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| | I will first rationalize the novel grouping of propositional forms in the last set of Tables, as that will extend a gentle invitation to the mathematical subject of group theory, and demonstrate its relevance to differential logic in a strikingly apt and useful way. The data for that account is contained in Table 5. | | I will first rationalize the novel grouping of propositional forms in the last set of Tables, as that will extend a gentle invitation to the mathematical subject of group theory, and demonstrate its relevance to differential logic in a strikingly apt and useful way. The data for that account is contained in Table 5. |
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| − | <pre>
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| − | Table 5. Ef Expanded Over Differential Features {dx, dy}
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| − | o------o------------o------------o------------o------------o------------o
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| − | | | | | | | |
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| − | | | f | T_11 f | T_10 f | T_01 f | T_00 f |
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| − | | | | | | | |
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| − | | | | Ef| dx dy | Ef| dx(dy) | Ef| (dx)dy | Ef|(dx)(dy)|
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| − | | | | | | | |
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| − | o------o------------o------------o------------o------------o------------o
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| − | | | | | | | |
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| − | | f_0 | () | () | () | () | () |
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| − | | | | | | | |
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| − | o------o------------o------------o------------o------------o------------o
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| − | | | | | | | |
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| − | | f_1 | (x)(y) | x y | x (y) | (x) y | (x)(y) |
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| − | | | | | | | |
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| − | | f_2 | (x) y | x (y) | x y | (x)(y) | (x) y |
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| − | | | | | | | |
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| − | | f_4 | x (y) | (x) y | (x)(y) | x y | x (y) |
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| − | | | | | | | |
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| − | | f_8 | x y | (x)(y) | (x) y | x (y) | x y |
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| − | | | | | | | |
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| − | o------o------------o------------o------------o------------o------------o
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| − | | | | | | | |
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| − | | f_3 | (x) | x | x | (x) | (x) |
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| − | | | | | | | |
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| − | | f_12 | x | (x) | (x) | x | x |
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| − | | | | | | | |
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| − | o------o------------o------------o------------o------------o------------o
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| − | | | | | | | |
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| − | | f_6 | (x, y) | (x, y) | ((x, y)) | ((x, y)) | (x, y) |
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| − | | | | | | | |
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| − | | f_9 | ((x, y)) | ((x, y)) | (x, y) | (x, y) | ((x, y)) |
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| − | | | | | | | |
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| − | o------o------------o------------o------------o------------o------------o
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| − | | | | | | | |
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| − | | f_5 | (y) | y | (y) | y | (y) |
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| − | | | | | | | |
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| − | | f_10 | y | (y) | y | (y) | y |
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| − | | | | | | | |
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| − | o------o------------o------------o------------o------------o------------o
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| − | | | | | | | |
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| − | | f_7 | (x y) | ((x)(y)) | ((x) y) | (x (y)) | (x y) |
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| − | | | | | | | |
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| − | | f_11 | (x (y)) | ((x) y) | ((x)(y)) | (x y) | (x (y)) |
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| − | | | | | | | |
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| − | | f_13 | ((x) y) | (x (y)) | (x y) | ((x)(y)) | ((x) y) |
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| − | | | | | | | |
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| − | | f_14 | ((x)(y)) | (x y) | (x (y)) | ((x) y) | ((x)(y)) |
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| − | | | | | | | |
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| − | o------o------------o------------o------------o------------o------------o
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| − | | | | | | | |
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| − | | f_15 | (()) | (()) | (()) | (()) | (()) |
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| − | | | | | | | |
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| − | o------o------------o------------o------------o------------o------------o
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| − | | | | | | |
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| − | | Fixed Point Total | 4 | 4 | 4 | 16 |
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| − | | | | | | |
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| − | o-------------------o------------o------------o------------o------------o
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| − | </pre>
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| | The shift operator ''E'' can be understood as enacting a substitution operation on the proposition that is given as its argument. In our immediate example, we have the following data and definition: | | The shift operator ''E'' can be understood as enacting a substitution operation on the proposition that is given as its argument. In our immediate example, we have the following data and definition: |