Changes

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 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 boolean function <math>F^{(0)}</math> on <math>0</math> variables is just an element of the boolean domain <math>\underline\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 <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.

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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.

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.

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

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

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F_{00}^{(1)}

F_{00}^{(1)}

& = &

& = &

0 ~:~ \underline\mathbb{B} \to \underline\mathbb{B}

0 ~:~ \mathbb{B} \to \mathbb{B}

\\

\\

\\

\\

F_{11}^{(1)}

F_{11}^{(1)}

& = &

& = &

1 ~:~ \underline\mathbb{B} \to \underline\mathbb{B}

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>\underline\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.

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.

Table&nbsp;18 presents the boolean functions on two variables, <math>F^{(2)} : \underline\mathbb{B}^2 \to \underline\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.

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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:

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:

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

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

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

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

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

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