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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
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===6.24. Literal Intensional Representations===
===6.24. Literal Intensional Representations===
<pre>
In this section I prepare the grounds for building bridges between ER's and IR's of sign relations. To establish an initial foothold on either side of the distinction and to gain a first march on connecting the two sites of the intended construction, I introduce an intermediate mode of description called a "literal intensional representation" (LIR).
Any LIR is a nominal form of IR that has exactly the same level of detail as an ER, merely shifting the interpretation of primitive terms from an extensional to an intensional modality, namely, from a frame of reference terminating in "points", "atomic elements", "elementary objects", or "real particulars" to a frame of reference terminating in "qualities", "basic features", "fundamental properties", or "simple propositions". This modification, that translates the entire set of elementary objects in an ER into a parallel set of fundamental properties in a LIR, constitutes a form of modulation that can be subtle or trivial, depending on one's point of view. Regarded as trivial, it tends to go unmarked, leaving it up to the judgment of the interpreter to decide whether the same sign is meant to denote a point, a particular, a property, or a proposition. An interpretive variance that goes unstated tends to be treated as final. It is always possible to bring in more signs in an attempt to signify the variants intended, but it needs to be noted that every effort to control the interpretive variance by means of these epithets and expletives only increases the level of liability for accidental errors, if not the actual probability of misinterpretation. For the sake of this introduction, and in spite of these risks, I treat the distinction between extensional and intensional modes of interpretation as worthy of note and deserving of an explicit notation.
Table 50 ...
Tables 51.1, 51.2, and 51.3 ...
Tables 52.1, 52.2, and 52.3 ...
Table 50. Notation for Objects & Their Signs
Object Sign of Object
A A w1 <A> <A> <w1>
B B w2 <B> <B> <w2>
"A" <A> w3 <"A"> <<A>> <w3>
"B" <B> w4 <"B"> <<B>> <w4>
"i" <i> w5 <"i"> <<i>> <w5>
"u" <u> w6 <"u"> <<u>> <w6>
Table 51.1 Notation for Properties & Their Signs (1)
Property Sign of Property
{A} {A} {w1} <{A}> <{A}> <{w1}>
{B} {B} {w2} <{B}> <{B}> <{w2}>
{"A"} {<A>} {w3} <{"A"}> <{<A>}> <{w3}>
{"B"} {<B>} {w4} <{"B"}> <{<B>}> <{w4}>
{"i"} {<i>} {w5} <{"i"}> <{<i>}> <{w5}>
{"u"} {<u>} {w6} <{"u"}> <{<u>}> <{w6}>
Table 51.2 Notation for Properties & Their Signs (2)
Property Sign of Property
A A w1 <A> <A> <w1>
B B w2 <B> <B> <w2>
"A" <A> w3 <"A"> <<A>> <w3>
"B" <B> w4 <"B"> <<B>> <w4>
"i" <i> w5 <"i"> <<i>> <w5>
"u" <u> w6 <"u"> <<u>> <w6>
Table 51.3 Notation for Properties & Their Signs (3)
Property Sign of Property
A o1 w1 <A> <o1> <w1>
B o2 w2 <B> <o2> <w2>
a s1 w3 <a> <s1> <w3>
b s2 w4 <b> <s2> <w4>
i s3 w5 <i> <s3> <w5>
u s4 w6 <u> <s4> <w6>
Table 52.1 Notation for Instances & Their Signs (1)
Instance Sign of Instance
[A] [A] [w1] <[A]> <[A]> <[w1]>
[B] [B] [w2] <[B]> <[B]> <[w2]>
["A"] [<A>] [w3] <["A"]> <[<A>]> <[w3]>
["B"] [<B>] [w4] <["B"]> <[<B>]> <[w4]>
["i"] [<i>] [w5] <["i"]> <[<i>]> <[w5]>
["u"] [<u>] [w6] <["u"]> <[<u>]> <[w6]>
Table 52.2 Notation for Instances & Their Signs (2)
Instance Sign of Instance
A A w1 <A> <A> <w1>
B B w2 <B> <B> <w2>
"A" <A> w3 <"A"> <<A>> <w3>
"B" <B> w4 <"B"> <<B>> <w4>
"i" <i> w5 <"i"> <<i>> <w5>
"u" <u> w6 <"u"> <<u>> <w6>
Table 52.3 Notation for Instances & Their Signs (3)
Instance Sign of Instance
A o1 w1 <A> <o1> <w1>
B o2 w2 <B> <o2> <w2>
a s1 w3 <a> <s1> <w3>
b s2 w4 <b> <s2> <w4>
i s3 w5 <i> <s3> <w5>
u s4 w6 <u> <s4> <w6>
Using two different stategies of representation:
1. The first strategy is called the "literal coding", because it sticks to obvious features of each syntactic element to contrive its code, or the "O(n) coding", because it uses a number on the order of n logical features to represent a domain of n 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 O(n) 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 O = {A, B} and the signs in S = {"A", "B", "i", "u"}, 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.
