User:Jon Awbrey/ZOL

=Initial draft=

Zeroth-order logic is a term in popular use among practitioners for the subject matter otherwise known as Boolean functions, monadic predicate logic, propositional calculus, or sentential calculus. One of the advantages of this terminology is that it institutes a higher level of abstraction in which the more inessential differences between these various subjects can be subsumed under the pertinent isomorphisms.

By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions of concrete type X &times; Y &rarr; B and abstract type B &times; B &rarr; B in a number of different languages for zeroth order logic.

Table 1. 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     | |        |         |         |          |                  |          | o-o-o-o--o--o--o

=Working draft=

Zeroth-order logic is a term in popular use among practitioners for the subject matter otherwise known as boolean functions, monadic predicate logic, propositional calculus, or sentential calculus. One of the advantages of this terminology is that it institutes a higher level of abstraction in which the more inessential differences between these various subjects can be subsumed under the pertinent isomorphisms.

By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions of concrete type X &times; Y &rarr; B and abstract type B &times; B &rarr; B in a number of different languages for zeroth order logic.

Table 1. 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     | |        |         |         |          |                  |          | o-o-o-o--o--o--o