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

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

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

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

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

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

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

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

A boolean function <math>F^{(0)}</math> on zero variables is just an element of the boolean domain <math>\mathbb{B} = \{ 0, 1 \}.</math>&nbsp; The&nbsp;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.

A boolean function <math>F^{(0)}</math> on zero variables is just an element of the boolean domain <math>\mathbb{B} = \{ 0, 1 \}.</math>&nbsp; The&nbsp;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" cellpadding="0" cellspacing="0" style="text-align:center"

 

|+ style="height:25px; font-size:large" | <math>\text{Table 16. Boolean Functions on Zero Variables}</math>

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|+ style="height:30px" | <math>\text{Table 16. Boolean Functions on Zero Variables}</math>

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

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| <math>1</math>

| [[File:Boolean Functions on Zero Variables.png|600px]]

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

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<ul>

<ul>

The next Table shows the four boolean functions on one variable, <math>F^{(1)} : \mathbb{B} \to \mathbb{B}.</math>

The next Table shows the four boolean functions on one variable, <math>F^{(1)} : \mathbb{B} \to \mathbb{B}.</math>

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|+ style="height:25px; font-size:large" | <math>\text{Table 17. Boolean Functions on One Variable}</math>

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|+ style="height:30px" | <math>\text{Table 17. Boolean Functions on One Variable}</math>

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| <math>F_0^{(1)}</math>

| [[File:Boolean Functions on One Variable.png|600px]]

| <math>F_{00}^{(1)}</math>

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<ul><li>Column&nbsp;1 lists the contents of Column&nbsp;2 in a more concise form, converting the lists of boolean values in the subscript strings to their decimal equivalents.&nbsp; Naturally, the boolean constants reprise themselves in this new setting as constant functions on one variable.&nbsp; The constant functions are thus expressible in the following equivalent ways.</li></ul>

<ul><li>Column&nbsp;1 lists the contents of Column&nbsp;2 in a more concise form, converting the lists of boolean values in the subscript strings to their decimal equivalents.&nbsp; Naturally, the boolean constants reprise themselves in this new setting as constant functions on one variable.&nbsp; The constant functions are thus expressible in the following equivalent ways.</li></ul>

<ul><li>The other two functions in the Table are easily recognized as the one&#8209;place logical connectives or the monadic operators on <math>\mathbb{B}.</math>&nbsp; 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>

<ul><li>The other two functions in the Table are easily recognized as the one&#8209;place logical connectives or the monadic operators on <math>\mathbb{B}.</math>&nbsp; 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>

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

The 16 boolean functions on two variables, <math>F^{(2)} : \mathbb{B}^2 \to \mathbb{B},</math> are shown in the following Table.

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|+ style="height:25px; font-size:large" | <math>\text{Table 18. Boolean Functions on Two Variables}</math>

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|+ style="height:30px" | <math>\text{Table 18. Boolean Functions on Two Variables}</math>

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| <math>F_{0}^{(2)}</math>

| [[File:Boolean Functions on Two Variables &bull; Truth Table.png|600px]]

| <math>F_{0000}^{(2)}</math>

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<br>

 

<ul>

 

<li>The constant function <math>1 ~:~ \mathbb{B}^2 \to \mathbb{B}</math> appears under the name <math>F_{15}^{(2)}.</math></li>

As before, all of the boolean functions of fewer variables are subsumed in this Table, though under a set of alternative names and possibly different interpretations.&nbsp; Just to acknowledge a few of the more notable pseudonyms:

<li>The function expressing the assertion of the first variable is <math>F_{12}^{(2)}.</math></li>

: The constant function <math>0 ~:~ \mathbb{B}^2 \to \mathbb{B}</math> appears under the name <math>F_{0}^{(2)}.</math>

<li>The function expressing the negation of the first variable is <math>F_{3}^{(2)}.</math></li>

: The constant function <math>1 ~:~ \mathbb{B}^2 \to \mathbb{B}</math> appears under the name <math>F_{15}^{(2)}.</math>

<li>The function expressing the assertion of the second variable is <math>F_{10}^{(2)}.</math></li>

: The negation and identity of the first variable are <math>F_{3}^{(2)}</math> and <math>F_{12}^{(2)},</math> respectively.

<li>The function expressing the negation of the second variable is <math>F_{5}^{(2)}.</math></li>

</ul>

: The negation and identity of the second variable are <math>F_{5}^{(2)}</math> and <math>F_{10}^{(2)},</math> respectively.

Next come the functions on two variables whose output values change depending on changes in both input variables.&nbsp; Notable among them are the following examples.

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

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

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

<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&#8209;then</i> statements appear as follows.

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

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

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

<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&#8209;and&#8209;only&#8209;if</i> statement appears in the following form.

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

<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.&nbsp; 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.

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

<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.&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.