Special Classes of Propositions
The full set of propositions 
contains a number of smaller classes deserving of special attention.
A basic proposition in the universe of discourse ![[a_1, \ldots, a_n]](https://s0.wp.com/latex.php?latex=%5Ba_1%2C+%5Cldots%2C+a_n%5D&bg=ffffff&fg=000000&s=0&c=20201002)
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.
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.
Resources
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