Differential Propositional Calculus • 21

There was never any more inception than there is now,
Nor any more youth or age than there is now;
And will never be any more perfection than there is now,
Nor any more heaven or hell than there is now.

— Walt Whitman • Leaves of Grass

A One‑Dimensional Universe

Let $\mathcal{X} = \{ A \}$ be a logical basis containing one boolean variable or logical feature $A.$ The basis element $A$ may be regarded as a simple proposition or coordinate projection $A : \mathbb{B} \to \mathbb{B}.$ Corresponding to the basis $\mathcal{X} = \{ A \}$ is the alphabet $\mathfrak{X} = \{ ``A" \}$ which serves whenever we need to make explicit mention of the symbols used in our formulas and representations.

The space $X = \langle A \rangle = \{ \texttt{(} A \texttt{)}, A \}$ of points (cells, vectors, interpretations) has cardinality $2^n = 2^1 = 2$ and is isomorphic to $\mathbb{B} = \{ 0, 1 \}.$ Moreover, $X$ may be identified with the set of singular propositions $\{ x : \mathbb{B} \xrightarrow{s} \mathbb{B} \}.$

The space of linear propositions $X^* = \{ \mathrm{hom} : \mathbb{B} \xrightarrow{\ell} \mathbb{B} \} = \{ 0, A \}$ is algebraically dual to $X$ and also has cardinality $2.$ Here, $``0"$ is interpreted as denoting the constant function $0 : \mathbb{B} \to \mathbb{B},$ 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 $\{ \mathrm{pos} : \mathbb{B} \xrightarrow{p} \mathbb{B} \} = \{ A, 1 \}$ of rank $1$ and $0,$ respectively, where $``1"$ is understood as denoting the constant function $1 : \mathbb{B} \to \mathbb{B}.$

All told there are $2^{2^n} = 2^{2^1} = 4$ propositions in the universe of discourse $\mathcal{X}^\bullet = [\mathcal{X}],$ collectively forming the set $X^\uparrow = \{ f : X \to \mathbb{B} \} = \{ 0, \texttt{(} A \texttt{)}, A, 1 \} \cong (\mathbb{B} \to \mathbb{B}).$

Resources

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