Changes

Among the <math>2^{2^n}</math> propositions in <math>[a_1, \ldots, a_n]</math> are several families of <math>2^n\!</math> propositions each that take on special forms with respect to the basis <math>\{ a_1, \ldots, a_n \}.</math>  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 <math>n\!</math>-tuples in <math>\mathbb{B}^n</math> and falls into <math>n + 1\!</math> ranks, with a binomial coefficient <math>\tbinom{n}{k}</math> giving the number of propositions that have rank or weight <math>k.\!</math>

Among the <math>2^{2^n}</math> propositions in <math>[a_1, \ldots, a_n]</math> are several families of <math>2^n\!</math> propositions each that take on special forms with respect to the basis <math>\{ a_1, \ldots, a_n \}.</math>  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 <math>n\!</math>-tuples in <math>\mathbb{B}^n</math> and falls into <math>n + 1\!</math> ranks, with a binomial coefficient <math>\tbinom{n}{k}</math> giving the number of propositions that have rank or weight <math>k.\!</math>

:* <p>The ''linear propositions'', <math>\{ \ell : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{\ell} \mathbb{B}),</math> may be written as sums:</p><blockquote><math>\sum_{i=1}^n e_i = e_1 + \ldots + e_n</math>&nbsp;&nbsp;where&nbsp;&nbsp;<math>e_i = a_i\!</math>&nbsp;&nbsp;or&nbsp;&nbsp;<math>e_i = 0\!</math>&nbsp;&nbsp;for&nbsp;&nbsp;<math>i = 1\!</math>&nbsp;&nbsp;to&nbsp;&nbsp;<math>n.\!</math></blockquote>

<ul>

:* <p>The ''positive propositions'', <math>\{ p : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{p} \mathbb{B}),</math> may be written as products:</p><blockquote><math>\prod_{i=1}^n e_i = e_1 \cdot \ldots \cdot e_n</math>&nbsp;&nbsp;where&nbsp;&nbsp;<math>e_i = a_i\!</math>&nbsp;&nbsp;or&nbsp;&nbsp;<math>e_i = 1\!</math>&nbsp;&nbsp;for&nbsp;&nbsp;<math>i = 1\!</math>&nbsp;&nbsp;to&nbsp;&nbsp;<math>n.\!</math></blockquote>

<p>The ''linear propositions'', <math>\{ \ell : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{\ell} \mathbb{B}),\!</math> may be written as sums:</p>

:* <p>The ''singular propositions'', <math>\{ \mathbf{x} : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{s} \mathbb{B}),</math> may be written as products:</p><blockquote><math>\prod_{i=1}^n e_i = e_1 \cdot \ldots \cdot e_n</math>&nbsp;&nbsp;where&nbsp;&nbsp;<math>e_i = a_i\!</math>&nbsp;&nbsp;or&nbsp;&nbsp;<math>e_i = (a_i)\!</math>&nbsp;&nbsp;for&nbsp;&nbsp;<math>i = 1\!</math>&nbsp;&nbsp;to&nbsp;&nbsp;<math>n.\!</math></blockquote>

 

<p>The ''positive propositions'', <math>\{ p : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{p} \mathbb{B}),\!</math> may be written as products:</p>

 

<blockquote>

<math>\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.\!</math>

 

<p>The ''singular propositions'', <math>\{ \mathbf{x} : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{s} \mathbb{B}),\!</math> may be written as products:</p>

 

<math>\prod_{i=1}^n e_i ~=~ e_1 \cdot \ldots \cdot e_n

~\text{for}~ i = 1 ~\text{to}~ n.\!</math>

</blockquote>

</li>

 

</ul>

In each case the rank <math>k\!</math> ranges from <math>0\!</math> to <math>n\!</math> and counts the number of positive appearances of the coordinate propositions <math>a_1, \ldots, a_n\!</math> in the resulting expression.  For example, for <math>n = 3,\!</math> the linear proposition of rank <math>0\!</math> is <math>0,\!</math> the positive proposition of rank <math>0\!</math> is <math>1,\!</math> and the singular proposition of rank <math>0\!</math> is <math>(a_1)(a_2)(a_3).\!</math>

In each case the rank <math>k\!</math> ranges from <math>0\!</math> to <math>n\!</math> and counts the number of positive appearances of the coordinate propositions <math>a_1, \ldots, a_n\!</math> in the resulting expression.  For example, for <math>n = 3,\!</math> the linear proposition of rank <math>0\!</math> is <math>0,\!</math> the positive proposition of rank <math>0\!</math> is <math>1,\!</math> and the singular proposition of rank <math>0\!</math> is <math>(a_1)(a_2)(a_3).\!</math>

The basic propositions <math>a_i : \mathbb{B}^n \to \mathbb{B}</math> 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.

The basic propositions <math>a_i : \mathbb{B}^n \to \mathbb{B}\!</math> 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 <math>\mathcal{A} = \{ a_1, \ldots, a_n \}.</math>  For example, a singular proposition with respect to the basis <math>\mathcal{A}</math> will not remain singular if <math>\mathcal{A}</math> is extended by a number of new and independent features.  Even if one keeps to the original set of pairwise options <math>\{ a_i \} \cup \{ (a_i) \}</math> 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.

Finally, it is important to note that all of the above distinctions are relative to the choice of a particular logical basis <math>\mathcal{A} = \{ a_1, \ldots, a_n \}.\!</math>  For example, a singular proposition with respect to the basis <math>\mathcal{A}\!</math> will not remain singular if <math>\mathcal{A}\!</math> is extended by a number of new and independent features.  Even if one keeps to the original set of pairwise options <math>\{ a_i \} \cup \{ \texttt{(} a_i \texttt{)} \}\!</math> 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.

===3.3. Differential Extensions===

===3.3. Differential Extensions===