Adapted from Differential Propositional Calculus • Special Classes of Propositions

A basic proposition, coordinate proposition, or simple proposition in the universe of discourse is one of the propositions in the set

Among the propositions in are several families of propositions each that take on special forms with respect to the logical basis Three of these families are especially prominent in the present context, the linear, the positive, and the singular propositions. Each family is naturally parameterized by the coordinate -tuples in and falls into ranks, with a binomial coefficient giving the number of propositions that have rank or weight

• The linear propositions, may be written as sums:

  

• The positive propositions, may be written as products:

  

• The singular propositions, may be written as products:

  

In each case the rank ranges from to and counts the number of positive appearances of the coordinate propositions in the resulting expression. For example, for the linear proposition of rank is the positive proposition of rank is and the singular proposition of rank is

The basic propositions 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.

Finally, it is important to note that all of the above distinctions are relative to the choice of a particular logical basis For example, a singular proposition with respect to the basis will not remain singular if is extended by a number of new and independent features. Even if one keeps to the original set of pairwise options to pick out a new basis, the sets of linear propositions and positive propositions are both determined by the choice of basic propositions, and this whole determination is tantamount to the purely conventional choice of a cell as origin.