MyWikiBiz, Author Your Legacy — Monday October 20, 2025
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| The basic propositions <math>a_i : \mathbb{B}^n \to \mathbb{B}</math> are both linear and positive. So these two kinds of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions. | | The basic propositions <math>a_i : \mathbb{B}^n \to \mathbb{B}</math> are both linear and positive. So these two kinds of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions. |
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− | The linear and the positive propositions are generated by taking boolean sums and products, respectively, over selected subsets of the basic propositions in {''a''<sub>''i''</sub>}. Therefore, each set of functions can be parameterized by the subsets ''J'' of the basic index set <font face="lucida calligraphy">I</font> = {1, …, ''n''}. | + | The linear and the positive propositions are generated by taking boolean sums and products, respectively, over selected subsets of the basic propositions in <math>\{a_i\}.\!</math> Therefore, each set of functions can be parameterized by the subsets <math>J\!</math> of the basic index set <math>\mathcal{I} = \{1, \ldots, n\}.\!</math> |
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− | Let us define <font face="lucida calligraphy">A</font><sub>''J''</sub> as the subset of <font face="lucida calligraphy">A</font> that is given by {''a''<sub>''i''</sub> : ''i'' ∈ ''J''}. Then we may comprehend the action of the linear and the positive propositions in the following terms: | + | Let us define <math>\mathcal{A}_J</math> as the subset of <math>\mathcal{A}</math> that is given by <math>\{a_i : i \in J\}.\!</math> Then we may comprehend the action of the linear and the positive propositions in the following terms: |
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| * The linear proposition <font face="mt extra">l</font><sub>''J''</sub> : '''B'''<sup>''n''</sup> → '''B''' evaluates each cell ''x'' of '''B'''<sup>''n''</sup> by looking at the coefficients of ''x'' with respect to the features that <font face="mt extra">l</font><sub>''J''</sub> "likes", namely those in <font face="lucida calligraphy">A</font><sub>''J''</sub>, and then adds them up in '''B'''. Thus, <font face="mt extra">l</font><sub>''J''</sub>(''x'') computes the parity of the number of features that ''x'' has in <font face="lucida calligraphy">A</font><sub>''J''</sub>, yielding one for odd and zero for even. Expressed in this idiom, <font face="mt extra">l</font><sub>''J''</sub>(''x'') = 1 says that ''x'' seems ''odd'' (or ''oddly true'') to <font face="lucida calligraphy">A</font><sub>''J''</sub>, whereas <font face="mt extra">l</font><sub>''J''</sub>(''x'') = 0 says that ''x'' seems ''even'' (or ''evenly true'') to <font face="lucida calligraphy">A</font><sub>''J''</sub>, so long as we recall that ''zero times'' is evenly often, too. | | * The linear proposition <font face="mt extra">l</font><sub>''J''</sub> : '''B'''<sup>''n''</sup> → '''B''' evaluates each cell ''x'' of '''B'''<sup>''n''</sup> by looking at the coefficients of ''x'' with respect to the features that <font face="mt extra">l</font><sub>''J''</sub> "likes", namely those in <font face="lucida calligraphy">A</font><sub>''J''</sub>, and then adds them up in '''B'''. Thus, <font face="mt extra">l</font><sub>''J''</sub>(''x'') computes the parity of the number of features that ''x'' has in <font face="lucida calligraphy">A</font><sub>''J''</sub>, yielding one for odd and zero for even. Expressed in this idiom, <font face="mt extra">l</font><sub>''J''</sub>(''x'') = 1 says that ''x'' seems ''odd'' (or ''oddly true'') to <font face="lucida calligraphy">A</font><sub>''J''</sub>, whereas <font face="mt extra">l</font><sub>''J''</sub>(''x'') = 0 says that ''x'' seems ''even'' (or ''evenly true'') to <font face="lucida calligraphy">A</font><sub>''J''</sub>, so long as we recall that ''zero times'' is evenly often, too. |