Differential Propositional Calculus • 9

Special Classes of Propositions

The full set of propositions $f : A \to \mathbb{B}$ contains a number of smaller classes deserving of special attention.

A basic proposition in the universe of discourse $[a_1, \ldots, a_n]$ is one of the propositions in the set $\{ a_1, \ldots, a_n \}.$ There are of course exactly $n$ of these. Depending on context, basic propositions may also be called coordinate propositions or simple propositions.

Among the $2^{2^n}$ propositions in $[a_1, \ldots, a_n]$ are several families numbering $2^n$ propositions each which take on special forms with respect to the basis $\{ a_1, \ldots, a_n \}.$ Three of those families are especially prominent in the present context, the linear, the positive, and the singular propositions. Each family is naturally parameterized by the coordinate $n$ -tuples in $\mathbb{B}^n$ and falls into $n + 1$ ranks, with a binomial coefficient $\tbinom{n}{k}$ giving the number of propositions having rank or weight $k$ in their class.

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

$\sum_{i=1}^n e_i ~=~ e_1 + \ldots + e_n ~\text{where}~ \left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 0 \end{matrix}\right\} ~\text{for}~ i = 1 ~\text{to}~ n.$

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

$\prod_{i=1}^n e_i ~=~ e_1 \cdot \ldots \cdot e_n ~\text{where}~ \left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 1 \end{matrix}\right\} ~\text{for}~ i = 1 ~\text{to}~ n.$

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

$\prod_{i=1}^n e_i ~=~ e_1 \cdot \ldots \cdot e_n ~\text{where}~ \left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = \texttt{(} a_i \texttt{)} \end{matrix}\right\} ~\text{for}~ i = 1 ~\text{to}~ n.$

In each case the rank $k$ ranges from $0$ to $n$ and counts the number of positive appearances of the coordinate propositions $a_1, \ldots, a_n$ in the resulting expression. For example, when $n = 3$ the linear proposition of rank $0$ is $0,$ the positive proposition of rank $0$ is $1,$ and the singular proposition of rank $0$ is $\texttt{(} a_1 \texttt{)} \texttt{(} a_2 \texttt{)} \texttt{(} a_3 \texttt{)}.$

The basic propositions $a_i : \mathbb{B}^n \to \mathbb{B}$ are both linear and positive. So those two families of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions.

It is important to note that all of the above distinctions are relative to the choice of a particular basis $\mathcal{A} = \{ a_1, \ldots, a_n \}.$ A singular proposition with respect to the basis $\mathcal{A}$ will not remain singular if $\mathcal{A}$ is extended by a number of new and independent features. Even if one keeps to the original set of pairwise options $\{ a_i \} \cup \{ \texttt{(} a_i \texttt{)} \}$ to pick out a new basis, the sets of linear propositions and positive propositions are both determined by the choice of basic propositions, and that entire determination is tantamount to the purely conventional choice of a cell as origin.

Resources

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