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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 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>

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.

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

 

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%"

{| align="center" ccellpadding="10" cellspacing="0" style="text-align:center"

|+ style="height:30px" | <math>\text{Table 14. Semantic Translation : Functional Form}</math>

| style="height:25px; font-size:large" | <math>\text{Table 14. Semantic Translation}</math> &bull; <math>\text{Functional Form}</math>

|- style="height:40px; background:ghostwhite"

| width="20%" | <math>\mathrm{Sentence}</math>

| width="20%" | <math>\xrightarrow[\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}]{\mathrm{Parse}}</math>

|-

|-

|

| [[File:Cactus Language Semantic Translation Functional Form.png|600px]]

{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"

|}

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<br>

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{| align="center" ccellpadding="10" cellspacing="0" style="text-align:center"

|+ style="height:30px" | <math>\text{Table 15. Semantic Translation : Equational Form}</math>

| style="height:25px; font-size:large" | <math>\text{Table 15. Semantic Translation}</math> &bull; <math>\text{Equational Form}</math>

|- style="height:40px; background:ghostwhite"

| width="20%" | <math>\downharpoonleft \mathrm{Sentence} \downharpoonright</math>

| width="20%" | <math>\downharpoonleft \mathrm{Graph} \downharpoonright</math>

|-

|-

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| [[File:Cactus Language Semantic Translation Equational Form.png|600px]]

{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"

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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 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.