## Special Classes of Propositions

A basic proposition, coordinate proposition, or simple proposition in the universe of discourse $\mathcal{X}^\bullet = \lbrack x_1, \ldots, x_k \rbrack$ is one of the propositions in the set $\mathcal{X} = \lbrace x_1, \ldots, x_k \rbrace.$

Among the $2^{2^k}$ propositions in $\lbrack x_1, \ldots, x_k \rbrack$ are several families of $2^k$ propositions each that take on special forms with respect to the logical basis $\lbrace x_1, \ldots, x_k \rbrace.$ 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 $k$-tuples in $\mathbb{B}^k$ and falls into $k + 1$ ranks, with a binomial coefficient $\dbinom{k}{j}$ giving the number of propositions that have rank or weight $j.$

• The linear propositions, $\lbrace \ell : \mathbb{B}^k \to \mathbb{B} \rbrace = (\mathbb{B}^k \xrightarrow{\ell} \mathbb{B}),$ may be written as sums:

$\begin{array}{llll} \displaystyle\sum_{i=1}^k e_i ~=~ e_1 + \ldots + e_k & \text{where} & \left\{ \begin{matrix} e_i = x_i \\ \text{or} \\ e_i = 0 \end{matrix} \right\} & \text{for}~ i=1 ~\text{to}~ k. \end{array}$

• The positive propositions, $\lbrace p : \mathbb{B}^k \to \mathbb{B} \rbrace = (\mathbb{B}^k \xrightarrow{p} \mathbb{B}),$ may be written as products:

$\begin{array}{llll} \displaystyle\prod_{i=1}^k e_i ~=~ e_1 \cdot \ldots \cdot e_k & \text{where} & \left\{ \begin{matrix} e_i = x_i \\ \text{or} \\ e_i = 1 \end{matrix} \right\} & \text{for}~ i=1 ~\text{to}~ k. \end{array}$

• The singular propositions, $\lbrace \mathbf{x} : \mathbb{B}^k \to \mathbb{B} \rbrace = (\mathbb{B}^k \xrightarrow{s} \mathbb{B}),$ may be written as products:

$\begin{array}{llll} \displaystyle\prod_{i=1}^k e_i ~=~ e_1 \cdot \ldots \cdot e_k & \text{where} & \left\{ \begin{matrix} e_i = x_i \\ \text{or} \\ e_i = \texttt{(}x_i\texttt{)} \end{matrix} \right\} & \text{for}~ i=1 ~\text{to}~ k. \end{array}$

In each case the rank $j$ ranges from $0$ to $k$ and counts the number of positive appearances of the coordinate propositions $x_1, \ldots, x_k$ in the resulting expression. For example, for $k = 3$ the linear proposition of rank $0$ is $0,$ the positive proposition of rank $0$ is $1,$ and the singular proposition of rank $0$ is $\texttt{(} x_1 \texttt{)(} x_2 \texttt{)(} x_3 \texttt{)}.$

The basic propositions $x_i : \mathbb{B}^k \to \mathbb{B}$ 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 $\mathcal{X} = \lbrace x_1, \ldots, x_k \rbrace.$ For example, a singular proposition with respect to the basis $\mathcal{X}$ will not remain singular if $\mathcal{X}$ is extended by a number of new and independent features. Even if one keeps to the original set of pairwise options $\lbrace x_i \rbrace \cup \lbrace \texttt{(} x_i \texttt{)} \rbrace$ 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.