Changes

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 <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>

Linear propositions and positive propositions are generated by taking boolean sums and products, respectively, over selected subsets of basic propositions, so both families of propositions are parameterized by the powerset <math>\mathcal{P}(\mathcal{I}),</math> that is, the set of all subsets <math>J\!</math> of the basic index set <math>\mathcal{I} = \{1, \ldots, n\}.\!</math>

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:

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: