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Directory:Jon Awbrey/Papers/Differential Propositional Calculus (view source)
Revision as of 19:26, 24 June 2008
, 19:26, 24 June 2008→Special classes of propositions: add item
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 expressed as sums:</p><blockquote><math>\sum_{i=1}^n e_i = e_1 + \ldots + e_n</math> where <math>e_i = a_i\!</math> or <math>e_i = 0\!</math> for all <math>i = 1\!</math> to <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><blockquote><math>\sum_{i=1}^n e_i = e_1 + \ldots + e_n</math> where <math>e_i = a_i\!</math> or <math>e_i = 0\!</math> for <math>i = 1\!</math> to <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</math> where <math>e_i = a_i\!</math> or <math>e_i = 1\!</math> for <math>i = 1\!</math> to <math>n.\!</math></blockquote>
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