Changes

\end{array}</math></p>

\end{array}</math></p>

In the above transcription, the operator bracket of the form <math>(\!| \ldots\ ,\ \ldots |\!)</math> is a ''cactus lobe'', in general signifying that just one of the arguments listed is false.  In the case of two arguments this is the same thing as saying that the arguments are not equal.  The plus sign signifies boolean addition, in the sense of addition in GF(2), and thus means the same thing in this context as the minus sign, in the sense of adding the additive inverse.

In the above transcription, the operator bracket of the form <math>(\!| \ldots\ ,\ \ldots |\!)</math> is a ''cactus lobe'', in general signifying that just one of the arguments listed is false.  In the case of two arguments this is the same thing as saying that the arguments are not equal.  The plus sign signifies boolean addition, in the sense of addition in <math>\operatorname{GF}(2),</math> and thus means the same thing in this context as the minus sign, in the sense of adding the additive inverse.

The above definition of <math>\operatorname{d}x_i : X^2 \to \mathbb{B}</math> is equivalent to defining <math>\operatorname{d}x_i : (\mathbb{B} \to X) \to \mathbb{B}</math> in the following way:

The above definition of <math>\operatorname{d}x_i : X^2 \to \mathbb{B}</math> is equivalent to defining <math>\operatorname{d}x_i : (\mathbb{B} \to X) \to \mathbb{B}</math> in the following way:

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>

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 <math>X\!</math> that 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.

===The Extended Universe of Discourse===

===The Extended Universe of Discourse===