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>

* 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>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><math>\begin{matrix}

\sum_{i=1}^n e_i & = & e_1 + \ldots + e_n &

\operatorname{where}\ e_i = a_i\ \operatorname{or}\ e_i = 0 &

\operatorname{for}\ i = 1\ \operatorname{to}\ n.

\end{matrix}</math></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>

* 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><math>\begin{matrix}

\prod_{i=1}^n e_i & = & e_1 \cdot \ldots \cdot e_n &

\operatorname{where}\ e_i = a_i\ \operatorname{or}\ e_i = 1 &

\end{matrix}</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>\begin{matrix}

\prod_{i=1}^n e_i & = & e_1 \cdot \ldots \cdot e_n &

\operatorname{where}\ e_i = a_i\ \operatorname{or}\ e_i = (a_i) &

\operatorname{for}\ i = 1\ \operatorname{to}\ n.

\end{matrix}</math></p>

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>

Let us define <math>\mathcal{A}_J</math> as the subset of <math>\mathcal{A}</math> that is given by <math>\{a_i : i \in J\}.\!</math>  Then we may comprehend the action of the linear and the positive propositions in the following terms:

Let us define <math>\mathcal{A}_J</math> as the subset of <math>\mathcal{A}</math> that is given by <math>\{a_i : i \in J\}.\!</math>  Then we may comprehend the action of the linear and the positive propositions in the following terms:

:* The linear proposition <math>\ell_J : \mathbb{B}^n \to \mathbb{B}</math> evaluates each cell <math>\mathbf{x}</math> of <math>\mathbb{B}^n</math> by looking at the coefficients of <math>\mathbf{x}</math> with respect to the features that <math>\ell_J</math> "likes", namely those in <math>\mathcal{A}_J,</math> and then adds them up in <math>\mathbb{B}.</math>  Thus, <math>\ell_J(\mathbf{x})</math> computes the parity of the number of features that <math>\mathbf{x}</math> has in <math>\mathcal{A}_J,</math> yielding one for odd and zero for even.  Expressed in this idiom, <math>\ell_J(\mathbf{x}) = 1</math> says that <math>\mathbf{x}</math> seems ''odd'' (or ''oddly true'') to <math>\mathcal{A}_J,</math> whereas <math>\ell_J(\mathbf{x}) = 0</math> says that <math>\mathbf{x}</math> seems ''even'' (or ''evenly true'') to <math>\mathcal{A}_J,</math> so long as we recall that ''zero times'' is evenly often, too.

* The linear proposition <math>\ell_J : \mathbb{B}^n \to \mathbb{B}</math> evaluates each cell <math>\mathbf{x}</math> of <math>\mathbb{B}^n</math> by looking at the coefficients of <math>\mathbf{x}</math> with respect to the features that <math>\ell_J</math> "likes", namely those in <math>\mathcal{A}_J,</math> and then adds them up in <math>\mathbb{B}.</math>  Thus, <math>\ell_J(\mathbf{x})</math> computes the parity of the number of features that <math>\mathbf{x}</math> has in <math>\mathcal{A}_J,</math> yielding one for odd and zero for even.  Expressed in this idiom, <math>\ell_J(\mathbf{x}) = 1</math> says that <math>\mathbf{x}</math> seems ''odd'' (or ''oddly true'') to <math>\mathcal{A}_J,</math> whereas <math>\ell_J(\mathbf{x}) = 0</math> says that <math>\mathbf{x}</math> seems ''even'' (or ''evenly true'') to <math>\mathcal{A}_J,</math> so long as we recall that ''zero times'' is evenly often, too.

:* The positive proposition <math>p_J : \mathbb{B}^n \to \mathbb{B}</math> evaluates each cell <math>\mathbf{x}</math> of <math>\mathbb{B}^n</math> by looking at the coefficients of <math>\mathbf{x}</math> with regard to the features that <math>p_J\!</math> "likes", namely those in <math>\mathcal{A}_J,</math> and then takes their product in <math>\mathbb{B}.</math>  Thus, <math>p_J(\mathbf{x})</math> assesses the unanimity of the multitude of features that <math>\mathbf{x}</math> has in <math>\mathcal{A}_J,</math> yielding one for all and aught for else.  In these consensual or contractual terms, <math>p_J(\mathbf{x}) = 1</math> means that <math>\mathbf{x}</math> is ''AOK'' or congruent with all of the conditions of <math>\mathcal{A}_J,</math> while <math>p_J(\mathbf{x}) = 0</math> means that <math>\mathbf{x}</math> defaults or dissents from some condition of <math>\mathcal{A}_J.</math>

* The positive proposition <math>p_J : \mathbb{B}^n \to \mathbb{B}</math> evaluates each cell <math>\mathbf{x}</math> of <math>\mathbb{B}^n</math> by looking at the coefficients of <math>\mathbf{x}</math> with regard to the features that <math>p_J\!</math> "likes", namely those in <math>\mathcal{A}_J,</math> and then takes their product in <math>\mathbb{B}.</math>  Thus, <math>p_J(\mathbf{x})</math> assesses the unanimity of the multitude of features that <math>\mathbf{x}</math> has in <math>\mathcal{A}_J,</math> yielding one for all and aught for else.  In these consensual or contractual terms, <math>p_J(\mathbf{x}) = 1</math> means that <math>\mathbf{x}</math> is ''AOK'' or congruent with all of the conditions of <math>\mathcal{A}_J,</math> while <math>p_J(\mathbf{x}) = 0</math> means that <math>\mathbf{x}</math> defaults or dissents from some condition of <math>\mathcal{A}_J.</math>

===Basis Relativity and Type Ambiguity===

===Basis Relativity and Type Ambiguity===