User:Jon Awbrey/DIF

Thematic Extensions
DLOG. Note D32 Thematization: Truth Tables (cont.) Table 27 summarizes the thematic extensions of all propositions on two variables. Column 4 lists the equations of form (( f^¢, f^¢ [u, v] )) and Column 5 simplifies these equations into the form of algebraic expressions. (As always, "+" refers to exclusive disjunction, and "f" should be read as "[f_i]^¢" in the body of the Table.)

Table 27. Thematization of Bivariate Propositions o-o-o--ooo |      u : 1 1 0 0 |    f     |     theta (f)      |     theta (f)      | |      v : 1 0 1 0 |          |                    |                    | o-o-o--ooo |        |         |          |                    |                    | | f_0     | 0 0 0 0 |        | (( f,        )) | f              + 1 | |        |         |          |                    |                    | | f_1     | 0 0 0 1 |  (u)(v)  | (( f,  (u)(v)  )) | f + u + v + uv     | |        |         |          |                    |                    | | f_2     | 0 0 1 0 |  (u) v   | (( f,  (u) v   )) | f     + v + uv + 1 | |        |         |          |                    |                    | | f_3     | 0 0 1 1 |  (u)     | (( f,  (u)     )) | f + u              | |        |         |          |                    |                    | | f_4     | 0 1 0 0 |   u (v)  | (( f,   u (v)  )) | f + u     + uv + 1 | |        |         |          |                    |                    | | f_5     | 0 1 0 1 |     (v)  | (( f,     (v)  )) | f     + v          | |        |         |          |                    |                    | | f_6     | 0 1 1 0 |  (u, v)  | (( f,  (u, v)  )) | f + u + v      + 1 | |        |         |          |                    |                    | | f_7     | 0 1 1 1 |  (u  v)  | (( f,  (u  v)  )) | f         + uv     | |        |         |          |                    |                    | o-o-o--ooo |        |         |          |                    |                    | | f_8     | 1 0 0 0 |   u  v   | (( f,   u  v   )) | f         + uv + 1 | |        |         |          |                    |                    | | f_9     | 1 0 0 1 | ((u, v)) | (( f, ((u, v)) )) | f + u + v          | |        |         |          |                    |                    | | f_10    | 1 0 1 0 |      v   | (( f,      v   )) | f     + v      + 1 | |        |         |          |                    |                    | | f_11    | 1 0 1 1 |  (u (v)) | (( f,  (u (v)) )) | f + u     + uv     | |        |         |          |                    |                    | | f_12    | 1 1 0 0 |   u      | (( f,   u      )) | f + u          + 1 | |        |         |          |                    |                    | | f_13    | 1 1 0 1 | ((u) v)  | (( f, ((u) v)  )) | f     + v + uv     | |        |         |          |                    |                    | | f_14    | 1 1 1 0 | ((u)(v)) | (( f, ((u)(v)) )) | f + u + v + uv + 1 | |        |         |          |                    |                    | | f_15    | 1 1 1 1 |   ()   | (( f,   ()   )) | f                  | |        |         |          |                    |                    | o-o-o--ooo