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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.
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>
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:
* 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.