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.

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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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| 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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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.

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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.</li>

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.

<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;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;3 lists the values each function takes for each combination of its domain 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;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>

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

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

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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:

<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>

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<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>

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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.

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.