1a. literal coding
For the domain W = {A, B, "A", "B", "i", "u"} of six elements one needs to use six logical features, in effect, elevating each individual object to the status of an exclusive ontological category in its own right. The easiest way to do this is simply to reuse the world syntactic domain O as a logical alphabet W, taking element wise identifications as follows:
W = {o1, o2, s1, s2, s3, s4} = {A, B, "A", "B", "i", "u"}.
W = {w1, w2, w3, w4, w5, w6} = {A, B, a, b, i, u}.
Tables 53.1 and 53.2 show three different ways of coding the elements of an ER and the features of a LIR, respectively, for the world set W = W (A, B), that is, for the set of objects, signs, and interpretants that are common to the sign relations A and B. Successive columns of these Tables give the "mnemonic code", the "pragmatic code", and the "abstract code" for each element.
Table 53.1 Elements of ER (W)
Mnemonic Element Pragmatic Element Abstract Element
w C W w C W wi C W
A o1 w1
B o2 w2
"A" s1 w3
"B" s2 w4
"i" s3 w5
"u" s4 w6
Table 53.2 Features of LIR (W)
Mnemonic Feature Pragmatic Feature Abstract Feature
w C W w C W wi C W
A o1 w1
B o2 w2
a s1 w3
b s2 w4
i s3 w5
u s3 w6
If the world of A and B, the set W = {A, B, "A", "B", "i", "u"}, is viewed abstractly, as an arbitrary set of six atomic points, then there are exactly 26 = 64 "abstract properties" (AP's) or "potential attributes" (PA's) that might be applied to or recognized in these points. The extensions of these AP's are the subsets of W, otherwise known as members of the "power set" Pow (W). 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 {}. 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 W as vectors in the coordinate space W and as singular propositions in the universe of discourse W[]. Altogether, these Tables present the "literal" codes for the elements of W and W[] in their "mnemonic", "pragmatic", and "abstract" versions, respectively. In each Table, Column 1 lists the element w C W, while Column 2 gives the corresponding coordinate vector w C W in the form of a bit string. The next two Columns represent each w C W as a proposition in W[], in effect, reconstituting it as a function w : W >B. 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.
Table 54.1 Mnemonic Literal Codes for Interpreters A & B
Element Vector Conjunct Term Code
A 100000 A (B)(a)(b)(i)(u) <A>W
B 010000 (A) B (a)(b)(i)(u) <B>W
"A" 001000 (A)(B) a (b)(i)(u) <a>W
"B" 000100 (A)(B)(a) b (i)(u) <b>W
"i" 000010 (A)(B)(a)(b) i (u) <i>W
"u" 000001 (A)(B)(a)(b)(i) u <u>W
Table 54.2 Pragmatic Literal Codes for Interpreters A & B
Element Vector Conjunct Term Code
A 100000 o1 (o2)(s1)(s2)(s3)(s4) <o1>W
B 010000 (o1) o2 (s1)(s2)(s3)(s4) <o2>W
"A" 001000 (o1)(o2) s1 (s2)(s3)(s4) <s1>W
"B" 000100 (o1)(o2)(s1) s2 (s3)(s4) <s2>W
"i" 000010 (o1)(o2)(s1)(s2) s3 (s4) <s3>W
"u" 000001 (o1)(o2)(s1)(s2)(s3) s4 <s4>W
Table 54.3 Abstract Literal Codes for Interpreters A & B
Element Vector Conjunct Term Code
A 100000 w1 (w2)(w3)(w4)(w5)(w6) <w1>W
B 010000 (w1) w2 (w3)(w4)(w5)(w6) <w2>W
"A" 001000 (w1)(w2) w3 (w4)(w5)(w6) <w3>W
"B" 000100 (w1)(w2)(w3) w4 (w5)(w6) <w4>W
"i" 000010 (w1)(w2)(w3)(w4) w5 (w6) <w5>W
"u" 000001 (w1)(w2)(w3)(w4)(w5) w6 <w6>W
Table 55.1 LIR1 (A): Literal Representation of A
Object Sign Interpretant
<A>W <a>W <a>W
<A>W <a>W <i>W
<A>W <i>W <a>W
<A>W <i>W <i>W
<B>W <b>W <b>W
<B>W <b>W <u>W
<B>W <u>W <b>W
<B>W <u>W <u>W
Table 55.2 LIR1 (Den A): Denotative Component of A
Object Sign Transition
<A>W <a>W <<a>W, <A>W>
<A>W <i>W <<i>W, <A>W>
<B>W <b>W <<b>W, <B>W>
<B>W <u>W <<u>W, <B>W>
Table 55.3 LIR1 (Con A): Connotative Component of A
Sign Interpretant Transition
<a>W <a>W <>dW
<a>W <i>W <da di>dW
<i>W <a>W <da di>dW
<i>W <i>W <>dW
<b>W <b>W <>dW
<b>W <u>W <db du>dW
<u>W <b>W <db du>dW
<u>W <u>W <>dW
Table 56.1 LIR1 (B): Literal Representation of B
Object Sign Interpretant
<A>W <a>W <a>W
<A>W <a>W <u>W
<A>W <u>W <a>W
<A>W <u>W <u>W
<B>W <b>W <b>W
<B>W <b>W <i>W
<B>W <i>W <b>W
<B>W <i>W <i>W
Table 56.2 LIR1 (Den B): Denotative Component of B
Object Sign Transition
<A>W <a>W <<a>W, <A>W>
<A>W <u>W <<u>W, <A>W>
<B>W <b>W <<b>W, <B>W>
<B>W <i>W <<i>W, <B>W>
Table 56.3 LIR1 (Con B): Connotative Component of B
Sign Interpretant Transition
<a>W <a>W <>dW
<a>W <u>W <da du>dW
<u>W <a>W <da du>dW
<u>W <u>W <>dW
<b>W <b>W <>dW
<b>W <i>W <db di>dW
<i>W <b>W <db di>dW
<i>W <i>W <>dW
1b. lateral coding
For the domain O = {A, B} of two elements
X = {o1, o2} = {A, B} = O.
For the domain S = I = {"A", "B", "i", "u"} of four elements one needs to use four logical features, in effect, elevating each individual sign to the status of an exclusive grammatical category in its own right. The easiest way to do this is simply to reuse the syntactic domain S = I as a logical alphabet Y, taking element wise identifications as follows:
Y = {s1, s2, s3, s4} = {"A", "B", "i", "u"}.
Y = {s1, s2, s3, s4} = {"A", "B", "i", "u"}.
Tables 57.1, 57.2, and 57.3 show several ways of representing the elements of O and S, presenting the "lateral" codes for world elements in their "mnemonic", "pragmatic", and "abstract" versions, respectively. In each Table, Column 2 gives the coordinate vector x C X or y C Y as a bit string, using a subscript to indicate the relevant space, X or Y. Column 3 lists 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, using doubly underlined capital letters for literal features of objects and doubly underlined lower case letters for literal features of quoted signs. Finally, Column 4 shows the compact code for each element, using a conjunction of positive features in subscripted angle brackets to represent the corresponding conjunct term as a singular proposition.
Table 57.1 Mnemonic Lateral Codes for Interpreters A & B
Element Vector Conjunct Term Code
A 10X A (B) <A>X
B 01X (A) B <B>X
"A" 1000Y a (b)(i)(u) <a>Y
"B" 0100Y (a) b (i)(u) <b>Y
"i" 0010Y (a)(b) i (u) <i>Y
"u" 0001Y (a)(b)(i) u <u>Y
Table 57.2 Pragmatic Lateral Codes for Interpreters A & B
Element Vector Conjunct Term Code
A 10X o1 (o2) <o1>X
B 01X (o1) o2 <o2>X
"A" 1000Y s1 (s2)(s3)(s4) <s1>Y
"B" 0100Y (s1) s2 (s3)(s4) <s2>Y
"i" 0010Y (s1)(s2) s3 (s4) <s3>Y
"u" 0001Y (s1)(s2)(s3) s4 <s4>Y
Table 57.3 Abstract Lateral Codes for Interpreters A & B
Element Vector Conjunct Term Code
A 10X x1 (x2) <x1>X
B 01X (x1) x2 <x2>X
"A" 1000Y y1 (y2)(y3)(y4) <y1>Y
"B" 0100Y (y1) y2 (y3)(y4) <y2>Y
