Changes

# The logical denotation of a lobe is the logical surjunction of that lobe's arguments, which are defined as the logical denotations of that lobe's accoutrements.  As a corollary, the logical denotation of the parse graph of <math>\underline{(} \underline{)},</math> otherwise called a ''needle'', is the boolean value <math>\underline{0} = \operatorname{false}.</math>

# The logical denotation of a lobe is the logical surjunction of that lobe's arguments, which are defined as the logical denotations of that lobe's accoutrements.  As a corollary, the logical denotation of the parse graph of <math>\underline{(} \underline{)},</math> otherwise called a ''needle'', is the boolean value <math>\underline{0} = \operatorname{false}.</math>

If one takes the point of view that PARCs and PARCEs amount to a pair of intertranslatable languages for the same domain of objects, then denotation brackets of the form <math>\downharpoonleft \ldots \downharpoonright</math> can be used to indicate the logical denotation <math>\downharpoonleft C_j \downharpoonright</math> of a cactus <math>C_j\!</math> or the logical denotation <math>\downharpoonleft s_j \downharpoonright</math> of a sentence <math>s_j.\!</math>

If one takes the point of view that PARC's and PARCE's amount to a

pair of intertranslatable languages for the same domain of objects,

then the "spiny bracket" notation, as in "-[C_j]-" or "-[S_j]-",

can be used on either domain of signs to indicate the logical

denotation of a cactus C_j or the logical denotation of

a sentence S_j, respectively.

Tables 13.1 and 13.2 summarize the relations that serve to connect the

Tables&nbsp;14.1 and 14.2 summarize the relations that serve to connect the formal language of sentences with the logical language of propositions. Between these two realms of expression there is a family of graphical data structures that arise in parsing the sentences and that serve to facilitate the performance of computations on the indicator functions. The graphical language supplies an intermediate form of representation between the formal sentences and the indicator functions, and the form of mediation that it provides is very useful in rendering the possible connections between the other two languages conceivable in fact, not to mention in carrying out the necessary translations on a practical basis. These Tables include this intermediate domain in their Central Columns. Between their First and Middle Columns they illustrate the mechanics of parsing the abstract sentences of the cactus language into the graphical data structures of the corresponding species.  Between their Middle and Final Columns they summarize the semantics of interpreting the graphical forms of representation for the purposes of reasoning with propositions.

formal language of sentences with the logical language of propositions.

Between these two realms of expression there is a family of graphical

data structures that arise in parsing the sentences and that serve to

facilitate the performance of computations on the indicator functions.

The graphical language supplies an intermediate form of representation

between the formal sentences and the indicator functions, and the form

of mediation that it provides is very useful in rendering the possible

connections between the other two languages conceivable in fact, not to

mention in carrying out the necessary translations on a practical basis.

These Tables include this intermediate domain in their Central Columns.

Between their First and Middle Columns they illustrate the mechanics of

parsing the abstract sentences of the cactus language into the graphical

data structures of the corresponding species.  Between their Middle and

Final Columns they summarize the semantics of interpreting the graphical

forms of representation for the purposes of reasoning with propositions.

Table 13.1  Semantic Translations: Functional Form

Table 14.1  Semantic Translations : Functional Form

o-------------------o-----o-------------------o-----o-------------------o

o-------------------o-----o-------------------o-----o-------------------o

|                  | Par |                  | Den |                  |

|                  | Par |                  | Den |                  |

o-------------------o-----o-------------------o-----o-------------------o

o-------------------o-----o-------------------o-----o-------------------o

Table 13.2  Semantic Translations: Equational Form

Table 14.2  Semantic Translations : Equational Form

o-------------------o-----o-------------------o-----o-------------------o

o-------------------o-----o-------------------o-----o-------------------o

|                  | Par |                  | Den |                  |

|                  | Par |                  | Den |                  |

|                  |    |                  |    |                  |

|                  |    |                  |    |                  |

o-------------------o-----o-------------------o-----o-------------------o

o-------------------o-----o-------------------o-----o-------------------o

Aside from their common topic, the two Tables present slightly different

Aside from their common topic, the two Tables present slightly different ways of conceptualizing the operations that go to establish their maps. Table&nbsp;14.1 records the functional associations that connect each domain with the next, taking the triplings of a sentence <math>s_j,\!</math> a cactus <math>C_j,\!</math> and a proposition <math>q_j</math> as basic data, and fixing the rest by recursion on these. Table 14.2 records these associations in the form of equations, treating sentences and graphs as alternative kinds of signs, and generalizing the denotation bracket operator to indicate the proposition that either denotes. It should be clear at this point that either scheme of translation puts the sentences, the graphs, and the propositions that it associates with each other roughly in the roles of the signs, the interpretants, and the objects, respectively, whose triples define an appropriate sign relation. Indeed, the "roughly" can be made "exactly" as soon as the domains of a suitable sign relation are specified precisely.

ways of conceptualizing the operations that go to establish their maps.

Table 13.1 records the functional associations that connect each domain

with the next, taking the triplings of a sentence S_j, a cactus C_j, and

a proposition Q_j as basic data, and fixing the rest by recursion on these.

Table 13.2 records these associations in the form of equations, treating

sentences and graphs as alternative kinds of signs, and generalizing the

spiny bracket operator to indicate the proposition that either denotes.

It should be clear at this point that either scheme of translation puts

the sentences, the graphs, and the propositions that it associates with

each other roughly in the roles of the signs, the interpretants, and the

objects, respectively, whose triples define an appropriate sign relation.

Indeed, the "roughly" can be made "exactly" as soon as the domains of

a suitable sign relation are specified precisely.

A good way to illustrate the action of the conjunction and surjunction

A good way to illustrate the action of the conjunction and surjunction

operators is to demonstate how they can be used to construct all of the

operators is to demonstate how they can be used to construct all of the