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<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>

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</dl>

: The logical conjunction is given by the function <math>F_{8}^{(2)} (x, y) = x \cdot y.</math>

: The logical conjunction is given by the function <math>F_{8}^{(2)} (x, y) = x \cdot y.</math>

: The logical disjunction is given by the function <math>F_{14}^{(2)} (x, y) = \underline{((} ~x~ \underline{)(} ~y~ \underline{))}.</math>

: The logical disjunction is given by the function <math>F_{14}^{(2)} (x, y) = \texttt{((} ~x~ \texttt{)(} ~y~ \texttt{))}.</math>

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 are given in the following ways:

: <math>[x \Rightarrow y] = F_{11}^{(2)} (x, y) = \underline{(} ~x~ \underline{(} ~y~ \underline{))} = [\mathrm{not}~ x ~\mathrm{without}~ y].</math>

: <math>[x \Rightarrow y] = F_{11}^{(2)} (x, y) = \texttt{(} ~x~ \texttt{(} ~y~ \texttt{))} = [\mathrm{not}~ x ~\mathrm{without}~ y].</math>

: <math>[x \Leftarrow y] = F_{13}^{(2)} (x, y) = \underline{((} ~x~ \underline{)} ~y~ \underline{)} = [\mathrm{not}~ y ~\mathrm{without}~ x].</math>

: <math>[x \Leftarrow y] = F_{13}^{(2)} (x, y) = \texttt{((} ~x~ \texttt{)} ~y~ \texttt{)} = [\mathrm{not}~ y ~\mathrm{without}~ x].</math>

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 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:

: <math>[x \Leftrightarrow y] = [x = y] = F_{9}^{(2)} (x, y) = \underline{((} ~x~,~y~ \underline{))}.</math>

: <math>[x \Leftrightarrow y] = [x = y] = F_{9}^{(2)} (x, y) = \texttt{((} ~x~,~y~ \texttt{))}.</math>

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.&nbsp; This function is given by:

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.&nbsp; This function is given by:

: <math>[x \neq y] = [x + y] = F_{6}^{(2)} (x, y) = \underline{(} ~x~,~y~ \underline{)}.</math>

: <math>[x \neq y] = [x + y] = F_{6}^{(2)} (x, y) = \texttt{(} ~x~,~y~ \texttt{)}.</math>

Let me now address one last question that may have occurred to some.&nbsp; 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:&nbsp; 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>?&nbsp; My answer is this:&nbsp; 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.&nbsp; 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:&nbsp; 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>?&nbsp; My answer is this:&nbsp; 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.