Changes

Aside from their common topic, the two Tables present slightly different ways of conceptualizing the operations that go to establish their maps.  Table&nbsp;14.1 records the functional associations that connect each domain with the next, taking the triplings of a sentence <math>s_j,\!</math> a cactus <math>C_j,\!</math> and a proposition <math>q_j\!</math> as basic data, and fixing the rest by recursion on these.  Table&nbsp;14.2 records these associations in the form of equations, treating sentences and graphs as alternative kinds of signs, and generalizing the denotation bracket operator to indicate the proposition that either denotes.  It should be clear at this point that either scheme of translation puts the sentences, the graphs, and the propositions that it associates with each other roughly in the roles of the signs, the interpretants, and the objects, respectively, whose triples define an appropriate sign relation.  Indeed, the "roughly" can be made "exactly" as soon as the domains of a suitable sign relation are specified precisely.

Aside from their common topic, the two Tables present slightly different ways of conceptualizing the operations that go to establish their maps.  Table&nbsp;14.1 records the functional associations that connect each domain with the next, taking the triplings of a sentence <math>s_j,\!</math> a cactus <math>C_j,\!</math> and a proposition <math>q_j\!</math> as basic data, and fixing the rest by recursion on these.  Table&nbsp;14.2 records these associations in the form of equations, treating sentences and graphs as alternative kinds of signs, and generalizing the denotation bracket operator to indicate the proposition that either denotes.  It should be clear at this point that either scheme of translation puts the sentences, the graphs, and the propositions that it associates with each other roughly in the roles of the signs, the interpretants, and the objects, respectively, whose triples define an appropriate sign relation.  Indeed, the "roughly" can be made "exactly" as soon as the domains of a suitable sign relation are specified precisely.

A good way to illustrate the action of the conjunction and surjunction operators is to demonstate how they can be used to construct all of the boolean functions on <math>k\!</math> variables, just now, let us say, for <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.  Let us begin by doing this for the first three cases, <math>k = 0, 1, 2.\!</math>

A boolean function on 0 variables is just a boolean constant <math>F^{(0)}\!</math> in the boolean domain <math>\underline\mathbb{B} = \{ \underline{0}, \underline{1} \}.</math>  Table&nbsp;15 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 appears in the immediate case:

A boolean function on 0 variables is just a boolean constant <math>F^{(0)}\!</math> in the boolean domain <math>\underline\mathbb{B} = \{ \underline{0}, \underline{1} \}.</math>  Table&nbsp;15 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.

 

 

 

 

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.  Here I illustrate also the useful convention of using the expression <math>^{\backprime\backprime} ((~)) ^{\prime\prime}</math> as a visible stand-in for the expression of a constantly "true" truth value, one that would otherwise be represented by a blank expression, and tend to elude our giving it much notice in the context of more demonstrative texts.

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|          |          |                                          |          |

|          |          |                                          |          |

o----------o----------o-------------------------------------------o----------o

o----------o----------o-------------------------------------------o----------o

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

of which there are precisely four.  Here, Column 1 codes the contents of

Column 2 in a more concise form, compressing the lists of boolean values,

Naturally, the boolean constants reprise themselves in this new setting

as constant functions on one variable. Thus, one has the synonymous

| F^1_0 = F^1_00 = %0% : %B% -> %B%

Column&nbsp;2 lists each boolean function in a style of function name <math>F^i_j\!</math> that is constructed as follows: The superscript <math>i\!</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.

 

| F^1_3  = F^1_11  = %1% : %B% -> %B%

 

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. Here I illustrate also the useful convention of using the expression <math>^{\backprime\backprime} ((~)) ^{\prime\prime}</math> as a visible stand-in for the expression of a constantly "true" truth value, one that would otherwise be represented by a blank expression, and tend to elude our giving it much notice in the context of more demonstrative texts.

 

Table 16 presents the boolean functions on one variable, <math>F^{(1)} : \underline\mathbb{B} \to \underline\mathbb{B},</math> of which there are precisely four. 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.  Naturally, the boolean constants reprise themselves in this new setting as constant functions on one variable.  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%"

F^{(1)}_0 & = & F^{(1)}_{00} & = & \underline{0} : \underline\mathbb{B} \to \underline\mathbb{B}

F^{(1)}_3 & = & F^{(1)}_{11} & = & \underline{1} : \underline\mathbb{B} \to \underline\mathbb{B}

\end{matrix}</math>

As for the rest, the other two functions are easily recognized as corresponding

As for the rest, the other two functions are easily recognized as corresponding

to the one-place logical connectives, or the monadic operators on %B%.  Thus,

to the one-place logical connectives, or the monadic operators on %B%.  Thus,