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Directory:Jon Awbrey/Papers/Inquiry Driven Systems (view source)
Revision as of 20:17, 13 January 2009
, 20:17, 13 January 2009→1.3.10.3. Propositions and Sentences: mathematical markup
A ''boolean connection'' of degree <math>k,\!</math> also known as a ''boolean function'' on <math>k\!</math> variables, is a map of the form <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}.</math> In other words, a boolean connection of degree <math>k\!</math> is a proposition about things in the universe <math>X = \underline\mathbb{B}^k.</math>
A ''boolean connection'' of degree <math>k,\!</math> also known as a ''boolean function'' on <math>k\!</math> variables, is a map of the form <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}.</math> In other words, a boolean connection of degree <math>k\!</math> is a proposition about things in the universe <math>X = \underline\mathbb{B}^k.</math>
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.
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The "play of images" that is determined by f and u, more specifically, the play of the imagination f = <f1, ..., fk> that has to with the element u of U, is the k�tuple v = <v1, ..., vk> of values in B that satisfies the equations vj = fj(u), for all j = 1 to k.
The "play of images" that is determined by f and u, more specifically, the play of the imagination f = <f1, ..., fk> that has to with the element u of U, is the k�tuple v = <v1, ..., vk> of values in B that satisfies the equations vj = fj(u), for all j = 1 to k.
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: