height:30px" |

−

<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} ~~ \operatorname{ER}(\operatorname{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}~\!</math>

|- style="height:40px; background:#f0f0ff"

| <math>\text{Object}\!</math>

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\\

({}^{\langle} \text{u} {}^{\rangle}, \text{A})

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|-

| valign="bottom" width="33%" |

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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] For a variety of foundational reasons that I do not fully understand, perhaps most of all because theoretically given structures have their real foundations outside the realm of theory, in empirically given structures, it is best to regard points, properties, and propositions as equally primitive elements, related to each other but not defined in terms of each other, analogous to the undefined elements of a geometry.

+

'''[Variant]''' For a variety of foundational reasons that I do not fully understand, perhaps most of all because theoretically given structures have their real foundations outside the realm of theory, in empirically given structures, it is best to regard points, properties, and propositions as equally primitive elements, related to each other but not defined in terms of each other, analogous to the undefined elements of a geometry.

−

[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 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.

# The second strategy is called the ''analytic coding'', because it attends to the nuances of each sign's interpretation to fashion its code, or the ''<math>\log (n)\!</math> coding'', because it uses roughly <math>\log_2 (n)\!</math> binary features to represent a domain of <math>n\!</math> elements.

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Using two different strategies of representation:

−

'''Literal Coding.''' 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.

+

'''Literal Coding.''' 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.

Being superficial as a matter of principle, or adhering to the surface appearances of signs, enjoys the initial advantage that the very same codes can be used by any interpreter that is capable of observing them. The down side of resorting to this technique is that it typically uses an excessive number of logical dimensions to get each point of the intended space across.

−

Even while operating within the general lines of the literal, superficial, or <math>\mathcal{O}(n)\!</math> strategy, there are still a number of choices to be made in the style of coding to be employed. For example, if there is an obvious distinction between different components of the world, like that between the objects in <math>O = \{ \text{A}, \text{B} \}\!</math> and the signs in <math>S = \{ {}^{\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> then it is common to let this distinction go formally unmarked in the LIR, that is, to omit the requirement of declaring an explicit logical feature to make a note of it in the formal coding. The distinction itself, as a property of reality, is in no danger of being obliterated or permanently erased, but it can be obscured and temporarily ignored. In practice, the distinction is not so much ignored as it is casually observed and informally attended to, usually being marked by incidental indices in the context of the representation.

+

Even while operating within the general lines of the literal, superficial, or <math>{\mathcal{O}(n)}\!</math> strategy, there are still a number of choices to be made in the style of coding to be employed. For example, if there is an obvious distinction between different components of the world, like that between the objects in <math>O = \{ \text{A}, \text{B} \}\!</math> and the signs in <math>S = \{ {}^{\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> then it is common to let this distinction go formally unmarked in the LIR, that is, to omit the requirement of declaring an explicit logical feature to make a note of it in the formal coding. The distinction itself, as a property of reality, is in no danger of being obliterated or permanently erased, but it can be obscured and temporarily ignored. In practice, the distinction is not so much ignored as it is casually observed and informally attended to, usually being marked by incidental indices in the context of the representation.

'''Literal Coding'''

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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.

−

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^\~~circ~~\!.</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.

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\\[4pt]

{\langle\underline{\underline{\text{u}}}\rangle}_W

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|}

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\\[4pt]

{\langle\underline{\underline{\text{i}}}\rangle}_W

−

\end{matrix}</math>

+

\end{matrix}\!</math>

| valign="bottom" |

<math>\begin{matrix}

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\\[4pt]

{\langle\underline{\underline{\text{u}}}\rangle}_W

−

\end{matrix}</math>

+

\end{matrix}\!</math>

| valign="bottom" |

<math>\begin{matrix}

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\\[4pt]

{\langle\underline{\underline{\text{i}}}\rangle}_W

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|}

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\\[4pt]

{\langle\underline{\underline{\text{i}}}\rangle}_W

−

\end{matrix}</math>

+

\end{matrix}\!</math>

| valign="bottom" |

<math>\begin{matrix}

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\underline{\underline{{}^{\backprime\backprime} \text{u} {}^{\prime\prime}}}

& \}

−

\end{array}</math>

+

\end{array}\!</math>

|}

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~\underline{\underline{\text{i}}}~

(\underline{\underline{\text{u}}})

−

\end{matrix}</math>

+

\end{matrix}\!</math>

| valign="bottom" |

<math>\begin{matrix}

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(\underline{\underline{\text{i}}})

~\underline{\underline{\text{u}}}~

−

\end{matrix}</math>

+

\end{matrix}\!</math>

| valign="bottom" |

<math>\begin{matrix}

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~\underline{\underline{\text{i}}}~

(\underline{\underline{\text{u}}})

