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First, <math>L\!</math> can be associated with a logical predicate or a proposition that says something about the space of effects, being true of certain effects and false of all others. In this way, <math>{}^{\backprime\backprime} L {}^{\prime\prime}</math> can be interpreted as naming a function from <math>\textstyle\prod_i X_i</math> to the domain of truth values <math>\mathbb{B} = \{ 0, 1 \}.</math> With the appropriate understanding, it is permissible to let the notation <math>{}^{\backprime\backprime} L : X_1 \times \ldots \times X_k \to \mathbb{B} {}^{\prime\prime}</math> indicate this interpretation.

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Second, <math>L\!</math> can be associated with a piece of information that allows one to complete various sorts of partial data sets in the space of effects. In particular, if one is given a partial effect or an incomplete <math>k\!</math>-tuple, say, one that is missing a value in the <math>j^\text{th}\!</math> place, as indicated by the notation <math>{}^{\backprime\backprime} (x_1, \ldots, \hat{j}, \ldots, x_k) {}^{\prime\prime},</math> then <math>{}^{\backprime\backprime} L {}^{\prime\prime}</math> can be interpreted as naming a function from the cartesian product of the domains at the filled places to the power set of the domain at the missing place. With this in mind, it is permissible to let <math>{}^{\backprime\backprime} L : X_1 \times \ldots \times \hat{j} \times \ldots \times X_k \to \~~operatorname~~{Pow}(X_j) {}^{\prime\prime}</math> indicate this use of <math>{}^{\backprime\backprime} L {}^{\prime\prime}.</math> If the sets in the range of this function are all singletons, then it is permissible to let <math>{}^{\backprime\backprime} L : X_1 \times \ldots \times \hat{j} \times \ldots \times X_k \to X_j {}^{\prime\prime}</math> specify the corresponding use of <math>{}^{\backprime\backprime} L {}^{\prime\prime}.</math>

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Second, <math>L\!</math> can be associated with a piece of information that allows one to complete various sorts of partial data sets in the space of effects. In particular, if one is given a partial effect or an incomplete <math>k\!</math>-tuple, say, one that is missing a value in the <math>j^\text{th}\!</math> place, as indicated by the notation <math>{}^{\backprime\backprime} (x_1, \ldots, \hat{j}, \ldots, x_k) {}^{\prime\prime},</math> then <math>{}^{\backprime\backprime} L {}^{\prime\prime}</math> can be interpreted as naming a function from the cartesian product of the domains at the filled places to the power set of the domain at the missing place. With this in mind, it is permissible to let <math>{}^{\backprime\backprime} L : X_1 \times \ldots \times \hat{j} \times \ldots \times X_k \to \mathrm{Pow}(X_j) {}^{\prime\prime}</math> indicate this use of <math>{}^{\backprime\backprime} L {}^{\prime\prime}.</math> If the sets in the range of this function are all singletons, then it is permissible to let <math>{}^{\backprime\backprime} L : X_1 \times \ldots \times \hat{j} \times \ldots \times X_k \to X_j {}^{\prime\prime}</math> specify the corresponding use of <math>{}^{\backprime\backprime} L {}^{\prime\prime}.</math>

In general, the indicated degrees of freedom in the interpretation of relation symbols can be exploited properly only if one understands the consequences of this interpretive liberality and is prepared to deal with the problems that arise from its “polymorphic” practices — from using the same sign in different contexts to refer to different types of objects. For example, one should consider what happens, and what sort of IF is demanded to deal with it, when the name <math>{}^{\backprime\backprime} L {}^{\prime\prime}</math> is used equivocally in a statement like <math>L = L^{-1}(1),\!</math> where a sensible reading requires it to denote the relational set <math>L \subseteq \textstyle\prod_i X_i</math> on the first appearance and the propositional function <math>L : \textstyle\prod_i X_i \to \mathbb{B}</math> on the second appearance.

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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 \mathrm{Pow}(Z),\!</math> <math>X \times Z \to \mathrm{Pow}(Y),\!</math> and <math>Y \times Z \to \mathrm{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:

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

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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>\mathrm{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>\mathrm{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>\mathrm{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>\mathrm{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>\mathrm{e}\!</math>

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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>\mathrm{g}\!</math>

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

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

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

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

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

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

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

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

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| <math>(\mathrm{h}, \mathrm{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>\mathrm{h}\!</math>

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

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

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

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

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

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

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

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

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| <math>(\mathrm{h}, \mathrm{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" |

+

<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"

−

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

+

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

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

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

|- style="height:50px"

−

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

+

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

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

−

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

+

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

−

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

+

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

−

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

+

| <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>\mathrm{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>\mathrm{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>

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

|- style="height:50px"

−

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

+

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

−

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

+

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

−

| <math>\~~operatorname~~{a}</math>

+

| <math>\mathrm{a}</math>

−

| <math>\~~operatorname~~{b}</math>

+

| <math>\mathrm{b}</math>

−

| <math>\~~operatorname~~{c}</math>

+

| <math>\mathrm{c}</math>

|- style="height:50px"

−

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

+

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

−

| <math>\~~operatorname~~{a}</math>

+

| <math>\mathrm{a}</math>

−

| <math>\~~operatorname~~{b}</math>

+

| <math>\mathrm{b}</math>

−

| <math>\~~operatorname~~{c}</math>

+

| <math>\mathrm{c}</math>

−

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

+

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

|- style="height:50px"

−

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

+

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

−

| <math>\~~operatorname~~{b}</math>

+

| <math>\mathrm{b}</math>

−

| <math>\~~operatorname~~{c}</math>

+

| <math>\mathrm{c}</math>

−

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

+

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

−

| <math>\~~operatorname~~{a}</math>

+

| <math>\mathrm{a}</math>

|- style="height:50px"

−

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

+

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

−

| <math>\~~operatorname~~{c}</math>

+

| <math>\mathrm{c}</math>

−

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

+

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

−

| <math>\~~operatorname~~{a}</math>

+

| <math>\mathrm{a}</math>

−

| <math>\~~operatorname~~{b}</math>

+

| <math>\mathrm{b}</math>

|}

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>\mathrm{1}\!</math>

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

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

|- style="height:50px"

−

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

+

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

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

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

|- style="height:50px"

−

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

+

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

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

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

|- style="height:50px"

−

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

+

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

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

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

|}

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>\mathrm{0}\!</math>

−

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

+

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

−

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

+

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

−

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

+

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

|- style="height:50px"

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

|- style="height:50px"

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

|- style="height:50px"

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

|- style="height:50px"

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

|}

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>\mathrm{0}\!</math>

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

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

|- style="height:50px"

−

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

+

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

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

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

|- style="height:50px"

−

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

+

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

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

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

|- style="height:50px"

−

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

+

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

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

−

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

+

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

−

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

+

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

−

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

+

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

−

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

+

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

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

|}

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 1,482: Line 1,492:

By way of definition, a sign <math>q\!</math> in a sign relation <math>L \subseteq O \times S \times I\!</math> is said to be, to constitute, or to make a '''plural indefinite reference''' ('''PIR''') to (every element in) a set of objects, <math>X \subseteq O,\!</math> if and only if <math>q\!</math> denotes every element of <math>X.\!</math> This relationship can be expressed in a succinct formula by making use of one additional definition.

−

The '''denotation''' of <math>q\!</math> in <math>L,\!</math> written <math>\~~operatorname~~{De}(q, L),\!</math> is defined as follows:

+

The '''denotation''' of <math>q\!</math> in <math>L,\!</math> written <math>\mathrm{De}(q, L),\!</math> is defined as follows:

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

−

| <math>\~~operatorname~~{De}(q, L) ~=~ \~~operatorname~~{Den}(L) \cdot q ~=~ L_{OS} \cdot q ~=~ \{ o \in O : (o, q, i) \in L, ~\text{for some}~ i \in I \}.</math>

+

| <math>\mathrm{De}(q, L) ~=~ \mathrm{Den}(L) \cdot q ~=~ L_{OS} \cdot q ~=~ \{ o \in O : (o, q, i) \in L, ~\text{for some}~ i \in I \}.</math>

|}

−

Then <math>q\!</math> makes a PIR to <math>X\!</math> in <math>L\!</math> if and only if <math>X \subseteq \~~operatorname~~{De}(q, L).\!</math> Of course, this includes the limiting case where <math>X\!</math> is a singleton, say <math>X = \{ o \}.\!</math> In this case the reference is neither plural nor indefinite, properly speaking, but <math>q\!</math> denotes <math>o\!</math> uniquely.

+

Then <math>q\!</math> makes a PIR to <math>X\!</math> in <math>L\!</math> if and only if <math>X \subseteq \mathrm{De}(q, L).\!</math> Of course, this includes the limiting case where <math>X\!</math> is a singleton, say <math>X = \{ o \}.\!</math> In this case the reference is neither plural nor indefinite, properly speaking, but <math>q\!</math> denotes <math>o\!</math> uniquely.

The proper exploitation of PIRs in sign relations makes it possible to finesse the distinction between HI signs and HU signs, in other words, to provide a ready means of translating between the two kinds of signs that preserves all the relevant information, at least, for many purposes. This is accomplished by connecting the sides of the distinction in two directions. First, a HI sign that makes a PIR to many triples of the form <math>(o, s, i)\!</math> can be taken as tantamount to a HU sign that denotes the corresponding sign relation. Second, a HU sign that denotes a singleton sign relation can be taken as tantamount to a HI sign that denotes its single triple. The relation of one sign being “tantamount to” another is not exactly a full-fledged semantic equivalence, but it is a reasonable approximation to it, and one that serves a number of practical purposes.

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|}

−

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.

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In ordinary discourse HA signs are usually generated by means of linguistic devices for quoting pieces of text. In computational frameworks these quoting mechanisms are implemented as functions that map syntactic arguments into numerical or syntactic values. A quoting function, given a sign or expression as its single argument, needs to accomplish two things: first, to defer the reference of that sign, in other words, to inhibit, delay, or prevent the evaluation of its argument expression, and then, to exhibit or produce another sign whose object is precisely that argument expression.

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The rest of this section considers the development of sign relations that have moderate capacities to reference their own signs as objects. In each case, these extensions are assumed to begin with sign relations like <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> that have disjoint sets of objects and signs and thus have no reflective capacity at the outset. The status of <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> as the reflective origins of the associated reflective developments is recalled by saying that <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> themselves are the ''zeroth order reflective extensions'' of <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> in symbols, <math>L(\text{A}) = \~~operatorname~~{Ref}^0 L(\text{A})\!</math> and <math>L(\text{B}) = \~~operatorname~~{Ref}^0 L(\text{B}).\!</math>

+

The rest of this section considers the development of sign relations that have moderate capacities to reference their own signs as objects. In each case, these extensions are assumed to begin with sign relations like <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> that have disjoint sets of objects and signs and thus have no reflective capacity at the outset. The status of <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> as the reflective origins of the associated reflective developments is recalled by saying that <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> themselves are the ''zeroth order reflective extensions'' of <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> in symbols, <math>L(\text{A}) = \mathrm{Ref}^0 L(\text{A})\!</math> and <math>L(\text{B}) = \mathrm{Ref}^0 L(\text{B}).\!</math>

The next set of Tables illustrates a few the most common ways that sign relations can begin to develop reflective extensions. For ease of reference, Tables 40 and 41 repeat the contents of Tables 1 and 2, respectively, merely replacing ordinary quotes with arch quotes.

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

−

|+ style="height:30px" | <math>\text{Table 40.} ~~ \text{Reflective Origin} ~ \~~operatorname~~{Ref}^0 L(\text{A})\!</math>

+

|+ style="height:30px" | <math>\text{Table 40.} ~~ \text{Reflective Origin} ~ \mathrm{Ref}^0 L(\text{A})\!</math>

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

| <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 41.} ~~ \text{Reflective Origin} ~ \~~operatorname~~{Ref}^0 L(\text{B})\!</math>

+

|+ style="height:30px" | <math>\text{Table 41.} ~~ \text{Reflective Origin} ~ \mathrm{Ref}^0 L(\text{B})\!</math>

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

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

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−

Tables 42 and 43 show one way that the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> can be extended in a reflective sense through the use of quotational devices, yielding the ''first order reflective extensions'', <math>\~~operatorname~~{Ref}^1 L(\text{A})\!</math> and <math>\~~operatorname~~{Ref}^1 L(\text{B}).\!</math> These extensions add one layer of HA signs and their objects to the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> respectively. The new triples specify that, for each <math>{}^{\langle} x {}^{\rangle}\!</math> in the set <math>\{ {}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{i} {}^{\rangle}, {}^{\langle} \text{u} {}^{\rangle} \},\!</math> the HA sign of the form <math>{}^{\langle\langle} x {}^{\rangle\rangle}\!</math> connotes itself while denoting <math>{}^{\langle} x {}^{\rangle}.\!</math>

+

Tables 42 and 43 show one way that the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> can be extended in a reflective sense through the use of quotational devices, yielding the ''first order reflective extensions'', <math>\mathrm{Ref}^1 L(\text{A})\!</math> and <math>\mathrm{Ref}^1 L(\text{B}).\!</math> These extensions add one layer of HA signs and their objects to the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> respectively. The new triples specify that, for each <math>{}^{\langle} x {}^{\rangle}\!</math> in the set <math>\{ {}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{i} {}^{\rangle}, {}^{\langle} \text{u} {}^{\rangle} \},\!</math> the HA sign of the form <math>{}^{\langle\langle} x {}^{\rangle\rangle}\!</math> connotes itself while denoting <math>{}^{\langle} x {}^{\rangle}.\!</math>

Notice that the semantic equivalences of nouns and pronouns referring to each interpreter do not extend to semantic equivalences of their higher order signs, exactly as demanded by the literal character of quotations. Also notice that the reflective extensions of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> coincide in their reflective parts, since exactly the same triples were added to each set.

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

−

|+ style="height:30px" | <math>\text{Table 42.} ~~ \text{Higher Ascent Sign Relation} ~ \~~operatorname~~{Ref}^1 L(\text{A})\!</math>

+

|+ style="height:30px" | <math>\text{Table 42.} ~~ \text{Higher Ascent Sign Relation} ~ \mathrm{Ref}^1 L(\text{A})\!</math>

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

| <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 43.} ~~ \text{Higher Ascent Sign Relation} ~ \~~operatorname~~{Ref}^1 L(\text{B})\!</math>

+

|+ style="height:30px" | <math>\text{Table 43.} ~~ \text{Higher Ascent Sign Relation} ~ \mathrm{Ref}^1 L(\text{B})\!</math>

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

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

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−

There are many ways to extend sign relations in an effort to develop their reflective capacities. The implicit goal of a reflective project is to reach a condition of ''reflective closure'', a configuration satisfying the inclusion <math>S \subseteq O,\!</math> where every sign is an object. It is important to note that not every process of reflective extension can achieve a reflective closure in a finite sign relation. This can only happen if there are additional equivalence relations that keep the effective orders of signs within finite bounds. As long as there are higher order signs that remain distinct from all lower order signs, the sign relation driven by a reflective process is forced to keep expanding. In particular, the process that is ''freely'' suggested by the formation of <math>\~~operatorname~~{Ref}^1 L(\text{A})\!</math> and <math>\~~operatorname~~{Ref}^1 L(\text{B})\!</math> cannot reach closure if it continues as indicated, without further constraints.

+

There are many ways to extend sign relations in an effort to develop their reflective capacities. The implicit goal of a reflective project is to reach a condition of ''reflective closure'', a configuration satisfying the inclusion <math>S \subseteq O,\!</math> where every sign is an object. It is important to note that not every process of reflective extension can achieve a reflective closure in a finite sign relation. This can only happen if there are additional equivalence relations that keep the effective orders of signs within finite bounds. As long as there are higher order signs that remain distinct from all lower order signs, the sign relation driven by a reflective process is forced to keep expanding. In particular, the process that is ''freely'' suggested by the formation of <math>\mathrm{Ref}^1 L(\text{A})~\!</math> and <math>\mathrm{Ref}^1 L(\text{B})~\!</math> cannot reach closure if it continues as indicated, without further constraints.

Tables 44 and 45 present ''higher import extensions'' of <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> respectively. These are just higher order sign relations that add selections of higher import signs and their objects to the underlying set of triples in <math>L(\text{A})\!</math> and <math>L(\text{B}).\!</math> One way to understand these extensions is as follows. The interpreters <math>\text{A}\!</math> and <math>\text{B}\!</math> each use nouns and pronouns just as before, except that the nouns are given additional denotations that refer to the interpretive conduct of the interpreter named. In this form of development, using a noun as a canonical form that refers indifferently to all the <math>(o, s, i)\!</math> triples of a sign relation is a pragmatic way that a sign relation can refer to itself and to other sign relations.

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

−

|+ style="height:30px" | <math>\text{Table 44.} ~~ \text{Higher Import Sign Relation} ~ \~~operatorname~~{HI}^1 L(\text{A})\!</math>

+

|+ style="height:30px" | <math>\text{Table 44.} ~~ \text{Higher Import Sign Relation} ~ \mathrm{HI}^1 L(\text{A})\!</math>

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

| <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 45.} ~~ \text{Higher Import Sign Relation} ~ \~~operatorname~~{HI}^1 L(\text{B})\!</math>

+

|+ style="height:30px" | <math>\text{Table 45.} ~~ \text{Higher Import Sign Relation} ~ \mathrm{HI}^1 L(\text{B})\!</math>

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

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

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−

Several important facts about the class of higher order sign relations in general are illustrated by these examples. First, the notations appearing in the object columns of <math>\~~operatorname~~{HI}^1 L(\text{A})\!</math> and <math>\~~operatorname~~{HI}^1 L(\text{B})\!</math> are not the terms that these newly extended interpreters are depicted as using to describe their objects, but the kinds of language that you and I, or other external observers, would typically make available to distinguish them. The sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> as extended by the transactions of <math>\~~operatorname~~{HI}^1 L(\text{A})\!</math> and <math>\~~operatorname~~{HI}^1 L(\text{B}),\!</math> respectively, are still restricted to their original syntactic domain <math>\{ {}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{i} {}^{\rangle}, {}^{\langle} \text{u} {}^{\rangle} \}.\!</math> This means that there need be nothing especially articulate about a HI sign relation just because it qualifies as higher order. Indeed, the sign relations <math>\~~operatorname~~{HI}^1 L(\text{A})\!</math> and <math>\~~operatorname~~{HI}^1 L(\text{B})\!</math> are not very discriminating in their descriptions of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> referring to many different things under the very same signs that you and I and others would explicitly distinguish, especially in marking the distinction between an interpretive agent and any one of its individual transactions.

+

Several important facts about the class of higher order sign relations in general are illustrated by these examples. First, the notations appearing in the object columns of <math>\mathrm{HI}^1 L(\text{A})\!</math> and <math>\mathrm{HI}^1 L(\text{B})\!</math> are not the terms that these newly extended interpreters are depicted as using to describe their objects, but the kinds of language that you and I, or other external observers, would typically make available to distinguish them. The sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> as extended by the transactions of <math>\mathrm{HI}^1 L(\text{A})\!</math> and <math>\mathrm{HI}^1 L(\text{B}),\!</math> respectively, are still restricted to their original syntactic domain <math>\{ {}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{i} {}^{\rangle}, {}^{\langle} \text{u} {}^{\rangle} \}.\!</math> This means that there need be nothing especially articulate about a HI sign relation just because it qualifies as higher order. Indeed, the sign relations <math>\mathrm{HI}^1 L(\text{A})\!</math> and <math>\mathrm{HI}^1 L(\text{B})\!</math> are not very discriminating in their descriptions of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> referring to many different things under the very same signs that you and I and others would explicitly distinguish, especially in marking the distinction between an interpretive agent and any one of its individual transactions.

In practice, it does an interpreter little good to have the higher import signs for referring to triples of objects, signs, and interpretants if it does not also have the higher ascent signs for referring to each triple's syntactic portions. Consequently, the higher order sign relations that one is likely to observe in practice are typically a mixed bag, having both higher ascent and higher import sections. Moreover, the ambiguity involved in having signs that refer equivocally to simple world elements and also to complex structures formed from these ingredients would most likely be resolved by drawing additional information from context and fashioning more distinctive signs.

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The technique illustrated here represents a general strategy, one that can be exploited to derive certain benefits of set theory without having to pay the overhead that is needed to maintain sets as abstract objects. Using an identified type of a sign as a canonical form that can refer indifferently to all the members of a set is a pragmatic way of making plural reference to the members of a set without invoking the set itself as an abstract object. Of course, it is not that one can get something for nothing by these means. One is merely banking on one's recurring investment in the setting of a certain sign relation, a particular set of elementary transactions that is taken for granted as already funded.

−

As a rule, it is desirable for the grammatical system that one uses to construct and interpret higher order signs, that is, signs for referring to signs as objects, to mesh in a comfortable fashion with the overall pragmatic system that one uses to assign syntactic codes to objects in general. For future reference, I call this requirement the problem of creating a ''conformally reflective extension'' (CRE) for a given sign relation. A good way to think about this task is to imagine oneself beginning with a sign relation <math>L \subseteq O \times S \times I,\!</math> and to consider its denotative component <math>\~~operatorname~~{Den}_L = L_{OS} \subseteq O \times S.\!</math> Typically one has a ''naming function'', say <math>\~~operatorname~~{Nom},\!</math> that maps objects into signs:

+

As a rule, it is desirable for the grammatical system that one uses to construct and interpret higher order signs, that is, signs for referring to signs as objects, to mesh in a comfortable fashion with the overall pragmatic system that one uses to assign syntactic codes to objects in general. For future reference, I call this requirement the problem of creating a ''conformally reflective extension'' (CRE) for a given sign relation. A good way to think about this task is to imagine oneself beginning with a sign relation <math>L \subseteq O \times S \times I,\!</math> and to consider its denotative component <math>\mathrm{Den}_L = L_{OS} \subseteq O \times S.\!</math> Typically one has a ''naming function'', say <math>\mathrm{Nom},\!</math> that maps objects into signs:

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

−

| <math>\~~operatorname~~{Nom} \subseteq \~~operatorname~~{Den}_L \subseteq O \times S ~\text{such that}~ \~~operatorname~~{Nom} : O \to S.\!</math>

+

| <math>\mathrm{Nom} \subseteq \mathrm{Den}_L \subseteq O \times S ~\text{such that}~ \mathrm{Nom} : O \to S.\!</math>

|}

−

Part of the task of making a sign relation more reflective is to extend it in ways that turn more of its signs into objects. This is the reason for creating higher order signs, which are just signs for making objects out of signs. One effect of progressive reflection is to extend the initial naming function <math>\~~operatorname~~{Nom}\!</math> through a succession of new naming functions <math>\~~operatorname~~{Nom}',\!</math> <math>\~~operatorname~~{Nom}'',\!</math> and so on, assigning unique names to larger allotments of the original and subsequent signs. With respect to the difficulties of construction, the ''hard core'' or ''adamant part'' of creating extended naming functions resides in the initial portion <math>\~~operatorname~~{Nom}\!</math> that maps objects of the “external world” to signs in the “internal world”. The subsequent task of assigning conventional names to signs is supposed to be comparatively natural and ''easy'', perhaps on account of the ''nominal'' nature of signs themselves.

+

Part of the task of making a sign relation more reflective is to extend it in ways that turn more of its signs into objects. This is the reason for creating higher order signs, which are just signs for making objects out of signs. One effect of progressive reflection is to extend the initial naming function <math>\mathrm{Nom}\!</math> through a succession of new naming functions <math>\mathrm{Nom}',\!</math> <math>\mathrm{Nom}'',\!</math> and so on, assigning unique names to larger allotments of the original and subsequent signs. With respect to the difficulties of construction, the ''hard core'' or ''adamant part'' of creating extended naming functions resides in the initial portion <math>\mathrm{Nom}\!</math> that maps objects of the “external world” to signs in the “internal world”. The subsequent task of assigning conventional names to signs is supposed to be comparatively natural and ''easy'', perhaps on account of the ''nominal'' nature of signs themselves.

The effect of reflection on the original sign relation <math>L \subseteq O \times S \times I\!</math> can be analyzed as follows. Suppose that a step of reflection creates higher order signs for a subset of <math>S.\!</math> Then this step involves the construction of a newly extended sign relation:

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

−

| <math>\~~operatorname~~{Nom}_1 : O_1 \to S_1 ~\text{such that}~ \~~operatorname~~{Nom}_1 : x \mapsto {}^{\langle} x {}^{\rangle}.\!</math>

+

| <math>\mathrm{Nom}_1 : O_1 \to S_1 ~\text{such that}~ \mathrm{Nom}_1 : x \mapsto {}^{\langle} x {}^{\rangle}.\!</math>

|}

−

Finally, the reflectively extended naming function <math>\~~operatorname~~{Nom}' : O' \to S'\!</math> is defined as <math>\~~operatorname~~{Nom}' = \~~operatorname~~{Nom} \cup \~~operatorname~~{Nom}_1.\!</math>

+

Finally, the reflectively extended naming function <math>\mathrm{Nom}' : O' \to S'\!</math> is defined as <math>\mathrm{Nom}' = \mathrm{Nom} \cup \mathrm{Nom}_1.\!</math>

A few remarks are necessary to see how this way of defining a CRE can be regarded as legitimate.

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In the present context an application of the arch notation, for example, <math>{}^{\langle} x {}^{\rangle},\!</math> is read on analogy with the use of any other functional notation, for example, <math>f(x),\!</math> where <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> is the name of a function <math>f,\!</math> <math>{}^{\backprime\backprime} f(~) {}^{\prime\prime}\!</math> is the context of its application, <math>{}^{\backprime\backprime} x {}^{\prime\prime}\!</math> is the name of an argument <math>x,\!</math> and where the functional abstraction <math>{}^{\backprime\backprime} x \mapsto f(x) {}^{\prime\prime}\!</math> is just another name for the function <math>f.\!</math>

−

It is clear that some form of functional abstraction is being invoked in the above definition of <math>\~~operatorname~~{Nom}_1.\!</math> Otherwise, the expression <math>x \mapsto {}^{\langle} x {}^{\rangle}\!</math> would indicate a constant function, one that maps every <math>x\!</math> in its domain to the same code or sign for the letter <math>{}^{\backprime\backprime} x {}^{\prime\prime}.\!</math> But if this is allowed, then it appears to pose a dilemma, either to invoke a more powerful concept of functional abstraction than the concept being defined, or else to attempt an improper definition of the naming function in terms of itself.

+

It is clear that some form of functional abstraction is being invoked in the above definition of <math>\mathrm{Nom}_1.\!</math> Otherwise, the expression <math>x \mapsto {}^{\langle} x {}^{\rangle}\!</math> would indicate a constant function, one that maps every <math>x\!</math> in its domain to the same code or sign for the letter <math>{}^{\backprime\backprime} x {}^{\prime\prime}.\!</math> But if this is allowed, then it appears to pose a dilemma, either to invoke a more powerful concept of functional abstraction than the concept being defined, or else to attempt an improper definition of the naming function in terms of itself.

Although it appears that this form of functional abstraction is being used to define the CRE in terms of itself, trying to extend the definition of the naming function in terms of a definition that is already assumed to be available, in reality this only uses a finite function, a finite table look up, to define the naming function for an unlimited number of higher order signs.

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===6.11. Higher Order Sign Relations : Application===

−

Given the language in which a notation like <math>{}^{\backprime\backprime} \~~operatorname~~{De}(q, L) {}^{\prime\prime}\!</math> makes sense, or in prospect of being given such a language, it is instructive to ask: “What must be assumed about the context of interpretation in which this language is supposed to make sense?” According to the theory of signs that is being examined here, the relevant formal aspects of that context are embodied in a particular sign relation, call it <math>{}^{\backprime\backprime} Q {}^{\prime\prime}.\!</math> With respect to the hypothetical sign relation <math>Q,\!</math> commonly personified as the prospective reader or the ideal interpreter of the intended language, the denotation of the expression <math>{}^{\backprime\backprime} \~~operatorname~~{De}(q, L) {}^{\prime\prime}\!</math> is given by:

+

Given the language in which a notation like <math>{}^{\backprime\backprime} \mathrm{De}(q, L) {}^{\prime\prime}\!</math> makes sense, or in prospect of being given such a language, it is instructive to ask: “What must be assumed about the context of interpretation in which this language is supposed to make sense?” According to the theory of signs that is being examined here, the relevant formal aspects of that context are embodied in a particular sign relation, call it <math>{}^{\backprime\backprime} Q {}^{\prime\prime}.\!</math> With respect to the hypothetical sign relation <math>Q,\!</math> commonly personified as the prospective reader or the ideal interpreter of the intended language, the denotation of the expression <math>{}^{\backprime\backprime} \mathrm{De}(q, L) {}^{\prime\prime}\!</math> is given by:

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

−

| <math>\~~operatorname~~{De}( {}^{\backprime\backprime} \~~operatorname~~{De}(q, L) {}^{\prime\prime}, Q ).\!</math>

+

| <math>\mathrm{De}( {}^{\backprime\backprime} \mathrm{De}(q, L) {}^{\prime\prime}, Q ).\!</math>

|}

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|

<math>\begin{array}{lccc}

−

\~~operatorname~~{De}( & {}^{\backprime\backprime} \~~operatorname~~{De} {}^{\prime\prime} & , & Q)

+

\mathrm{De}( & {}^{\backprime\backprime} \mathrm{De} {}^{\prime\prime} & , & Q)

\\[6pt]

−

\~~operatorname~~{De}( & {}^{\backprime\backprime} q {}^{\prime\prime} & , & Q)

+

\mathrm{De}( & {}^{\backprime\backprime} q {}^{\prime\prime} & , & Q)

\\[6pt]

−

\~~operatorname~~{De}( & {}^{\backprime\backprime} L {}^{\prime\prime} & , & Q)

+

\mathrm{De}( & {}^{\backprime\backprime} L {}^{\prime\prime} & , & Q)

\end{array}</math>

|}

−

What are the roles of the signs <math>{}^{\backprime\backprime} \~~operatorname~~{De} {}^{\prime\prime},\!</math> <math>{}^{\backprime\backprime} q {}^{\prime\prime},\!</math> <math>{}^{\backprime\backprime} L {}^{\prime\prime}\!</math> and what are they supposed to mean to <math>Q\!</math>? Evidently, <math>{}^{\backprime\backprime} \~~operatorname~~{De} {}^{\prime\prime}\!</math> is a constant name that refers to a particular function, <math>{}^{\backprime\backprime} q {}^{\prime\prime}\!</math> is a variable name that makes a PIR to a collection of signs, and <math>{}^{\backprime\backprime} L {}^{\prime\prime}\!</math> is a variable name that makes a PIR to a collection of sign relations.

+

What are the roles of the signs <math>{}^{\backprime\backprime} \mathrm{De} {}^{\prime\prime},\!</math> <math>{}^{\backprime\backprime} q {}^{\prime\prime},\!</math> <math>{}^{\backprime\backprime} L {}^{\prime\prime}\!</math> and what are they supposed to mean to <math>Q\!</math>? Evidently, <math>{}^{\backprime\backprime} \mathrm{De} {}^{\prime\prime}\!</math> is a constant name that refers to a particular function, <math>{}^{\backprime\backprime} q {}^{\prime\prime}\!</math> is a variable name that makes a PIR to a collection of signs, and <math>{}^{\backprime\backprime} L {}^{\prime\prime}\!</math> is a variable name that makes a PIR to a collection of sign relations.

This is not the place to take up the possibility of an ideal, universal, or even a very comprehensive interpreter for the language indicated here, so I specialize the account to consider an interpreter <math>Q_{\text{AB}} = Q(\text{A}, \text{B})\!</math> that is competent to cover the initial level of reflections that arise from the dialogue of <math>\text{A}\!</math> and <math>\text{B}.\!</math>

−

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}

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\\

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

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|-

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

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\\

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

−

\end{matrix}</math>

+

\end{matrix}\!</math>

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

<math>\begin{matrix}

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\\

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

−

\end{matrix}</math>

+

\end{matrix}\!</math>

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

<math>\begin{matrix}

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\\

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

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|-

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

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\\

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

−

\end{matrix}</math>

+

\end{matrix}\!</math>

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

<math>\begin{matrix}

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\\

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

−

\end{matrix}</math>

+

\end{matrix}\!</math>

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

<math>\begin{matrix}

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\\

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

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|-

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

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\\

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

−

\end{matrix}</math>

+

\end{matrix}\!</math>

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

<math>\begin{matrix}

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\\

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

−

\end{matrix}</math>

+

\end{matrix}\!</math>

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

<math>\begin{matrix}

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\\

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

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|-

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

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\\

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

−

\end{matrix}</math>

+

\end{matrix}\!</math>

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

<math>\begin{matrix}

−

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

+

{}^{\langle} \mathrm{De} {}^{\rangle}

\\

−

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

+

{}^{\langle} \mathrm{De} {}^{\rangle}

\\

−

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

+

{}^{\langle} \mathrm{De} {}^{\rangle}

\\

−

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

+

{}^{\langle} \mathrm{De} {}^{\rangle}

\\

−

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

+

{}^{\langle} \mathrm{De} {}^{\rangle}

\\

−

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

+

{}^{\langle} \mathrm{De} {}^{\rangle}

\\

−

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

+

{}^{\langle} \mathrm{De} {}^{\rangle}

\\

−

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

+

{}^{\langle} \mathrm{De} {}^{\rangle}

−

\end{matrix}</math>

+

\end{matrix}\!</math>

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

<math>\begin{matrix}

−

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

+

{}^{\langle} \mathrm{De} {}^{\rangle}

\\

−

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

+

{}^{\langle} \mathrm{De} {}^{\rangle}

\\

−

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

+

{}^{\langle} \mathrm{De} {}^{\rangle}

\\

−

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

+

{}^{\langle} \mathrm{De} {}^{\rangle}

\\

−

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

+

{}^{\langle} \mathrm{De} {}^{\rangle}

\\

−

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

+

{}^{\langle} \mathrm{De} {}^{\rangle}

\\

−

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

+

{}^{\langle} \mathrm{De} {}^{\rangle}

\\

−

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

+

{}^{\langle} \mathrm{De} {}^{\rangle}

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|}

−

Following the manner of construction in this extremely reduced example, it is possible to see how answers to the above questions, concerning the meaning of <math>{}^{\backprime\backprime} \~~operatorname~~{De}(q, L) {}^{\prime\prime},\!</math> might be worked out. In the present instance:

+

Following the manner of construction in this extremely reduced example, it is possible to see how answers to the above questions, concerning the meaning of <math>{}^{\backprime\backprime} \mathrm{De}(q, L) {}^{\prime\prime},\!</math> might be worked out. In the present instance:

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

|

<math>\begin{array}{lll}

−

\~~operatorname~~{De} ({}^{\backprime\backprime} q {}^{\prime\prime}, Q_{\text{AB}})

+

\mathrm{De} ({}^{\backprime\backprime} q {}^{\prime\prime}, Q_{\text{AB}})

& = & S

\\[6pt]

−

\~~operatorname~~{De} ({}^{\backprime\backprime} L {}^{\prime\prime}, Q_{\text{AB}})

+

\mathrm{De} ({}^{\backprime\backprime} L {}^{\prime\prime}, Q_{\text{AB}})

& = & \{ L(\text{A}), L(\text{B}) \}

\end{array}</math>

Line 2,904: Line 2,914:

The ''nominal resource'' (''nominal alphabet'' or ''nominal lexicon'') for <math>X\!</math> is a set of signs that is notated and defined as follows:

−

<math>X^{\backprime\backprime\prime\prime} = \~~operatorname~~{Nom}(X) = \{ {}^{\backprime\backprime} x_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} x_n {}^{\prime\prime} \}.</math>

+

<math>X^{\backprime\backprime\prime\prime} = \mathrm{Nom}(X) = \{ {}^{\backprime\backprime} x_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} x_n {}^{\prime\prime} \}.</math>

This concept is intended to capture the ordinary usage of this set of signs in one familiar context or another.</li>

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The ''mediate resource'' (''mediate alphabet'' or ''mediate lexicon'') for <math>X\!</math> is a set of signs that is notated and defined as follows:

−

<math>X^{\langle\rangle} = \~~operatorname~~{Med}(X) = \{ {}^{\langle} x_1 {}^{\rangle}, \ldots, {}^{\langle} x_n {}^{\rangle} \}.</math>

+

<math>X^{\langle\rangle} = \mathrm{Med}(X) = \{ {}^{\langle} x_1 {}^{\rangle}, \ldots, {}^{\langle} x_n {}^{\rangle} \}.</math>

This concept provides a middle ground between the nominal resource above and the literal resource described next.</li>

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The ''literal resource'' (''literal alphabet'' or ''literal lexicon'') for <math>X\!</math> is a set of signs that is notated and defined as follows:

−

<math>X = \~~operatorname~~{Lit}(X) = \{ x_1, \ldots, x_n \}.</math>

+

<math>X = \mathrm{Lit}(X) = \{ x_1, \ldots, x_n \}.</math>

This concept is intended to supply a set of signs that can be used in ways analogous to familiar usages, but which are more subject to free variation and thematic control.</li></ol>

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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''.

Line 2,998: Line 3,008:

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

−

| <math>\underline{\underline{X}} = \~~operatorname~~{Lit}(X) = \{ \underline{\underline{x_1}}, \ldots, \underline{\underline{x_n}} \}.\!</math>

+

| <math>\underline{\underline{X}} = \mathrm{Lit}(X) = \{ \underline{\underline{x_1}}, \ldots, \underline{\underline{x_n}} \}.\!</math>

|}

Line 3,016: Line 3,026:

# 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”.

Line 3,029: Line 3,039:

<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{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]])} : \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>

Line 3,081: Line 3,091:

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>

Line 3,089: Line 3,099:

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>

+

Working under the functional perspective, the formal system known as ''propositional calculus'' is introduced as a general system of notations for referring to boolean functions. Typically, one takes a space <math>X\!</math> and a coordinate representation <math>T : X \to \mathbb{B}^n\!</math> as parameters of a particular system and speaks of the propositional calculus on a finite set of variables <math>\{ \underline{\underline{x_i}} \}.\!</math> In objective terms, this constitutes the ''domain of propositions'' on the basis <math>\{ \underline{\underline{x_i}} \},\!</math> notated as <math>\mathrm{DOP}\{ \underline{\underline{x_i}} \}.\!</math> Ideally, one does not want to become too fixed on a particular set of logical features or to let the momentary dimensions of the space be cast in stone. In practice, this means that the formalism and its computational implementation should allow for the automatic embedding of <math>\mathrm{DOP}(\underline{\underline{X}})\!</math> into <math>\mathrm{DOP}(\underline{\underline{Y}})\!</math> whenever <math>\underline{\underline{X}} \subseteq \underline{\underline{Y}}.\!</math>

The rest of this section presents the elements of a particular calculus for propositional logic. First, I establish the basic notations and summarize the axiomatic presentation of the calculus, and then I give special attention to its functional and geometric interpretations.

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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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A sign relation is a complex object and its representations, insofar as they faithfully preserve its structure, are complex signs. Accordingly, the problems of translating between ERs and IRs of sign relations, of detecting when representations alleged to be of sign relations do indeed represent objects of the specified character, and of recognizing whether different representations do or do not represent the same sign relation as their common object — these are the familiar questions that would be asked of the signs and interpretants in a simple sign relation, but this time asked at a higher level, in regard to the complex signs and complex interpretants that are posed by the different stripes of representation. At the same time, it should be obvious that these are also the natural questions to be faced in building a bridge between representations.

−

How many different sorts of entities are conceivably involved in translating between ERs and IRs of sign relations? To address this question it helps to introduce a system of type notations that can be used to keep track of the various sorts of things, or the varieties of objects of thought, that are generated in the process of answering it. Table 47.1 summarizes the basic types of things that are needed in this pursuit, while the rest can be derived by constructions of the form <math>X ~\~~operatorname~~{of}~ Y,\!</math> notated <math>X(Y)\!</math> or just <math>XY,\!</math> for any basic types <math>X\!</math> and <math>Y.\!</math> The constructed types of things involved in the ERs and IRs of sign relations are listed in Tables 47.2 and 47.3, respectively.

+

How many different sorts of entities are conceivably involved in translating between ERs and IRs of sign relations? To address this question it helps to introduce a system of type notations that can be used to keep track of the various sorts of things, or the varieties of objects of thought, that are generated in the process of answering it. Table 47.1 summarizes the basic types of things that are needed in this pursuit, while the rest can be derived by constructions of the form <math>X ~\mathrm{of}~ Y,\!</math> notated <math>X(Y)\!</math> or just <math>XY,\!</math> for any basic types <math>X\!</math> and <math>Y.\!</math> The constructed types of things involved in the ERs and IRs of sign relations are listed in Tables 47.2 and 47.3, respectively.

