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===6.25. Analytic Intensional Representations===
===6.25. Analytic Intensional Representations===
<pre>
In this section the ER's of A and B are translated into a variety of different IR's that actually accomplish some measure of analytic work. These are referred to as "analytic intensional representations" (AIR's). This strategy of representation is also called the "structural coding" or the "sensitive coding", because it pays attention to the structure of its object domain and attends to the nuances of each sign's interpretation to fashion its code, or the "log(n) coding", because it uses roughly log2(n) binary features to represent a domain of n elements.
For the domain O = {A, B} of two elements one needs to use a single logical feature. It is often convenient to use an object feature that is relative to the interpreter using it, for instance, telling whether the object described is the self or the other.
For the domain S = I = {"A", "B", "i", "u"} of four elements one needs to use two logical features. One possibility is to classify each element: according to its syntactic category, as being a noun or a pronoun, and according to its semantic category, as denoting the self or the other.
Tables 62.1 through 62.3 show several ways of representing these categories in terms of feature value pairs and propositional codes. In each Table, Column 1 describes the category in question, Column 2 gives the mnemonic form of a propositional expression for that category, and Column 3 gives the abbreviated form of that expression, using a notation for propositional calculus where parentheses circumscribing a term or expression are interpreted as forming its logical negation.
Table 62.1 Analytic Codes for Object Features
Category Mnemonic Code
Self self s
Other (self) (s)
Table 62.2 Analytic Codes for Semantic Features
Category Mnemonic Code
1st Person my m
2nd Person (my) (m)
Table 62.3 Analytic Codes for Syntactic Features
Category Mnemonic Code
Noun name n
Pronoun (name) (n)
Tables 63 and 64 list the codes for each element of the world domain W = { A, B, "A", "B", "i", "u"}, giving all features relative to the interpreters A and B, respectively.
Table 63. Analytic Codes for Interpreter A
Name Vector Conjunct Mnemonic Code
A 1X x1 self s
B 0X (x1) (self) (s)
"A" 11Y y1 y2 my name m n
"B" 01Y (y1) y2 (my) name (m) n
"i" 10Y y1 (y2) my (name) m (n)
"u" 00Y (y1)(y2) (my)(name) (m)(n)
Table 64. Analytic Codes for Interpreter B
Name Vector Conjunct Mnemonic Code
A 0X (x1) (self) (s)
B 1X x1 self s
"A" 01Y (y1) y2 (my) name (m) n
"B" 11Y y1 y2 my name m n
"i" 10Y y1 (y2) my (name) m (n)
"u" 00Y (y1)(y2) (my)(name) (m)(n)
Tables 65.1 and 66.1 transcribe the sign relations A and B, respectively, into the forms of the AIR just suggested. Tables 65.2 and 66.2 extract the denotative components of A and B, respectively, and isolate the transitions from signs to objects as ordered pairs of the form <s, o>. Tables 65.3 and 66.3 extract the connotative components of A and B, respectively, and represent the transitions from signs to interpretants in terms of differential features, in other words, as propositions in the differential extension of the syntactic domain.
Table 65.1 AIR1 (A): Analytic Representation of A
Object Sign Interpretant
s m n m n
s m n m (n)
s m (n) m n
s m (n) m (n)
(s) (m) n (m) n
(s) (m) n (m)(n)
(s) (m)(n) (m) n
(s) (m)(n) (m)(n)
Table 65.2 AIR1 (Den A): Denotative Component of A
Object Sign Transition
s m n < m n , s >
s m (n) < m (n), s >
(s) (m) n <(m) n , (s)>
(s) (m)(n) <(m)(n), (s)>
Table 65.3 AIR1 (Con A): Connotative Component of A
Sign Interpretant Transition
m n m n (dm)(dn)
m n m (n) (dm) dn
m (n) m n (dm) dn
m (n) m (n) (dm)(dn)
(m) n (m) n (dm)(dn)
(m) n (m)(n) (dm) dn
(m)(n) (m) n (dm) dn
(m)(n) (m)(n) (dm)(dn)
Table 66.1 AIR1 (B): Analytic Representation of B
Object Sign Interpretant
(s) (m) n (m) n
(s) (m) n (m)(n)
(s) (m)(n) (m) n
(s) (m)(n) (m)(n)
s m n m n
s m n m (n)
s m (n) m n
s m (n) m (n)
Table 66.2 AIR1 (Den B): Denotative Component of B
Object Sign Transition
(s) (m) n <(m) n , (s)>
(s) (m)(n) <(m)(n), (s)>
s m n < m n , s >
s m (n) < m (n), s >
Table 66.3 AIR1 (Con B): Connotative Component of B
Sign Interpretant Transition
(m) n (m) n (dm)(dn)
(m) n (m)(n) (dm) dn
(m)(n) (m) n (dm) dn
(m)(n) (m)(n) (dm)(dn)
m n m n (dm)(dn)
m n m (n) (dm) dn
m (n) m n (dm) dn
m (n) m (n) (dm)(dn)
Table 67.1 AIR2 (A): Analytic Representation of A
Object Sign Interpretant
<*>X <*>Y <*>Y
<*>X <*>Y <m>Y
<*>X <m>Y <*>Y
<*>X <m>Y <m>Y
<!>X <n>Y <n>Y
<!>X <n>Y <!>Y
<!>X <!>Y <n>Y
<!>X <!>Y <!>Y
Table 67.2 AIR2 (Den A): Denotative Component of A
Object Sign Transition
<*>X <*>Y <<*>Y, <*>X>
<*>X <m>Y <<m>Y, <*>X>
<!>X <n>Y <<n>Y, <!>X>
<!>X <!>Y <<!>Y, <!>X>
Table 67.3 AIR2 (Con A): Connotative Component of A
Sign Interpretant Transition
<*>Y <*>Y <d!>dY
<*>Y <m>Y <dn>dY
<m>Y <*>Y <dn>dY
<m>Y <m>Y <d!>dY
<n>Y <n>Y <d!>dY
<n>Y <!>Y <dn>dY
<!>Y <n>Y <dn>dY
<!>Y <!>Y <d!>dY
Table 68.1 AIR2 (B): Analytic Representation of B
Object Sign Interpretant
<!>X <n>Y <n>Y
<!>X <n>Y <!>Y
<!>X <!>Y <n>Y
<!>X <!>Y <!>Y
<*>X <*>Y <*>Y
<*>X <*>Y <m>Y
<*>X <m>Y <*>Y
<*>X <m>Y <m>Y
Table 68.2 AIR2 (Den B): Denotative Component of B
Object Sign Transition
<!>X <n>Y <<n>Y, <!>X>
<!>X <!>Y <<!>Y, <!>X>
<*>X <*>Y <<*>Y, <*>X>
<*>X <m>Y <<m>Y, <*>X>
Table 68.3 AIR2 (Con B): Connotative Component of B
Sign Interpretant Transition
<n>Y <n>Y <d!>dY
<n>Y <!>Y <dn>dY
<!>Y <n>Y <dn>dY
<!>Y <!>Y <d!>dY
<*>Y <*>Y <d!>dY
<*>Y <m>Y <dn>dY
<m>Y <*>Y <dn>dY
<m>Y <m>Y <d!>dY
</pre>
===6.26. Differential Logic and Directed Graphs===
===6.26. Differential Logic and Directed Graphs===
