ONT Re: Differential Logic
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DLOG. Note D11
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Tables of Propositional Forms
| To the scientist longing for non-quantitative techniques, then,
| mathematical logic brings hope. It provides explicit techniques
| for manipulating the most basic ingredients of discourse.
|
| W.V. Quine, 'Mathematical Logic', [Qui, 7-8]
To prepare for the next phase of discussion, Tables 6 and 7 collect
and summarize all of the propositional forms on one and two variables.
These propositional forms are represented over bases of boolean variables
as complete sets of boolean-valued functions. Adjacent to their names and
specifications are listed what are roughly the simplest expressions in the
"cactus language", the particular syntax for propositional calculus that
I use in formal and computational contexts. For the sake of orientation,
the English paraphrases and the more common notations are listed in the
last two columns. As simple and circumscribed as these low-dimensional
universes may appear to be, a careful exploration of their differential
extensions will involve us in complexities sufficient to demand our
attention for some time to come.
Propositional forms on one variable correspond to boolean functions f : B^1 -> B.
In Table 6 these functions are listed in a variant form of truth table, one which
rotates the axes of the usual arrangement. Each function f_i is indexed by the
string of values that it takes on the points of the universe X% = [x] ~=~ B^1.
The binary index generated in this way is converted to its decimal equivalent,
and these are used as conventional names for the f_i, as shown in the first
column of the Table. In their own right the 2^1 points of the universe X%
are coordinated as a space of type B^1, this in light of the universe X%
being a functional domain where the coordinate projection x takes on
its values in B.
Table 6. Propositional Forms on One Variable
o---------o---------o---------o----------o------------------o----------o
| L_1 | L_2 | L_3 | L_4 | L_5 | L_6 |
| | | | | | |
| Decimal | Binary | Vector | Cactus | English | Ordinary |
o---------o---------o---------o----------o------------------o----------o
| | x : 1 0 | | | |
o---------o---------o---------o----------o------------------o----------o
| | | | | | |
| f_0 | f_00 | 0 0 | ( ) | false | 0 |
| | | | | | |
| f_1 | f_01 | 0 1 | (x) | not x | ~x |
| | | | | | |
| f_2 | f_10 | 1 0 | x | x | x |
| | | | | | |
| f_3 | f_11 | 1 1 | (( )) | true | 1 |
| | | | | | |
o---------o---------o---------o----------o------------------o----------o
Propositional forms on two variables correspond to boolean functions f : B^2 -> B.
In Table 7 each function f_i is indexed by the values that it takes on the points
of the universe X% = [x, y] ~=~ B^2. Converting the binary index thus generated
to a decimal equivalent, we obtain the functional nicknames that are listed in
the first column. The 2^2 points of the universe X% are coordinated as a space
of type B^2, as indicated under the heading of the Table, where the coordinate
projections x and y run through the various combinations of their values in B.
Table 7. Propositional Forms On Two Variables
o---------o---------o---------o----------o------------------o----------o
| L_1 | L_2 | L_3 | L_4 | L_5 | L_6 |
| | | | | | |
| Decimal | Binary | Vector | Cactus | English | Ordinary |
o---------o---------o---------o----------o------------------o----------o
| | x : 1 1 0 0 | | | |
| | y : 1 0 1 0 | | | |
o---------o---------o---------o----------o------------------o----------o
| | | | | | |
| f_0 | f_0000 | 0 0 0 0 | () | false | 0 |
| | | | | | |
| f_1 | f_0001 | 0 0 0 1 | (x)(y) | neither x nor y | ~x & ~y |
| | | | | | |
| f_2 | f_0010 | 0 0 1 0 | (x) y | y and not x | ~x & y |
| | | | | | |
| f_3 | f_0011 | 0 0 1 1 | (x) | not x | ~x |
| | | | | | |
| f_4 | f_0100 | 0 1 0 0 | x (y) | x and not y | x & ~y |
| | | | | | |
| f_5 | f_0101 | 0 1 0 1 | (y) | not y | ~y |
| | | | | | |
| f_6 | f_0110 | 0 1 1 0 | (x, y) | x not equal to y | x + y |
| | | | | | |
| f_7 | f_0111 | 0 1 1 1 | (x y) | not both x and y | ~x v ~y |
| | | | | | |
| f_8 | f_1000 | 1 0 0 0 | x y | x and y | x & y |
| | | | | | |
| f_9 | f_1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y |
| | | | | | |
| f_10 | f_1010 | 1 0 1 0 | y | y | y |
| | | | | | |
| f_11 | f_1011 | 1 0 1 1 | (x (y)) | not x without y | x => y |
| | | | | | |
| f_12 | f_1100 | 1 1 0 0 | x | x | x |
| | | | | | |
| f_13 | f_1101 | 1 1 0 1 | ((x) y) | not y without x | x <= y |
| | | | | | |
| f_14 | f_1110 | 1 1 1 0 | ((x)(y)) | x or y | x v y |
| | | | | | |
| f_15 | f_1111 | 1 1 1 1 | (()) | true | 1 |
| | | | | | |
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Jon Awbrey
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