"i" 0010Y (y1)(y2) y3 (y4) <y3>Y
"u" 0001Y (y1)(y2)(y3) y4 <y4>Y
Table 58.1 LIR2 (A): Lateral Representation of A
Object Sign Interpretant
A (B) a (b)(i)(u) a (b)(i)(u)
A (B) a (b)(i)(u) (a)(b) i (u)
A (B) (a)(b) i (u) a (b)(i)(u)
A (B) (a)(b) i (u) (a)(b) i (u)
(A) B (a) b (i)(u) (a) b (i)(u)
(A) B (a) b (i)(u) (a)(b)(i) u
(A) B (a)(b)(i) u (a) b (i)(u)
(A) B (a)(b)(i) u (a)(b)(i) u
Table 58.2 LIR2 (Den A): Denotative Component of A
Object Sign Transition
A (B) a (b)(i)(u) <<a>Y, <A>X>
A (B) (a)(b) i (u) <<i>Y, <A>X>
(A) B (a) b (i)(u) <<b>Y, <B>X>
(A) B (a)(b)(i) u <<u>Y, <B>X>
Table 58.3 LIR2 (Con A): Connotative Component of A
Sign Interpretant Transition
a (b)(i)(u) a (b)(i)(u) (da)(db)(di)(du)
a (b)(i)(u) (a)(b) i (u) da (db) di (du)
(a)(b) i (u) a (b)(i)(u) da (db) di (du)
(a)(b) i (u) (a)(b) i (u) (da)(db)(di)(du)
(a) b (i)(u) (a) b (i)(u) (da)(db)(di)(du)
(a) b (i)(u) (a)(b)(i) u (da) db (di) du
(a)(b)(i) u (a) b (i)(u) (da) db (di) du
(a)(b)(i) u (a)(b)(i) u (da)(db)(di)(du)
Table 59.1 LIR2 (B): Lateral Representation of B
Object Sign Interpretant
A (B) a (b)(i)(u) a (b)(i)(u)
A (B) a (b)(i)(u) (a)(b)(i) u
A (B) (a)(b)(i) u a (b)(i)(u)
A (B) (a)(b)(i) u (a)(b)(i) u
(A) B (a) b (i)(u) (a) b (i)(u)
(A) B (a) b (i)(u) (a)(b) i (u)
(A) B (a)(b) i (u) (a) b (i)(u)
(A) B (a)(b) i (u) (a)(b) i (u)
Table 59.2 LIR2 (Den B): Denotative Component of B
Object Sign Transition
A (B) a (b)(i)(u) <<a>Y, <A>X>
A (B) (a)(b)(i) u <<u>Y, <A>X>
(A) B (a) b (i)(u) <<b>Y, <B>X>
(A) B (a)(b) i (u) <<i>Y, <B>X>
Table 59.3 LIR2 (Con B): Connotative Component of B
Sign Interpretant Transition
a (b)(i)(u) a (b)(i)(u) (da)(db)(di)(du)
a (b)(i)(u) (a)(b)(i) u da (db)(di) du
(a)(b)(i) u a (b)(i)(u) da (db)(di) du
(a)(b)(i) u (a)(b)(i) u (da)(db)(di)(du)
(a) b (i)(u) (a) b (i)(u) (da)(db)(di)(du)
(a) b (i)(u) (a)(b) i (u) (da) db di (du)
(a)(b) i (u) (a) b (i)(u) (da) db di (du)
(a)(b) i (u) (a)(b) i (u) (da)(db)(di)(du)
Table 60.1 LIR3 (A): Lateral Representation of A
Object Sign Interpretant
<A>X <a>Y <a>Y
<A>X <a>Y <i>Y
<A>X <i>Y <a>Y
<A>X <i>Y <i>Y
<B>X <b>Y <b>Y
<B>X <b>Y <u>Y
<B>X <u>Y <b>Y
<B>X <u>Y <u>Y
Table 60.2 LIR3 (Den A): Denotative Component of A
Object Sign Transition
<A>X <a>Y <<a>Y, <A>X>
<A>X <i>Y <<i>Y, <A>X>
<B>X <b>Y <<b>Y, <B>X>
<B>X <u>Y <<u>Y, <B>X>
Table 60.3 LIR3 (Con A): Connotative Component of A
Sign Interpretant Transition
<a>Y <a>Y <>dY
<a>Y <i>Y <da di>dY
<i>Y <a>Y <da di>dY
<i>Y <i>Y <>dY
<b>Y <b>Y <>dY
<b>Y <u>Y <db du>dY
<u>Y <b>Y <db du>dY
<u>Y <u>Y <>dY
Table 61.1 LIR3 (B): Lateral Representation of B
Object Sign Interpretant
<A>X <a>Y <a>Y
<A>X <a>Y <u>Y
<A>X <u>Y <a>Y
<A>X <u>Y <u>Y
<B>X <b>Y <b>Y
<B>X <b>Y <i>Y
<B>X <i>Y <b>Y
<B>X <i>Y <i>Y
Table 61.2 LIR3 (Den B): Denotative Component of B
Object Sign Transition
<A>X <a>Y <<a>Y, <A>X>
<A>X <u>Y <<u>Y, <A>X>
<B>X <b>Y <<b>Y, <B>X>
<B>X <i>Y <<i>Y, <B>X>
Table 61.3 LIR3 (Con B): Connotative Component of B
Sign Interpretant Transition
<a>Y <a>Y <>dY
<a>Y <u>Y <da du>dY
<u>Y <a>Y <da du>dY
<u>Y <u>Y <>dY
<b>Y <b>Y <>dY
<b>Y <i>Y <db di>dY
<i>Y <b>Y <db di>dY
<i>Y <i>Y <>dY
</pre>
===6.25. Analytic Intensional Representations===
===6.25. Analytic Intensional Representations===