−

\end{matrix}</math>

+

\end{matrix}\!</math>

| valign="bottom" |

<math>\begin{matrix}

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(\underline{\underline{\text{di}}})

(\underline{\underline{\text{du}}})

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|-

| valign="bottom" |

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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"

|+ style="height:30px" |

−

<math>\text{Table 70.3} ~~ \text{Group Representation} ~ \operatorname{Rep}^\text{C} (V_4)\!</math>

+

<math>{\text{Table 70.3} ~~ \text{Group Representation} ~ \operatorname{Rep}^\text{C} (V_4)}\!</math>

|- style="background:#f0f0ff"

| width="16%" | <math>\begin{matrix} \text{Abstract} \\ \text{Element} \end{matrix}</math>

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|- style="background:#f0f0ff"

| width="25%" | <math>\text{Group Coset}\!</math>

−

| width="25%" | <math>\text{Logical Coset}\!</math>

+

| width="25%" | <math>\text{Logical Coset}~\!</math>

| width="25%" | <math>\text{Logical Element}\!</math>

| width="25%" | <math>\text{Group Element}\!</math>

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|- style="background:#f0f0ff"

| width="25%" | <math>\text{Group Coset}\!</math>

−

| width="25%" | <math>\text{Logical Coset}\!</math>

+

| width="25%" | <math>\text{Logical Coset}~\!</math>

| width="25%" | <math>\text{Logical Element}\!</math>

| width="25%" | <math>\text{Group Element}\!</math>

|-

| <math>G_\text{n}\!</math>

−

| <math>(\operatorname{d}\text{n})\!</math>

+

| <math>({\operatorname{d}\text{n})}\!</math>

|

<math>\begin{matrix}

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Strictly speaking, the logical entity <math>p_S\!</math> is the intensional representation of the tribe, presiding at the highest level of abstraction, while <math>f_S\!</math> and <math>S\!</math> are its more concrete extensional representations, rendering its concept in functional and geometric materials, respectively. Whenever it is possible to do so without confusion, I try to use identical or similar names for the corresponding objects and species of each type, and I generally ignore the distinctions that otherwise set them apart. For instance, in moving toward computational settings, <math>f_S\!</math> makes the best computational proxy for <math>p_S,\!</math> so I commonly refer to the mapping <math>f_S : X \to \mathbb{B}\!</math> as a proposition on <math>X.\!</math>

−

Regarded as logical models, the elements of the contension <math>P\!\!\And\!\!Q</math> satisfy the proposition referred to as the ''conjunction of extensions'' <math>P'\!</math> and <math>Q'.\!</math>

+

Regarded as logical models, the elements of the contension <math>P\!\!\And\!\!Q</math> satisfy the proposition referred to as the ''conjunction of extensions'' <math>P^\prime\!</math> and <math>Q^\prime.\!</math>

Next, the ''composition'' of <math>P\!</math> and <math>Q\!</math> is a dyadic relation <math>R' \subseteq X \times Z\!</math> that is notated as <math>R' = P \circ Q\!</math> and defined as follows.

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In order to speak of generalized orders of relations I need to outline the dimensions of variation along which I intend the characters of already familiar orders of relations to be broadened. Generally speaking, the taxonomic features of <math>n\!</math>-place relations that I wish to liberalize can be read off from their ''local incidence properties'' (LIPs).

−

'''Definition.''' A ''local incidence property'' of a <math>k\!</math>-place relation <math>L \subseteq X_1 \times \ldots \times X_k\!</math> is one that is based on the following type of data. Pick an element <math>x\!</math> in one of the domains <math>X_j\!</math> of <math>L.\!</math> Let <math>L_{x \,\text{at}\, j}\!</math> be a subset of <math>L\!</math> called the ''flag of <math>L\!</math> with <math>x\!</math> at <math>j,\!</math>'' or the ''<math>x \,\text{at}\, j\!</math> flag of <math>L.\!</math>'' The ''local flag'' <math>L_{x \,\text{at}\, j} \subseteq L\!</math> is defined as follows.

+

'''Definition.''' A ''local incidence property'' of a <math>k\!</math>-place relation <math>L \subseteq X_1 \times \ldots \times X_k\!</math> is one that is based on the following type of data. Pick an element <math>x\!</math> in one of the domains <math>{X_j}\!</math> of <math>L.\!</math> Let <math>L_{x \,\text{at}\, j}\!</math> be a subset of <math>L\!</math> called the ''flag of <math>L\!</math> with <math>x\!</math> at <math>{j},\!</math>'' or the ''<math>x \,\text{at}\, j\!</math> flag of <math>L.\!</math>'' The ''local flag'' <math>L_{x \,\text{at}\, j} \subseteq L\!</math> is defined as follows.

{| align="center" cellspacing="8" width="90%"

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Any property <math>P\!</math> of <math>L_{x \,\text{at}\, j}\!</math> constitutes a ''local incidence property'' of <math>L\!</math> with reference to the locus <math>x \,\text{at}\, j.\!</math>

−

'''Definition.''' A <math>k\!</math>-place relation <math>L \subseteq X_1 \times \ldots \times X_k\!</math> is ''<math>P\!</math>-regular at <math>j\!</math>'' if and only if every flag of <math>L\!</math> with <math>x\!</math> at <math>j\!</math> is <math>P,\!</math> letting <math>x\!</math> range over the domain <math>X_j,\!</math> in symbols, if and only if <math>P(L_{x \,\text{at}\, j})\!</math> is true for all <math>x \in X_j.\!</math>

+

'''Definition.''' A <math>k\!</math>-place relation <math>L \subseteq X_1 \times \ldots \times X_k\!</math> is ''<math>P\!</math>-regular at <math>j\!</math>'' if and only if every flag of <math>L\!</math> with <math>x\!</math> at <math>j\!</math> is <math>P,\!</math> letting <math>x\!</math> range over the domain <math>X_j,\!</math> in symbols, if and only if <math>P(L_{x \,\text{at}\, j})\!</math> is true for all <math>{x \in X_j}.\!</math>

Of particular interest are the local incidence properties of relations that can be calculated from the cardinalities of their local flags, and these are naturally called ''numerical incidence properties'' (NIPs).

−

For example, <math>L\!</math> is <math>c\text{-regular at}~ j\!</math> if and only if the cardinality of the local flag <math>L_{x \,\text{at}\, j}\!</math> is equal to <math>c\!</math> for all <math>x \in X_j,\!</math> coded in symbols, if and only if <math>|L_{x \,\text{at}\, j}| = c\!</math> for all <math>x \in X_j.\!</math>

+

For example, <math>L\!</math> is <math>c\text{-regular at}~ j\!</math> if and only if the cardinality of the local flag <math>L_{x \,\text{at}\, j}\!</math> is equal to <math>c\!</math> for all <math>x \in X_j,\!</math> coded in symbols, if and only if <math>|L_{x \,\text{at}\, j}| = c\!</math> for all <math>{x \in X_j}.\!</math>

In a similar fashion, it is possible to define the numerical incidence properties <math>(< c)\text{-regular at}~ j,\!</math> <math>(> c)\text{-regular at}~ j,\!</math> and so on. For ease of reference, a few of these definitions are recorded below.

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& \iff &

|L_{x \,\text{at}\, j}| \ge c ~\text{for all}~ x \in X_j.

−

\end{array}</math>

+

\end{array}\!</math>

|}

−

The definition of local flags can be broadened to give a definition of ''regional flags''. Suppose <math>L \subseteq X_1 \times \ldots \times X_k\!</math> and choose a subset <math>M \subseteq X_j.\!</math> Let <math>L_{M \,\text{at}\, j}\!</math> be a subset of <math>L\!</math> called the ''flag of <math>L\!</math> with <math>M\!</math> at <math>j,\!</math>'' or the ''<math>M \,\text{at}\, j\!</math> flag of <math>L,\!</math>'' defined as follows.

+

The definition of local flags can be broadened to give a definition of ''regional flags''. Suppose <math>L \subseteq X_1 \times \ldots \times X_k\!</math> and choose a subset <math>M \subseteq X_j.\!</math> Let <math>L_{M \,\text{at}\, j}\!</math> be a subset of <math>L\!</math> called the ''flag of <math>L\!</math> with <math>M\!</math> at <math>{j},\!</math>'' or the ''<math>M \,\text{at}\, j\!</math> flag of <math>L,\!</math>'' defined as follows.

{| align="center" cellspacing="8" width="90%"

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& \iff &

L ~\text{is}~ 1\text{-regular at}~ Y.

−

\end{array}</math>

+

\end{array}\!</math>

|}

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# The notation <math>{}^{\backprime\backprime} \operatorname{Dom}_j (L) {}^{\prime\prime}\!</math> denotes the set <math>X_j,\!</math> called the ''domain of <math>L\!</math> at <math>j\!</math>'' or the ''<math>j^\text{th}\!</math> domain of <math>L.\!</math>''.

−

# The notation <math>{}^{\backprime\backprime} \operatorname{Quo}_j (L) {}^{\prime\prime}\!</math> denotes a subset of <math>X_j\!</math> called the ''quorum of <math>L\!</math> at <math>j\!</math>'' or the ''<math>j^\text{th}\!</math> quorum of <math>L,\!</math>'' defined as follows.

+

# The notation <math>{}^{\backprime\backprime} \operatorname{Quo}_j (L) {}^{\prime\prime}\!</math> denotes a subset of <math>{X_j}\!</math> called the ''quorum of <math>L\!</math> at <math>j\!</math>'' or the ''<math>j^\text{th}\!</math> quorum of <math>L,\!</math>'' defined as follows.

{| align="center" cellspacing="8" width="90%"

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In applying the equivalence class notation to a sign relation <math>L,\!</math> the definitions and examples considered so far cover only the case where the connotative component <math>L_{SI}\!</math> is a total equivalence relation on the whole syntactic domain <math>S.\!</math> The next job is to adapt this usage to PERs.

−

If <math>L\!</math> is a sign relation whose syntactic projection <math>L_{SI}\!</math> is a PER on <math>S\!</math> then we may still write <math>{}^{\backprime\backprime} [s]_L {}^{\prime\prime}\!</math> for the “equivalence class of <math>s\!</math> under <math>L_{SI}\!</math>”. But now, <math>[s]_L\!</math> can be empty if <math>s\!</math> has no interpretant, that is, if <math>s\!</math> lies outside the “adequately meaningful” subset of the syntactic domain, where synonymy and equivalence of meaning are defined. Otherwise, if <math>s\!</math> has an <math>i\!</math> then it also has an <math>o,\!</math> by the definition of <math>L_{SI}.\!</math> In this case, there is a triple <math>(o, s, i) \in L,\!</math> and it is permissible to let <math>[o]_L = [s]_L.\!</math>

+

If <math>L\!</math> is a sign relation whose syntactic projection <math>L_{SI}\!</math> is a PER on <math>S\!</math> then we may still write <math>{}^{\backprime\backprime} [s]_L {}^{\prime\prime}\!</math> for the “equivalence class of <math>s\!</math> under <math>L_{SI}\!</math>”. But now, <math>[s]_L\!</math> can be empty if <math>s\!</math> has no interpretant, that is, if <math>s\!</math> lies outside the “adequately meaningful” subset of the syntactic domain, where synonymy and equivalence of meaning are defined. Otherwise, if <math>s\!</math> has an <math>i\!</math> then it also has an <math>o,\!</math> by the definition of <math>L_{SI}.\!</math> In this case, there is a triple <math>{(o, s, i) \in L},\!</math> and it is permissible to let <math>[o]_L = [s]_L.\!</math>

===6.32. Partiality : Selective Operations===

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The smaller and shorter-term index sets, typically having the form <math>I = \{ 1, \ldots, n \},\!</math> are used to keep tabs on the terms of finite sets and sequences, unions and intersections, sums and products.

−

In this context and elsewhere, the notation <math>[n] = \{ 1, \ldots, n \}\!</math> will be used to refer to a ''standard segment'' (finite initial subset) of the natural numbers <math>\mathbb{N} = \{ 1, 2, 3, \ldots \}.\!</math></li>

+

In this context and elsewhere, the notation <math>{[n] = \{ 1, \ldots, n \}}\!</math> will be used to refer to a ''standard segment'' (finite initial subset) of the natural numbers <math>\mathbb{N} = \{ 1, 2, 3, \ldots \}.\!</math></li>

<li>

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<math>L\!</math> assigns a unique set of “local habitations” <math>L(s)\!</math> to each element <math>s\!</math> in the underlying set <math>S.\!</math>

−

'''Definition.''' A ''numbered set'' <math>(S, f),\!</math> based on the set <math>S\!</math> and the injective function <math>f : S \to \mathbb{N},</math> is defined as follows. …

+

'''Definition.''' A ''numbered set'' <math>(S, f),\!</math> based on the set <math>S\!</math> and the injective function <math>{f : S \to \mathbb{N}},</math> is defined as follows. …

'''Definition.''' An ''enumerated set'' <math>(S, f)\!</math> is a numbered set with a bijective <math>f.\!</math> …

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|}

−

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>\operatorname{Pow}(S),\!</math> in other words, as <math>L \in \operatorname{Pow}(X_1 \times \ldots \times X_n).\!</math>

The set of triples of dyadic relations, with pairwise cartesian products chosen in a pre-arranged order from a triple of three sets <math>(X, Y, Z),\!</math> is called the ''dyadic explosion'' of <math>X \times Y \times Z.\!</math> This object is denoted <math>\operatorname{Explo}(X, Y, Z ~|~ 2),\!</math> read as the ''explosion of <math>X \times Y \times Z\!</math> by twos'', or more simply as <math>X, Y, Z ~\operatorname{choose}~ 2,\!</math> and defined as follows:

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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"

−

|+ style="height:30px" | <math>\text{Table 72.1} ~~ \text{Sign Relation of Interpreter A}\!</math>

+

|+ style="height:30px" | <math>\text{Table 72.1} ~~ \text{Sign Relation of Interpreter A}~\!</math>

|- style="height:40px; background:#f0f0ff"

| width="33%" | <math>\text{Object}\!</math>

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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"

−

|+ style="height:30px" | <math>\text{Table 73.1} ~~ \text{Sign Relation of Interpreter B}\!</math>

+

|+ style="height:30px" | <math>\text{Table 73.1} ~~ \text{Sign Relation of Interpreter B}~\!</math>

|- style="height:40px; background:#f0f0ff"

| width="33%" | <math>\text{Object}\!</math>

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As it happens, each of the relations <math>L = L(\text{A})\!</math> or <math>L = L(\text{B})\!</math> is uniquely determined by its projective triple <math>\operatorname{Proj}^{(2)} L.\!</math> This can be seen as follows.

−

Consider any coordinate position <math>(s, i)\!</math> in the plane <math>S \times I.\!</math> If <math>(s, i)\!</math> is not in <math>L_{SI}\!</math> then there can be no element <math>(o, s, i)\!</math> in <math>L,\!</math> therefore we may restrict our attention to positions <math>(s, i)\!</math> in <math>L_{SI},\!</math> knowing that there exist at least <math>|L_{SI}| = 8\!</math> elements in <math>L,\!</math> and seeking only to determine what objects <math>o\!</math> exist such that <math>(o, s, i)\!</math> is an element in the objective ''fiber'' of <math>(s, i).\!</math> In other words, for what <math>o \in O\!</math> is <math>(o, s, i) \in \operatorname{proj}_{SI}^{-1}((s, i))?\!</math> The fact that <math>L_{OS}\!</math> has exactly one element <math>(o, s)\!</math> for each coordinate <math>s \in S\!</math> and that <math>L_{OI}\!</math> has exactly one element <math>(o, i)\!</math> for each coordinate <math>i \in I,\!</math> plus the “coincidence” of it being the same <math>o\!</math> at any one choice for <math>(s, i),\!</math> tells us that <math>L\!</math> has just the one element <math>(o, s, i)\!</math> over each point of <math>S \times I.\!</math> This proves that both <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> are reducible in an informational sense to triples of dyadic relations, that is, they are ''dyadically reducible''.

+

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''.

===6.36. Irreducibly Triadic Relations===

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The relations <math>L_0, L_1 \subseteq \mathbb{B}^3\!</math> are defined by the following equations, with algebraic operations taking place as in <math>\text{GF}(2),\!</math> that is, with <math>1 + 1 = 0.\!</math>

−

# The triple <math>(x, y, z)\!</math> in <math>\mathbb{B}^3\!</math> belongs to <math>L_0\!</math> if and only if <math>x + y + z = 0.\!</math> Thus, <math>L_0\!</math> is the set of even-parity bit vectors, with <math>x + y = z.\!</math>

+

# The triple <math>(x, y, z)\!</math> in <math>\mathbb{B}^3\!</math> belongs to <math>L_0\!</math> if and only if <math>{x + y + z = 0}.\!</math> Thus, <math>L_0\!</math> is the set of even-parity bit vectors, with <math>x + y = z.\!</math>

−

# The triple <math>(x, y, z)\!</math> in <math>\mathbb{B}^3\!</math> belongs to <math>L_1\!</math> if and only if <math>x + y + z = 1.\!</math> Thus, <math>L_1\!</math> is the set of odd-parity bit vectors, with <math>x + y = z + 1.\!</math>

+

# The triple <math>(x, y, z)\!</math> in <math>\mathbb{B}^3\!</math> belongs to <math>L_1\!</math> if and only if <math>{x + y + z = 1}.\!</math> Thus, <math>L_1\!</math> is the set of odd-parity bit vectors, with <math>x + y = z + 1.\!</math>

The corresponding projections of <math>\operatorname{Proj}^{(2)} L_0\!</math> and <math>\operatorname{Proj}^{(2)} L_1\!</math> are identical. In fact, all six projections, taken at the level of logical abstraction, constitute precisely the same dyadic relation, isomorphic to the whole of <math>\mathbb{B} \times \mathbb{B}\!</math> and expressed by the universal constant proposition <math>1 : \mathbb{B} \times \mathbb{B} \to \mathbb{B}.\!</math> In summary:

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With this last modification, angle quotes become like ascribed quotes or attributed remarks, indexed with the name of the interpretive agent that issued the message in question. In sum, the notation <math>{}^{\backprime\backprime ~ \langle\langle} \text{A} {}^\rangle \text{B} {}^{\rangle ~ \prime\prime}\!</math> is intended to situate the sign <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math> in the context of its contemplated use and to index the sign <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math> with the name of the interpreter that is considered to be using it on a given occasion.

