Differential Propositional Calculus • 17 • https://inquiryintoinquiry.com/2023/12/06/differential-propositional-calcul… Differential Propositions • Tangent Spaces — The “tangent space” to A at one of its points x, sometimes written Tₓ(A), takes the form dA = ⟨d†A†⟩ = ⟨da₁, …, daₙ⟩. Strictly speaking, the name “cotangent space” is probably more correct for this construction but since we take up spaces and their duals in pairs to form our universes of discourse it allows our language to be pliable here. Proceeding as we did with the base space A, the tangent space dA at a point of A may be analyzed as the following product of distinct and independent factors. • dA = ∏ dAₖ = dA₁ × … × dAₙ. Each factor dAₖ is a set consisting of two differential propositions, dAₖ = {(daₖ), daₖ}, where (daₖ) is a proposition with the logical value of ¬daₖ. Each component dAₖ has the type B, operating under the ordered correspondence {(daₖ), daₖ} ≅ {0, 1}. A measure of clarity is achieved, however, by acknowledging the differential usage with a superficially distinct type D, whose sense may be indicated as follows. • D = {(daₖ), daₖ} = {same, different} = {stay, change} = {stop, step}. Viewed within a coordinate representation, spaces of type Bⁿ and Dⁿ may appear to be identical sets of binary vectors, but taking a view at that level of abstraction would be like ignoring the qualitative units and the diverse dimensions that distinguish position and momentum, or the different roles of quantity and impulse. Resources — Differential Logic and Dynamic Systems • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Differential Propositions • Tangent Spaces • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Regards, Jon cc: https://www.academia.edu/community/5Rkm2V

Differential Propositional Calculus • 19 • https://inquiryintoinquiry.com/2023/12/10/differential-propositional-calcul… Failing to fetch me at first keep encouraged, Missing me one place search another, I stop some where waiting for you — Walt Whitman • Leaves of Grass Life on Easy Street — The finite character of the extended universe [E†A†] makes the task of solving differential propositions relatively straightforward. The solution set of the differential proposition q : EA → B is the set of models q⁻¹(1) in EA. Finding all models of q, the extended interpretations in EA which satisfy q, can be carried out by a finite search. Being in possession of complete algorithms for propositional calculus modeling, theorem checking, and theorem proving makes the analytic task fairly simple in principle, even if the question of efficiency in the face of arbitrary complexity remains another matter entirely. The NP‑completeness of propositional satisfiability may weigh against the prospects of a single efficient algorithm capable of covering the whole space [E†A†] with equal facility but there appears to be much room for improvement in classifying special forms and in developing algorithms tailored to their practical processing. Resources — Differential Logic and Dynamic Systems • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Differential Logic • Life on Easy Street • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Regards, Jon cc: https://www.academia.edu/community/LGZ3eV

Differential Propositional Calculus • 21 • 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 • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Differential Logic • One-Dimensional Universe • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Regards, Jon cc: https://www.academia.edu/community/lKDXw5

Differential Propositional Calculus • 23 • https://inquiryintoinquiry.com/2023/12/15/differential-propositional-calcul… A One-Dimensional Universe (concl.) ❝The clock indicates the moment . . . . but what does eternity indicate?❞ — Walt Whitman • Leaves of Grass It might be thought an independent time variable needs to be brought in at this point but it is an insight of fundamental importance to recognize the idea of process is logically prior to the notion of time. A time variable is a reference to a “clock” — a canonical, conventional process accepted or established as a standard of measurement but in essence no different than any other process. This raises the question of how different subsystems in a more global process can be brought into comparison and what it means for one process to serve the function of a local standard for others. But inquiries of that order serve but to wrap up puzzles in further riddles and are obviously too involved to be handled at our current level of approximation. Observe how the secular inference rules, used by themselves, involve a loss of information, since nothing in them tells whether the momenta {¬dA, dA} are changed or unchanged in the next moment. To know that one would have to determine d²A, and so on, pursuing an infinite regress. In order to rest with a finitely determinate system it is necessary to make an infinite assumption, for example, that dⁿA = 0 for all n greater than some fixed value M. Another way to escape the regress is through the provision of a dynamic law, in typical form making higher order differentials dependent on lower degrees and estates. Resources — Differential Logic and Dynamic Systems • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Differential Logic • One-Dimensional Universe • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Regards, Jon cc: https://www.academia.edu/community/lzJejl