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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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Starting from a standpoint in concrete constructions, the easiest way to begin developing an explicit treatment of ERs is to gather the relevant materials in the forms already presented, to fill out the missing details and expand the abbreviated contents of these forms, and to review their full structures in a more formal light. Consequently, this section inaugurates the formal discussion of ERs by taking a second look at the interpreters <math>\text{A}\!</math> and <math>\text{B},\!</math> recollecting the Tables of their sign relations and finishing up the Tables of their dyadic components. Since the form of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> no longer presents any novelty, I can exploit their second presentation as a first opportunity to examine a selection of finer points, previously overlooked. Also, in the process of reviewing this material it is useful to anticipate a number of incidental issues that are reaching the point of becoming critical within this discussion and to begin introducing the generic types of technical devices that are needed to deal with them.

−

The next set of Tables summarizes the ERs of <math>L(\text{A})\!</math> and <math>L(\text{B}).\!</math> For ease of reference, Tables 48.1 and 49.1 repeat the contents of Tables 1 and 2, respectively, the only difference being that appearances of ordinary quotation marks <math>({}^{\backprime\backprime} \ldots {}^{\prime\prime})\!</math> are transcribed as invocations of the ''arch operator'' <math>({}^{\langle} \ldots {}^{\rangle}).\!</math> The reason for this slight change of notation will be explained shortly. The denotative components <math>\~~operatorname~~{Den}(\text{A})\!</math> and <math>\~~operatorname~~{Den}(\text{B})\!</math> are shown in the first two columns of Tables 48.2 and 49.2, respectively, while the third column gives the transition from sign to object as an ordered pair <math>(s, o).\!</math> The connotative components <math>\~~operatorname~~{Con}(\text{A})\!</math> and <math>\~~operatorname~~{Con}(\text{B})\!</math> are shown in the first two columns of Tables 48.3 and 49.3, respectively, while the third column gives the transition from sign to interpretant as an ordered pair <math>(s, i).\!</math>

+

The next set of Tables summarizes the ERs of <math>L(\text{A})\!</math> and <math>L(\text{B}).\!</math> For ease of reference, Tables 48.1 and 49.1 repeat the contents of Tables 1 and 2, respectively, the only difference being that appearances of ordinary quotation marks <math>({}^{\backprime\backprime} \ldots {}^{\prime\prime})\!</math> are transcribed as invocations of the ''arch operator'' <math>({}^{\langle} \ldots {}^{\rangle}).\!</math> The reason for this slight change of notation will be explained shortly. The denotative components <math>\mathrm{Den}(\text{A})\!</math> and <math>\mathrm{Den}(\text{B})\!</math> are shown in the first two columns of Tables 48.2 and 49.2, respectively, while the third column gives the transition from sign to object as an ordered pair <math>(s, o).\!</math> The connotative components <math>\mathrm{Con}(\text{A})\!</math> and <math>\mathrm{Con}(\text{B})\!</math> are shown in the first two columns of Tables 48.3 and 49.3, respectively, while the third column gives the transition from sign to interpretant as an ordered pair <math>(s, i).\!</math>

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

|+ style="height:30px" |

−

<math>\text{Table 48.1} ~~ \~~operatorname~~{ER}(L_\text{A}) : \text{Extensional Representation of} ~ L_\text{A}\!</math>

+

<math>\text{Table 48.1} ~~ \mathrm{ER}(L_\text{A}) : \text{Extensional Representation of} ~ L_\text{A}\!</math>

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

| <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 48.2} ~~ \~~operatorname~~{ER}(\~~operatorname~~{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!</math>

+

<math>\text{Table 48.2} ~~ \mathrm{ER}(\mathrm{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!</math>

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

| <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 48.3} ~~ \~~operatorname~~{ER}(\~~operatorname~~{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!</math>

+

<math>\text{Table 48.3} ~~ \mathrm{ER}(\mathrm{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!</math>

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

| <math>\text{Sign}\!</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 49.1} ~~ \~~operatorname~~{ER}(L_\text{B}) : \text{Extensional Representation of} ~ L_\text{B}\!</math>

+

<math>\text{Table 49.1} ~~ \mathrm{ER}(L_\text{B}) : \text{Extensional Representation of} ~ L_\text{B}\!</math>

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

| <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 49.2} ~~ \~~operatorname~~{ER}(\~~operatorname~~{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!</math>

+

<math>\text{Table 49.2} ~~ \mathrm{ER}(\mathrm{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}~\!</math>

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

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

|+ style="height:30px" |

−

<math>\text{Table 49.3} ~~ \~~operatorname~~{ER}(\~~operatorname~~{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!</math>

+

<math>\text{Table 49.3} ~~ \mathrm{ER}(\mathrm{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!</math>

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

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

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For the sake of maximum clarity and reusability of results, I begin by articulating the abstract skeleton of the paradigm structure, treating the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> as sundry aspects of a single, unitary, but still uninterpreted object. Then I return at various successive stages to differentiate and individualize the two interpreters, to arrange more functional flesh on the basis provided by their structural bones, and to illustrate how their bare forms can be arrayed in many different styles of qualitative detail.

−

In building connections between ERs and IRs of sign relations the discussion turns on two types of partially ordered sets, or ''posets''. Suppose that <math>L\!</math> is one of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> and let <math>\~~operatorname~~{ER}(L)\!</math> be an ER of <math>L.\!</math>

+

In building connections between ERs and IRs of sign relations the discussion turns on two types of partially ordered sets, or ''posets''. Suppose that <math>L\!</math> is one of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> and let <math>\mathrm{ER}(L)\!</math> be an ER of <math>L.\!</math>

In the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> both of their ERs are based on a common world set:

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To devise an IR of any relation <math>L\!</math> one needs to describe <math>L\!</math> in terms of properties of its ingredients. Broadly speaking, the ingredients of a relation include its elementary relations or <math>n\!</math>-tuples and the elementary components of these <math>n\!</math>-tuples that reside in the relational domains.

−

The poset <math>\~~operatorname~~{Pos}(W)\!</math> of interest here is the power set <math>\mathcal{P}(W) = \~~operatorname~~{Pow}(W).\!</math>

+

The poset <math>\mathrm{Pos}(W)\!</math> of interest here is the power set <math>\mathcal{P}(W) = \mathrm{Pow}(W).\!</math>

The elements of these posets are abstractly regarded as ''properties'' or ''propositions'' that apply to the elements of <math>W.\!</math> These properties and propositions are independently given entities. In other words, they are primitive elements in their own right, and cannot in general be defined in terms of points, but they exist in relation to these points, and their extensions can be represented as sets of points.

−

[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 discussion is now specialized to consider the IRs of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> their denotative projections as the digraphs <math>\mathrm{Den}(L_\text{A})\!</math> and <math>\mathrm{Den}(L_\text{B}),\!</math> and their connotative projections as the digraphs <math>\mathrm{Con}(L_\text{A})\!</math> and <math>\mathrm{Con}(L_\text{B}).\!</math> In doing this I take up two different strategies of representation:

−

# The first strategy is called the ''literal coding'', because it sticks to obvious features of each syntactic element to contrive its code, or the ''<math>\mathcal{O}(n)\!</math> coding'', because it uses a number on the order of <math>n\!</math> logical features to represent a domain of <math>n\!</math> elements.

+

# The first strategy is called the ''literal coding'', because it sticks to obvious features of each syntactic element to contrive its code, or the ''<math>{\mathcal{O}(n)}\!</math> coding'', because it uses a number on the order of <math>n\!</math> logical features to represent a domain of <math>n\!</math> elements.

# 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'''

Line 4,431: Line 4,441:

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"

|+ style="height:30px" |

−

<math>\text{Table 53.1} ~~ \text{Elements of} ~ \~~operatorname~~{ER}(W)\!</math>

+

<math>\text{Table 53.1} ~~ \text{Elements of} ~ \mathrm{ER}(W)\!</math>

|- style="background:#f0f0ff"

| <math>\text{Mnemonic Element}\!</math> <math>w \in W\!</math>

Line 4,485: Line 4,495:

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"

|+ style="height:30px" |

−

<math>\text{Table 53.2} ~~ \text{Features of} ~ \~~operatorname~~{LIR}(W)\!</math>

+

<math>\text{Table 53.2} ~~ \text{Features of} ~ \mathrm{LIR}(W)\!</math>

|- style="background:#f0f0ff"

|

Line 4,543: Line 4,553:

−

If the world of <math>\text{A}\!</math> and <math>\text{B},\!</math> the set <math>W = \{ \text{A}, \text{B}, {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \},\!</math> is viewed abstractly, as an arbitrary set of six atomic points, then there are exactly <math>2^6 = 64\!</math> ''abstract properties'' or ''potential attributes'' that might be applied to or recognized in these points. The elements of <math>W\!</math> that possess a given property form a subset of <math>W\!</math> called the ''extension'' of that property. Thus the extensions of abstract properties are exactly the subsets of <math>W.\!</math> The set of all subsets of <math>W\!</math> is called the ''power set'' of <math>W,\!</math> notated as <math>\~~operatorname~~{Pow}(W)\!</math> or <math>\mathcal{P}(W).\!</math> In order to make this way of talking about properties consistent with the previous definition of reality, it is necessary to say that one potential property is never realized, since no point has it, and its extension is the empty set <math>\varnothing = \{ \}.\!</math> All the ''natural'' properties of points that one observes in a concrete situation, properties whose extensions are known as ''natural kinds'', can be recognized among the ''abstract'', ''arbitrary'', or ''set-theoretic'' properties that are systematically generated in this way. Typically, however, many of these abstract properties will not be recognized as falling among the more natural kinds.

+

If the world of <math>\text{A}\!</math> and <math>\text{B},\!</math> the set <math>W = \{ \text{A}, \text{B}, {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \},\!</math> is viewed abstractly, as an arbitrary set of six atomic points, then there are exactly <math>2^6 = 64\!</math> ''abstract properties'' or ''potential attributes'' that might be applied to or recognized in these points. The elements of <math>W\!</math> that possess a given property form a subset of <math>W\!</math> called the ''extension'' of that property. Thus the extensions of abstract properties are exactly the subsets of <math>W.\!</math> The set of all subsets of <math>W\!</math> is called the ''power set'' of <math>W,\!</math> notated as <math>\mathrm{Pow}(W)\!</math> or <math>\mathcal{P}(W).\!</math> In order to make this way of talking about properties consistent with the previous definition of reality, it is necessary to say that one potential property is never realized, since no point has it, and its extension is the empty set <math>\varnothing = \{ \}.\!</math> All the ''natural'' properties of points that one observes in a concrete situation, properties whose extensions are known as ''natural kinds'', can be recognized among the ''abstract'', ''arbitrary'', or ''set-theoretic'' properties that are systematically generated in this way. Typically, however, many of these abstract properties will not be recognized as falling among the more natural kinds.

−

Tables 54.1, 54.2, and 54.3 show three different ways of representing the elements of the world set <math>W\!</math> as vectors in the coordinate space <math>\underline{W}\!</math> and as singular propositions in the universe of discourse <math>W^\~~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.

Line 4,848: Line 4,858:

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

|+ style="height:30px" |

−

<math>\text{Table 55.1} ~~ \~~operatorname~~{LIR}_1 (L_\text{A}) : \text{Literal Representation of} ~ L_\text{A}\!</math>

+

<math>\text{Table 55.1} ~~ \mathrm{LIR}_1 (L_\text{A}) : \text{Literal Representation of} ~ L_\text{A}\!</math>

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

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

Line 4,914: Line 4,924:

\\[4pt]

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

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|}

Line 4,921: Line 4,931:

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

|+ style="height:30px" |

−

<math>\text{Table 55.2} ~~ \~~operatorname~~{LIR}_1 (\~~operatorname~~{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!</math>

+

<math>\text{Table 55.2} ~~ \mathrm{LIR}_1 (\mathrm{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!</math>

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

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

Line 4,938: Line 4,948:

\\[4pt]

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

−

\end{matrix}</math>

+

\end{matrix}\!</math>

| valign="bottom" |

<math>\begin{matrix}

Line 4,974: Line 4,984:

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

|+ style="height:30px" |

−

<math>\text{Table 55.3} ~~ \~~operatorname~~{LIR}_1 (\~~operatorname~~{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!</math>

+

<math>\text{Table 55.3} ~~ \mathrm{LIR}_1 (\mathrm{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!</math>

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

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

Line 5,002: Line 5,012:

| valign="bottom" |

<math>\begin{matrix}

−

0_{\~~operatorname~~{d}W}

+

0_{\mathrm{d}W}

\\[4pt]

{\langle

−

\~~operatorname~~{d}\underline{\underline{\text{a}}}

+

\mathrm{d}\underline{\underline{\text{a}}}

~

−

\~~operatorname~~{d}\underline{\underline{\text{i}}}

+

\mathrm{d}\underline{\underline{\text{i}}}

−

\rangle}_{\~~operatorname~~{d}W}

+

\rangle}_{\mathrm{d}W}

\\[4pt]

{\langle

−

\~~operatorname~~{d}\underline{\underline{\text{a}}}

+

\mathrm{d}\underline{\underline{\text{a}}}

~

−

\~~operatorname~~{d}\underline{\underline{\text{i}}}

+

\mathrm{d}\underline{\underline{\text{i}}}

−

\rangle}_{\~~operatorname~~{d}W}

+

\rangle}_{\mathrm{d}W}

\\[4pt]

−

0_{\~~operatorname~~{d}W}

+

0_{\mathrm{d}W}

\end{matrix}</math>

|-

Line 5,038: Line 5,048:

\\[4pt]

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

−

\end{matrix}</math>

+

\end{matrix}\!</math>

| valign="bottom" |

<math>\begin{matrix}

−

0_{\~~operatorname~~{d}W}

+

0_{\mathrm{d}W}

\\[4pt]

{\langle

−

\~~operatorname~~{d}\underline{\underline{\text{b}}}

+

\mathrm{d}\underline{\underline{\text{b}}}

~

−

\~~operatorname~~{d}\underline{\underline{\text{u}}}

+

\mathrm{d}\underline{\underline{\text{u}}}

−

\rangle}_{\~~operatorname~~{d}W}

+

\rangle}_{\mathrm{d}W}

\\[4pt]

{\langle

−

\~~operatorname~~{d}\underline{\underline{\text{b}}}

+

\mathrm{d}\underline{\underline{\text{b}}}

~

−

\~~operatorname~~{d}\underline{\underline{\text{u}}}

+

\mathrm{d}\underline{\underline{\text{u}}}

−

\rangle}_{\~~operatorname~~{d}W}

+

\rangle}_{\mathrm{d}W}

\\[4pt]

−

0_{\~~operatorname~~{d}W}

+

0_{\mathrm{d}W}

\end{matrix}</math>

|}

Line 5,063: Line 5,073:

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

|+ style="height:30px" |

−

<math>\text{Table 56.1} ~~ \~~operatorname~~{LIR}_1 (L_\text{B}) : \text{Literal Representation of} ~ L_\text{B}\!</math>

+

<math>\text{Table 56.1} ~~ \mathrm{LIR}_1 (L_\text{B}) : \text{Literal Representation of} ~ L_\text{B}\!</math>

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

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

Line 5,129: Line 5,139:

\\[4pt]

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

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|}

Line 5,136: Line 5,146:

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

|+ style="height:30px" |

−

<math>\text{Table 56.2} ~~ \~~operatorname~~{LIR}_1 (\~~operatorname~~{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!</math>

+

<math>\text{Table 56.2} ~~ \mathrm{LIR}_1 (\mathrm{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!</math>

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

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

Line 5,189: Line 5,199:

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

|+ style="height:30px" |

−

<math>\text{Table 56.3} ~~ \~~operatorname~~{LIR}_1 (\~~operatorname~~{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!</math>

+

<math>\text{Table 56.3} ~~ \mathrm{LIR}_1 (\mathrm{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!</math>

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

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

Line 5,217: Line 5,227:

| valign="bottom" |

<math>\begin{matrix}

−

0_{\~~operatorname~~{d}W}

+

0_{\mathrm{d}W}

\\[4pt]

{\langle

−

\~~operatorname~~{d}\underline{\underline{\text{a}}}

+

\mathrm{d}\underline{\underline{\text{a}}}

~

−

\~~operatorname~~{d}\underline{\underline{\text{u}}}

+

\mathrm{d}\underline{\underline{\text{u}}}

−

\rangle}_{\~~operatorname~~{d}W}

+

\rangle}_{\mathrm{d}W}

\\[4pt]

{\langle

−

\~~operatorname~~{d}\underline{\underline{\text{a}}}

+

\mathrm{d}\underline{\underline{\text{a}}}

~

−

\~~operatorname~~{d}\underline{\underline{\text{u}}}

+

\mathrm{d}\underline{\underline{\text{u}}}

−

\rangle}_{\~~operatorname~~{d}W}

+

\rangle}_{\mathrm{d}W}

\\[4pt]

−

0_{\~~operatorname~~{d}W}

+

0_{\mathrm{d}W}

\end{matrix}</math>

|-

Line 5,253: Line 5,263:

\\[4pt]

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

−

\end{matrix}</math>

+

\end{matrix}\!</math>

| valign="bottom" |

<math>\begin{matrix}

−

0_{\~~operatorname~~{d}W}

+

0_{\mathrm{d}W}

\\[4pt]

{\langle

−

\~~operatorname~~{d}\underline{\underline{\text{b}}}

+

\mathrm{d}\underline{\underline{\text{b}}}

~

−

\~~operatorname~~{d}\underline{\underline{\text{i}}}

+

\mathrm{d}\underline{\underline{\text{i}}}

−

\rangle}_{\~~operatorname~~{d}W}

+

\rangle}_{\mathrm{d}W}

\\[4pt]

{\langle

−

\~~operatorname~~{d}\underline{\underline{\text{b}}}

+

\mathrm{d}\underline{\underline{\text{b}}}

~

−

\~~operatorname~~{d}\underline{\underline{\text{i}}}

+

\mathrm{d}\underline{\underline{\text{i}}}

−

\rangle}_{\~~operatorname~~{d}W}

+

\rangle}_{\mathrm{d}W}

\\[4pt]

−

0_{\~~operatorname~~{d}W}

+

0_{\mathrm{d}W}

\end{matrix}</math>

|}

Line 5,336: Line 5,346:

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

& \}

−

\end{array}</math>

+

\end{array}\!</math>

|}

Line 5,594: Line 5,604:

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

|+ style="height:30px" |

−

<math>\text{Table 58.1} ~~ \~~operatorname~~{LIR}_2 (L_\text{A}) : \text{Lateral Representation of} ~ L_\text{A}\!</math>

+

<math>\text{Table 58.1} ~~ \mathrm{LIR}_2 (L_\text{A}) : \text{Lateral Representation of} ~ L_\text{A}\!</math>

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

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

Line 5,635: Line 5,645:

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

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

−

\end{matrix}</math>

+

\end{matrix}\!</math>

| valign="bottom" |

<math>\begin{matrix}

Line 5,723: Line 5,733:

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

|+ style="height:30px" |

−

<math>\text{Table 58.2} ~~ \~~operatorname~~{LIR}_2 (\~~operatorname~~{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!</math>

+

<math>\text{Table 58.2} ~~ \mathrm{LIR}_2 (\mathrm{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!</math>

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

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

Line 5,777: Line 5,787:

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

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

−

\end{matrix}</math>

+

\end{matrix}\!</math>

| valign="bottom" |

<math>\begin{matrix}

Line 5,792: Line 5,802:

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

|+ style="height:30px" |

−

<math>\text{Table 58.3} ~~ \~~operatorname~~{LIR}_2 (\~~operatorname~~{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!</math>

+

<math>\text{Table 58.3} ~~ \mathrm{LIR}_2 (\mathrm{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!</math>

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

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

Line 5,819: Line 5,829:

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

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

−

\end{matrix}</math>

+

\end{matrix}\!</math>

| valign="bottom" |

<math>\begin{matrix}

Line 5,937: Line 5,947:

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

|+ style="height:30px" |

−

<math>\text{Table 59.1} ~~ \~~operatorname~~{LIR}_2 (L_\text{B}) : \text{Lateral Representation of} ~ L_\text{B}\!</math>

+

<math>\text{Table 59.1} ~~ \mathrm{LIR}_2 (L_\text{B}) : \text{Lateral Representation of} ~ L_\text{B}\!</math>

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

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

Line 6,066: Line 6,076:

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

|+ style="height:30px" |

−

<math>\text{Table 59.2} ~~ \~~operatorname~~{LIR}_2 (\~~operatorname~~{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!</math>