−

The notation <math>{}^{\backprime\backprime ~ \langle\langle} \text{A} {}^\rangle \text{B} {}^{\rangle ~ \prime\prime},\!</math> read <math>{}^{\backprime\backprime ~ \langle} \text{A} {}^\rangle ~\text{quoth}~ \text{B} {}^{\prime\prime}\!</math> or <math>{}^{\backprime\backprime ~ \langle} \text{A} {}^\rangle ~\text{used by}~ \text{B} {}^{\prime\prime},\!</math> is an expression that indicates the use of the sign <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math> by the interpreter <math>\text{B}.\!</math> The expression inside the outer quotes is referred to as an ''indexed quotation'', since it is indexed by the name of the interpreter to which it is referred.

+

The notation <math>{}^{\backprime\backprime ~ \langle\langle} \text{A} {}^\rangle \text{B} {}^{\rangle ~ \prime\prime},~\!</math> read <math>{}^{\backprime\backprime ~ \langle} \text{A} {}^\rangle ~\text{quoth}~ \text{B} {}^{\prime\prime}\!</math> or <math>{}^{\backprime\backprime ~ \langle} \text{A} {}^\rangle ~\text{used by}~ \text{B} {}^{\prime\prime},\!</math> is an expression that indicates the use of the sign <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math> by the interpreter <math>\text{B}.\!</math> The expression inside the outer quotes is referred to as an ''indexed quotation'', since it is indexed by the name of the interpreter to which it is referred.

Since angle quotes with a blank index are equivalent to ordinary quotes, we have the following equivalence. [Not sure about this.]

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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"

−

|+ style="height:30px" | <math>\text{Table 80.} ~~ \text{Reflective Extension} ~ \operatorname{Ref}^1 (\text{A})\!</math>

+

|+ style="height:30px" |

+

<math>{\text{Table 80.} ~~ \text{Reflective Extension} ~ \operatorname{Ref}^1 (\text{A})}\!</math>

|- style="height:40px; background:#f0f0ff"

| width="33%" | <math>\text{Object}\!</math>

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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"

−

|+ style="height:30px" | <math>\text{Table 81.} ~~ \text{Reflective Extension} ~ \operatorname{Ref}^1 (\text{B})\!</math>

+

|+ style="height:30px" |

+

<math>{\text{Table 81.} ~~ \text{Reflective Extension} ~ \operatorname{Ref}^1 (\text{B})}\!</math>

|- style="height:40px; background:#f0f0ff"

| width="33%" | <math>\text{Object}\!</math>

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& = &

\operatorname{proj}_{SI} (\operatorname{Ref}^1 (L))

−

\end{array}</math>

+

\end{array}\!</math>

|}

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Taken as transition digraphs, <math>\operatorname{Den}^1 (L_\text{A})\!</math> and <math>\operatorname{Den}^1 (L_\text{B})\!</math> summarize the upshots, end results, or effective steps of computation that are involved in the respective evaluations of signs in <math>S\!</math> by <math>\operatorname{Ref}^1 (\text{A})\!</math> and <math>\operatorname{Ref}^1 (\text{B}).\!</math>

−

The connotative components <math>\operatorname{Con}^1 (L_\text{A})\!</math> and <math>\operatorname{Con}^1 (L_\text{B})\!</math> can be viewed as digraphs on the eight points of the syntactic domain <math>S.\!</math> The arcs of these digraphs are given as follows.

+

The connotative components <math>\operatorname{Con}^1 (L_\text{A})~\!</math> and <math>\operatorname{Con}^1 (L_\text{B})~\!</math> can be viewed as digraphs on the eight points of the syntactic domain <math>S.\!</math> The arcs of these digraphs are given as follows.

<ol>

−

<li><math>\operatorname{Con}^1 (L_\text{A})\!</math> inherits from <math>L_\text{A}\!</math> the structure of a semiotic equivalence relation on <math>S^{(1)},\!</math> having a sling on each point of <math>S^{(1)},\!</math> arcs in both directions between <math>{}^{\langle} \text{A} {}^{\rangle}\!</math> and <math>{}^{\langle} \text{i}{}^{\rangle},\!</math> and arcs in both directions between <math>{}^{\langle} \text{B} {}^{\rangle}\!</math> and <math>{}^{\langle} \text{u}{}^{\rangle}.\!</math> The reflective extension <math>\operatorname{Ref}^1 (L_\text{A})\!</math> adds a sling on each point of <math>S^{(2)},\!</math> creating a semiotic equivalence relation on <math>S.\!</math></li>

+

<li><math>\operatorname{Con}^1 (L_\text{A})\!</math> inherits from <math>L_\text{A}\!</math> the structure of a semiotic equivalence relation on <math>S^{(1)},\!</math> having a sling on each point of <math>S^{(1)},\!</math> arcs in both directions between <math>{}^{\langle} \text{A} {}^{\rangle}\!</math> and <math>{}^{\langle} \text{i}{}^{\rangle},\!</math> and arcs in both directions between <math>{}^{\langle} \text{B} {}^{\rangle}~\!</math> and <math>{}^{\langle} \text{u}{}^{\rangle}.~\!</math> The reflective extension <math>\operatorname{Ref}^1 (L_\text{A})\!</math> adds a sling on each point of <math>S^{(2)},\!</math> creating a semiotic equivalence relation on <math>S.\!</math></li>

−

<li><math>\operatorname{Con}^1 (L_\text{B})\!</math> inherits from <math>L_\text{B}\!</math> the structure of a semiotic equivalence relation on <math>S^{(1)},\!</math> having a sling on each point of <math>S^{(1)},\!</math> arcs in both directions between <math>{}^{\langle} \text{A} {}^{\rangle}\!</math> and <math>{}^{\langle} \text{u}{}^{\rangle},\!</math> and arcs in both directions between <math>{}^{\langle} \text{B} {}^{\rangle}\!</math> and <math>{}^{\langle} \text{i}{}^{\rangle}.\!</math> The reflective extension <math>\operatorname{Ref}^1 (L_\text{B})\!</math> adds a sling on each point of <math>S^{(2)},\!</math> creating a semiotic equivalence relation on <math>S.\!</math></li>

+

<li><math>\operatorname{Con}^1 (L_\text{B})~\!</math> inherits from <math>L_\text{B}\!</math> the structure of a semiotic equivalence relation on <math>S^{(1)},\!</math> having a sling on each point of <math>S^{(1)},\!</math> arcs in both directions between <math>{}^{\langle} \text{A} {}^{\rangle}\!</math> and <math>{}^{\langle} \text{u}{}^{\rangle},\!</math> and arcs in both directions between <math>{}^{\langle} \text{B} {}^{\rangle}~\!</math> and <math>{}^{\langle} \text{i}{}^{\rangle}.~\!</math> The reflective extension <math>\operatorname{Ref}^1 (L_\text{B})\!</math> adds a sling on each point of <math>S^{(2)},\!</math> creating a semiotic equivalence relation on <math>S.\!</math></li>

</ol>

−

Taken as transition digraphs, <math>\operatorname{Con}^1 (L_\text{A})\!</math> and <math>\operatorname{Con}^1 (L_\text{B})\!</math> highlight the associations between signs in <math>\operatorname{Ref}^1 (L_\text{A})\!</math> and <math>\operatorname{Ref}^1 (L_\text{B}),\!</math> respectively.

+

Taken as transition digraphs, <math>\operatorname{Con}^1 (L_\text{A})~\!</math> and <math>\operatorname{Con}^1 (L_\text{B})~\!</math> highlight the associations between signs in <math>\operatorname{Ref}^1 (L_\text{A})\!</math> and <math>\operatorname{Ref}^1 (L_\text{B}),\!</math> respectively.

The semiotic equivalence relation given by <math>\operatorname{Con}^1 (L_\text{A})\!</math> for interpreter <math>\text{A}\!</math> has the following semiotic equations.

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|-

| width="10%" | or

−

|  <math>{}^{\langle} \text{A} {}^{\rangle}\!</math>

+

|  <math>{}^{\langle} \text{A} {}^{\rangle}~\!</math>

| <math>=_\text{A}\!</math>

−

|  <math>{}^{\langle} \text{i} {}^{\rangle}\!</math>

+

|  <math>{}^{\langle} \text{i} {}^{\rangle}~\!</math>

| width="20%" |  

−

|  <math>{}^{\langle} \text{B} {}^{\rangle}\!</math>

+

|  <math>{}^{\langle} \text{B} {}^{\rangle}~\!</math>

| <math>=_\text{A}\!</math>

−

|  <math>{}^{\langle} \text{u} {}^{\rangle}\!</math>

+

|  <math>{}^{\langle} \text{u} {}^{\rangle}~\!</math>

|}

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|}

−

The semiotic equivalence relation given by <math>\operatorname{Con}^1 (L_\text{B})\!</math> for interpreter <math>\text{B}\!</math> has the following semiotic equations.

+

The semiotic equivalence relation given by <math>\operatorname{Con}^1 (L_\text{B})~\!</math> for interpreter <math>\text{B}\!</math> has the following semiotic equations.

