Differential Propositional Calculus • 21
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https://inquiryintoinquiry.com/2023/12/12/differential-propositional-calcul…
A One‑Dimensional Universe —
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
Let †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 : B → B. Corresponding to the
basis †X† = {A} is the alphabet ‡X‡ = {“A”} which serves whenever we
need to make explicit mention of the symbols used in our formulas and
representations.
The space X = ⟨A⟩ = {¬A, A} of points (cells, vectors, interpretations)
has cardinality 2ⁿ = 2¹ = 2 and is isomorphic to B = {0, 1}. Moreover,
X may be identified with the set of singular propositions {s : B → B}.
The space of linear propositions X* = {ℓ : B → 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 {p : B → B}
= {A, 1} of rank 1 and 0, respectively, where “1” is understood
as denoting the constant function 1 : B → B.
All told there are 2^{2ⁿ} = 2^{2¹} = 4 propositions in the
universe of discourse [†X†], collectively forming the set
X↑ = {f : X→B} = {0, ¬A, A, 1}.
Resources —
Differential Logic and Dynamic Systems
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https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part…
Differential Logic • One-Dimensional Universe
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https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part…
Regards,
Jon
cc:
https://www.academia.edu/community/lKDXw5