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To prepare for the next phase of discussion, Tables 6 and 7 collect and summarize all of the propositional forms on one and two variables.  These propositional forms are represented over bases of boolean variables as complete sets of boolean-valued functions.  Adjacent to their names and specifications are listed what are roughly the simplest expressions in the ''[[cactus language]]'', the particular syntax for propositional calculus that I use in formal and computational contexts.  For the sake of orientation, the English paraphrases and the more common notations are listed in the last two columns.  As simple and circumscribed as these low-dimensional universes may appear to be, a careful exploration of their differential extensions will involve us in complexities sufficient to demand our attention for some time to come.

To prepare for the next phase of discussion, Tables 6 and 7 collect and summarize all of the propositional forms on one and two variables.  These propositional forms are represented over bases of boolean variables as complete sets of boolean-valued functions.  Adjacent to their names and specifications are listed what are roughly the simplest expressions in the ''[[cactus language]]'', the particular syntax for propositional calculus that I use in formal and computational contexts.  For the sake of orientation, the English paraphrases and the more common notations are listed in the last two columns.  As simple and circumscribed as these low-dimensional universes may appear to be, a careful exploration of their differential extensions will involve us in complexities sufficient to demand our attention for some time to come.

Propositional forms on one variable correspond to boolean functions ''f''&nbsp;:&nbsp;'''B'''¹&nbsp;&rarr;&nbsp;'''B'''.  In Table 6 these functions are listed in a variant form of [[truth table]], one which rotates the axes of the usual arrangement.  Each function ''f''_''i'' is indexed by the string of values that it takes on the points of the universe ''X''^&nbsp;&bull;&nbsp;=&nbsp;[''x'']&nbsp;<math>\cong</math>&nbsp;'''B'''¹. The binary index generated in this way is converted to its decimal equivalent, and these are used as conventional names for the ''f''_''i''&nbsp;, as shown in the first column of the Table.  In their own right the 2¹ points of the universe ''X''^&nbsp;&bull; are coordinated as a space of type '''B'''¹, this in light of the universe ''X''^&nbsp;&bull; being a functional domain where the coordinate projection ''x'' takes on its values in '''B'''.

Propositional forms on one variable correspond to boolean functions <math>f : \mathbb{B}^1 \to \mathbb{B}.</math> In Table&nbsp;6 these functions are listed in a variant form of [[truth table]], one in which the axes of the usual arrangement are rotated through a right angle.  Each function <math>f_i\!</math> is indexed by the string of values that it takes on the points of the universe <math>X^\circ = [x] \cong \mathbb{B}^1.</math>  The binary index generated in this way is converted to its decimal equivalent and these are used as conventional names for the <math>f_i,\!</math> as shown in the first column of the Table.  In their own right the <math>2^1\!</math> points of the universe <math>X^\circ</math> are coordinated as a space of type <math>\mathbb{B}^1,</math> this in light of the universe <math>X^\circ</math> being a functional domain where the coordinate projection <math>x\!</math> takes on its values in <math>\mathbb{B}.</math>

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