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=={{anchor|Semantics}}Cactus Language • Semantics==
==Cactus Language • Semantics==
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{| align="center" cellpadding="0" cellspacing="0" width="90%"
<dd>The <i>bound connective</i> of <math>k</math> places is signified by the surcatenation of the <math>k</math> sentences filling those places.</dd>
<dd>The <i>bound connective</i> of <math>k</math> places is signified by the surcatenation of the <math>k</math> sentences filling those places.</dd>
<dd>For the initial case <math>k = 0,</math> the bound connective is an empty closure, an expression taking one of the forms <math>\texttt{()}, \texttt{(~)}, \texttt{(~~)}, \ldots</math> with any number of spaces between the parentheses, all of which have the same denotation among propositions.</dd>
<dd>For the initial case <math>k = 0,</math> the bound connective is an empty closure, an expression taking one of the forms <math>\texttt{()}, \texttt{( )}, \texttt{( )}, \ldots</math> with any number of spaces between the parentheses, all of which have the same denotation among propositions.</dd>
<dd>For the generic case <math>k > 0,</math> the bound connective takes the form <math>\texttt{(} s_1 \texttt{,} \ldots \texttt{,} s_k \texttt{)}.</math></dd>
<dd>For the generic case <math>k > 0,</math> the bound connective takes the form <math>\texttt{(} s_1 \texttt{,} \ldots \texttt{,} s_k \texttt{)}.</math></dd>
<dd>The <i>logical denotation of a node</i> is the logical conjunction of that node's arguments, which are defined as the logical denotations of that node's attachments.</dd>
<dd>The <i>logical denotation of a node</i> is the logical conjunction of that node's arguments, which are defined as the logical denotations of that node's attachments.</dd>
<dd>The logical denotation of either a blank symbol or empty node is the boolean value <math>\underline{1} = \mathrm{true}.</math></dd>
<dd>The logical denotation of either a blank symbol or empty node is the boolean value <math>1 = \mathrm{true}.</math></dd>
<dd>The logical denotation of the paint <math>\mathfrak{p}_j</math> is the proposition <math>p_j,</math> a proposition regarded as <i>primitive</i>, at least, with respect to the level of analysis represented in the current instance of <math>\mathfrak{C} (\mathfrak{P}).</math></dd>
<dd>The logical denotation of the paint <math>\mathfrak{p}_j</math> is the proposition <math>p_j,</math> a proposition regarded as <i>primitive</i>, at least, with respect to the level of analysis represented in the current instance of <math>\mathfrak{C} (\mathfrak{P}).</math></dd>
<dd>The <i>logical denotation of a lobe</i> is the logical surjunction of that lobe's arguments, which are defined as the logical denotations of that lobe's appendants.</dd>
<dd>The <i>logical denotation of a lobe</i> is the logical surjunction of that lobe's arguments, which are defined as the logical denotations of that lobe's appendants.</dd>
<dd>As a corollary, the logical denotation of the parse graph of <math>\texttt{()},</math> also known as a <i>needle</i>, is the boolean value <math>\underline{0} = \mathrm{false}.</math></dd>
<dd>As a corollary, the logical denotation of the parse graph of <math>\texttt{()},</math> also known as a <i>needle</i>, is the boolean value <math>0 = \mathrm{false}.</math></dd>
</dl>
</dl>
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 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.
{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"
|+ style="height:25px; font-size:large" | <math>\text{Table 14. Semantic Translation}</math> • <math>\text{Functional Form}</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%"
|+ style="height:30px" | <math>\text{Table 14. Semantic Translation : Functional Form}</math>
|-
|-
| width="20%" | <math>\mathrm{Surc}^k_j s_j</math>
| [[File:Cactus Language Semantic Translation Functional Form.png|600px]]
|}
|}
<br>
<br>
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{| align="center" cellpadding="0" 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> • <math>\text{Equational Form}</math>
|-
|-
|
| [[File:Cactus Language Semantic Translation Equational Form.png|600px]]
|}
|}
<br>
Between the sentences and the propositions run the graph‑theoretic data structures which arise in the process of parsing sentences and catalyze their potential for expressing logical propositions or indicator functions. The graph‑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.
In each Table the passage from the first to the middle column articulates the mechanics of parsing cactus language sentences into graph‑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 drawing the maps which go to make up the full semantic transformation.
<dl style="margin-left:28px">
<dt>Semantic Translation • 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 • 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 type.</dd>
</dl>
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. Indeed, the <i>roughly</i> can be rendered <i>exactly</i> as soon as the domains of a suitable sign relation are specified precisely.
A boolean function <math>F^{(0)}</math> on <math>0</math> variables is just an element of the boolean domain <math>\mathbb{B} = \{ 0, 1 \}.</math> Table 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.
A good way to illustrate the action of the conjunction and surjunction operators is to show how they can be used to construct the boolean functions on any finite number of variables. Though it's not much to look at let's start with the case of zero variables, boolean constants by any other word, partly for completeness and partly to supply an anchor for the cases in its train.