{| cellpadding="10"

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|-

| width="10%" | or

−

|  <math>{}^{\langle} \text{A} {}^{\rangle}\!</math>

+

|  <math>{}^{\langle} \text{A} {}^{\rangle}~\!</math>

| <math>=_\text{B}\!</math>

−

|  <math>{}^{\langle} \text{u} {}^{\rangle}\!</math>

+

|  <math>{}^{\langle} \text{u} {}^{\rangle}~\!</math>

| width="20%" |  

−

|  <math>{}^{\langle} \text{B} {}^{\rangle}\!</math>

+

|  <math>{}^{\langle} \text{B} {}^{\rangle}~\!</math>

| <math>=_\text{B}\!</math>

−

|  <math>{}^{\langle} \text{i} {}^{\rangle}\!</math>

+

|  <math>{}^{\langle} \text{i} {}^{\rangle}~\!</math>

|}

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<ol>

−

<li>A sign relation <math>L\!</math> has a non-deterministic denotation if its dyadic component <math>L_{SO}\!</math> is not a function <math>L_{SO} : S \to O,\!</math> in other words, if there are signs in <math>S\!</math> with missing or multiple objects in <math>O.\!</math></li>

+

<li>A sign relation <math>L\!</math> has a non-deterministic denotation if its dyadic component <math>{L_{SO}}\!</math> is not a function <math>L_{SO} : S \to O,\!</math> in other words, if there are signs in <math>S\!</math> with missing or multiple objects in <math>O.\!</math></li>

<li>A sign relation <math>L\!</math> has a non-deterministic connotation if its dyadic component <math>L_{SI}\!</math> is not a function <math>L_{SI} : S \to I,\!</math> in other words, if there are signs in <math>S\!</math> with missing or multiple interpretants in <math>I.\!</math> As a rule, sign relations are rife with this variety of non-determinism, but it is usually felt to be under control so long as <math>L_{SI}\!</math> remains close to being an equivalence relation.</li>

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Against this background, other varieties of reflective extension can be specified by means of semantic equations that are considered to be imposed on the elements of <math>F.\!</math> Taking the reflective extensions <math>\operatorname{Ref}^1 (\text{A})\!</math> and <math>\operatorname{Ref}^1 (\text{B})\!</math> as the first orders of a “free” project toward reflective closure, variant extensions can be described by relating their entries with those of comparable members in the standard sequences <math>\operatorname{Ref}^n (\text{A})\!</math> and <math>\operatorname{Ref}^n (\text{B}).\!</math>

−

A variant pair of reflective extensions, <math>\operatorname{Ref}^1 (\text{A} | E_1)\!</math> and <math>\operatorname{Ref}^1 (\text{B} | E_1),\!</math> is presented in Tables 82 and 83, respectively. These are identical to the corresponding free variants, <math>\operatorname{Ref}^1 (\text{A})\!</math> and <math>\operatorname{Ref}^1 (\text{B}),\!</math> with the exception of those entries that are constrained by the following system of semantic equations.

+

A variant pair of reflective extensions, <math>\operatorname{Ref}^1 (\text{A} | E_1)\!</math> and <math>\operatorname{Ref}^1 (\text{B} | E_1),\!</math> is presented in Tables 82 and 83, respectively. These are identical to the corresponding free variants, <math>\operatorname{Ref}^1 (\text{A})~\!</math> and <math>\operatorname{Ref}^1 (\text{B}),~\!</math> with the exception of those entries that are constrained by the following system of semantic equations.

{| align="center" cellspacing="8" width="90%"

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−

Another pair of reflective extensions, <math>\operatorname{Ref}^1 (\text{A} | E_2)\!</math> and <math>\operatorname{Ref}^1 (\text{B} | E_2),\!</math> is presented in Tables 84 and 85, respectively. These are identical to the corresponding free variants, <math>\operatorname{Ref}^1 (\text{A})\!</math> and <math>\operatorname{Ref}^1 (\text{B}),\!</math> except for the entries constrained by the following semantic equations.

+

Another pair of reflective extensions, <math>\operatorname{Ref}^1 (\text{A} | E_2)\!</math> and <math>\operatorname{Ref}^1 (\text{B} | E_2),\!</math> is presented in Tables 84 and 85, respectively. These are identical to the corresponding free variants, <math>\operatorname{Ref}^1 (\text{A})~\!</math> and <math>\operatorname{Ref}^1 (\text{B}),~\!</math> except for the entries constrained by the following semantic equations.

{| align="center" cellspacing="8" width="90%"

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The mark of an intelligent interpreter that is relevant in this context is the ability to face (encounter, countenance) a non-deterministic juncture of choices in a sign relation and to respond to it as such with actions appropriate to the uncertain nature of the situation.

−

[Variants]

+

'''[Variants]'''

An intelligent interpreter is one that can follow up several different interpretations at once, experimenting with the denotations and connotations that are available in a non-deterministic sign relation, …

Jon Awbrey

12,080

edits