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Revision as of 15:22, 17 January 2009
, 15:22, 17 January 2009→1.3.10.6. Stretching Principles: mathematical markup
In particular, a connection <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}</math> can be understood to indicate a relation among boolean values, namely, the <math>k\!</math>-ary relation <math>F^{-1} (\underline{1}) \subseteq \underline\mathbb{B}^k.</math> If these <math>k\!</math> values are values of things in a universe <math>X,</math> that is, if one imagines each value in a <math>k\!</math>-tuple of values to be the functional image that results from evaluating an element of <math>X\!</math> under one of its possible aspects of value, then one has in mind the <math>k\!</math> propositions <math>f_j : X \to \underline\mathbb{B},</math> for <math>j = 1 ~\text{to}~ k,</math> in sum, one embodies the imagination <math>\underline{f} = (f_1, \ldots, f_k).</math> Together, the imagination <math>\underline{f} \in (X \to \underline\mathbb{B})^k</math> and the connection <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}</math> stretch each other to cover the universe <math>X,\!</math> yielding a new proposition <math>p : X \to \underline\mathbb{B}.</math>
In particular, a connection <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}</math> can be understood to indicate a relation among boolean values, namely, the <math>k\!</math>-ary relation <math>F^{-1} (\underline{1}) \subseteq \underline\mathbb{B}^k.</math> If these <math>k\!</math> values are values of things in a universe <math>X,</math> that is, if one imagines each value in a <math>k\!</math>-tuple of values to be the functional image that results from evaluating an element of <math>X\!</math> under one of its possible aspects of value, then one has in mind the <math>k\!</math> propositions <math>f_j : X \to \underline\mathbb{B},</math> for <math>j = 1 ~\text{to}~ k,</math> in sum, one embodies the imagination <math>\underline{f} = (f_1, \ldots, f_k).</math> Together, the imagination <math>\underline{f} \in (X \to \underline\mathbb{B})^k</math> and the connection <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}</math> stretch each other to cover the universe <math>X,\!</math> yielding a new proposition <math>p : X \to \underline\mathbb{B}.</math>
To encapsulate the form of this general result, I define a composition that takes an imagination <math>\underline{f} = (f_1, \ldots, f_k) \in (X \to \underline\mathbb{B})^k</math> and a boolean connection <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}</math> and gives a proposition <math>p : X \to \underline\mathbb{B}.</math> Depending on the situation, specifically, according to whether many <math>F\!</math> and many <math>\underline{f},</math> a single <math>F\!</math> and many <math>\underline{f},</math> or many <math>F\!</math> and a single <math>\underline{f}</math> are being considered, respectively, the proposition <math>p\!</math> thus constructed may be referred to under one of three descriptions:
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1. In a general setting, where the connection F and the imagination f are both permitted to take up a variety of concrete possibilities, call P the "stretch of F and f from U to B", and write it in the style of a composition as "F $ f". This is meant to suggest that the symbol "$", here read as "stretch", denotes an operator of the form
1. In a general setting, where the connection F and the imagination f are both permitted to take up a variety of concrete possibilities, call P the "stretch of F and f from U to B", and write it in the style of a composition as "F $ f". This is meant to suggest that the symbol "$", here read as "stretch", denotes an operator of the form
$ : (Bk �> B) x (U �> B)k �> (U �> B).
$ : (Bk -> B) x (U -> B)k -> (U -> B).
2. In a setting where the connection F is fixed but the imagination f is allowed to vary over a wide range of possibilities, call P the "stretch of F to f on U", and write it in the style "F$f", exactly as if "F$" denotes an operator F$ : (U �> B)k �> (U �> B) that is derived from F and applied to f, ultimately yielding a proposition F$f : U �> B.
2. In a setting where the connection F is fixed but the imagination f is allowed to vary over a wide range of possibilities, call P the "stretch of F to f on U", and write it in the style "F$f", exactly as if "F$" denotes an operator F$ : (U -> B)k -> (U -> B) that is derived from F and applied to f, ultimately yielding a proposition F$f : U -> B.
3. In a setting where the imagination f is fixed but the connection F is allowed to range over wide variety of possibilities, call P the "stretch of f by F to B", and write it in the style "f$F", exactly as if "f$" denotes an operator f$ : (Bk �> B) �> (U �> B) that is derived from f and applied to F, ultimately yielding a proposition f$F : U �> B.
3. In a setting where the imagination f is fixed but the connection F is allowed to range over wide variety of possibilities, call P the "stretch of f by F to B", and write it in the style "f$F", exactly as if "f$" denotes an operator f$ : (Bk �> B) �> (U �> B) that is derived from f and applied to F, ultimately yielding a proposition f$F : U �> B.