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Let <font face="lucida calligraphy">X</font> = {''x''<sub>1</sub>} = {''A''} be an alphabet that represents one boolean variable or a single logical feature. In this example I am using the capital letter "''A''" in a more usual informal way, to name a feature and not a space, at variance with my formerly stated formal conventions. At any rate, the basis element
Let <math>\mathcal{X} = \{ x_1 \} = \{ A \}</math> be an alphabet that represents one boolean variable or a single logical feature. In this example I am using the capital letter "<math>A\!</math>" in a more usual informal way, to name a feature and not a space, in departure from my formerly stated formal conventions. At any rate, the basis element <math>A = x_1\!</math> may be interpreted as a simple proposition or a coordinate projection <math>A = x_1 : \mathbb{B} \xrightarrow{i} \mathbb{B}.</math> The space <math>X = \langle A \rangle = \{ (A), A \}</math> of points (cells, vectors, interpretations) has cardinality 2<sup>''n''</sup> = 2<sup>1</sup> = 2 and is isomorphic to '''B''' = {0, 1}. Moreover, ''X'' may be identified with the set of singular propositions {''x'' : '''B''' <font face=symbol>'''××>'''</font> '''B'''}. The space of linear propositions ''X''* = {hom : '''B''' <font face=symbol>'''+>'''</font> '''B'''} = {0, ''A''} is algebraically dual to ''X'' and also has cardinality 2. Here, "0" is interpreted as denoting the constant function 0 : '''B''' → '''B''', amounting to the linear proposition of rank 0, while ''A'' is the linear proposition of rank 1. Last but not least we have the positive propositions {pos : '''B''' <font face=symbol>'''¥>'''</font> '''B'''} = {''A'', 1}, of rank 1 and 0, respectively, where "1" is understood as denoting the constant function 1 : '''B''' → '''B'''. In sum, there are <math>2^{2^n} = 2^{2^1} = 4</math> propositions altogether in the universe of discourse, comprising the set ''X''^ = {f : ''X'' → '''B'''} = {0, (''A''), ''A'', 1} <math>\cong</math> ('''B''' → '''B''').
The first order differential extension of <font face="lucida calligraphy">X</font> is E<font face="lucida calligraphy">X</font> = {''x''<sub>1</sub>, d''x''<sub>1</sub>} = {''A'', d''A''}. If the feature "''A''" is understood as applying to some object or state, then the feature "d''A''" may be interpreted as an attribute of the same object or state that says that it is changing ''significantly'' with respect to the property ''A'', or that it has an ''escape velocity'' with respect to the state ''A''. In practice, differential features acquire their logical meaning through a class of ''temporal inference rules''.
The first order differential extension of <font face="lucida calligraphy">X</font> is E<font face="lucida calligraphy">X</font> = {''x''<sub>1</sub>, d''x''<sub>1</sub>} = {''A'', d''A''}. If the feature "''A''" is understood as applying to some object or state, then the feature "d''A''" may be interpreted as an attribute of the same object or state that says that it is changing ''significantly'' with respect to the property ''A'', or that it has an ''escape velocity'' with respect to the state ''A''. In practice, differential features acquire their logical meaning through a class of ''temporal inference rules''.