Differential Propositional Calculus • 3 • https://inquiryintoinquiry.com/2024/12/01/differential-propositional-calcul… Casual Introduction (cont.) Figure 3 returns to the situation in Figure 1, but this time interpolates a new quality specifically tailored to account for the relation between Figure 1 and Figure 2. Figure 3. Back, To The Future • https://inquiryintoinquiry.files.wordpress.com/2023/11/differential-proposi… The new quality, dq, is called a “differential quality” by virtue of the fact its absence or presence qualifies the absence or presence of change occurring in another quality. As with any quality, it is represented in the venn diagram by means of a “circle” distinguishing two halves of the universe of discourse, in this case, the portions of X outside and inside the region dQ. Figure 1 represents a universe of discourse X together with a basis of discussion {q} for expressing propositions about the contents of that universe. Once the quality q is given a name, say, the symbol “q”, we have the basis for a formal language specifically cut out for discussing X in terms of q. That language is more formally known as the “propositional calculus” with alphabet {“q”}. In the context marked by X and {q} there are just four distinct pieces of information which can be expressed in the corresponding propositional calculus, namely, the constant proposition False, the negative proposition ¬q, the positive proposition q, and the constant proposition True. For example, referring to the points in Figure 1, the constant proposition False holds of no points, the negative proposition ¬q holds of a and d, the positive proposition q holds of b and c, and the constant proposition True holds of all points in the sample. Figure 3 preserves the same universe of discourse and extends the basis of discussion to a set of two qualities, {q, dq}. In corresponding fashion, the initial propositional calculus is extended by means of the enlarged alphabet, {“q”, “dq”}. Resources — Logic Syllabus • https://inquiryintoinquiry.com/logic-syllabus/ Survey of Differential Logic • https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7 Regards, Jon cc: https://www.academia.edu/community/lypQnm cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

Differential Propositional Calculus • 5 • https://inquiryintoinquiry.com/2024/12/03/differential-propositional-calcul… Casual Introduction (concl.) Table 5 exhibits the rules of inference responsible for giving the differential proposition dq its meaning in practice. Table 5. Differential Inference Rules • https://inquiryintoinquiry.files.wordpress.com/2020/02/differential-proposi… If the feature q is interpreted as applying to an object in the universe of discourse X then the differential feature dq may be taken as an attribute of the same object which tells it is changing “significantly” with respect to the property q — as if the object bore an “escape velocity” with respect to the condition q. For example, relative to a frame of observation to be made more explicit later on, if q and dq are true at a given moment, it would be reasonable to assume ¬q will be true in the next moment of observation. Taken all together we have the fourfold scheme of inference shown above. Resources — Logic Syllabus • https://inquiryintoinquiry.com/logic-syllabus/ Survey of Differential Logic • https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7 Regards, Jon cc: https://www.academia.edu/community/VBqdBD cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

Differential Propositional Calculus • 7 • https://inquiryintoinquiry.com/2024/12/06/differential-propositional-calcul… Note. Please see the blog post linked above for the proper formats of the notations used below, as they depend on numerous typographical distinctions lost in the following transcript. Formal Development — The preceding discussion outlined the ideas leading to the differential extension of propositional logic. The next task is to lay out the concepts and terminology needed to describe various orders of differential propositional calculi. Elementary Notions — Logical description of a universe of discourse begins with a collection of logical signs. For simplicity in a first approach we assume the signs are collected in the form of a finite alphabet, ‡A‡ = {“a₁”, …, “aₙ”}. The signs are interpreted as denoting logical features, for example, properties of objects in the universe of discourse or simple propositions about those objects. Corresponding to the alphabet ‡A‡ there is then a set of logical features, †A† = {a₁, …, aₙ}. A set of logical features †A† = {a₁, …, aₙ} affords a basis for generating an n‑dimensional universe of discourse, written A° = [†A†] = [a₁, …, aₙ]. It is useful to consider a universe of discourse as a categorical object incorporating both the set of points A = <a₁, …, aₙ> and the set of propositions A↑ = {f : A → B} implicit with the ordinary picture of a venn diagram on n features. Accordingly, the universe of discourse A° may be regarded as an ordered pair (A, A↑) bearing the type (Bⁿ, (Bⁿ → B)), which type designation may be abbreviated as Bⁿ +→ B or even more succinctly as [Bⁿ]. For convenience, the data type of a finite set on n elements may be indicated by either one of the equivalent notations [n] or *n*. Table 7 summarizes the basic notations needed to describe ordinary propositional calculi in a systematic fashion. Table 7. Propositional Calculus • Basic Notation • https://inquiryintoinquiry.files.wordpress.com/2020/02/propositional-calcul… Resources — Logic Syllabus • https://inquiryintoinquiry.com/logic-syllabus/ Survey of Differential Logic • https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7 Regards, Jon cc: https://www.academia.edu/community/LxkZB2 cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

Differential Propositional Calculus • 9 • https://inquiryintoinquiry.com/2024/12/08/differential-propositional-calcul… Special Classes of Propositions — The full set of propositions f : A → B contains a number of smaller classes deserving of special attention. A “basic proposition” in the universe of discourse [a₁, …, aₙ] is one of the propositions in the set {a₁, …, aₙ}. There are of course exactly n of these. Depending on context, basic propositions may also be called “coordinate propositions” or “simple propositions”. Among the 2^(2ⁿ) propositions in [a₁, …, aₙ] are several families numbering 2ⁿ propositions each which take on special forms with respect to the basis {a₁, …, aₙ}. Three of those families are especially prominent in the present context, the “linear”, the “positive”, and the “singular” propositions. Each family is naturally parameterized by the coordinate n‑tuples in Bⁿ and falls into n+1 ranks, with a binomial coefficient (n choose k) giving the number of propositions having rank or weight k in their class. Linear Propositions ℓ : Bⁿ → B may be written as sums. • https://inquiryintoinquiry.files.wordpress.com/2020/02/linear-propositions-… Positive Propositions p : Bⁿ → B may be written as products. • https://inquiryintoinquiry.files.wordpress.com/2020/02/positive-proposition… Singular Propositions s : Bⁿ → B may be written as products. • https://inquiryintoinquiry.files.wordpress.com/2020/02/singular-proposition… In each case the rank k ranges from 0 to n and counts the number of positive appearances of the coordinate propositions a₁, …, aₙ in the resulting expression. For example, when n = 3 the linear proposition of rank 0 is 0, the positive proposition of rank 0 is 1, and the singular proposition of rank 0 is (a₁)(a₂)(a₃), that is, ¬a₁ ∧ ¬a₂ ∧ ¬a₃. The basic propositions a_i : Bⁿ → B are both linear and positive. So those two families of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions. It is important to note that all the above distinctions are relative to the choice of a particular logical basis †A† = {a₁, …, aₙ}. A singular proposition with respect to the basis †A† will not remain singular if †A† is extended by a number of new and independent features. Even if one keeps to the original set of pairwise options {a_i} ∪ {¬a_i} to pick out a new basis, the sets of linear propositions and positive propositions are both determined by the choice of basic propositions, and that entire determination is tantamount to the purely conventional choice of a cell as origin. Resources — Logic Syllabus • https://inquiryintoinquiry.com/logic-syllabus/ Survey of Differential Logic • https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7 Regards, Jon cc: https://www.academia.edu/community/VjKj3b cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

Differential Propositional Calculus • 11 • https://inquiryintoinquiry.com/2024/12/10/differential-propositional-calcul… Special Classes of Propositions (cont.) Next we take up the family of positive propositions and follow the same plan as before, tracing the rule of their formation in the case of a 3‑dimensional universe of discourse. Positive Propositions — The positive propositions p : Bⁿ → B may be written as products: • https://inquiryintoinquiry.files.wordpress.com/2020/02/positive-proposition… In a universe of discourse based on three boolean variables, p, q, r, there are 2³ = 8 positive propositions, taking the shapes shown in Figure 9. Figure 9. Positive Propositions on Three Variables • https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagrams-e280a2… At the top is the venn diagram for the positive proposition of rank 3, corresponding to the boolean product or logical conjunction pqr. Next are the venn diagrams for the three positive propositions of rank 2, corresponding to the three boolean products, pr, qr, pq, respectively. Next are the three positive propositions of rank 1, which are none other than the three basic propositions, p, q, r. At the bottom is the positive proposition of rank 0, the everywhere true proposition or the constant 1 function, which may be expressed by the form (( )) or by a simple 1. Resources — Logic Syllabus • https://inquiryintoinquiry.com/logic-syllabus/ Survey of Differential Logic • https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7 Regards, Jon cc: https://www.academia.edu/community/VoqgZr cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

