Changes

The relations connecting sentences, graphs, and propositions are shown in the next two Tables.

The relations connecting sentences, graphs, and propositions are shown in the next two Tables.

{| align="center" ccellpadding="10" cellspacing="0" style="text-align:center"

{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"

| 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>

|-

|-

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

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

<br>

<br>

{| align="center" ccellpadding="10" cellspacing="0" style="text-align:center"

{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"

| 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>

|-

|-

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

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

<dt>Semantic Translation &bull; Equational Form</dt>

<dt>Semantic Translation &bull; 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.</dd>

<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>

</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.&nbsp; Indeed, the <i>roughly</i> can be rendered <i>exactly</i> as soon as the domains of a suitable sign&nbsp;relation are specified precisely.

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.&nbsp; Indeed, the <i>roughly</i> can be rendered <i>exactly</i> as soon as the domains of a suitable sign&nbsp;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.&nbsp; 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 show how they can be used to construct the boolean functions on any finite number of variables.&nbsp; 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.

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>&nbsp; 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.

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>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%"

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

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

|-

|-

| <math>1</math>

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

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

|}

|}

<br>

<ul>

 

<li>Column&nbsp;1 lists each boolean element or boolean function under its ordinary constant name or under a succinct nickname, respectively.</li>

Column&nbsp;1 lists each boolean element or boolean function under its ordinary constant name or under a succinct nickname, respectively.

 

Column&nbsp;2 lists each boolean function in a style of function name <math>F_j^{(k)}</math> that is constructed as follows:&nbsp; The superscript <math>(k)</math> gives the dimension of the functional domain, that is, the number of its functional variables, and the subscript <math>j</math> is a binary string that recapitulates the functional values, using the obvious translation of boolean values into binary values.

Column&nbsp;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&nbsp;2 lists each boolean function by means of a function name <math>F_j^{(k)}</math> of the following form.&nbsp; 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&nbsp;4 shows the usual expressions of these elements in the cactus language, conforming to the practice of omitting the underlines in display formats.&nbsp; 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&nbsp;3 lists the values each function takes for each combination of its domain values.</li>

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

<li>Column&nbsp;4 lists the ordinary cactus expressions for each boolean function.&nbsp; Here, as usual, the expression <math>\text{“} \texttt{(( ))} \text{”}</math> renders the blank expression for logical truth more visible in context.</li>

<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>x</math>

|-

|-

| <math>F_3^{(1)}</math>

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

| <math>F_{11}^{(1)}</math>

|}

|}

<br>

<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>

 

Here, Column&nbsp;1 codes the contents of Column&nbsp;2 in a more concise form, compressing the lists of boolean values, recorded as bits in the subscript string, into their decimal equivalents.&nbsp; Naturally, the boolean constants reprise themselves in this new setting as constant functions on one variable.&nbsp; Thus, one has the synonymous expressions for constant functions that are expressed in the next two chains of equations:

{| 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>

|}

|}

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>\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.

<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.

<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>\texttt{((} x \texttt{)(} y \texttt{))}</math>

|-

|-

| <math>F_{15}^{(2)}</math>

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

| <math>F_{1111}^{(2)}</math>

|}

|}

<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.