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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 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>

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.

Tables&nbsp;14 and 15 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.

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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"

|+ '''Table 14.1 Semantic Translation : Functional Form'''

|+ '''Table 14.  Semantic Translation : Functional Form'''

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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"

|+ '''Table 14.2 Semantic Translation : Equational Form'''

|+ '''Table 15.  Semantic Translation : Equational Form'''

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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.  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&nbsp;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.

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 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&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.  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 operators is to demonstrate how they can be used to construct the boolean functions on any finite number of variables.  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.  Let us begin by doing this for the first three cases, <math>k = 0, 1, 2.\!</math>

A boolean function <math>F^{(0)}\!</math> on <math>0\!</math> variables is just an element of the boolean domain <math>\underline\mathbb{B} = \{ \underline{0}, \underline{1} \}.</math>  Table&nbsp;15 shows several different ways of referring to these elements, just for the sake of consistency using the same format that will be used in subsequent Tables, no matter how degenerate it tends to appear in the initial case.

A boolean function <math>F^{(0)}\!</math> on <math>0\!</math> variables is just an element of the boolean domain <math>\underline\mathbb{B} = \{ \underline{0}, \underline{1} \}.</math>  Table&nbsp;16 shows several different ways of referring to these elements, just for the sake of consistency using the same format that will be used in subsequent Tables, no matter how degenerate it tends to appear in the initial case.

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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"

|+ '''Table 15.  Boolean Functions on Zero Variables'''

|+ '''Table 16.  Boolean Functions on Zero Variables'''

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| width="14%" | <math>F\!</math>

| width="14%" | <math>F\!</math>

Column&nbsp;4 shows the usual expressions of these elements in the cactus language, conforming to the practice of omitting the underlines in display formats.  Here I illustrate also the convention of using the expression <math>^{\backprime\backprime} ((~)) ^{\prime\prime}</math> as a visible stand-in for the expression of the logical value <math>\operatorname{true},</math> a value that is minimally represented by a blank expression that tends to elude our giving it much notice in the context of more demonstrative texts.

Column&nbsp;4 shows the usual expressions of these elements in the cactus language, conforming to the practice of omitting the underlines in display formats.  Here I illustrate also the convention of using the expression <math>^{\backprime\backprime} ((~)) ^{\prime\prime}</math> as a visible stand-in for the expression of the logical value <math>\operatorname{true},</math> a value that is minimally represented by a blank expression that tends to elude our giving it much notice in the context of more demonstrative texts.

Table 16 presents the boolean functions on one variable, <math>F^{(1)} : \underline\mathbb{B} \to \underline\mathbb{B},</math> of which there are precisely four.

Table 17 presents the boolean functions on one variable, <math>F^{(1)} : \underline\mathbb{B} \to \underline\mathbb{B},</math> of which there are precisely four.

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{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"

{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"

|+ '''Table 16.  Boolean Functions on One Variable'''

|+ '''Table 17.  Boolean Functions on One Variable'''

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| width="14%" | <math>F\!</math>

| width="14%" | <math>F\!</math>

As for the rest, the other two functions are easily recognized as corresponding to the one-place logical connectives, or the monadic operators on <math>\underline\mathbb{B}.</math>  Thus, the function <math>F_1^{(1)} = F_{01}^{(1)}</math> is recognizable as the negation operation, and the function <math>F_2^{(1)} = F_{10}^{(1)}</math> is obviously the identity operation.

As for the rest, the other two functions are easily recognized as corresponding to the one-place logical connectives, or the monadic operators on <math>\underline\mathbb{B}.</math>  Thus, the function <math>F_1^{(1)} = F_{01}^{(1)}</math> is recognizable as the negation operation, and the function <math>F_2^{(1)} = F_{10}^{(1)}</math> is obviously the identity operation.

Table&nbsp;17 presents the boolean functions on two variables, <math>F^{(2)} : \underline\mathbb{B}^2 \to \underline\mathbb{B},</math> of which there are precisely sixteen.

Table&nbsp;18 presents the boolean functions on two variables, <math>F^{(2)} : \underline\mathbb{B}^2 \to \underline\mathbb{B},</math> of which there are precisely sixteen.

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{| align="center" border="1" cellpadding="4" cellspacing="0" style="text-align:center; width:90%"

{| align="center" border="1" cellpadding="4" cellspacing="0" style="text-align:center; width:90%"

|+ '''Table 17.  Boolean Functions on Two Variables'''

|+ '''Table 18.  Boolean Functions on Two Variables'''

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| width="14%" | <math>F\!</math>

| width="14%" | <math>F\!</math>

Taking up the preceding arrays of particular connections, namely, the boolean functions on two or less variables, it possible to illustrate the use of the stretch operation in a variety of concrete cases.

Taking up the preceding arrays of particular connections, namely, the boolean functions on two or less variables, it possible to illustrate the use of the stretch operation in a variety of concrete cases.

For example, suppose that <math>F\!</math> is a connection of the form <math>F : \underline\mathbb{B}^2 \to \underline\mathbb{B},</math> that is, any one of the sixteen possibilities in Table&nbsp;17, while <math>p\!</math> and <math>q\!</math> are propositions of the form <math>p, q : X \to \underline\mathbb{B},</math> that is, propositions about things in the universe <math>X,\!</math> or else the indicators of sets contained in <math>X.\!</math>

For example, suppose that <math>F\!</math> is a connection of the form <math>F : \underline\mathbb{B}^2 \to \underline\mathbb{B},</math> that is, any one of the sixteen possibilities in Table&nbsp;18, while <math>p\!</math> and <math>q\!</math> are propositions of the form <math>p, q : X \to \underline\mathbb{B},</math> that is, propositions about things in the universe <math>X,\!</math> or else the indicators of sets contained in <math>X.\!</math>

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