Changes

</pre>

</pre>

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 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 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 appears in the immediate case:

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

Column 1 lists each boolean element or boolean function under its

ordinary constant name or under a succinct nickname, respectively.

Column 2 lists each boolean function in a style of function name "F^i_j"

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.

that is constructed as follows:  The superscript "i" gives the dimension

of the functional domain, that is, the number of its functional variables,

and the subscript "j" is a binary string that recapitulates the functional

values, using the obvious translation of boolean values into binary values.

Column 3 lists the functional values for each boolean function, or possibly

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.

a boolean element appearing in the guise of a function, for each combination

of its domain values.

Column 4 shows the usual expressions of these elements in the cactus language,

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.

conforming to the practice of omitting the strike-throughs in display formats.

Here I illustrate also the useful convention of sending the expression "(())"

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 14.  Boolean Functions on Zero Variables

Table 15.  Boolean Functions on Zero Variables

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

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

| Constant | Function |                    F()                    | Function |

| Constant | Function |                    F()                    | Function |

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

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

Table 15 presents the boolean functions on one variable, F^1 : %B% -> %B%,

Table 16 presents the boolean functions on one variable, F^1 : %B% -> %B%,

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

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,

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

the function F^1_2  =  F^1_10 is obviously the identity operation.

the function F^1_2  =  F^1_10 is obviously the identity operation.

Table 15.  Boolean Functions on One Variable

Table 16.  Boolean Functions on One Variable

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

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

| Function | Function |                  F(x)                    | Function |

| Function | Function |                  F(x)                    | Function |

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

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

Table 16 presents the boolean functions on two variables, F^2 : %B%^2 -> %B%,

Table 17 presents the boolean functions on two variables, F^2 : %B%^2 -> %B%,

of which there are precisely sixteen in number.  As before, all of the boolean

of which there are precisely sixteen in number.  As before, all of the boolean

functions of fewer variables are subsumed in this Table, though under a set of

functions of fewer variables are subsumed in this Table, though under a set of

[x =/= y]  =  [x + y]  =  F^2_06 (x, y)  =  (x , y).

[x =/= y]  =  [x + y]  =  F^2_06 (x, y)  =  (x , y).

Table 16.  Boolean Functions on Two Variables

Table 17.  Boolean Functions on Two Variables

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

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

| Function | Function |                  F(x, y)                  | Function |

| Function | Function |                  F(x, y)                  | Function |