+

<math>\text{Table 59.2} ~~ \mathrm{LIR}_2 (\mathrm{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!</math>

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

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

Line 6,135: Line 6,145:

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

|+ style="height:30px" |

−

<math>\text{Table 59.3} ~~ \~~operatorname~~{LIR}_2 (\~~operatorname~~{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!</math>

+

<math>\text{Table 59.3} ~~ \mathrm{LIR}_2 (\mathrm{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!</math>

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

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

Line 6,206: Line 6,216:

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

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

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|-

| valign="bottom" |

Line 6,280: Line 6,290:

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

|+ style="height:30px" |

−

<math>\text{Table 60.1} ~~ \~~operatorname~~{LIR}_3 (L_\text{A}) : \text{Lateral Representation of} ~ L_\text{A}\!</math>

+

<math>\text{Table 60.1} ~~ \mathrm{LIR}_3 (L_\text{A}) : \text{Lateral Representation of} ~ L_\text{A}\!</math>

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

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

Line 6,353: Line 6,363:

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

|+ style="height:30px" |

−

<math>\text{Table 60.2} ~~ \~~operatorname~~{LIR}_3 (\~~operatorname~~{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!</math>

+

<math>\text{Table 60.2} ~~ \mathrm{LIR}_3 (\mathrm{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!</math>

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

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

Line 6,406: Line 6,416:

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

|+ style="height:30px" |

−

<math>\text{Table 60.3} ~~ \~~operatorname~~{LIR}_3 (\~~operatorname~~{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!</math>

+

<math>\text{Table 60.3} ~~ \mathrm{LIR}_3 (\mathrm{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!</math>

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

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

Line 6,434: Line 6,444:

| valign="bottom" |

<math>\begin{matrix}

−

0_{\~~operatorname~~{d}Y}

+

0_{\mathrm{d}Y}

\\[4pt]

{\langle

−

\~~operatorname~~{d}\underline{\underline{\text{a}}}

+

\mathrm{d}\underline{\underline{\text{a}}}

~

−

\~~operatorname~~{d}\underline{\underline{\text{i}}}

+

\mathrm{d}\underline{\underline{\text{i}}}

−

\rangle}_{\~~operatorname~~{d}Y}

+

\rangle}_{\mathrm{d}Y}

\\[4pt]

{\langle

−

\~~operatorname~~{d}\underline{\underline{\text{a}}}

+

\mathrm{d}\underline{\underline{\text{a}}}

~

−

\~~operatorname~~{d}\underline{\underline{\text{i}}}

+

\mathrm{d}\underline{\underline{\text{i}}}

−

\rangle}_{\~~operatorname~~{d}Y}

+

\rangle}_{\mathrm{d}Y}

\\[4pt]

−

0_{\~~operatorname~~{d}Y}

+

0_{\mathrm{d}Y}

\end{matrix}</math>

|-

Line 6,473: Line 6,483:

| valign="bottom" |

<math>\begin{matrix}

−

0_{\~~operatorname~~{d}Y}

+

0_{\mathrm{d}Y}

\\[4pt]

{\langle

−

\~~operatorname~~{d}\underline{\underline{\text{b}}}

+

\mathrm{d}\underline{\underline{\text{b}}}

~

−

\~~operatorname~~{d}\underline{\underline{\text{u}}}

+

\mathrm{d}\underline{\underline{\text{u}}}

−

\rangle}_{\~~operatorname~~{d}Y}

+

\rangle}_{\mathrm{d}Y}

\\[4pt]

{\langle

−

\~~operatorname~~{d}\underline{\underline{\text{b}}}

+

\mathrm{d}\underline{\underline{\text{b}}}

~

−

\~~operatorname~~{d}\underline{\underline{\text{u}}}

+

\mathrm{d}\underline{\underline{\text{u}}}

−

\rangle}_{\~~operatorname~~{d}Y}

+

\rangle}_{\mathrm{d}Y}

\\[4pt]

−

0_{\~~operatorname~~{d}Y}

+

0_{\mathrm{d}Y}

\end{matrix}</math>

|}

Line 6,495: Line 6,505:

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

|+ style="height:30px" |

−

<math>\text{Table 61.1} ~~ \~~operatorname~~{LIR}_3 (L_\text{B}) : \text{Lateral Representation of} ~ L_\text{B}\!</math>

+

<math>\text{Table 61.1} ~~ \mathrm{LIR}_3 (L_\text{B}) : \text{Lateral Representation of} ~ L_\text{B}\!</math>

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

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

Line 6,568: Line 6,578:

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

|+ style="height:30px" |

−

<math>\text{Table 61.2} ~~ \~~operatorname~~{LIR}_3 (\~~operatorname~~{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!</math>

+

<math>\text{Table 61.2} ~~ \mathrm{LIR}_3 (\mathrm{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!</math>

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

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

Line 6,621: Line 6,631:

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

|+ style="height:30px" |

−

<math>\text{Table 61.3} ~~ \~~operatorname~~{LIR}_3 (\~~operatorname~~{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!</math>

+

<math>\text{Table 61.3} ~~ \mathrm{LIR}_3 (\mathrm{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!</math>

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

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

Line 6,649: Line 6,659:

| valign="bottom" |

<math>\begin{matrix}

−

0_{\~~operatorname~~{d}Y}

+

0_{\mathrm{d}Y}

\\[4pt]

{\langle

−

\~~operatorname~~{d}\underline{\underline{\text{a}}}

+

\mathrm{d}\underline{\underline{\text{a}}}

~

−

\~~operatorname~~{d}\underline{\underline{\text{u}}}

+

\mathrm{d}\underline{\underline{\text{u}}}

−

\rangle}_{\~~operatorname~~{d}Y}

+

\rangle}_{\mathrm{d}Y}

\\[4pt]

{\langle

−

\~~operatorname~~{d}\underline{\underline{\text{a}}}

+

\mathrm{d}\underline{\underline{\text{a}}}

~

−

\~~operatorname~~{d}\underline{\underline{\text{u}}}

+

\mathrm{d}\underline{\underline{\text{u}}}

−

\rangle}_{\~~operatorname~~{d}Y}

+

\rangle}_{\mathrm{d}Y}

\\[4pt]

−

0_{\~~operatorname~~{d}Y}

+

0_{\mathrm{d}Y}

\end{matrix}</math>

|-

Line 6,688: Line 6,698:

| valign="bottom" |

<math>\begin{matrix}

−

0_{\~~operatorname~~{d}Y}

+

0_{\mathrm{d}Y}

\\[4pt]

{\langle

−

\~~operatorname~~{d}\underline{\underline{\text{b}}}

+

\mathrm{d}\underline{\underline{\text{b}}}

~

−

\~~operatorname~~{d}\underline{\underline{\text{i}}}

+

\mathrm{d}\underline{\underline{\text{i}}}

−

\rangle}_{\~~operatorname~~{d}Y}

+

\rangle}_{\mathrm{d}Y}

\\[4pt]

{\langle

−

\~~operatorname~~{d}\underline{\underline{\text{b}}}

+

\mathrm{d}\underline{\underline{\text{b}}}

~

−

\~~operatorname~~{d}\underline{\underline{\text{i}}}

+

\mathrm{d}\underline{\underline{\text{i}}}

−

\rangle}_{\~~operatorname~~{d}Y}

+

\rangle}_{\mathrm{d}Y}

\\[4pt]

−

0_{\~~operatorname~~{d}Y}

+

0_{\mathrm{d}Y}

\end{matrix}</math>

|}

Line 6,989: Line 6,999:

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

|+ style="height:30px" |

−

<math>\text{Table 65.1} ~~ \~~operatorname~~{AIR}_1 (L_\text{A}) : \text{Analytic Representation of} ~ L_\text{A}\!</math>

+

<math>\text{Table 65.1} ~~ \mathrm{AIR}_1 (L_\text{A}) : \text{Analytic Representation of} ~ L_\text{A}\!</math>

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

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

Line 7,062: Line 7,072:

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

|+ style="height:30px" |

−

<math>\text{Table 65.2} ~~ \~~operatorname~~{AIR}_1 (\~~operatorname~~{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!</math>

+

<math>\text{Table 65.2} ~~ \mathrm{AIR}_1 (\mathrm{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!</math>

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

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

Line 7,111: Line 7,121:

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

|+ style="height:30px" |

−

<math>\text{Table 65.3} ~~ \~~operatorname~~{AIR}_1 (\~~operatorname~~{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!</math>

+

<math>\text{Table 65.3} ~~ \mathrm{AIR}_1 (\mathrm{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!</math>

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

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

Line 7,184: Line 7,194:

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

|+ style="height:30px" |

−

<math>\text{Table 66.1} ~~ \~~operatorname~~{AIR}_1 (L_\text{B}) : \text{Analytic Representation of} ~ L_\text{B}\!</math>

+

<math>\text{Table 66.1} ~~ \mathrm{AIR}_1 (L_\text{B}) : \text{Analytic Representation of} ~ L_\text{B}\!</math>

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

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

Line 7,257: Line 7,267:

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

|+ style="height:30px" |

−

<math>\text{Table 66.2} ~~ \~~operatorname~~{AIR}_1 (\~~operatorname~~{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!</math>

+

<math>\text{Table 66.2} ~~ \mathrm{AIR}_1 (\mathrm{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!</math>

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

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

Line 7,306: Line 7,316:

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

|+ style="height:30px" |

−

<math>\text{Table 66.3} ~~ \~~operatorname~~{AIR}_1 (\~~operatorname~~{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!</math>

+

<math>\text{Table 66.3} ~~ \mathrm{AIR}_1 (\mathrm{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!</math>

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

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

Line 7,379: Line 7,389:

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

|+ style="height:30px" |

−

<math>\text{Table 67.1} ~~ \~~operatorname~~{AIR}_2 (L_\text{A}) : \text{Analytic Representation of} ~ L_\text{A}\!</math>

+

<math>\text{Table 67.1} ~~ \mathrm{AIR}_2 (L_\text{A}) : \text{Analytic Representation of} ~ L_\text{A}\!</math>

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

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

Line 7,452: Line 7,462:

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

|+ style="height:30px" |

−

<math>\text{Table 67.2} ~~ \~~operatorname~~{AIR}_2 (\~~operatorname~~{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!</math>

+

<math>\text{Table 67.2} ~~ \mathrm{AIR}_2 (\mathrm{Den}(L_\text{A})) : \text{Denotative Component of} ~ L_\text{A}\!</math>

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

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

Line 7,501: Line 7,511:

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

|+ style="height:30px" |

−

<math>\text{Table 67.3} ~~ \~~operatorname~~{AIR}_2 (\~~operatorname~~{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!</math>

+

<math>\text{Table 67.3} ~~ \mathrm{AIR}_2 (\mathrm{Con}(L_\text{A})) : \text{Connotative Component of} ~ L_\text{A}\!</math>

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

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

Line 7,529: Line 7,539:

| valign="bottom" |

<math>\begin{matrix}

−

{\langle\~~operatorname~~{d}!\rangle}_{\~~operatorname~~{d}Y}

+

{\langle\mathrm{d}!\rangle}_{\mathrm{d}Y}

\\[4pt]

−

{\langle\~~operatorname~~{d}\text{n}\rangle}_{\~~operatorname~~{d}Y}

+

{\langle\mathrm{d}\text{n}\rangle}_{\mathrm{d}Y}

\\[4pt]

−

{\langle\~~operatorname~~{d}\text{n}\rangle}_{\~~operatorname~~{d}Y}

+

{\langle\mathrm{d}\text{n}\rangle}_{\mathrm{d}Y}

\\[4pt]

−

{\langle\~~operatorname~~{d}!\rangle}_{\~~operatorname~~{d}Y}

+

{\langle\mathrm{d}!\rangle}_{\mathrm{d}Y}

\end{matrix}</math>

|-

Line 7,560: Line 7,570:

| valign="bottom" |

<math>\begin{matrix}

−

{\langle\~~operatorname~~{d}!\rangle}_{\~~operatorname~~{d}Y}

+

{\langle\mathrm{d}!\rangle}_{\mathrm{d}Y}

\\[4pt]

−

{\langle\~~operatorname~~{d}\text{n}\rangle}_{\~~operatorname~~{d}Y}

+

{\langle\mathrm{d}\text{n}\rangle}_{\mathrm{d}Y}

\\[4pt]

−

{\langle\~~operatorname~~{d}\text{n}\rangle}_{\~~operatorname~~{d}Y}

+

{\langle\mathrm{d}\text{n}\rangle}_{\mathrm{d}Y}

\\[4pt]

−

{\langle\~~operatorname~~{d}!\rangle}_{\~~operatorname~~{d}Y}

+

{\langle\mathrm{d}!\rangle}_{\mathrm{d}Y}

\end{matrix}</math>

|}

Line 7,574: Line 7,584:

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

|+ style="height:30px" |

−

<math>\text{Table 68.1} ~~ \~~operatorname~~{AIR}_2 (L_\text{B}) : \text{Analytic Representation of} ~ L_\text{B}\!</math>

+

<math>\text{Table 68.1} ~~ \mathrm{AIR}_2 (L_\text{B}) : \text{Analytic Representation of} ~ L_\text{B}\!</math>

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

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

Line 7,647: Line 7,657:

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

|+ style="height:30px" |

−

<math>\text{Table 68.2} ~~ \~~operatorname~~{AIR}_2 (\~~operatorname~~{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!</math>

+

<math>\text{Table 68.2} ~~ \mathrm{AIR}_2 (\mathrm{Den}(L_\text{B})) : \text{Denotative Component of} ~ L_\text{B}\!</math>

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

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

Line 7,696: Line 7,706:

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

|+ style="height:30px" |

−

<math>\text{Table 68.3} ~~ \~~operatorname~~{AIR}_2 (\~~operatorname~~{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!</math>

+

<math>\text{Table 68.3} ~~ \mathrm{AIR}_2 (\mathrm{Con}(L_\text{B})) : \text{Connotative Component of} ~ L_\text{B}\!</math>

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

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

Line 7,724: Line 7,734:

| valign="bottom" |

<math>\begin{matrix}

−

{\langle\~~operatorname~~{d}!\rangle}_{\~~operatorname~~{d}Y}

+

{\langle\mathrm{d}!\rangle}_{\mathrm{d}Y}

\\[4pt]

−

{\langle\~~operatorname~~{d}\text{n}\rangle}_{\~~operatorname~~{d}Y}

+

{\langle\mathrm{d}\text{n}\rangle}_{\mathrm{d}Y}

\\[4pt]

−

{\langle\~~operatorname~~{d}\text{n}\rangle}_{\~~operatorname~~{d}Y}

+

{\langle\mathrm{d}\text{n}\rangle}_{\mathrm{d}Y}

\\[4pt]

−

{\langle\~~operatorname~~{d}!\rangle}_{\~~operatorname~~{d}Y}

+

{\langle\mathrm{d}!\rangle}_{\mathrm{d}Y}

\end{matrix}</math>

|-

Line 7,755: Line 7,765:

| valign="bottom" |

<math>\begin{matrix}

−

{\langle\~~operatorname~~{d}!\rangle}_{\~~operatorname~~{d}Y}

+

{\langle\mathrm{d}!\rangle}_{\mathrm{d}Y}

\\[4pt]

−

{\langle\~~operatorname~~{d}\text{n}\rangle}_{\~~operatorname~~{d}Y}

+

{\langle\mathrm{d}\text{n}\rangle}_{\mathrm{d}Y}

\\[4pt]

−

{\langle\~~operatorname~~{d}\text{n}\rangle}_{\~~operatorname~~{d}Y}

+

{\langle\mathrm{d}\text{n}\rangle}_{\mathrm{d}Y}

\\[4pt]

−

{\langle\~~operatorname~~{d}!\rangle}_{\~~operatorname~~{d}Y}

+

{\langle\mathrm{d}!\rangle}_{\mathrm{d}Y}

\end{matrix}</math>

|}

Line 7,799: Line 7,809:

| valign="bottom" |

<math>\begin{matrix}

−

~x~ ~\~~operatorname~~{at}~ t

+

~x~ ~\mathrm{at}~ t

\\[4pt]

−

~x~ ~\~~operatorname~~{at}~ t

+

~x~ ~\mathrm{at}~ t

\\[4pt]

−

(x) ~\~~operatorname~~{at}~ t

+

(x) ~\mathrm{at}~ t

\\[4pt]

−

(x) ~\~~operatorname~~{at}~ t

+

(x) ~\mathrm{at}~ t

\end{matrix}</math>

| valign="bottom" |

<math>\begin{matrix}

−

~\~~operatorname~~{d}x~ ~\~~operatorname~~{at}~ t

+

~\mathrm{d}x~ ~\mathrm{at}~ t

\\[4pt]

−

(\~~operatorname~~{d}x) ~\~~operatorname~~{at}~ t

+

(\mathrm{d}x) ~\mathrm{at}~ t

\\[4pt]

−

~\~~operatorname~~{d}x~ ~\~~operatorname~~{at}~ t

+

~\mathrm{d}x~ ~\mathrm{at}~ t

\\[4pt]

−

(\~~operatorname~~{d}x) ~\~~operatorname~~{at}~ t

+

(\mathrm{d}x) ~\mathrm{at}~ t

\end{matrix}</math>

| valign="bottom" |

<math>\begin{matrix}

−

(x) ~\~~operatorname~~{at}~ t'

+

(x) ~\mathrm{at}~ t'

\\[4pt]

−

~x~ ~\~~operatorname~~{at}~ t'

+

~x~ ~\mathrm{at}~ t'

\\[4pt]

−

~x~ ~\~~operatorname~~{at}~ t'

+

~x~ ~\mathrm{at}~ t'

\\[4pt]

−

(x) ~\~~operatorname~~{at}~ t'

+

(x) ~\mathrm{at}~ t'

\end{matrix}</math>

|}

Line 7,833: Line 7,843:

It might be thought that a notion of real time <math>(t \in \mathbb{R})\!</math> is needed at this point to fund the account of sequential processes. From a logical point of view, however, I think it will be found that it is precisely out of such data that the notion of time has to be constructed.

−

The symbol <math>{}^{\backprime\backprime} \ominus\!\!- {}^{\prime\prime},</math> read ''thus'', ''then'', or ''yields'', can be used to mark sequential inferences, allowing for expressions like <math>x \land \~~operatorname~~{d}x \ominus\!\!-~ (x).\!</math> In each case, a suitable context of temporal moments <math>(t, t')\!</math> is understood to underlie the inference.

+

The symbol <math>{}^{\backprime\backprime} \ominus\!\!- {}^{\prime\prime},</math> read ''thus'', ''then'', or ''yields'', can be used to mark sequential inferences, allowing for expressions like <math>x \land \mathrm{d}x \ominus\!\!-~ (x).\!</math> In each case, a suitable context of temporal moments <math>(t, t')\!</math> is understood to underlie the inference.

−

A ''sequential inference constraint'' is a logical condition that applies to a temporal system, providing information about the kinds of sequential inference that apply to the system in a hopefully large number of situations. Typically, a sequential inference constraint is formulated in intensional terms and expressed by means of a collection of sequential inference rules or schemata that tell what sequential inferences apply to the system in particular situations. Since it has the status of logical theory about an empirical system, a sequential inference constraint is subject to being reformulated in terms of its set-theoretic extension, and it can be established as existing in the customary sort of dual relationship with this extension. Logically, it determines, and, empirically, it is determined by the corresponding set of ''sequential inference triples'', the <math>(x, y, z)\!</math> such that <math>x \land y \ominus\!\!-~ z.\!</math> The set-theoretic extension of a sequential inference constraint is thus a triadic relation, generically notated as <math>\ominus,\!</math> where <math>\ominus \subseteq X \times \~~operatorname~~{d}X \times X\!</math> is defined as follows.

+

A ''sequential inference constraint'' is a logical condition that applies to a temporal system, providing information about the kinds of sequential inference that apply to the system in a hopefully large number of situations. Typically, a sequential inference constraint is formulated in intensional terms and expressed by means of a collection of sequential inference rules or schemata that tell what sequential inferences apply to the system in particular situations. Since it has the status of logical theory about an empirical system, a sequential inference constraint is subject to being reformulated in terms of its set-theoretic extension, and it can be established as existing in the customary sort of dual relationship with this extension. Logically, it determines, and, empirically, it is determined by the corresponding set of ''sequential inference triples'', the <math>(x, y, z)\!</math> such that <math>x \land y \ominus\!\!-~ z.\!</math> The set-theoretic extension of a sequential inference constraint is thus a triadic relation, generically notated as <math>\ominus,\!</math> where <math>\ominus \subseteq X \times \mathrm{d}X \times X\!</math> is defined as follows.

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

−

| <math>\ominus ~=~ \{ (x, y, z) \in X \times \~~operatorname~~{d}X \times X : x \land y \ominus\!\!-~ z \}.\!</math>

+

| <math>\ominus ~=~ \{ (x, y, z) \in X \times \mathrm{d}X \times X : x \land y \ominus\!\!-~ z \}.\!</math>

|}

−

Using the appropriate isomorphisms, or recognizing how, in terms of the information given, that each of several descriptions is tantamount to the same object, the triadic relation <math>\ominus \subseteq X \times \~~operatorname~~{d}X \times X\!</math> constituted by a sequential inference constraint can be interpreted as a proposition <math>\ominus : X \times \~~operatorname~~{d}X \times X \to \mathbb{B}\!</math> about sequential inference triples, and thus as a map <math>\ominus : \~~operatorname~~{d}X \to (X \times X \to \mathbb{B})\!</math> from the space <math>\~~operatorname~~{d}X\!</math> of differential states to the space of propositions about transitions in <math>X.\!</math>

+

Using the appropriate isomorphisms, or recognizing how, in terms of the information given, that each of several descriptions is tantamount to the same object, the triadic relation <math>\ominus \subseteq X \times \mathrm{d}X \times X\!</math> constituted by a sequential inference constraint can be interpreted as a proposition <math>\ominus : X \times \mathrm{d}X \times X \to \mathbb{B}\!</math> about sequential inference triples, and thus as a map <math>\ominus : \mathrm{d}X \to (X \times X \to \mathbb{B})\!</math> from the space <math>\mathrm{d}X\!</math> of differential states to the space of propositions about transitions in <math>X.\!</math>

−

'''Question.''' Group Actions? <math>r : \~~operatorname~~{d}X \to (X \to X)\!</math>

+

'''Question.''' Group Actions? <math>r : \mathrm{d}X \to (X \to X)\!</math>

Line 7,851: Line 7,861:

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

|+ style="height:30px" |

−

<math>\text{Table 70.1} ~~ \text{Group Representation} ~ \~~operatorname~~{Rep}^\text{A} (V_4)\!</math>

+

<math>\text{Table 70.1} ~~ \text{Group Representation} ~ \mathrm{Rep}^\text{A} (V_4)\!</math>

|- style="background:#f0f0ff"

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

Line 7,871: Line 7,881:

| valign="bottom" |

<math>\begin{matrix}

−

(\~~operatorname~~{d}\underline{\underline{\text{a}}})

+

(\mathrm{d}\underline{\underline{\text{a}}})

−

(\~~operatorname~~{d}\underline{\underline{\text{b}}})

+

(\mathrm{d}\underline{\underline{\text{b}}})

−

(\~~operatorname~~{d}\underline{\underline{\text{i}}})

+

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

−

(\~~operatorname~~{d}\underline{\underline{\text{u}}})

+

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

\\[4pt]

−

~\~~operatorname~~{d}\underline{\underline{\text{a}}}~

+

~\mathrm{d}\underline{\underline{\text{a}}}~

−

(\~~operatorname~~{d}\underline{\underline{\text{b}}})

+

(\mathrm{d}\underline{\underline{\text{b}}})

−

~\~~operatorname~~{d}\underline{\underline{\text{i}}}~

+

~\mathrm{d}\underline{\underline{\text{i}}}~

−

(\~~operatorname~~{d}\underline{\underline{\text{u}}})

+

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

\\[4pt]

−

(\~~operatorname~~{d}\underline{\underline{\text{a}}})

+

(\mathrm{d}\underline{\underline{\text{a}}})

−

~\~~operatorname~~{d}\underline{\underline{\text{b}}}~

+

~\mathrm{d}\underline{\underline{\text{b}}}~

−

(\~~operatorname~~{d}\underline{\underline{\text{i}}})

+

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

−

~\~~operatorname~~{d}\underline{\underline{\text{u}}}~

+

~\mathrm{d}\underline{\underline{\text{u}}}~

\\[4pt]

−

~\~~operatorname~~{d}\underline{\underline{\text{a}}}~

+

~\mathrm{d}\underline{\underline{\text{a}}}~

−

~\~~operatorname~~{d}\underline{\underline{\text{b}}}~

+

~\mathrm{d}\underline{\underline{\text{b}}}~

−

~\~~operatorname~~{d}\underline{\underline{\text{i}}}~

+

~\mathrm{d}\underline{\underline{\text{i}}}~

−

~\~~operatorname~~{d}\underline{\underline{\text{u}}}~

+

~\mathrm{d}\underline{\underline{\text{u}}}~

\end{matrix}</math>

| valign="bottom" |

<math>\begin{matrix}

−

\langle \~~operatorname~~{d}! \rangle

+

\langle \mathrm{d}! \rangle

\\[4pt]

\langle

−

\~~operatorname~~{d}\underline{\underline{\text{a}}} ~

+

\mathrm{d}\underline{\underline{\text{a}}} ~

−

\~~operatorname~~{d}\underline{\underline{\text{i}}}

+

\mathrm{d}\underline{\underline{\text{i}}}

\rangle

\\[4pt]

\langle

−

\~~operatorname~~{d}\underline{\underline{\text{b}}} ~

+

\mathrm{d}\underline{\underline{\text{b}}} ~

−

\~~operatorname~~{d}\underline{\underline{\text{u}}}

+

\mathrm{d}\underline{\underline{\text{u}}}

\rangle

\\[4pt]

−

\langle \~~operatorname~~{d}* \rangle

+

\langle \mathrm{d}* \rangle

\end{matrix}</math>

| valign="bottom" |

<math>\begin{matrix}

−

\~~operatorname~~{d}!

+

\mathrm{d}!

\\[4pt]

−

\~~operatorname~~{d}\underline{\underline{\text{a}}} \cdot

+

\mathrm{d}\underline{\underline{\text{a}}} \cdot

−

\~~operatorname~~{d}\underline{\underline{\text{i}}} ~ !

+

\mathrm{d}\underline{\underline{\text{i}}} ~ !

\\[4pt]

−

\~~operatorname~~{d}\underline{\underline{\text{b}}} \cdot

+

\mathrm{d}\underline{\underline{\text{b}}} \cdot

−

\~~operatorname~~{d}\underline{\underline{\text{u}}} ~ !

+

\mathrm{d}\underline{\underline{\text{u}}} ~ !

\\[4pt]

−

\~~operatorname~~{d}*

+

\mathrm{d}*

\end{matrix}</math>

| valign="bottom" |

Line 7,923: Line 7,933:

1

\\[4pt]

−

\~~operatorname~~{d}_{\text{ai}}

+

\mathrm{d}_{\text{ai}}

\\[4pt]

−

\~~operatorname~~{d}_{\text{bu}}

+

\mathrm{d}_{\text{bu}}

\\[4pt]

−

\~~operatorname~~{d}_{\text{ai}} * \~~operatorname~~{d}_{\text{bu}}

+

\mathrm{d}_{\text{ai}} * \mathrm{d}_{\text{bu}}

\end{matrix}</math>

|}

Line 7,935: Line 7,945:

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

|+ style="height:30px" |

−

<math>\text{Table 70.2} ~~ \text{Group Representation} ~ \~~operatorname~~{Rep}^\text{B} (V_4)\!</math>

+

<math>\text{Table 70.2} ~~ \text{Group Representation} ~ \mathrm{Rep}^\text{B} (V_4)\!</math>

|- style="background:#f0f0ff"

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

Line 7,955: Line 7,965:

| valign="bottom" |

<math>\begin{matrix}

−

(\~~operatorname~~{d}\underline{\underline{\text{a}}})

+

(\mathrm{d}\underline{\underline{\text{a}}})

−

(\~~operatorname~~{d}\underline{\underline{\text{b}}})

+

(\mathrm{d}\underline{\underline{\text{b}}})

−

(\~~operatorname~~{d}\underline{\underline{\text{i}}})

+

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

−

(\~~operatorname~~{d}\underline{\underline{\text{u}}})

+

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

\\[4pt]

−

~\~~operatorname~~{d}\underline{\underline{\text{a}}}~

+

~\mathrm{d}\underline{\underline{\text{a}}}~

−

(\~~operatorname~~{d}\underline{\underline{\text{b}}})

+

(\mathrm{d}\underline{\underline{\text{b}}})

−

(\~~operatorname~~{d}\underline{\underline{\text{i}}})

+

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

−

~\~~operatorname~~{d}\underline{\underline{\text{u}}}~

+

~\mathrm{d}\underline{\underline{\text{u}}}~

\\[4pt]

−

(\~~operatorname~~{d}\underline{\underline{\text{a}}})

+

(\mathrm{d}\underline{\underline{\text{a}}})

−

~\~~operatorname~~{d}\underline{\underline{\text{b}}}~

+

~\mathrm{d}\underline{\underline{\text{b}}}~

−

~\~~operatorname~~{d}\underline{\underline{\text{i}}}~

+

~\mathrm{d}\underline{\underline{\text{i}}}~

−

(\~~operatorname~~{d}\underline{\underline{\text{u}}})

+

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

\\[4pt]

−

~\~~operatorname~~{d}\underline{\underline{\text{a}}}~

+

~\mathrm{d}\underline{\underline{\text{a}}}~

−

~\~~operatorname~~{d}\underline{\underline{\text{b}}}~

+

~\mathrm{d}\underline{\underline{\text{b}}}~

−

~\~~operatorname~~{d}\underline{\underline{\text{i}}}~

+

~\mathrm{d}\underline{\underline{\text{i}}}~

−

~\~~operatorname~~{d}\underline{\underline{\text{u}}}~

+

~\mathrm{d}\underline{\underline{\text{u}}}~

\end{matrix}</math>

| valign="bottom" |

<math>\begin{matrix}

−

\langle \~~operatorname~~{d}! \rangle

+

\langle \mathrm{d}! \rangle

\\[4pt]

\langle

−

\~~operatorname~~{d}\underline{\underline{\text{a}}} ~

+

\mathrm{d}\underline{\underline{\text{a}}} ~

−

\~~operatorname~~{d}\underline{\underline{\text{u}}}

+

\mathrm{d}\underline{\underline{\text{u}}}

\rangle

\\[4pt]

\langle

−

\~~operatorname~~{d}\underline{\underline{\text{b}}} ~

+

\mathrm{d}\underline{\underline{\text{b}}} ~

−

\~~operatorname~~{d}\underline{\underline{\text{i}}}

+

\mathrm{d}\underline{\underline{\text{i}}}

\rangle

\\[4pt]

−

\langle \~~operatorname~~{d}* \rangle

+

\langle \mathrm{d}* \rangle

\end{matrix}</math>

| valign="bottom" |

<math>\begin{matrix}

−

\~~operatorname~~{d}!

+

\mathrm{d}!

\\[4pt]

−

\~~operatorname~~{d}\underline{\underline{\text{a}}} \cdot

+

\mathrm{d}\underline{\underline{\text{a}}} \cdot

−

\~~operatorname~~{d}\underline{\underline{\text{u}}} ~ !

+

\mathrm{d}\underline{\underline{\text{u}}} ~ !

\\[4pt]

−

\~~operatorname~~{d}\underline{\underline{\text{b}}} \cdot

+

\mathrm{d}\underline{\underline{\text{b}}} \cdot

−

\~~operatorname~~{d}\underline{\underline{\text{i}}} ~ !

+

\mathrm{d}\underline{\underline{\text{i}}} ~ !

\\[4pt]

−

\~~operatorname~~{d}*

+

\mathrm{d}*

\end{matrix}</math>

| valign="bottom" |

Line 8,007: Line 8,017:

1

\\[4pt]

−

\~~operatorname~~{d}_{\text{au}}

+

\mathrm{d}_{\text{au}}

\\[4pt]

−

\~~operatorname~~{d}_{\text{bi}}

+

\mathrm{d}_{\text{bi}}

\\[4pt]

−

\~~operatorname~~{d}_{\text{au}} * \~~operatorname~~{d}_{\text{bi}}

+

\mathrm{d}_{\text{au}} * \mathrm{d}_{\text{bi}}

\end{matrix}</math>

|}

Line 8,019: Line 8,029:

{| 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} ~ \mathrm{Rep}^\text{C} (V_4)}\!</math>

|- style="background:#f0f0ff"

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

Line 8,039: Line 8,049:

| valign="bottom" |

<math>\begin{matrix}

−

(\~~operatorname~~{d}\text{m})

+

(\mathrm{d}\text{m})

−

(\~~operatorname~~{d}\text{n})

+

(\mathrm{d}\text{n})

\\[4pt]

−

~\~~operatorname~~{d}\text{m}~

+

~\mathrm{d}\text{m}~

−

(\~~operatorname~~{d}\text{n})

+

(\mathrm{d}\text{n})

\\[4pt]

−

(\~~operatorname~~{d}\text{m})

+

(\mathrm{d}\text{m})

−

~\~~operatorname~~{d}\text{n}~

+

~\mathrm{d}\text{n}~

\\[4pt]

−

~\~~operatorname~~{d}\text{m}~

+

~\mathrm{d}\text{m}~

−

~\~~operatorname~~{d}\text{n}~

+

~\mathrm{d}\text{n}~

\end{matrix}</math>

| valign="bottom" |

<math>\begin{matrix}

−

\langle\~~operatorname~~{d}!\rangle

+

\langle\mathrm{d}!\rangle

\\[4pt]

−

\langle\~~operatorname~~{d}\text{m}\rangle

+

\langle\mathrm{d}\text{m}\rangle

\\[4pt]

−

\langle\~~operatorname~~{d}\text{n}\rangle

+

\langle\mathrm{d}\text{n}\rangle

\\[4pt]

−

\langle\~~operatorname~~{d}*\rangle

+

\langle\mathrm{d}*\rangle

\end{matrix}</math>

| valign="bottom" |

<math>\begin{matrix}

−

\~~operatorname~~{d}!

+

\mathrm{d}!

\\[4pt]

−

\~~operatorname~~{d}\text{m}!

+

\mathrm{d}\text{m}!

\\[4pt]

−

\~~operatorname~~{d}\text{n}!

+

\mathrm{d}\text{n}!

\\[4pt]

−

\~~operatorname~~{d}*

+

\mathrm{d}*

\end{matrix}</math>

| valign="bottom" |

Line 8,075: Line 8,085:

1

\\[4pt]

−

\~~operatorname~~{d}_{\text{m}}

+

\mathrm{d}_{\text{m}}

\\[4pt]

−

\~~operatorname~~{d}_{\text{n}}

+

\mathrm{d}_{\text{n}}

\\[4pt]

−

\~~operatorname~~{d}_{\text{m}} * \~~operatorname~~{d}_{\text{n}}

+

\mathrm{d}_{\text{m}} * \mathrm{d}_{\text{n}}

\end{matrix}</math>

|}

Line 8,107: Line 8,117:

| valign="bottom" |

<math>\begin{matrix}

−

(\~~operatorname~~{d}\text{m})

+

(\mathrm{d}\text{m})

−

(\~~operatorname~~{d}\text{n})

+

(\mathrm{d}\text{n})

\\[4pt]

−

~\~~operatorname~~{d}\text{m}~

+

~\mathrm{d}\text{m}~

−

(\~~operatorname~~{d}\text{n})

+

(\mathrm{d}\text{n})

\\[4pt]

−

(\~~operatorname~~{d}\text{m})

+

(\mathrm{d}\text{m})

−

~\~~operatorname~~{d}\text{n}~

+

~\mathrm{d}\text{n}~

\\[4pt]

−

~\~~operatorname~~{d}\text{m}~

+

~\mathrm{d}\text{m}~

−

~\~~operatorname~~{d}\text{n}~

+

~\mathrm{d}\text{n}~

\end{matrix}</math>

| valign="bottom" |

<math>\begin{matrix}

−

\langle\~~operatorname~~{d}!\rangle

+

\langle\mathrm{d}!\rangle

\\[4pt]

−

\langle\~~operatorname~~{d}\text{m}\rangle

+

\langle\mathrm{d}\text{m}\rangle

\\[4pt]

−

\langle\~~operatorname~~{d}\text{n}\rangle

+

\langle\mathrm{d}\text{n}\rangle

\\[4pt]

−

\langle\~~operatorname~~{d}*\rangle

+

\langle\mathrm{d}*\rangle

\end{matrix}</math>

| valign="bottom" |

<math>\begin{matrix}

−

\~~operatorname~~{d}!

+

\mathrm{d}!

\\[4pt]

−

\~~operatorname~~{d}\text{m}!

+

\mathrm{d}\text{m}!

\\[4pt]

−

\~~operatorname~~{d}\text{n}!

+

\mathrm{d}\text{n}!

\\[4pt]

−

\~~operatorname~~{d}*

+

\mathrm{d}*

\end{matrix}</math>

| valign="bottom" |

Line 8,143: Line 8,153:

1

\\[4pt]

−

\~~operatorname~~{d}_{\text{m}}

+

\mathrm{d}_{\text{m}}

\\[4pt]

−

\~~operatorname~~{d}_{\text{n}}

+

\mathrm{d}_{\text{n}}

\\[4pt]

−

\~~operatorname~~{d}_{\text{m}} * \~~operatorname~~{d}_{\text{n}}

+

\mathrm{d}_{\text{m}} * \mathrm{d}_{\text{n}}

\end{matrix}</math>

|}

Line 8,158: Line 8,168:

|- 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{m}\!</math>

−

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

+

| <math>(\mathrm{d}\text{m})\!</math>

|

<math>\begin{matrix}

−

(\~~operatorname~~{d}\text{m})(\~~operatorname~~{d}\text{n})

+

(\mathrm{d}\text{m})(\mathrm{d}\text{n})

\\[4pt]

−

(\~~operatorname~~{d}\text{m})~\~~operatorname~~{d}\text{n}~

+

(\mathrm{d}\text{m})~\mathrm{d}\text{n}~

\end{matrix}</math>

|

Line 8,174: Line 8,184:

1

\\[4pt]

−

\~~operatorname~~{d}_\text{n}

+

\mathrm{d}_\text{n}

\end{matrix}</math>

|-

−

| <math>G_\text{m} * \~~operatorname~~{d}_\text{m}\!</math>

+

| <math>G_\text{m} * \mathrm{d}_\text{m}\!</math>

−

| <math>\~~operatorname~~{d}\text{m}\!</math>

+

| <math>\mathrm{d}\text{m}\!</math>

|

<math>\begin{matrix}

−

~\~~operatorname~~{d}\text{m}~(\~~operatorname~~{d}\text{n})

+

~\mathrm{d}\text{m}~(\mathrm{d}\text{n})

\\[4pt]

−

~\~~operatorname~~{d}\text{m}~~\~~operatorname~~{d}\text{n}~

+

~\mathrm{d}\text{m}~~\mathrm{d}\text{n}~

\end{matrix}</math>

|

<math>\begin{matrix}

−

\~~operatorname~~{d}_\text{m}

+

\mathrm{d}_\text{m}

\\[4pt]

−

\~~operatorname~~{d}_\text{n} * \~~operatorname~~{d}_\text{m}

+

\mathrm{d}_\text{n} * \mathrm{d}_\text{m}

\end{matrix}</math>

|}

Line 8,200: Line 8,210:

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

|

<math>\begin{matrix}

−

(\~~operatorname~~{d}\text{m})(\~~operatorname~~{d}\text{n})

+

(\mathrm{d}\text{m})(\mathrm{d}\text{n})

\\[4pt]

−

~\~~operatorname~~{d}\text{m}~(\~~operatorname~~{d}\text{n})

+

~\mathrm{d}\text{m}~(\mathrm{d}\text{n})

\end{matrix}</math>

|

Line 8,216: Line 8,226:

1

\\[4pt]

−

\~~operatorname~~{d}_\text{m}

+

\mathrm{d}_\text{m}

\end{matrix}</math>

|-

−

| <math>G_\text{n} * \~~operatorname~~{d}_\text{n}\!</math>

+

| <math>G_\text{n} * \mathrm{d}_\text{n}\!</math>

−

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

+

| <math>\mathrm{d}\text{n}\!</math>

|

<math>\begin{matrix}

−

(\~~operatorname~~{d}\text{m})~\~~operatorname~~{d}\text{n}~

+

(\mathrm{d}\text{m})~\mathrm{d}\text{n}~

\\[4pt]

−

~\~~operatorname~~{d}\text{m}~~\~~operatorname~~{d}\text{n}~

+

~\mathrm{d}\text{m}~~\mathrm{d}\text{n}~

\end{matrix}</math>

|

<math>\begin{matrix}

−

\~~operatorname~~{d}_\text{n}

+

\mathrm{d}_\text{n}

\\[4pt]

−

\~~operatorname~~{d}_\text{m} * \~~operatorname~~{d}_\text{n}

+

\mathrm{d}_\text{m} * \mathrm{d}_\text{n}

\end{matrix}</math>

|}

Line 8,259: Line 8,269:

|}

−

In other words, <math>P\!\!\And\!\!Q</math> is the intersection of the ''inverse projections'' <math>P' = \~~operatorname~~{Pr}_{12}^{-1}(P)\!</math> and <math>Q' = \~~operatorname~~{Pr}_{23}^{-1}(Q),\!</math> which are defined as follows:

+

In other words, <math>P\!\!\And\!\!Q</math> is the intersection of the ''inverse projections'' <math>P' = \mathrm{Pr}_{12}^{-1}(P)\!</math> and <math>Q' = \mathrm{Pr}_{23}^{-1}(Q),\!</math> which are defined as follows:

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

|

<math>\begin{matrix}

−

\~~operatorname~~{Pr}_{12}^{-1}(P) & = & P \times Z & = & \{ (x, y, z) \in X \times Y \times Z : (x, y) \in P \}.

+

\mathrm{Pr}_{12}^{-1}(P) & = & P \times Z & = & \{ (x, y, z) \in X \times Y \times Z : (x, y) \in P \}.

\\[4pt]

−

\~~operatorname~~{Pr}_{23}^{-1}(Q) & = & X \times Q & = & \{ (x, y, z) \in X \times Y \times Z : (y, z) \in Q \}.

+

\mathrm{Pr}_{23}^{-1}(Q) & = & X \times Q & = & \{ (x, y, z) \in X \times Y \times Z : (y, z) \in Q \}.

\end{matrix}</math>

|}

Line 8,276: Line 8,286:

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.

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

−

| <math>P \circ Q ~=~ \~~operatorname~~{Pr}_{13} (P\!\!\And\!\!Q) ~=~ \{ (x, z) \in X \times Z : (x, y, z) \in P\!\!\And\!\!Q \}.</math>

+

| <math>P \circ Q ~=~ \mathrm{Pr}_{13} (P\!\!\And\!\!Q) ~=~ \{ (x, z) \in X \times Z : (x, y, z) \in P\!\!\And\!\!Q \}.</math>

|}

Line 8,398: Line 8,408:

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%"

Line 8,406: Line 8,416:

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.

Line 8,436: Line 8,446:

& \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%"

Line 8,495: Line 8,505:

& \iff &

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

−

\end{array}</math>

+

\end{array}\!</math>

|}

Line 8,529: Line 8,539:

For a <math>k\!</math>-place relation <math>L \subseteq X_1 \times \ldots \times X_k,\!</math> we have the following usages.

−

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

+

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

−

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

+

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

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

|

<math>\begin{array}{lll}

−

\~~operatorname~~{Quo}_j (L)

+

\mathrm{Quo}_j (L)

& = &

\text{the largest}~ Q \subseteq X_j ~\text{such that}~ ~L_{Q \,\text{at}\, j}~ ~\text{is}~ (> 1)\text{-regular at}~ j,

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# The arbitrarily designated domains <math>X_1 = X\!</math> and <math>X_2 = Y\!</math> that form the widest sets admitted to the dyadic relation are referred to as the ''domain'' or ''source'' and the ''codomain'' or ''target'', respectively, of the relation in question.

−

# The terms ''quota'' and ''range'' are reserved for those uniquely defined sets whose elements actually appear as the first and second members, respectively, of the ordered pairs in that relation. Thus, for a dyadic relation <math>L \subseteq X \times Y,\!</math> we identify <math>\~~operatorname~~{Quo} (L) = \~~operatorname~~{Quo}_1 (L) \subseteq X\!</math> with what is usually called the ''domain of definition'' of <math>L\!</math> and we identify <math>\~~operatorname~~{Ran} (L) = \~~operatorname~~{Quo}_2 (L) \subseteq Y\!</math> with the usual ''range'' of <math>L.\!</math>

+

# The terms ''quota'' and ''range'' are reserved for those uniquely defined sets whose elements actually appear as the first and second members, respectively, of the ordered pairs in that relation. Thus, for a dyadic relation <math>L \subseteq X \times Y,\!</math> we identify <math>\mathrm{Quo} (L) = \mathrm{Quo}_1 (L) \subseteq X\!</math> with what is usually called the ''domain of definition'' of <math>L\!</math> and we identify <math>\mathrm{Ran} (L) = \mathrm{Quo}_2 (L) \subseteq Y\!</math> with the usual ''range'' of <math>L.\!</math>

−

A ''partial equivalence relation'' (PER) on a set <math>X\!</math> is a relation <math>L \subseteq X \times X\!</math> that is an equivalence relation on its domain of definition <math>\~~operatorname~~{Quo} (L) \subseteq X.\!</math> In this situation, <math>[x]_L\!</math> is empty for each <math>x\!</math> in <math>X\!</math> that is not in <math>\~~operatorname~~{Quo} (L).\!</math> Another way of reaching the same concept is to call a PER a dyadic relation that is symmetric and transitive, but not necessarily reflexive. Like the “self-identical elements” of old that epitomized the very definition of self-consistent existence in classical logic, the property of being a self-related or self-equivalent element in the purview of a PER on <math>X\!</math> singles out the members of <math>\~~operatorname~~{Quo} (L)\!</math> as those for which a properly meaningful existence can be contemplated.

+

A ''partial equivalence relation'' (PER) on a set <math>X\!</math> is a relation <math>L \subseteq X \times X\!</math> that is an equivalence relation on its domain of definition <math>\mathrm{Quo} (L) \subseteq X.\!</math> In this situation, <math>[x]_L\!</math> is empty for each <math>x\!</math> in <math>X\!</math> that is not in <math>\mathrm{Quo} (L).\!</math> Another way of reaching the same concept is to call a PER a dyadic relation that is symmetric and transitive, but not necessarily reflexive. Like the “self-identical elements” of old that epitomized the very definition of self-consistent existence in classical logic, the property of being a self-related or self-equivalent element in the purview of a PER on <math>X\!</math> singles out the members of <math>\mathrm{Quo} (L)\!</math> as those for which a properly meaningful existence can be contemplated.

A ''moderate equivalence relation'' (MER) on the ''modus'' <math>M \subseteq X\!</math> is a relation on <math>X\!</math> whose restriction to <math>M\!</math> is an equivalence relation on <math>M.\!</math> In symbols, <math>L \subseteq X \times X\!</math> such that <math>L|M \subseteq M \times M\!</math> is an equivalence relation. Notice that the subset of restriction, or modus <math>M,\!</math> is a part of the definition, so the same relation <math>L\!</math> on <math>X\!</math> could be a MER or not depending on the choice of <math>M.\!</math> In spite of how it sounds, a moderate equivalence relation can have more ordered pairs in it than the ordinary sort of equivalence relation on the same set.

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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>

Line 8,762: Line 8,772:

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

−

| <math>\~~operatorname~~{Proj}^{(2)} L ~=~ (\~~operatorname~~{proj}_{12} L, ~ \~~operatorname~~{proj}_{13} L, ~ \~~operatorname~~{proj}_{23} L).\!</math>

+

| <math>\mathrm{Proj}^{(2)} L ~=~ (\mathrm{proj}_{12} L, ~ \mathrm{proj}_{13} L, ~ \mathrm{proj}_{23} L).\!</math>

|}

−

If <math>L\!</math> is visualized as a solid body in the 3-dimensional space <math>X \times Y \times Z,\!</math> then <math>\~~operatorname~~{Proj}^{(2)} L\!</math> can be visualized as the arrangement or ordered collection of shadows it throws on the <math>XY, ~ XZ, ~ YZ\!</math> planes, respectively.

+

If <math>L\!</math> is visualized as a solid body in the 3-dimensional space <math>X \times Y \times Z,\!</math> then <math>\mathrm{Proj}^{(2)} L\!</math> can be visualized as the arrangement or ordered collection of shadows it throws on the <math>XY, ~ XZ, ~ YZ\!</math> planes, respectively.

−

Two more set-theoretic constructions are worth introducing at this point, in particular for describing the source and target domains of the projection operator <math>\~~operatorname~~{Proj}^{(2)}.\!</math>

+

Two more set-theoretic constructions are worth introducing at this point, in particular for describing the source and target domains of the projection operator <math>\mathrm{Proj}^{(2)}.\!</math>

−

The set of subsets of a set <math>S\!</math> is called the ''power set'' of <math>S.\!</math> This object is denoted by either of the forms <math>\~~operatorname~~{Pow}(S)\!</math> or <math>2^S\!</math> and defined as follows:

+

The set of subsets of a set <math>S\!</math> is called the ''power set'' of <math>S.\!</math> This object is denoted by either of the forms <math>\mathrm{Pow}(S)\!</math> or <math>2^S\!</math> and defined as follows:

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

−

| <math>\~~operatorname~~{Pow}(S) ~=~ 2^S ~=~ \{ T : T \subseteq S \}.\!</math>

+

| <math>\mathrm{Pow}(S) ~=~ 2^S ~=~ \{ T : T \subseteq S \}.\!</math>

|}

−

The power set notation can be used to provide an alternative description of relations. In the case where <math>S\!</math> is a cartesian product, say <math>S = X_1 \times \ldots \times X_n,\!</math> then each <math>n\!</math>-place relation <math>L\!</math> described as a subset of <math>S,\!</math> say <math>L \subseteq X_1 \times \ldots \times X_n,\!</math> is equally well described as an element of <math>\~~operatorname~~{Pow}(S),\!</math> in other words, as <math>L \in \~~operatorname~~{Pow}(X_1 \times \ldots \times X_n).\!</math>

+

The power set notation can be used to provide an alternative description of relations. In the case where <math>S\!</math> is a cartesian product, say <math>{S = X_1 \times \ldots \times X_n},\!</math> then each <math>n\!</math>-place relation <math>L\!</math> described as a subset of <math>S,\!</math> say <math>L \subseteq X_1 \times \ldots \times X_n,\!</math> is equally well described as an element of <math>\mathrm{Pow}(S),\!</math> in other words, as <math>L \in \mathrm{Pow}(X_1 \times \ldots \times X_n).\!</math>

−

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

+

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

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

−

| <math>\~~operatorname~~{Explo}(X, Y, Z ~|~ 2) ~=~ \~~operatorname~~{Pow}(X \times Y) \times \~~operatorname~~{Pow}(X \times Z) \times \~~operatorname~~{Pow}(Y \times Z).\!</math>

+

| <math>\mathrm{Explo}(X, Y, Z ~|~ 2) ~=~ \mathrm{Pow}(X \times Y) \times \mathrm{Pow}(X \times Z) \times \mathrm{Pow}(Y \times Z).\!</math>

|}

This domain is defined well enough to serve the immediate purposes of this section, but later it will become necessary to examine its construction more closely.

−

By means of these constructions the operation that forms <math>\~~operatorname~~{Proj}^{(2)} L\!</math> for each triadic relation <math>L \subseteq X \times Y \times Z\!</math> can be expressed as a function:

+

By means of these constructions the operation that forms <math>\mathrm{Proj}^{(2)} L\!</math> for each triadic relation <math>L \subseteq X \times Y \times Z\!</math> can be expressed as a function:

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

−

| <math>\~~operatorname~~{Proj}^{(2)} : \~~operatorname~~{Pow}(X \times Y \times Z) \to \~~operatorname~~{Explo}(X, Y, Z ~|~ 2).\!</math>

+

| <math>\mathrm{Proj}^{(2)} : \mathrm{Pow}(X \times Y \times Z) \to \mathrm{Explo}(X, Y, Z ~|~ 2).\!</math>

|}

−

In this setting the issue of whether triadic relations are ''reducible to'' or ''reconstructible from'' their dyadic projections, both in general and in specific cases, can be identified with the question of whether <math>\~~operatorname~~{Proj}^{(2)}\!</math> is injective. The mapping <math>\~~operatorname~~{Proj}^{(2)}\!</math> is said to ''preserve information'' about the triadic relations <math>L \in \~~operatorname~~{Pow}(X \times Y \times Z)\!</math> if and only if it is injective, otherwise one says that some loss of information has occurred in taking the projections. Given a specific instance of a triadic relation <math>L \in \~~operatorname~~{Pow}(X \times Y \times Z),\!</math> it can be said that <math>L\!</math> is ''determined by'' (''reducible to'' or ''reconstructible from'') its dyadic projections if and only if <math>(\~~operatorname~~{Proj}^{(2)})^{-1}(\~~operatorname~~{Proj}^{(2)}L)\!</math> is the singleton set <math>\{ L \}.\!</math> Otherwise, there exists an <math>L'\!</math> such that <math>\~~operatorname~~{Proj}^{(2)}L = \~~operatorname~~{Proj}^{(2)}L',\!</math> and in this case <math>L\!</math> is said to be ''irreducibly triadic'' or ''genuinely triadic''. Notice that irreducible or genuine triadic relations, when they exist, naturally occur in sets of two or more, the whole collection of them being equated or confounded with one another under <math>\~~operatorname~~{Proj}^{(2)}.\!</math>

+

In this setting the issue of whether triadic relations are ''reducible to'' or ''reconstructible from'' their dyadic projections, both in general and in specific cases, can be identified with the question of whether <math>\mathrm{Proj}^{(2)}\!</math> is injective. The mapping <math>\mathrm{Proj}^{(2)}\!</math> is said to ''preserve information'' about the triadic relations <math>L \in \mathrm{Pow}(X \times Y \times Z)\!</math> if and only if it is injective, otherwise one says that some loss of information has occurred in taking the projections. Given a specific instance of a triadic relation <math>L \in \mathrm{Pow}(X \times Y \times Z),\!</math> it can be said that <math>L\!</math> is ''determined by'' (''reducible to'' or ''reconstructible from'') its dyadic projections if and only if <math>(\mathrm{Proj}^{(2)})^{-1}(\mathrm{Proj}^{(2)}L)\!</math> is the singleton set <math>\{ L \}.\!</math> Otherwise, there exists an <math>L'\!</math> such that <math>\mathrm{Proj}^{(2)}L = \mathrm{Proj}^{(2)}L',\!</math> and in this case <math>L\!</math> is said to be ''irreducibly triadic'' or ''genuinely triadic''. Notice that irreducible or genuine triadic relations, when they exist, naturally occur in sets of two or more, the whole collection of them being equated or confounded with one another under <math>\mathrm{Proj}^{(2)}.\!</math>

−

The next series of Tables illustrates the operation of <math>\~~operatorname~~{Proj}^{(2)}\!</math> by means of its actions on the sign relations <math>L_\text{A}\!</math> and <math>L_\text{B}.\!</math> For ease of reference, Tables 72.1 and 73.1 repeat the contents of Tables 1 and 2, respectively, while the dyadic relations comprising <math>\~~operatorname~~{Proj}^{(2)}L_\text{A}\!</math> and <math>\~~operatorname~~{Proj}^{(2)}L_\text{B}\!</math> are shown in Tables 72.2 to 72.4 and Tables 73.2 to 73.4, respectively.

+

The next series of Tables illustrates the operation of <math>\mathrm{Proj}^{(2)}\!</math> by means of its actions on the sign relations <math>L_\text{A}\!</math> and <math>L_\text{B}.\!</math> For ease of reference, Tables 72.1 and 73.1 repeat the contents of Tables 1 and 2, respectively, while the dyadic relations comprising <math>\mathrm{Proj}^{(2)}L_\text{A}\!</math> and <math>\mathrm{Proj}^{(2)}L_\text{B}\!</math> are shown in Tables 72.2 to 72.4 and Tables 73.2 to 73.4, respectively.

{| 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>

Line 9,216: Line 9,226:

−

A comparison of the corresponding projections in <math>\~~operatorname~~{Proj}^{(2)} L(\text{A})\!</math> and <math>\~~operatorname~~{Proj}^{(2)} L(\text{B})\!</math> shows that the distinction between the triadic relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> is preserved by <math>\~~operatorname~~{Proj}^{(2)},\!</math> and this circumstance allows one to say that this much information, at least, can be derived from the dyadic projections. However, to say that a triadic relation <math>L \in \~~operatorname~~{Pow} (O \times S \times I)\!</math> is reducible in this sense it is necessary to show that no distinct <math>L' \in \~~operatorname~~{Pow} (O \times S \times I)\!</math> exists such that <math>\~~operatorname~~{Proj}^{(2)} L = \~~operatorname~~{Proj}^{(2)} L',\!</math> and this can take a rather more exhaustive or comprehensive investigation of the space <math>\~~operatorname~~{Pow} (O \times S \times I).\!</math>

+

A comparison of the corresponding projections in <math>\mathrm{Proj}^{(2)} L(\text{A})\!</math> and <math>\mathrm{Proj}^{(2)} L(\text{B})\!</math> shows that the distinction between the triadic relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> is preserved by <math>\mathrm{Proj}^{(2)},\!</math> and this circumstance allows one to say that this much information, at least, can be derived from the dyadic projections. However, to say that a triadic relation <math>L \in \mathrm{Pow} (O \times S \times I)\!</math> is reducible in this sense it is necessary to show that no distinct <math>L' \in \mathrm{Pow} (O \times S \times I)\!</math> exists such that <math>\mathrm{Proj}^{(2)} L = \mathrm{Proj}^{(2)} L',\!</math> and this can take a rather more exhaustive or comprehensive investigation of the space <math>\mathrm{Pow} (O \times S \times I).\!</math>

−

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

+

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

−

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

+

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

===6.36. Irreducibly Triadic Relations===

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In order to show what an irreducibly triadic relation looks like, this Section presents a pair of triadic relations that have the same dyadic projections, and thus cannot be distinguished from each other on this basis alone. As it happens, these examples of triadic relations can be discussed independently of sign relational concerns, but structures of their general ilk are frequently found arising in signal-theoretic applications, and they are undoubtedly closely associated with problems of reliable coding and communication.

−

Tables 74.1 and 75.1 show a pair of irreducibly triadic relations <math>L_0\!</math> and <math>L_1,\!</math> respectively. Tables 74.2 to 74.4 and Tables 75.2 to 75.4 show the dyadic relations comprising <math>\~~operatorname~~{Proj}^{(2)} L_0\!</math> and <math>\~~operatorname~~{Proj}^{(2)} L_1,\!</math> respectively.

+

Tables 74.1 and 75.1 show a pair of irreducibly triadic relations <math>L_0\!</math> and <math>L_1,\!</math> respectively. Tables 74.2 to 74.4 and Tables 75.2 to 75.4 show the dyadic relations comprising <math>\mathrm{Proj}^{(2)} L_0\!</math> and <math>\mathrm{Proj}^{(2)} L_1,\!</math> respectively.

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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:

+

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

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

Line 9,386: Line 9,396:

[The following piece occurs in § 6.35.]

−

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

+

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

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

−

| <math>\~~operatorname~~{Explo}(X, Y, Z ~|~ 2) ~=~ \~~operatorname~~{Pow}(X \times Y) \times \~~operatorname~~{Pow}(X \times Z) \times \~~operatorname~~{Pow}(Y \times Z)\!</math>

+

| <math>\mathrm{Explo}(X, Y, Z ~|~ 2) ~=~ \mathrm{Pow}(X \times Y) \times \mathrm{Pow}(X \times Z) \times \mathrm{Pow}(Y \times Z)\!</math>

|}

Line 9,450: Line 9,460:

|}

−

Table 76 displays the results of indexing every sign of the <math>\text{A}\!</math> and <math>\text{B}\!</math> example with a superscript indicating its source or ''exponent'', namely, the interpreter who actively communicates or transmits the sign. The operation of attribution produces two new sign relations, but it turns out that both sign relations have the same form and content, so a single Table will do. The new sign relation generated by this operation will be denoted <math>\~~operatorname~~{At} (\text{A}, \text{B})\!</math> and called the ''attributed sign relation'' for the <math>\text{A}\!</math> and <math>\text{B}\!</math> example.