<br>
A boolean function <math>F^{(0)}</math> on zero variables is just an element of the boolean domain <math>\mathbb{B} = \{ 0, 1 \}.</math> The following Table shows several ways of referring to those elements, for the sake of consistency using the same format we'll use in subsequent Tables, however degenerate it appears in this case.
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%"
{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"
|+ style="height:30px" | <math>\text{Table 16. Boolean Functions on Zero Variables}</math>
|+ style="height:25px; font-size:large" | <math>\text{Table 16. Boolean Functions on Zero Variables}</math>
|-
|-
| <math>1</math>
| [[File:Boolean Functions on Zero Variables.png|600px]]
| <math>F_1^{(0)}</math>
|}
|}
<br>
<ul>
<li>Column 1 lists each boolean element or boolean function under its ordinary constant name or under a succinct nickname, respectively.</li>
Column 1 lists each boolean element or boolean function under its ordinary constant name or under a succinct nickname, respectively.
Column 3 lists the functional values for each boolean function, or possibly a boolean element appearing in the guise of a function, for each combination of its domain values.
<li>Column 2 lists each boolean function by means of a function name <math>F_j^{(k)}</math> of the following form. The superscript <math>(k)</math> gives the dimension of the functional domain, in effect, the number of variables, and the subscript <math>j</math> is a binary string formed from the functional values, using the obvious coding of boolean values into binary values.</li>
Column 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>\text{“} ((~)) \text{”}</math> as a visible stand-in for the expression of the logical value <math>\mathrm{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.
<li>Column 3 lists the values each function takes for each combination of its domain values.</li>
<li>Column 4 lists the ordinary cactus expressions for each boolean function. Here, as usual, the expression <math>\text{“} \texttt{(( ))} \text{”}</math> renders the blank expression for logical truth more visible in context.</li>
</ul>
<br>
The next Table shows the four boolean functions on one variable, <math>F^{(1)} : \mathbb{B} \to \mathbb{B}.</math>
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:60%"
{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"
|+ style="height:30px" | <math>\text{Table 17. Boolean Functions on One Variable}</math>
|+ style="height:25px; font-size:large" | <math>\text{Table 17. Boolean Functions on One Variable}</math>
|-
|-
| <math>F_3^{(1)}</math>
| [[File:Boolean Functions on One Variable.png|600px]]
| <math>F_{11}^{(1)}</math>
|}
|}
<br>
<ul><li>Column 1 lists the contents of Column 2 in a more concise form, converting the lists of boolean values in the subscript strings to their decimal equivalents. Naturally, the boolean constants reprise themselves in this new setting as constant functions on one variable. The constant functions are thus expressible in the following equivalent ways.</li></ul>
{| align="center" cellpadding="8" width="90%"
{| align="center" cellpadding="8"
|
|
<math>\begin{matrix}
<math>\begin{matrix}
F_0^{(1)}
F_0^{(1)} & = & F_{00}^{(1)} & = & 0 ~:~ \mathbb{B} \to \mathbb{B}.
& = &
\\[4pt]
F_{00}^{(1)}
F_3^{(1)} & = & F_{11}^{(1)} & = & 1 ~:~ \mathbb{B} \to \mathbb{B}.
& = &
0 ~:~ \mathbb{B} \to \mathbb{B}
\\
F_3^{(1)}
& = &
F_{11}^{(1)}
& = &
1 ~:~ \mathbb{B} \to \mathbb{B}
\end{matrix}</math>
\end{matrix}</math>
|}
|}
<ul><li>The other two functions in the Table are easily recognized as the one‑place logical connectives or the monadic operators on <math>\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.</li></ul>
<br>
The 16 boolean functions on two variables, <math>F^{(2)} : \mathbb{B}^2 \to \mathbb{B},</math> are shown in the following Table.
{| align="center" border="1" cellpadding="4" cellspacing="0" style="text-align:center; width:60%"
{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"
|+ style="height:30px" | <math>\text{Table 18. Boolean Functions on Two Variables}</math>
|+ style="height:25px; font-size:large" | <math>\text{Table 18. Boolean Functions on Two Variables}</math>
|-
|-
| <math>F_{2}^{(2)}</math>
| [[File:Boolean Functions on Two Variables • Truth Table.png|600px]]
| <math>F_{0010}^{(2)}</math>
|}
|}
<br>
As before, all boolean functions on proper subsets of the current variables are subsumed in the Table at hand. In particular, we have the following inclusions.
<ul>
<li>The constant function <math>0 ~:~ \mathbb{B}^2 \to \mathbb{B}</math> appears under the name <math>F_{0}^{(2)}.</math></li>
<li>The constant function <math>1 ~:~ \mathbb{B}^2 \to \mathbb{B}</math> appears under the name <math>F_{15}^{(2)}.</math></li>
<li>The function expressing the assertion of the first variable is <math>F_{12}^{(2)}.</math></li>
<li>The function expressing the negation of the first variable is <math>F_{3}^{(2)}.</math></li>
<li>The function expressing the assertion of the second variable is <math>F_{10}^{(2)}.</math></li>
<li>The function expressing the negation of the second variable is <math>F_{5}^{(2)}.</math></li>
</ul>
Next come the functions on two variables whose output values change depending on changes in both input variables. Notable among them are the following examples.
<ul>
<li>The logical conjunction is given by the function <math>F_{8}^{(2)} (x, y) ~=~ x \cdot y.</math></li>
<li>The logical disjunction is given by the function <math>F_{14}^{(2)} (x, y) ~=~ \texttt{((} ~x~ \texttt{)(} ~y~ \texttt{))}.</math></li>
</ul>
Functions expressing the <i>conditionals</i>, <i>implications</i>, or <i>if-then</i> statements are given in the following ways:
Functions expressing the <i>conditionals</i>, <i>implications</i>, or <i>if‑then</i> statements appear as follows.
<ul>
<li><math>[x \Rightarrow y] ~=~ F_{11}^{(2)} (x, y) ~=~ \texttt{(} ~x~ \texttt{(} ~y~ \texttt{))} ~=~ [\mathrm{not}~ x ~\mathrm{without}~ y].</math></li>
<li><math>[x \Leftarrow y] ~=~ F_{13}^{(2)} (x, y) ~=~ \texttt{((} ~x~ \texttt{)} ~y~ \texttt{)} ~=~ [\mathrm{not}~ y ~\mathrm{without}~ x].</math></li>
</ul>
The function that corresponds to the <i>biconditional</i>, the <i>equivalence</i>, or the <i>if and only</i> statement is exhibited in the following fashion:
The function expressing the <i>biconditional</i>, <i>equivalence</i>, or <i>if‑and‑only‑if</i> statement appears in the following form.
<ul><li><math>[x \Leftrightarrow y] ~=~ [x = y] ~=~ F_{9}^{(2)} (x, y) ~=~ \texttt{((} ~x~ \texttt{,} ~y~ \texttt{))}.</math></li></ul>
Finally, there is a boolean function that is logically associated with the <i>exclusive disjunction</i>, <i>inequivalence</i>, or <i>not equals</i> statement, algebraically associated with the <i>binary sum</i> operation, and geometrically associated with the <i>symmetric difference</i> of sets. This function is given by:
Finally, the boolean function expressing the <i>exclusive disjunction</i>, <i>inequivalence</i>, or <i>not equals</i> statement, algebraically associated with the <i>binary sum</i> operation, and geometrically associated with the <i>symmetric difference</i> of sets, appears as follows.
<ul><li><math>[x \neq y] ~=~ [x + y] ~=~ F_{6}^{(2)} (x, y) ~=~ \texttt{(} ~x~ \texttt{,} ~y~ \texttt{)}.</math></li></ul>
Let me now address one last question that may have occurred to some. What has happened, in this suggested scheme of functional reasoning, to the distinction that is quite pointedly made by careful logicians between (1) the connectives called <i>conditionals</i> and symbolized by the signs <math>(\rightarrow)</math> and <math>(\leftarrow),</math> and (2) the assertions called <i>implications</i> and symbolized by the signs <math>(\Rightarrow)</math> and <math>(\Leftarrow)</math>, and, in a related question: What has happened to the distinction that is equally insistently made between (3) the connective called the <i>biconditional</i> and signified by the sign <math>(\leftrightarrow)</math> and (4) the assertion that is called an <i>equivalence</i> and signified by the sign <math>(\Leftrightarrow)</math>? My answer is this: For my part, I am deliberately avoiding making these distinctions at the level of syntax, preferring to treat them instead as distinctions in the use of boolean functions, turning on whether the function is mentioned directly and used to compute values on arguments, or whether its inverse is being invoked to indicate the fibers of truth or untruth under the propositional function in question.
Let me now address one last question that may have occurred to some. What has happened, in this suggested scheme of functional reasoning, to the distinction that is quite pointedly made by careful logicians between (1) the connectives called <i>conditionals</i> and symbolized by the signs <math>(\rightarrow)</math> and <math>(\leftarrow),</math> and (2) the assertions called <i>implications</i> and symbolized by the signs <math>(\Rightarrow)</math> and <math>(\Leftarrow)</math>, and, in a related question: What has happened to the distinction that is equally insistently made between (3) the connective called the <i>biconditional</i> and signified by the sign <math>(\leftrightarrow)</math> and (4) the assertion that is called an <i>equivalence</i> and signified by the sign <math>(\Leftrightarrow)</math>? My answer is this: For my part, I am deliberately avoiding making these distinctions at the level of syntax, preferring to treat them instead as distinctions in the use of boolean functions, turning on whether the function is mentioned directly and used to compute values on arguments, or whether its inverse is being invoked to indicate the fibers of truth or untruth under the propositional function in question.