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Revision as of 21:25, 13 January 2009
, 21:25, 13 January 2009→1.3.10.3. Propositions and Sentences: mathematical markup
An ''imagination'' of degree <math>k\!</math> on <math>X\!</math> is a <math>k\!</math>-tuple of propositions about things in the universe <math>X.\!</math> By way of displaying the kinds of notation that are used to express this idea, the imagination <math>\underline{f} = (f_1, \ldots, f_k)</math> is given as a sequence of indicator functions <math>f_j : X \to \underline\mathbb{B},</math> for <math>j = {}_1^k.</math> All of these features of the typical imagination <math>\underline{f}</math> can be summed up in either one of two ways: either in the form of a membership statement, to the effect that <math>\underline{f} \in (X \to \underline\mathbb{B})^k,</math> or in the form of a type statement, to the effect that <math>\underline{f} : (X \to \underline\mathbb{B})^k,</math> though perhaps the latter form is slightly more precise than the former.
An ''imagination'' of degree <math>k\!</math> on <math>X\!</math> is a <math>k\!</math>-tuple of propositions about things in the universe <math>X.\!</math> By way of displaying the kinds of notation that are used to express this idea, the imagination <math>\underline{f} = (f_1, \ldots, f_k)</math> is given as a sequence of indicator functions <math>f_j : X \to \underline\mathbb{B},</math> for <math>j = {}_1^k.</math> All of these features of the typical imagination <math>\underline{f}</math> can be summed up in either one of two ways: either in the form of a membership statement, to the effect that <math>\underline{f} \in (X \to \underline\mathbb{B})^k,</math> or in the form of a type statement, to the effect that <math>\underline{f} : (X \to \underline\mathbb{B})^k,</math> though perhaps the latter form is slightly more precise than the former.
The ''play of images'' determined by <math>\underline{f}</math> and <math>x,\!</math> more specifically, the play of the imagination <math>\underline{f} = (f_1, \ldots, f_k)</math> that has to do with the element <math>x \in X,</math> is the <math>k\!</math>-tuple <math>\underline{b} = (b_1, \ldots, b_k)</math> of values in <math>\underline\mathbb{B}</math> that satisfies the equations <math>b_j = f_j (x),\!</math> for <math>j = 1 ~\text{to}~ k.</math>
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A "projection" of Bk, typically denoted by "pj" or "prj", is one of the maps pj : Bk �> B, for j = 1 to k, that is defined as follows:
A "projection" of Bk, typically denoted by "pj" or "prj", is one of the maps pj : Bk �> B, for j = 1 to k, that is defined as follows: