Special Classes of Propositions
Before moving on, let’s unpack some of the assumptions, conventions, and implications involved in the array of concepts and notations introduced above.
A universe of discourse based on the logical features is a set plus the set of all possible functions from the space to the boolean domain There are elements in often pictured as the cells of a venn diagram or the nodes of a hypercube. There are possible functions from to accordingly pictured as all the ways of painting the cells of a venn diagram or the nodes of a hypercube with a palette of two colors.
A logical proposition about the elements of is either true or false of each element in while a function evaluates to or on each element of The analogy between logical propositions and boolean-valued functions is close enough to adopt the latter as models of the former and simply refer to the functions as propositions about the elements of
The full set of propositions contains a number of smaller classes deserving of special attention.
A basic proposition in the universe of discourse is one of the propositions in the set There are of course exactly of these. Depending on the context, basic propositions may also be called coordinate propositions or simple propositions.
Among the propositions in are several families numbering propositions each which take on special forms with respect to the 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 having rank or weight in their class.
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, when 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 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.