Changes

<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>

| <math>F_0^{(0)}</math>

| <math>F_0^{(0)}</math>

| <math>0</math>

| <math>0</math>

| <math>\texttt{(~)}</math>

| <math>\texttt{( )}</math>

|-

|-

| <math>1</math>

| <math>1</math>

| <math>F_1^{(0)}</math>

| <math>F_1^{(0)}</math>

| <math>1</math>

| <math>1</math>

| <math>\texttt{((~))}</math>

| <math>\texttt{(( ))}</math>

|}

|}

| <math>0</math>

| <math>0</math>

| <math>0</math>

| <math>0</math>

| <math>\texttt{(~)}</math>

| <math>\texttt{( )}</math>

|-

|-

| <math>F_1^{(1)}</math>

| <math>F_1^{(1)}</math>

| <math>1</math>

| <math>1</math>

| <math>1</math>

| <math>1</math>

| <math>\texttt{((~))}</math>

| <math>\texttt{(( ))}</math>

|}

|}

| <math>0</math>

| <math>0</math>

| <math>0</math>

| <math>0</math>

| <math>\texttt{(~)}</math>

| <math>\texttt{( )}</math>

|-

|-

| <math>F_{1}^{(2)}</math>

| <math>F_{1}^{(2)}</math>

| <math>1</math>

| <math>1</math>

| <math>1</math>

| <math>1</math>

| <math>\texttt{((~))}</math>

| <math>\texttt{(( ))}</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) = \texttt{((} ~x~,~y~ \texttt{))}.</math>

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

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