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If one takes the 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 view that PARCEs and PARCs amount to a pair of intertranslatable representations 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 s_j \downharpoonright</math> of a sentence <math>s_j</math> or the logical denotation <math>\downharpoonleft C_j \downharpoonright</math> of a cactus <math>C_j.</math>

The relations connecting sentences, graphs, and propositions are shown in the next two Tables.

The relations connecting sentences, graphs, and propositions are shown in the next two Tables.

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Between the sentences and the propositions run the graph&#8209;theoretic data structures which arise in the process of parsing sentences and serve to catalyze their potential for logical applications.

Between the sentences and the propositions run the graph&#8209;theoretic data structures which arise in the process of parsing sentences and catalyze their potential for expressing logical propositions or indicator functions.&nbsp; The graph&#8209;theoretic medium supplies an intermediate form of representation between the linguistic sentences and the indicator functions, not only rendering the possibilities of connection between them more readily conceivable in fact but facilitating the necessary translations on a practical basis.

Tables&nbsp;14 and 15 summarize the relations that serve to connect the formal language of sentences with the logical language of propositions.&nbsp; 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.&nbsp; 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.&nbsp; These Tables include this intermediate domain in their Central Columns.&nbsp; 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.&nbsp; Between their Middle and Final Columns they summarize the semantics of interpreting the graphical forms of representation for the purposes of reasoning with propositions.

In each Table the passage from the first to the middle column articulates the mechanics of parsing cactus language sentences into graph&#8209;theoretic data structures while the passage from the middle to the last column articulates the semantics of interpreting cactus graphs as logical propositions or indicator functions.

Aside from their common topic, the two Tables present slightly different ways of conceptualizing the operations that go to establish their maps.&nbsp; Table&nbsp;14 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.&nbsp; Table&nbsp;15 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.&nbsp; 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.&nbsp; Indeed, the "roughly" can be made "exactly" as soon as the domains of a suitable sign relation are specified precisely.

Aside from their common topic, the two Tables present slightly different ways of drawing the maps which go to make up the full semantic transformation.

 

<dt>Semantic Translation &bull; Functional Form</dt>

<dd>The first Table shows the functional associations connecting each domain with the next, taking the triple 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 those ingredients.</dd>

 

<dt>Semantic Translation &bull; Equational Form</dt>

<dd>The second Table records the transitions in the form of equations, treating sentences and graphs as alternative types of signs and generalizing the denotation bracket to indicate the proposition denoted by either.</dd>

 

It should be clear at this point that either scheme of translation puts the triples of sentences, graphs, and propositions roughly in the roles of signs, interpretants, and objects, respectively, of a triadic sign relation.&nbsp; Indeed, the <i>roughly</i> can be rendered <i>exactly</i> as soon as the domains of a suitable sign&nbsp;relation are specified precisely.

A good way to illustrate the action of the conjunction and surjunction operators is to demonstrate how they can be used to construct the boolean functions on any finite number of variables.&nbsp; Let us begin by doing this for the first three cases, <math>k = 0, 1, 2.</math>

A good way to illustrate the action of the conjunction and surjunction operators is to demonstrate how they can be used to construct the boolean functions on any finite number of variables.&nbsp; Let us begin by doing this for the first three cases, <math>k = 0, 1, 2.</math>