ONT Re: Differential Logic
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DLOG. Note D32
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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
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| u : 1 1 0 0 | f | theta (f) | theta (f) |
| v : 1 0 1 0 | | | |
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| | | | | |
| 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 |
| | | | | |
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| | | | | |
| 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 |
| | | | | |
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Jon Awbrey
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