Differential Propositional Calculus • 13 • https://inquiryintoinquiry.com/2024/12/11/differential-propositional-calcul… Note. Please see the blog post linked above for the proper formats of the notations used below, as they depend on many typographical distinctions lost in the following transcript. Differential Extensions — An initial universe of discourse A° supplies the groundwork for any number of further extensions, beginning with the first order differential extension EA°. The construction of EA° can be described in the following stages. The initial alphabet ‡A‡ = {“a₁”, …, “aₙ”} is extended by a first order differential alphabet d‡A‡ = {“da₁”, …, “daₙ”} resulting in a first order extended alphabet E‡A‡ defined as follows. • E‡A‡ = ‡A‡ ∪ d‡A‡ = {“a₁”, …, “aₙ”, “da₁”, …, “daₙ”}. The initial basis †A† = {a₁, …, aₙ} is extended by a first order differential basis d†A† = {da₁, …, daₙ} resulting in a first order extended basis E†A† defined as follows. • E†A† = †A† ∪ d†A† = {a₁, …, aₙ, da₁, …, daₙ}. The initial space A = ⟨a₁, …, aₙ⟩ is extended by a first order differential space or tangent space dA = ⟨da₁, …, daₙ⟩ at each point of A, resulting in a first order extended space EA defined as follows. • EA = A × dA = ⟨E†A†⟩ = ⟨†A† ∪ d†A†⟩ = ⟨a₁, …, aₙ, da₁, …, daₙ⟩. Finally, the initial universe A° = [a₁, …, aₙ] is extended by a first order differential universe or tangent universe dA° = [da₁, …, daₙ] at each point of A°, resulting in a first order extended universe EA° defined as follows. • EA° = [E†A†] = [†A† ∪ d†A†] = [a₁, …, aₙ, da₁, …, daₙ]. This gives EA° a type defined as follows. • [Bⁿ × Dⁿ] = (Bⁿ × Dⁿ +→ B) = (Bⁿ × Dⁿ, Bⁿ × Dⁿ → B). A proposition in a differential extension of a universe of discourse is called a “differential proposition” and forms the analogue of a system of differential equations in ordinary calculus. With the construction of the first order extended universe EA° and the first order differential propositions f : EA→B we arrive at the foothills of differential logic. Resources — Logic Syllabus • https://inquiryintoinquiry.com/logic-syllabus/ Survey of Differential Logic • https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7 Regards, Jon cc: https://www.academia.edu/community/laYDqx cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

Differential Propositional Calculus • 15 • https://inquiryintoinquiry.com/2024/12/13/differential-propositional-calcul… Fire over water: The image of the condition before transition. Thus the superior man is careful In the differentiation of things, So that each finds its place. — I Ching ䷿ Hexagram 64 Differential Extension of Propositional Calculus — This much preparation is enough to begin introducing my subject, if I excuse myself from giving full arguments for my definitional choices until a later stage. To express the goal in a turn of phrase, the aim is to develop a “differential theory of qualitative equations”, one which can parallel the application of differential geometry to dynamical systems. The idea of a tangent vector is key to the work and a major goal is to find the right logical analogues of tangent spaces, bundles, and functors. The strategy is taken of looking for the simplest versions of those constructions which can be discovered within the realm of propositional calculus, so long as they serve to fill out the general theme. Reference — Wilhelm, R., and Baynes, C.F. (trans.), The I Ching, or Book of Changes, Foreword by C.G. Jung, Preface by H. Wilhelm, 3rd edition, Bollingen Series XIX, Princeton University Press, Princeton, NJ, 1967. Resources — Differential Logic and Dynamic Systems • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Differential Extension of Propositional Calculus • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Regards, Jon cc: https://www.academia.edu/community/V3KyY4 cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

Differential Propositional Calculus • 17 • https://inquiryintoinquiry.com/2024/12/15/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/lQ1nxp cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

Differential Propositional Calculus • 19 • https://inquiryintoinquiry.com/2024/12/17/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 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/LxkGPy cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

Differential Propositional Calculus • 21 • https://inquiryintoinquiry.com/2024/12/18/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/laY4w3 cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

Differential Propositional Calculus • 23 • https://inquiryintoinquiry.com/2024/12/20/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 realize 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. That raises questions 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. Inquiries of that order serve but to wrap up our present puzzles in further riddles and are far too involved to be handled at our current level of approximation. We'll return to them another time. 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/5AqmbK cc: https://www.researchgate.net/post/Differential_Propositional_Calculus