Changes

In this definition <math>q_b = q(b),\!</math> for each <math>b\!</math> in <math>\mathbb{B}.</math>  Thus, the proposition <math>\operatorname{d}x_i</math> is true of the path <math>q = (u, v)\!</math> exactly if the terms of <math>q,\!</math> the endpoints <math>u\!</math> and <math>v,\!</math> lie on different sides of the question <math>x_i.\!</math>

In this definition <math>q_b = q(b),\!</math> for each <math>b\!</math> in <math>\mathbb{B}.</math>  Thus, the proposition <math>\operatorname{d}x_i</math> is true of the path <math>q = (u, v)\!</math> exactly if the terms of <math>q,\!</math> the endpoints <math>u\!</math> and <math>v,\!</math> lie on different sides of the question <math>x_i.\!</math>

Now we can use the language of features in 〈d<font face="lucida calligraphy">X</font>〉, indeed the whole calculus of propositions in [d<font face="lucida calligraphy">X</font>], to classify paths and sets of paths.  In other words, the paths can be taken as models of the propositions ''g''&nbsp;:&nbsp;d''X''&nbsp;&rarr;&nbsp;'''B'''.  For example, the paths corresponding to ''Diag''(''X'') fall under the description <font face=system>(</font>d''x''₁<font face=system>)</font>&hellip;<font face=system>(</font>d''x''_''n''<font face=system>)</font>, which says that nothing changes among the set of features {''x''₁,&nbsp;&hellip;,&nbsp;''x''_''n''}.

The language of features in <math>\langle \operatorname{d}\mathcal{X} \rangle,</math> indeed the whole calculus of propositions in <math>[\operatorname{d}\mathcal{X}],</math> may now be used to classify paths and sets of paths.  In other words, the paths can be taken as models of the propositions <math>g : \operatorname{d}X \to \mathbb{B}.</math> For example, the paths corresponding to <math>\operatorname{diag}(X)</math> fall under the description <math>(\!| \operatorname{d}x_1 |\!) \cdots (\!| \operatorname{d}x_n |\!),</math> which says that nothing changes against the backdrop of the coordinate frame <math>\{ x_1, \ldots, x_n \}.</math>

Finally, a few words of explanation may be in order.  If this concept of a path appears to be described in a roundabout fashion, it is because I am trying to avoid using any assumption of vector space properties for the space ''X'' which contains its range.  In many ways the treatment is still unsatisfactory, but improvements will have to wait for the introduction of substitution operators acting on singular propositions.

Finally, a few words of explanation may be in order.  If this concept of a path appears to be described in a roundabout fashion, it is because I am trying to avoid using any assumption of vector space properties for the space ''X'' which contains its range.  In many ways the treatment is still unsatisfactory, but improvements will have to wait for the introduction of substitution operators acting on singular propositions.