Differential Propositional Calculus • 25 • https://inquiryintoinquiry.com/2023/12/17/differential-propositional-calcul… Example 1. A Square Rigging (cont.) Because the initial space X = ⟨A⟩ is one‑dimensional we can easily fit the second order extension E²X = ⟨A, dA, d²A⟩ within the compass of a single venn diagram, charting the pair of converging trajectories as shown in the following Figure. Example 1. The Anchor • https://inquiryintoinquiry.files.wordpress.com/2023/12/differential-logic-e… Resources — Differential Logic and Dynamic Systems • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Differential Logic • A Square Rigging • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Regards, Jon cc: https://www.academia.edu/community/LZvM3L

Differential Propositional Calculus • 27 • https://inquiryintoinquiry.com/2023/12/19/differential-propositional-calcul… Commentary On Small Models — One reason for engaging in our present order of extremely reduced but explicitly controlled case study is to throw light on the general study of languages, formal and natural, in their full array of syntactic, semantic, and pragmatic aspects. Propositional calculus is one of the last points of departure where it is possible to see that trio of aspects interacting in a non‑trivial way without being immediately and totally overwhelmed by the complexity they generate. Often that complexity leads investigators of formal and natural languages to adopt the strategy of focusing on a single aspect and abandoning all hope of understanding the whole, whether it’s the still living natural language or the dynamics of inquiry crystallized in formal logic. From the perspective I find most useful here, a language is a syntactic system designed or evolved in part to express a set of descriptions. When the explicit symbols of a language have extensions in its object world which are actually infinite, or when the implicit categories and generative devices of a linguistic theory have extensions in its subject matter which are potentially infinite, then the finite characters of terms, statements, arguments, grammars, logics, and rhetorics force a surplus intension to color all its symbols and functions, across the spectrum from object language to metalinguistic reflection. In the aphorism of W. von Humboldt often cited by Chomsky, language requires “the infinite use of finite means”. That is necessarily true when the extensions are infinite, when the referential symbols and grammatical categories of a language possess infinite sets of models and instances. But it also voices a practical truth when the extensions, though finite at every stage, tend to grow at exponential rates. Resources — Differential Logic and Dynamic Systems • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Differential Logic • Commentary On Small Models • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Regards, Jon cc: https://www.academia.edu/community/l8m8z5

Differential Propositional Calculus • 29 • https://inquiryintoinquiry.com/2023/12/21/differential-propositional-calcul… Back to the Feature — ❝I guess it must be the flag of my disposition, out of hopeful green stuff woven.❞ — Walt Whitman • Leaves of Grass Let's assume the sense intended for differential features is well enough established in the intuition for now that we may continue outlining the structure of the differential extension [E†X†] = [A, dA]. Over the extended alphabet E†X† = {x₁, dx₁} = {A, dA} of cardinality 2ⁿ = 2 we generate the set of points EX of cardinality 2²ⁿ = 4 which bears the following chain of equivalent descriptions. • EX = ⟨A, dA⟩ = {(A), A} × {(dA), dA} = {(A)(dA), (A)dA, A(dA), A∙dA}. The space EX may be given the nominal type B×D, at root isomorphic to B×B = B². An element of EX may be regarded as a “disposition at a point” or a “situated direction”, in effect, a singular mode of change occurring at a single point in the universe of discourse. In practice the modality of those changes may be interpreted in various ways, for example, as expectations, intentions, or observations with respect to the behavior of a system. To complete the construction of the extended universe of discourse EX• = [A, dA] the basic dispositions in EX need to be extended to the full set of differential propositions EX↑ = {g : EX → B}, each of type B×D → B. There are 2^{2²ⁿ} = 16 propositions in EX↑, as detailed in the following Table. Table. Differential Propositions • https://inquiryintoinquiry.files.wordpress.com/2023/12/differential-logic-e… Aside from changing the names of variables and shuffling the order of rows, the Table follows the format previously used for boolean functions of two variables. The rows are grouped to reflect natural similarity classes holding among the propositions. In a future discussion the classes will be given additional explanation and motivation as the orbits of a certain transformation group acting on the set of 16 propositions. Notice that four of the propositions, in their logical expressions, resemble those given in the table for X↑. Thus the first set of propositions {fₖ} is automatically embedded in the present set {gₘ} and the corresponding inclusions are indicated at the far left margin of the Table. Resources — Differential Logic and Dynamic Systems • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Differential Logic • Back to the Feature • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Regards, Jon cc: https://www.academia.edu/community/5kzK0l

