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It may help to get a sense of the relation between '''B''' and '''D''' by considering the ''path classifier'' (or equivalence class of curves) approach to tangent vectors.  As if by reflex, the thought of physical motion makes us cross over to a universe marked by the nominal character [<font face="lucida calligraphy">X</font>]. Given the boolean value system, a path in the space ''X'' = 〈<font face="lucida calligraphy">X</font>〉 is a map ''q'' : '''B''' &rarr; ''X''. In this case, the set of paths ('''B''' &rarr; ''X'') is isomorphic to the cartesian square ''X''² = ''X'' &times; ''X'', or the set of ordered pairs from ''X''.

A sense of the relation between <math>\mathbb{B}</math> and <math>\mathbb{D}</math> may be obtained by considering the ''path classifier'' (or ''equivalence class of curves'') approach to tangent vectors.  Consider a universe <math>[\mathcal{X}].</math>  Given the boolean value system, a path in the space <math>X = \langle \mathcal{X} \rangle</math> is a map <math>q : \mathbb{B} \to X.</math>    In this context the set of paths <math>(\mathbb{B} \to X)</math> is isomorphic to the cartesian square <math>X^2 = X \times X,</math> or the set of ordered pairs chosen from <math>X.\!</math>

We may analyze ''X''² = {‹''u'', ''v''› : ''u'', ''v'' &isin; ''X''} into two parts, specifically, the pairs that lie on and off the diagonal:

We may analyze ''X''² = {‹''u'', ''v''› : ''u'', ''v'' &isin; ''X''} into two parts, specifically, the pairs that lie on and off the diagonal: