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===Special Classes of Propositions===

===Special Classes of Propositions===

It is important to remember that the coordinate propositions <math>\{a_i\},</math> besides being projection maps <math>a_i : \mathbb{B}^n \to \mathbb{B},</math> are propositions on an equal footing with all others, even though employed as a basis in a particular moment.  This set of <mat>n\!</math> propositions may sometimes be referred to as the ''basic'' or ''simple'' propositions that found the universe of discourse.  As typical and collective notations, we may use the forms <math>\{a_i : \mathbb{B}^n \to \mathbb{B}\} = (\mathbb{B}^n \xrightarrow{i} \mathbb{B})</math> to indicate the adoption of a set of <math>a_i\!</math> as a basis for discourse.

It is important to remember that the coordinate propositions <math>\{a_i\},\!</math> besides being projection maps <math>a_i : \mathbb{B}^n \to \mathbb{B},</math> are propositions on an equal footing with all others, even though employed as a basis in a particular moment.  This set of <math>n\!</math> propositions may sometimes be referred to as the ''basic propositions'', the ''coordinate propositions'', or the ''simple propositions'' that found a universe of discourse.  Either one of the equivalent notations, <math>\{a_i : \mathbb{B}^n \to \mathbb{B}\}</math> or <math>(\mathbb{B}^n \xrightarrow{i} \mathbb{B}),</math> may be used to indicate the adoption of the propositions <math>a_i\!</math> as a basis for describing a universe of discourse.

Among the <math>2^{2^n}</math> propositions or functions in ('''B'''^''n''&nbsp;&rarr;&nbsp;'''B''') are several fundamental sets of 2^''n'' propositions each that take on special forms with respect to a given basis <font face="lucida calligraphy">A</font>&nbsp;=&nbsp;{''a''_''i''}. Three of these forms are especially common, the ''linear'', the ''positive'', and the ''singular''

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>

propositions.  Each set is naturally parameterized by the coordinate vectors in '''B'''^''n'' and falls into ''n''+1 ranks, with a binomial coefficient <math>\tbinom{n}{k}</math> giving the number of propositions that have rank or weight ''k''.

The ''linear propositions'', {hom&nbsp;:&nbsp;'''B'''^''n''&nbsp;&rarr;&nbsp;'''B'''} = ('''B'''^''n''&nbsp;<font face=symbol>'''+>'''</font>&nbsp;'''B'''), may be expressed as sums of the following form:

:* <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>

: <math>\textstyle \sum_{i=1}^n e_i = e_1 + \ldots + e_n \ \mbox{where} \ \forall_{i=1}^n \ e_i = a_i \ \mbox{or} \ e_i = 0.</math>

:* <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>

The ''positive propositions'', {pos&nbsp;:&nbsp;'''B'''^''n''&nbsp;&rarr;&nbsp;'''B'''} = ('''B'''^''n''&nbsp;<font face=symbol>'''¥>'''</font>&nbsp;'''B'''), may be expressed as products of the following form:

:* <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>

: <math>\textstyle \prod_{i=1}^n e_i = e_1 \cdot \ldots \cdot e_n \ \mbox{where} \ \forall_{i=1}^n \ e_i = a_i \ \mbox{or} \ e_i = 1.</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 ''singular propositions'', {''x''&nbsp;:&nbsp;'''B'''^''n''&nbsp;&rarr;&nbsp;'''B'''} = ('''B'''^''n''&nbsp;<font face=symbol>'''××>'''</font>&nbsp;'''B'''), may be expressed as products of the following form:

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.

 

: <math>\textstyle \prod_{i=1}^n e_i = e_1 \cdot \ldots \cdot e_n \ \mbox{where} \ \forall_{i=1}^n \ e_i = a_i \ \mbox{or} \ e_i = (a_i) = \lnot a_i.</math>

 

 

 

The coordinate projections or simple propositions ''a''_''i''&nbsp;:&nbsp;'''B'''^''n''&nbsp;&rarr;&nbsp;'''B''' are both linear and positive.  So these two kinds of propositions, the linear or the positive, may be viewed as two different ways of generalizing the class of simple projections.

The linear and the positive propositions are generated by taking boolean sums and products, respectively, over selected subsets of the basic propositions in {''a''_''i''}.  Therefore, each set of functions can be parameterized by the subsets ''J'' of the basic index set <font face="lucida calligraphy">I</font>&nbsp;=&nbsp;{1,&nbsp;&hellip;,&nbsp;''n''}.

The linear and the positive propositions are generated by taking boolean sums and products, respectively, over selected subsets of the basic propositions in {''a''_''i''}.  Therefore, each set of functions can be parameterized by the subsets ''J'' of the basic index set <font face="lucida calligraphy">I</font>&nbsp;=&nbsp;{1,&nbsp;&hellip;,&nbsp;''n''}.