+

Table 76 displays the results of indexing every sign of the <math>\text{A}\!</math> and <math>\text{B}\!</math> example with a superscript indicating its source or ''exponent'', namely, the interpreter who actively communicates or transmits the sign. The operation of attribution produces two new sign relations, but it turns out that both sign relations have the same form and content, so a single Table will do. The new sign relation generated by this operation will be denoted <math>\mathrm{At} (\text{A}, \text{B})\!</math> and called the ''attributed sign relation'' for the <math>\text{A}\!</math> and <math>\text{B}\!</math> example.

Line 9,825: Line 9,835:

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.]

Line 10,210: Line 10,220:

Working from these principles alone, there are numerous ways that a plausible dynamics can be invented for a given sign relation. I will concentrate on two principal forms of dynamic realization, or two ways of interpreting and augmenting sign relations as sign processes.

−

One form of realization lets each element of the object domain <math>O\!</math> correspond to the observed presence of an object in the environment of the systematic agent. In this interpretation, the object <math>x\!</math> acts as an input datum that causes the system <math>Y\!</math> to shift from whatever sign state it happens to occupy at a given moment to a random sign state in <math>[x]_Y.\!</math> Expressed in a cognitive vein, <math>{}^{\backprime\backprime} Y ~\~~operatorname~~{notes}~ x {}^{\prime\prime}.</math>

+

One form of realization lets each element of the object domain <math>O\!</math> correspond to the observed presence of an object in the environment of the systematic agent. In this interpretation, the object <math>x\!</math> acts as an input datum that causes the system <math>Y\!</math> to shift from whatever sign state it happens to occupy at a given moment to a random sign state in <math>[x]_Y.\!</math> Expressed in a cognitive vein, <math>{}^{\backprime\backprime} Y ~\mathrm{notes}~ x {}^{\prime\prime}.</math>

Another form of realization lets each element of the object domain <math>O\!</math> correspond to the autonomous intention of the systematic agent to denote an object, achieve an objective, or broadly speaking to accomplish any other purpose with respect to an object in its domain. In this interpretation, the object <math>x\!</math> is a control parameter that brings the system <math>Y\!</math> into line with realizing a target set <math>[x]_Y.\!</math>

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Treated in accord with these interpretations, the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> constitute partially degenerate cases of dynamic processes, in which the transitions are totally non-deterministic up to semantic equivalence classes but still manage to preserve those classes. Whether construed as present observation or projective speculation, the most significant feature to note about a sign process is how the contemplation of an object or objective leads the system from a less determined to a more determined condition.

−

On reflection, one observes that these processes are not completely trivial since they preserve the structure of their semantic partitions. In fact, each sign process preserves the entire topology — the family of sets closed under finite intersections and arbitrary unions — that is generated by its semantic equivalence classes. These topologies, <math>\~~operatorname~~{Top}(\text{A})\!</math> and <math>\~~operatorname~~{Top}(\text{B}),\!</math> can be viewed as partially ordered sets, <math>\~~operatorname~~{Poset}(\text{A})\!</math> and <math>\~~operatorname~~{Poset}(\text{B}),\!</math> by taking the inclusion ordering <math>(\subseteq)\!</math> as <math>(\le).\!</math> For each of the interpreters <math>\text{A}\!</math> and <math>\text{B},\!</math> as things stand in their respective orderings <math>\~~operatorname~~{Poset}(\text{A})\!</math> and <math>\~~operatorname~~{Poset}(\text{B}),\!</math> the semantic equivalence classes of <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \text{B} {}^{\prime\prime}\!</math> are situated as intermediate elements that are incomparable to each other.

+

On reflection, one observes that these processes are not completely trivial since they preserve the structure of their semantic partitions. In fact, each sign process preserves the entire topology — the family of sets closed under finite intersections and arbitrary unions — that is generated by its semantic equivalence classes. These topologies, <math>\mathrm{Top}(\text{A})\!</math> and <math>\mathrm{Top}(\text{B}),\!</math> can be viewed as partially ordered sets, <math>\mathrm{Poset}(\text{A})\!</math> and <math>\mathrm{Poset}(\text{B}),\!</math> by taking the inclusion ordering <math>(\subseteq)\!</math> as <math>(\le).\!</math> For each of the interpreters <math>\text{A}\!</math> and <math>\text{B},\!</math> as things stand in their respective orderings <math>\mathrm{Poset}(\text{A})\!</math> and <math>\mathrm{Poset}(\text{B}),\!</math> the semantic equivalence classes of <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \text{B} {}^{\prime\prime}\!</math> are situated as intermediate elements that are incomparable to each other.

{| align="center" cellspacing="6" width="90%"

|

<math>\begin{array}{lllll}

−

\~~operatorname~~{Top}(\text{A})

+

\mathrm{Top}(\text{A})

& = &

−

\~~operatorname~~{Poset}(\text{A})

+

\mathrm{Poset}(\text{A})

& = &

\{

Line 10,482: Line 10,492:

\}.

\\[6pt]

−

\~~operatorname~~{Top}(\text{B})

+

\mathrm{Top}(\text{B})

& = &

−

\~~operatorname~~{Poset}(\text{B})

+

\mathrm{Poset}(\text{B})

& = &

\{ \varnothing,

Line 10,512: Line 10,522:

{| align="center" cellspacing="6" width="90%"

|

−

<math>Y ~\text{at}~ x ~=~ \~~operatorname~~{At}[x]_Y ~=~ [x]_Y \cup \{ \text{arcs into}~ [x]_Y \}.</math>

+

<math>Y ~\text{at}~ x ~=~ \mathrm{At}[x]_Y ~=~ [x]_Y \cup \{ \text{arcs into}~ [x]_Y \}.</math>

|}

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This section takes up the topic of reflective extensions in a more systematic fashion, starting from the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> once again and keeping its focus within their vicinity, but exploring the space of nearby extensions in greater detail.

−

Tables 80 and 81 show one way that the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> can be extended in a reflective sense through the use of quotational devices, yielding the ''first order reflective extensions'', <math>\~~operatorname~~{Ref}^1 (\text{A})\!</math> and <math>\~~operatorname~~{Ref}^1 (\text{B}).\!</math>

+

Tables 80 and 81 show one way that the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> can be extended in a reflective sense through the use of quotational devices, yielding the ''first order reflective extensions'', <math>\mathrm{Ref}^1 (\text{A})\!</math> and <math>\mathrm{Ref}^1 (\text{B}).\!</math>

{| 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} ~ \mathrm{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} ~ \mathrm{Ref}^1 (\text{B})}\!</math>

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

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

Line 10,731: Line 10,743:

−

The common ''world'' <math>W\!</math> of the reflective extensions <math>\~~operatorname~~{Ref}^1 (\text{A})\!</math> and <math>\~~operatorname~~{Ref}^1 (\text{B})\!</math> is the totality of objects and signs they contain, namely, the following set of 10 elements.

+

The common ''world'' <math>W\!</math> of the reflective extensions <math>\mathrm{Ref}^1 (\text{A})\!</math> and <math>\mathrm{Ref}^1 (\text{B})\!</math> is the totality of objects and signs they contain, namely, the following set of 10 elements.

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

Line 10,739: Line 10,751:

Raised angle brackets or ''supercilia'' <math>({}^{\langle} \ldots {}^{\rangle})\!</math> are here being used on a par with ordinary quotation marks <math>({}^{\backprime\backprime} \ldots {}^{\prime\prime})\!</math> to construct a new sign whose object is precisely the sign they enclose.

−

Regarded as sign relations in their own right, <math>\~~operatorname~~{Ref}^1 (\text{A})\!</math> and <math>\~~operatorname~~{Ref}^1 (\text{B})\!</math> are formed on the following relational domains.

+

Regarded as sign relations in their own right, <math>\mathrm{Ref}^1 (\text{A})\!</math> and <math>\mathrm{Ref}^1 (\text{B})\!</math> are formed on the following relational domains.

{| align="center" cellspacing="6" width="90%"

Line 10,793: Line 10,805:

|

<math>\begin{array}{lllll}

−

\~~operatorname~~{Den}^1 (L)

+

\mathrm{Den}^1 (L)

& = &

−

(\~~operatorname~~{Ref}^1 (L))_{SO}

+

(\mathrm{Ref}^1 (L))_{SO}

& = &

−

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

+

\mathrm{proj}_{OS} (\mathrm{Ref}^1 (L))

\\[6pt]

−

\~~operatorname~~{Con}^1 (L)

+

\mathrm{Con}^1 (L)

& = &

−

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

+

(\mathrm{Ref}^1 (L))_{SI}

& = &

−

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

+

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

−

\end{array}</math>

+

\end{array}\!</math>

|}

The dyadic components of sign relations can be given graph-theoretic representations, namely, as ''digraphs'' (directed graphs), that provide concise pictures of their structural and potential dynamic properties. By way of terminology, a directed edge <math>(x, y)\!</math> is called an ''arc'' from point <math>x\!</math> to point <math>y,\!</math> and a self-loop <math>(x, x)\!</math> is called a ''sling'' at <math>x.\!</math>

−

The denotative components <math>\~~operatorname~~{Den}^1 (L_\text{A})\!</math> and <math>\~~operatorname~~{Den}^1 (L_\text{B})\!</math> can be viewed as digraphs on the 10 points of the world set <math>W.\!</math> The arcs of these digraphs are given as follows.

+

The denotative components <math>\mathrm{Den}^1 (L_\text{A})\!</math> and <math>\mathrm{Den}^1 (L_\text{B})\!</math> can be viewed as digraphs on the 10 points of the world set <math>W.\!</math> The arcs of these digraphs are given as follows.

<ol>

−

<li><math>\~~operatorname~~{Den}^1 (L_\text{A})\!</math> has an arc from each point of <math>[\text{A}]_\text{A} = \{ {}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{i}{}^{\rangle} \}\!</math> to <math>\text{A}\!</math> and from each point of <math>[\text{B}]_\text{A} = \{ {}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{u} {}^{\rangle} \}\!</math> to <math>\text{B}.\!</math></li>

+

<li><math>\mathrm{Den}^1 (L_\text{A})\!</math> has an arc from each point of <math>[\text{A}]_\text{A} = \{ {}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{i}{}^{\rangle} \}\!</math> to <math>\text{A}\!</math> and from each point of <math>[\text{B}]_\text{A} = \{ {}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{u} {}^{\rangle} \}\!</math> to <math>\text{B}.\!</math></li>

−

<li><math>\~~operatorname~~{Den}^1 (L_\text{B})\!</math> has an arc from each point of <math>[\text{A}]_\text{B} = \{ {}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{u}{}^{\rangle} \}\!</math> to <math>\text{A}\!</math> and from each point of <math>[\text{B}]_\text{B} = \{ {}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{i} {}^{\rangle} \}\!</math> to <math>\text{B}.\!</math></li>

+

<li><math>\mathrm{Den}^1 (L_\text{B})\!</math> has an arc from each point of <math>[\text{A}]_\text{B} = \{ {}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{u}{}^{\rangle} \}\!</math> to <math>\text{A}\!</math> and from each point of <math>[\text{B}]_\text{B} = \{ {}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{i} {}^{\rangle} \}\!</math> to <math>\text{B}.\!</math></li>

−

<li>In the parts added by reflective extension <math>\~~operatorname~~{Den}^1 (L_\text{A})\!</math> and <math>\~~operatorname~~{Den}^1 (L_\text{B})\!</math> both have arcs from <math>{}^{\langle} s {}^{\rangle}\!</math> to <math>s,\!</math> for each <math>s \in S^{(1)}.\!</math></li>

+

<li>In the parts added by reflective extension <math>\mathrm{Den}^1 (L_\text{A})\!</math> and <math>\mathrm{Den}^1 (L_\text{B})\!</math> both have arcs from <math>{}^{\langle} s {}^{\rangle}\!</math> to <math>s,\!</math> for each <math>s \in S^{(1)}.\!</math></li>

</ol>

−

Taken as transition digraphs, <math>\~~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>

+

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

−

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

<ol>

−

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

−

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

</ol>

−

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

−

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

+

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

{| cellpadding="10"

Line 10,844: Line 10,856:

|-

| 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>

|}

Line 10,869: Line 10,881:

|}

−

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>\mathrm{Con}^1 (L_\text{B})~\!</math> for interpreter <math>\text{B}\!</math> has the following semiotic equations.

{| cellpadding="10"

Line 10,882: Line 10,894:

|-

| 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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There are many ways to extend sign relations in an effort to increase their reflective capacities. The implicit goal of a reflective project is to achieve ''reflective closure'', <math>S \subseteq O,\!</math> where every sign is an object.

−

Considered as reflective extensions, there is nothing unique about the constructions of <math>\~~operatorname~~{Ref}^1 (\text{A})\!</math> and <math>\~~operatorname~~{Ref}^1 (\text{B})\!</math> but their common pattern of development illustrates a typical approach toward reflective closure. In a sense it epitomizes the project of ''free'', ''naive'', or ''uncritical'' reflection, since continuing this mode of production to its closure would generate an infinite sign relation, passing through infinitely many higher orders of signs, but without examining critically to what purpose the effort is directed or evaluating alternative constraints that might be imposed on the initial generators toward this end.

+

Considered as reflective extensions, there is nothing unique about the constructions of <math>\mathrm{Ref}^1 (\text{A})\!</math> and <math>\mathrm{Ref}^1 (\text{B})\!</math> but their common pattern of development illustrates a typical approach toward reflective closure. In a sense it epitomizes the project of ''free'', ''naive'', or ''uncritical'' reflection, since continuing this mode of production to its closure would generate an infinite sign relation, passing through infinitely many higher orders of signs, but without examining critically to what purpose the effort is directed or evaluating alternative constraints that might be imposed on the initial generators toward this end.

At first sight it seems as though the imposition of reflective closure has multiplied a finite sign relation into an infinite profusion of highly distracting and largely redundant signs, all by itself and all in one step. But this explosion of orders happens only with the complicity of another requirement, that of deterministic interpretation.

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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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As a flexible and fairly general strategy for describing reflective extensions, it is convenient to take the following tack. Given a syntactic domain <math>S,\!</math> there is an independent formal language <math>F = F(S) = S \langle {}^{\langle\rangle} \rangle,\!</math> called the ''free quotational extension of <math>S,\!</math>'' that can be generated from <math>S\!</math> by embedding each of its signs to any depth of quotation marks. Within <math>F,\!</math> the quoting operation can be regarded as a syntactic generator that is inherently free of constraining relations. In other words, for every <math>s \in S,\!</math> the sequence <math>s, {}^{\langle} s {}^{\rangle}, {}^{\langle\langle} s {}^{\rangle\rangle}, \ldots\!</math> contains nothing but pairwise distinct elements in <math>F\!</math> no matter how far it is produced. The set <math>F(s) = s \langle {}^{\langle\rangle} \rangle \subseteq F\!</math> that collects the elements of this sequence is called the ''subset of <math>F\!</math> generated from <math>s\!</math> by quotation''.

−

Against this background, other varieties of reflective extension can be specified by means of semantic equations that are considered to be imposed on the elements of <math>F.\!</math> Taking the reflective extensions <math>\~~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>

+

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

−

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

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

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

−

|+ style="height:30px" | <math>\text{Table 82.} ~~ \text{Reflective Extension} ~ \~~operatorname~~{Ref}^1 (\text{A} | E_1)\!</math>

+

|+ style="height:30px" | <math>\text{Table 82.} ~~ \text{Reflective Extension} ~ \mathrm{Ref}^1 (\text{A} | E_1)\!</math>

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

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

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

−

|+ style="height:30px" | <math>\text{Table 83.} ~~ \text{Reflective Extension} ~ \~~operatorname~~{Ref}^1 (\text{B} | E_1)\!</math>

+

|+ style="height:30px" | <math>\text{Table 83.} ~~ \text{Reflective Extension} ~ \mathrm{Ref}^1 (\text{B} | E_1)\!</math>

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

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

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

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

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

−

|+ style="height:30px" | <math>\text{Table 84.} ~~ \text{Reflective Extension} ~ \~~operatorname~~{Ref}^1 (\text{A} | E_2)\!</math>

+

|+ style="height:30px" | <math>\text{Table 84.} ~~ \text{Reflective Extension} ~ \mathrm{Ref}^1 (\text{A} | E_2)\!</math>

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

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

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

−

|+ style="height:30px" | <math>\text{Table 85.} ~~ \text{Reflective Extension} ~ \~~operatorname~~{Ref}^1 (\text{B} | E_2)\!</math>

+

|+ style="height:30px" | <math>\text{Table 85.} ~~ \text{Reflective Extension} ~ \mathrm{Ref}^1 (\text{B} | E_2)\!</math>

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

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

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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, …

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<ol>

−

<li>

+

<li>The way I have chosen to deal with this issue in the present case is not by injecting more features of the informal discussion into the dialogue of <math>\text{A}\!</math> and <math>\text{B},\!</math> but by trying to imagine how agents like <math>\text{A}\!</math> and <math>\text{B}\!</math> might be enabled to reflect on these aspects of their own discussion.</li>

−

The way I have chosen to deal with this issue in the present case is not by injecting more features of the informal discussion into the dialogue of <math>\text{A}\!</math> and <math>\text{B},\!</math> but by trying to imagine how agents like <math>\text{A}\!</math> and <math>\text{B}\!</math> might be enabled to reflect on these aspects of their own discussion.

−

</li>

<li>

Line 11,459: Line 11,469:

−

<li>First, I employ the sign relations A and B to illustrate two basic kinds of set theoretic merges, the ordinary or "simple" union and the indexed or "situated" union of extensional relations. On review, both forms of combination are observed to fall short of what is needed to constitute the desired characteristics of a shared sign relation.</li>

+

<li>First, I employ the sign relations <math>L_\text{A}\!</math> and <math>L_\text{B}\!</math> to illustrate two basic kinds of set theoretic merges, the ordinary or ''simple'' union and the indexed or ''situated'' union of extensional relations. On review, both forms of combination are observed to fall short of what is needed to constitute the desired characteristics of a shared sign relation.</li>

−

<li>Next, I present two other ways of extending the sign relations A and B into a common ~~SOI~~. These extensions succeed in capturing further aspects of what interpreters know about their shared language use. Although motivated on different grounds, the alternative constructions that develop coincide in exactly the same abstract structure.</li>

+

<li>Next, I present two other ways of extending the sign relations <math>L_\text{A}\!</math> and <math>L_\text{B}\!</math> into a common system of interpretation. These extensions succeed in capturing further aspects of what interpreters know about their shared language use. Although motivated on different grounds, the alternative constructions that develop coincide in exactly the same abstract structure.</li>

</ol>

</li>

</ol>

</li>

+

<li>As this project begins to take on sign relations that are complex enough to convey the impression of genuine inquiry processes, a fuller explication of this issue will become mandatory. Eventually, this will demand a concept of ''higher-order sign relations'', whose objects, signs, and interpretants can all be complete sign relations in their own rights.</li>

</ol>

−

~~<pre>~~

−

3. As this project begins to take on sign relations that are complex enough to convey the impression of genuine inquiry processes, a fuller explication of this issue will become mandatory. Eventually, this will demand a concept of "higher order sign relations", whose objects, signs, and interpretants can all be complete sign relations in their own rights.

In principle, the successive grades of complexity enumerated above could be ascended in a straightforward way, if only the steps did not go straight up the cliffs of abstraction. As always, the kinds of intentional objects that are the toughest to face are those whose realization is so distant that even the gear needed to approach their construction is not yet in existence.

−

~~</pre>~~

−

~~===6.47. Mutually Intelligible Codes===~~

−

~~<pre>~~

−

Before this complex of relationships can be formalized in much detail, I must introduce linguistic devices for generating "higher order signs", used to indicate other signs, and "situated signs", indexed by the names of their users, their contexts of use, and other types of information incidental to their usage in general. This leads to the consideration of SOIs that maintain recursive mechanisms for naming everything within their purview. This "nominal generosity" gives them a new order of generative capacity, producing a sufficient number of distinctive signs to name all the objects and then name the names that are needed in a given discussion.

−

Symbolic systems for quoting inscriptions and ascribing quotations are associated in metamathematics with "godel numberings" of formal objects, enumerative functions that provide systematic but ostensibly arbitrary reference numbers for the signs and expressions in a formal language. Assuming these signs and expressions denote anything at all, their formal enumerations become the "codes" of formal objects, just as programs taken literally are code names for certain mathematical objects known as computable functions. Partial forms of specification not withstanding, these codes are the only complete modes of representation that formal objects can have in the medium of mechanical activity.

−

In the dialogue of A and B there happens to be an exact coincidence between signs and states. That is, the states of the interpretive systems A and B are not distinguished from the signs in S that are imagined to be mediating, moment by moment, the attentions of the interpretive agents A and B toward their respective objects in O. So the question arises: Is this identity bound to be a general property of all useful sign relations, or is it only a degenerate feature occurring by chance or unconscious design in the immediate example?

−

To move toward a resolution of this question I reason as follows. In one direction, it seems obvious that a "sign in use" (SIU) by a particular interpreter constitutes a component of that agent's state. In other words, the very notion of an identifiable SIU refers to numerous instances of a particular interpreter's state that share in the abstract property of being such instances, whether or not anyone can give a more concise or illuminating characterization of the concept under which these momentary states are gathered. Conversely, it is at least conceivable that the whole state of a system, constituting its transitory response to the entirety of its environment, history, and goals, can be interpreted as a sign of something to someone. In sum, there remains an outside chance of signs and states being precisely the same things, since nothing precludes the existence of an IF that could make it so.

−

Still, if the question about the distinction or coincidence between signs and states is restricted to the domains where existential realizations are conceivable, no matter whether in biological or computational media, then the prerequisites of the task become more severe, due to the narrower scope of materials that are admitted to answer them. In focussing on this arena the problem is threefold:

−

1. The crucial point is not just whether it is possible to imagine an ideal SOI, an external perspective or an independent POV, for which all states are signs, but whether this is so for the prospective SOI of the very agent that passes through these states.

−

~~2. To what extent can the transient states and persistent conduct of each agent in a community of interpretation take on a moderately public and objective aspect in relation to the other participants?~~

−

~~3. How far in this respect, in the common regard for this species of outward demeanor, can each agent's behavior act as a sign of genuine objects in the eyes of other interpreters?~~

−

The special task of a nuanced hermeneutic approach to computational interpretation is to realize the relativity of all formal codes to their formal coders, and to seek ways of facilitating mutual intelligibility among interpreters whose internal codes can be thoroughly private, synchronistically keyed to external events, and even a bit idiosyncratic.

−

Ultimately, working through this maze of "meta" questions, as posed on the tentative grounds of the present project, leads to a question about the "logical reference frames" or "metamathematical coordinate systems" that are supposed to distinguish "objective" from "symbolic" entities and are imagined to discriminate a range of gradations along their lines. The question is: Whether any gague of objectivity or scale of virtuality has invariant properties discoverable by all independent interpreters, or whether all is vanity and inane relativism, and everything concerning a subjective point of view is sheer caprice?

−

Thus, the problem of mutual intelligibility turns on the question of "common significance": How can there be signs that are truly public, when the most natural signs that distinct agents can know, their own internal states, have no guarantee and very little likelihood of being related in systematically fathomable ways? As a partial answer to this, I am willing to contemplate certain forms of pre established harmony, like the common evolution of a biological species or the shared culture of an interpretive community, but my experience has been that harmony, once established, quickly corrupts unless active means are available to maintain it. So there still remains the task of identifying these means. With or without the benefit of a prior consensus, or the assumption of an initial, but possibly fragile equilibrium, an explanation of robust harmony must detail the modes of maintaining communication that enable coordinated action to persist in the meanest of times.

−

The formal character of these questions, in the potential complexities that can be forced on contemplation in the pursuit of their answers, is independent of the species of interpreters that are chosen for the termini of comparison, whether person to person, person to computer, or computer to computer. As always, the truth of this kind of thesis is formal, all too formal. What it brings is a new refrain of an old motif: Are there meaningful, if necessarily formal series of analogies that can be strung from the patterns of whizzing electrons and humming protons, whose controlled modes of collective excitation form and inform the conducts of computers, all the way to the rather different patterns of wizened electrons and humbled protons, whose deliberate energies of communal striving substantiate the forms of life known to be intelligible?

−

A full consideration of the geometries available for the spaces in which these levels of reflective abstraction are commonly imagined to reside leads to the conclusion that familiar distinctions of "top down" versus "bottom up" are being taken for granted in an arena that has not even been established to be orientable. Thus, it needs to be recognized that the distinction between objects and signs is relative to a definite SOI. The pragmatic theory of signs is designed, in part, precisely to deal with the circumstance that thoroughly objective states of systems can be signs of each other, undermining any pretended distinction between objects and signs that one might propose to draw on essential grounds.

−

From now on, I will reuse the ancient term "gnomon" in a technical sense to refer to the godel numbers or code names of formal objects. In other words, a gnomon is a godel numbering or enumeration function that maps a domain of objects into a domain of signs, Gno : O > S. When the syntactic domain S is contained within the object domain O, then the part of the gnomon that maps S into S, providing names for signs and expressions, is usually regarded as a "quoting function".

−

In the pluralistic contexts that go with pragmatic theories of signs, it is no longer entirely appropriate to refer to "the" gnomon of any object. At any moment of discussion, I can only have so and so's gnomon or code word for each thing under the sun. Thus, apparent references to a uniquely determined gnomon only make sense if taken as enthymemic invocations of the ordinary context and of all that is comprehended to be implied in it, promising to convert tacit common sense into definite articulations of what is understood. Actually achieving this requires each elliptic reference to the gnomon to be explicitly grounded in the context of informal discussion, interpreted with respect to the conventional basis of understanding assumed in it, and relayed to the indexing function taken for granted by all parties to it.

−

In computational terms, this brand of pluralism means that neither the gnomon nor the quoting function that forms a part of it can be viewed as well defined unless it is indexed, explicity or implicitly, by the name of a particular interpreter. I will use the notations "Gnoi(x)" = "<x, i>" to indicate the gnomon of the object x with respect to the interpreter i. The value Gnoi(x) = <x, i> C S is the "nominal sign in use" or the "name in use" (NIU) of the object x with respect to the interpreter i, and thus it constitutes a component of i's state.

−

In the special case where x is a sign or expression in the syntactic domain, then Gnoi(x) = <x, i> is tantamount to the quotation of x by and for the use of the ith interpreter, in short, the nominal sign to i that makes x an object for i. For signs and expressions, it is usually only the quoting function that makes them objects. But nothing is an object in any sense for an interpreter unless it is an object of a sign relation for that interpreter. Therefore, ...

−

If it is now asked what measure of invariant understanding can be enjoyed by diverse parties of interpretive agents, then the discussion has come upon an issue with a familiar echo in mathematical analysis. The organization of many local coordinate frames into systems capable of supporting communicative references to relatively "objective" objects is usually handled by means of the concept of a "manifold". Therefore, the analogous task that is suggested for this project is to arrive at a workable definition of "sign relational manifolds".

−

The discrete nature of the A and B dialogue renders moot the larger share of issues of interest in continuous and differentiable manifolds. However, it is still possible to get things moving in this direction by looking at simple structural analogies that connect the pragmatic theory of sign relations with the basic notions of analysis on manifolds.

−

~~</pre>~~

−

~~===6.48. Discourse Analysis : Ways and Means===~~

−

~~<pre>~~

−

Before the discussion of the A and B dialogue can proceed to richer veins of semantic structure it will be necessary to extract the relevant traces of embedded sign relations from their environments of informally interpreted syntax.

−

On the substantive front, sign relations serving as raw materials of discourse need to be refined and their content assayed, but first their identifying signatures must be sounded out, carved out, and lifted from their embroiling inclusions in the dense strata of obscure intuitions that sediment ordinary discussion. On the instrumental front, sign relations serving as primitive tools of discourse analysis need to be identified and improved by a deliberate examination of their designs and purposes.

−

So far, the models and methods made available to formal treatment were borrowed outright, with little hesitation and less recognition, from the context of casual discussion. Thus, these materials and mechanisms have come to the threshold of critical reflection already in play, devoid of concern for the presuppositions and consequences associated with their use, and only belatedly turned to the effortful work and odious formalities of self conscious exposition.

−

To reflect on the properties of complex and higher order sign relations with any degree of clarity it is necessary to arrange a clearer field of investigation and a less cluttered staging area for analytic work than is commonly provided. Habitual processes of interpretation that typically operate as automatic routines and uncritical defaults in the informal context of discussion have to be selectively inhibited, slowed down, and critically examined as objective possibilities, instead of being taken for granted as absolute necessities.

−

In other words, an apparatus for critical reflection does not merely add more mirrors to the kaleidoscopic fun house of interpretive discourse, but it provides transient moments of equanimity, or balanced neutrality, and a moderately detached perspective on alternative points of view. A scope so limited does not by any means grant a God's Eye View (GEV), but permits a sufficient quantity of light to consider how the original array of sights and reflections might have been created otherwise.

−

Ordinarily, the extra degree of attention to syntax that is needed for critical reflection on interpretive processes is called into play by means of syntactic operators and diacritical devices acting at the level of individual signs and elementary expressions. For example, quotation marks are used to force one type of "semantic ascent", causing signs to be treated as objects and marking points of interpretive shift as they occur in the syntactic medium. But these operators and devices must be symbolized, and these symbols must be interpreted. Consequently, there is no way to avoid the invocation of a cohering interpretive framework, one that needs to be specialized for analytic purposes.

−

The best way to achieve the desired type of reflective capacity is by attaching a parameter to the IF used as an instrument of formal study, specifying certain choices or interpretive presumptions that affect the entire context of discussion. The aesthetic distance needed to arrive at a formal perspective on sign relations is maintained, not by jury rigging ordinary discussion with locally effective syntactic devices, but by asking the reader to consider certain dimensions of parametric variation in the global IFs used to comprehend the sign relations under study.

−

The interpretive parameter of paramount importance to this work is one that is critical to reflection. It can be presented as a choice between two alternative conventions, affecting the way one reflexively regards each sign in a text: (1) as a sign provoking interest only in passing, exchanged for the sake of a meaningful object it is always taken for granted to have, or (2) as a sign comprising an interest in and of itself, a state of a system or a modification of a medium that can signify an external value but does not necessarily denote anything else at all. I will name these options for responding to signs according to the aspects of character that are most appreciated in their net effects, whether signs for the sake of objects, or signs for their own sake, respectively.

−

The first option I call the "object convention", recognizing it as the natural default of informal language use. In the ordinary language context it is the automatic assumption that signs and expressions are intended to denote something external to themselves, and even though it is quite obvious to all interpreters that the medium is filled with the appearances of signs and not with the objects themselves, this fact passes for little more than transitory interest in the rush to cash out tokens for their indicated values.

−

The object convention, as appropriate to an introduction that needs to begin in the context of ordinary discussion, is the parametric choice that was left in force throughout the treatment of the A and B example. Doing things this way is like trying to roller skate in a buffalo herd, that is, it attempts to formalize a fragment of discussion on a patchwork of local scales without interrupting the automatic routines and default assumptions that prevail on a global basis in the informal context. Ultimately, one cannot avoid stumbling over the hoofprints ("...") of overly cited and opaquely enthymemic textual deposits.

−

The second option I call the "sign convention", observing it to be the treatment of choice in programming and formal language studies. In the formal language context it is necessary to consider the possibility that not all signs and expressions are assured to denote or even connote much of anything at all. This danger is amplified in computational frameworks where it resonates with a related theme, that not all programs are guaranteed to terminate normally with a definite result. In order to deal with these eventualities, a more cautious approach to sign relations is demanded to cover the risk of generating nonsense, in other words, to guard against degenerate forms of sign relations that fail to serve any significant purpose in communication or inquiry.

−

Whenever a greater degree of care is required, it becomes necessary to replace the object convention with the sign convention, which presumes to take for granted only what can be obvious to all observers, namely, the phenomenal appearances and temporal occurrences of objectified states of systems. To be sure, these modulations of media are still presented as signs, but only potentially as signs of other things. It goes with the territory of the formal language context to constantly check the inveterate impulses of the literate mind, to reflect on its automatic reflex toward meaning, to inhibit its uncontrolled operation, and to pause long enough in the rush to judgment to question whether its constant presumption of a motive is itself innocent.

−

In order to deal with these issues of discourse analysis in an explicit way, it is necessary to have in place a technical notation for marking the very kinds of interpretive assumptions that normally go unmarked. Thus, I will describe a set of devices for annotating certain kinds of interpretive contingencies, called the "discourse analysis frames" (DAFs) or the "global interpretive frames" (GIFs), that can be operative at any given moment in a particular context of discussion.

−

To mark a context of discussion where a particular set J of interpretive conventions is being maintained, I use labeled brackets of the following two forms: "unitary", as "{J| ... |J}, or "divided", as {J| ... | ... |J}. The unitary form encloses a context of discussion by delimiting a range of text whose reading is subject to the interpretive constraints J. The divided form specifies the objects, signs, and interpretive information in accord with which a species of discussion is generated. Labeled brackets enclosing contexts can be nested in their scopes, with interpretive data on each outer envelope applying to every inclusion. Labeled brackets arranging the "conversation pieces" or the "generators and relations" of a topic can lead to discussions that spill outside their frames, and thus are permitted to constitute overlapping contexts.

−

~~For the present, I will consider two types of interpretive parameters to be used as indices of labeled brackets.~~

−

1. Names of interpreters or other references to context can be used to indicate the provenance of the objects and signs that make up the assorted contents of brackets. On occasion, I will use the first person singular pronoun to signify the immediate context of informal discussion, as in "{I| ... |I}", but more often than not this context goes unmarked.

−

2. Two other modifiers can be used to toggle between the options of the object convention, more common in casual or ordinary contexts, and the sign convention, more useful in formal or sign theoretic contexts.

−

a. The brackets "{o| ... |o}" mark a context of informal language use or ordinary discussion, where the object convention applies. To specify the elements of a sign relation under these conditions, I use a form of presentation like the following:

−

~~{o| A, B ||| "A", "B", "i", "u" |o}.~~

−

Here, the names of objects are placed on the left side and the names of signs on the right side of the central divide, and the outer brackets stipulate that the object convention is in force throughout the discussion of a sign relation that is generated on these elements.

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b. The brackets "{s| ... |s}" mark a context of formal language use or controlled discussion, where the sign convention applies. To specify the elements of a sign relation in this case, I use a form like:

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~~{s| [A], [B] ||| A, B, i, u |s}.~~

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Again, expressions for objects are placed on the left and expressions of signs on the right, but formal language conventions are now invoked to let the alphabet letters and the lexical items of a formal vocabulary stand for themselves, and denotation brackets "[]" are placed around signs to indicate the corresponding objects, when they exist.

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When the information carried by labeled brackets becomes more involved and more extensive, a set of convenient abbreviations and suggestions for "pretty printing" can be followed. When the bracket labels become too long to bother repeating, I will leave the last label blank or use ditto marks, as with {a, b, c| ... |"}. When it is necessary to break labeled brackets over several lines, multiple dividers "|" and dittos """ can be used to fill out corresponding columns, as in the following text.

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~~{I, o| A , B~~

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~~|||||| "A", "B", "i", "u"~~

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~~|""""}~~

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A notation for discourse analysis ought to find a crucial test of its usefulness in whether it can help to disclose structural properties of interpretive frameworks that would otherwise escape the attention due. If the dimensions of interpretive choice that are represented by these devices are to serve a useful function, then ...

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Although these devices for discourse analysis are bound to seem a bit ad hoc at this point, they have been designed with a sign relational bootstrap in mind, that is, with a view to being formalized and recognized as a species within the domain of sign relations itself, where this is the very domain that is laid out as their field of application.

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One note of caution may help to prevent a common misunderstanding. It is futile to imagine that any system of interpretive markers for discourse can become totally self sufficient, like the Worm Uroboros, determining all aspects of interpretation and eliminating all ambiguity. The ultimate appeal of signs, and signs upon signs, is always to an intelligent interpreter, a reader who knows there are more interpretive choices to make than could ever be surrendered to signs, and whose free responsibility to appropriate interpretations cannot be abdicated to any text or abridged by any gloss on it, no matter how fit or finished.

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In a sense, at least at first, nothing is being created that could not have been noticed without signs. It is merely that actions are being articulated that were not articulated before, and hopefully in ways that make transient insights easier to remember and reuse on new occasions. Instead, the requirement here is to devise a language, the marks of which can reflect the ambient light of observation on its own process. It is not unusual to succeed at this in artificial environments crafted especially for the purpose, but to achieve the critical angle in vivo, in the living context of a natural language, takes more art.

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~~</pre>~~

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~~===6.49. Combinations of Sign Relations===~~

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~~<pre>~~

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At a point like this in the development of a formal subject matter, it is customary to introduce elements of a logical calculus that can be used to describe relevant aspects of the formal structures involved and to expedite reasoning about their manifold combinations and decompositions. I will hold off from doing this for sign relations in any formal way at present. Instead, I consider the informal requirements and the forseeable ends that a suitable calculus for sign relations might be expected to meet, and I present as tentative alternatives a few different ways of proceeding to formalize these intentions.

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The first order of business for the "comparative anatomy" and the "developmental biology" of sign relations is to undertake a pair of closely related tasks: (1) to examine the structural articulation of highly complex sign relations in terms of the primitive constituents that are found available, and (2) to explain the functional genesis of formal (that is, reflectively considered and critically regarded) sign relations as they naturally arise within the informal context of representational and communicational activities.

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Converting to a political metaphor, how does the "republic" constituted by a sign relation — the representational community of agents invested with a congeries of legislative, executive, and interpretive powers, employing a consensual body of conventional languages, encompassing a commonwealth of comprehensible meanings, diversely but flexibly manifested in the practical administration of abiding and shared representations — how does all of this first come into being?

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~~... and their development from primitive/ rudimentary to highly structured ...~~

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The grasp of the discussion between A and B that is represented in the separate sign relations given for them can best be described as fragmentary. It fails to capture what everyone knows A and B would know about each other's language use.

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How can the fragmentary system of interpretation (SOI) constituted by the juxtaposition of individual sign relations A and B be combined or developed into a new SOI that represents what agents like A and B are sure to know about each other's language use? In order to make it clear that this is a non trivial question, and in the process to illustrate different ways of combining sign relations, I begin by considering a couple of obvious suggestions for their integration that immediate reflection will show to miss the mark.

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The first thing to try is the set theoretic union of the sign relations. This commonly leads to a "confused" or "confounded" combination of the component sign relations. For example, the sign relation defined as C = A U B is shown in Table 86. Interpreted as a transition digraph on the four points of the syntactic domain S = {"A", "B", "i", "u"}, the sign relation C specifies the following behavior for the conduct of its interpreter:

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~~1. AC has a sling at each point of {"A", "i", "u"} and two way arcs on the pairs {"A", "i"} and {"A", "u"}.~~

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~~2. BC has a sling at each point of {"B", "i", "u"} and two way arcs on the pairs {"B", "i"} and {"B", "u"}.~~

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These sub-relations do not form equivalence relations on the relevant sets of signs. If closed up under transitive compositions, then {"A", "i", "u"} are all equivalent in the presence of object A, but {"B", "i", "u"} are all equivalent in the presence of object B. This may accurately represent certain types of political thinking, but it does not constitute the kind of sign relation that is wanted here.

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Reflecting on this disappointing experience with using simple unions to combine sign relations, it appears that some type of indexed union or categorical co product might be demanded. Table 87 presents the results of taking the disjoint union D = A U B to constitute a new sign relation.

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~~</pre>~~

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~~ ~~

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~~<pre>~~

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~~Table 86. Confounded Sign Relation C~~

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~~Object Sign Interpretant~~

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~~A "A" "A"~~

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~~A "A" "i"~~

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~~A "A" "u"~~

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~~A "i" "A"~~

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~~A "i" "i"~~

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~~A "u" "A"~~

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~~A "u" "u"~~

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~~B "B" "B"~~

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~~B "B" "i"~~

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~~B "B" "u"~~

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~~B "i" "B"~~

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~~B "i" "i"~~

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~~B "u" "B"~~

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~~B "u" "u"~~

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~~</pre>~~

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~~ ~~

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~~<pre>~~

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~~Table 87. Disjointed Sign Relation D~~

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~~Object Sign Interpretant~~

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~~AA "A"A "A"A~~

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~~AA "A"A "i"A~~

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~~AA "i"A "A"A~~

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~~AA "i"A "i"A~~

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~~AB "A"B "A"B~~

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~~AB "A"B "u"B~~

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~~AB "u"B "A"B~~

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~~AB "u"B "u"B~~

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~~BA "B"A "B"A~~

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~~BA "B"A "u"A~~

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~~BA "u"A "B"A~~

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~~BA "u"A "u"A~~

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~~BB "B"B "B"B~~

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~~BB "B"B "i"B~~

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~~BB "i"B "B"B~~

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~~BB "i"B "i"B~~

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~~</pre>~~

===6.50. Revisiting the Source===

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[[Category:Artificial Intelligence]]

Jon Awbrey

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