Differential Propositional Calculus • 31 • https://inquiryintoinquiry.com/2023/12/23/differential-propositional-calcul… Tacit Extensions — Re: Table of Differential Propositions • https://inquiryintoinquiry.files.wordpress.com/2023/12/differential-logic-e… Returning to the Table of Differential Propositions, let's examine how the general concept of a tacit extension applies to the differential extension of a one‑dimensional universe of discourse, where †X† = {A} and †Y† = E†X† = {A, dA}. Each proposition fₖ : X → B has a canonical expression eₖ in the set {0, (A), A, 1}. The tacit extension εfₖ : EX → B may then be expressed as a logical conjunction of two factors, fₖ = eₖ∙τ, where τ is a logical tautology using all the variables of †Y† − †X†. The following Table shows how the tacit extensions εfₖ of the propositions fₖ may be expressed in terms of the extended basis {A, dA}. Table. Tacit Extension of [A] to [A, dA] • https://inquiryintoinquiry.files.wordpress.com/2023/12/differential-logic-e… In its bearing on the singular propositions over a universe of discourse X the above analysis has an interesting interpretation. The tacit extension takes us from thinking about a particular state, like A or (A), to considering the collection of outcomes, the outgoing changes or singular dispositions, springing or stemming from that state. Resources — Differential Logic and Dynamic Systems • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Differential Logic • Tacit Extensions • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Regards, Jon cc: https://www.academia.edu/community/laNmp5

Differential Propositional Calculus • 33 • https://inquiryintoinquiry.com/2023/12/27/differential-propositional-calcul… Example 2. Drives and Their Vicissitudes (cont.) Expressed in terms of “drives” and “gears” our next Example may be described as the family of 4‑th‑gear curves in the fourth extension E⁴X = ⟨A, dA, d²A, d³A, d⁴A⟩. Those are the trajectories generated subject to the dynamic law d⁴A = 1, where it's understood all higher differences are equal to 0. Since d⁴A and all higher differences dⁿA are fixed, the state vectors vary only with respect to their projections as points of E³X = ⟨A, dA, d²A, d³A⟩. Thus there is just enough space in a planar venn diagram to plot all the orbits and to show how they partition the points of E³X. It turns out there are exactly two possible orbits, of eight points each, as shown in the following Figure. Example 2. Fourth Gear Orbits • https://inquiryintoinquiry.files.wordpress.com/2023/12/differential-logic-e… Resources — Differential Logic and Dynamic Systems • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Differential Logic • Drives and Their Vicissitudes • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Regards, Jon cc: https://www.academia.edu/community/VXAx1L

Differential Propositional Calculus • 35 • https://inquiryintoinquiry.com/2023/12/29/differential-propositional-calcul… Example 2. Drives and Their Vicissitudes (concl.) Applied to the example of 4‑th‑gear curves, the indexing scheme results in the data of the next two Tables, showing one period for each orbit. Fourth Gear Orbits 1 and 2 • https://inquiryintoinquiry.files.wordpress.com/2023/12/differential-logic-e… The states in each orbit are listed as ordered pairs (p_i, q_j), where p_i may be read as a temporal parameter indicating the present time of the state and where j is the decimal equivalent of the binary numeral s. Grasped more intuitively, the Tables show each state q_s with a subscript s equal to the numerator of its rational index, taking for granted the constant denominator of 2⁴=16. In that way the temporal succession of states can be reckoned by a “parallel round‑up rule”. Namely, if (dₖ, dₖ₊₁) is any pair of adjacent digits in the state index r then the value of dₖ in the next state is dₖ′=dₖ+dₖ₊₁. Resources — Differential Logic and Dynamic Systems • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Differential Logic • Drives and Their Vicissitudes • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Regards, Jon cc: https://www.academia.edu/community/Vj44Wl

Differential Propositional Calculus • 37 • https://inquiryintoinquiry.com/2024/01/01/differential-propositional-calcul… Foreshadowing Transformations • Extensions and Projections of Discourse — ❝And, despite the care which she took to look behind her at every moment, she failed to see a shadow which followed her like her own shadow, which stopped when she stopped, which started again when she did, and which made no more noise than a well‑conducted shadow should.❞ — Gaston Leroux • The Phantom of the Opera Many times in our discussion we have occasion to place one universe of discourse in the context of a larger universe of discourse. An embedding of the type [†X†] → [†Y†] is implied any time we make use of one basis †X† which happens to be included in another basis †Y†. When discussing differential relations we usually have in mind the extended alphabet ‡Y‡ has a special construction or a specific lexical relation with respect to the initial alphabet ‡X‡, one which is marked by characteristic types of accents, indices, or inflected forms. Resources — Differential Logic and Dynamic Systems • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Differential Logic • Foreshadowing Transformations • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Regards, Jon cc: https://www.academia.edu/community/V9WrEl