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.
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