Differential Logic • 3 • https://inquiryintoinquiry.com/2024/11/02/differential-logic-3-a/ Table 1 shows the cactus graphs, the corresponding cactus expressions, their logical meanings under the so‑called “existential interpretation”, and their translations into conventional notations for a sample of basic propositional forms. Table 1. Syntax and Semantics of a Calculus for Propositional Logic • https://inquiryintoinquiry.files.wordpress.com/2022/10/syntax-and-semantics… The simplest expression for logical truth is the empty word, typically denoted by ε or λ in formal languages, where it is the identity element for concatenation. To make it visible in context, it may be denoted by the equivalent expression “(())” or, especially if operating in an algebraic context, by a simple “1”. Also when working in an algebraic mode, the plus sign “+” may be used for exclusive disjunction. Thus we have the following translations of algebraic expressions into cactus expressions. • a + b = (a, b) • a + b + c = (a, (b, c)) = ((a, b), c) It is important to note the last expressions are not equivalent to the 3‑place form (a, b, c). 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/l8D7KG cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_an…

Differential Logic • 5 • https://inquiryintoinquiry.com/2024/11/04/differential-logic-5-a/ Differential Expansions of Propositions — Worm's Eye View — Let's run through the initial example again, keeping an eye on the meanings of the formulas which develop along the way. We begin with a proposition or a boolean function f(p, q) = pq whose venn diagram and cactus graph are shown below. Venn Diagram f = pq • https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagram-f-p-and… Cactus Graph f = pq • https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-f-p-and… A function like f has an abstract type and a concrete type. The abstract type is what we invoke when we write things like f : B×B→B or f : B²→B. The concrete type takes into account the qualitative dimensions or the “units” of the case, which can be explained as follows. • Let P be the set of values {(p), p} = {not p, p} isomorphic to B = {0, 1}. • Let Q be the set of values {(q), q} = {not q, q} isomorphic to B = {0, 1}. Then interpret the usual propositions about p, q as functions of the concrete type f : P×Q→B. We are going to consider various operators on these functions. An operator F is a function which takes one function f into another function Ff. The first couple of operators we need are logical analogues of two which play a founding role in the classical finite difference calculus, namely, the following. • The “difference operator” ∆, written here as D. • The “enlargement operator”, written here as E. These days, E is more often called the “shift operator”. In order to describe the universe in which these operators operate, it is necessary to enlarge the original universe of discourse. Starting from the initial space X = P×Q, its “(first order) differential extension” EX is constructed according to the following specifications. • EX = X×dX where: • X = P×Q • dX = dP×dQ • dP = {(dp), dp} • dQ = {(dq), dq} The interpretations of these new symbols can be diverse, but the easiest option for now is just to say dp means “change p” and dq means “change q”. Drawing a venn diagram for the differential extension EX = X×dX requires four logical dimensions, P, Q, dP, dQ, but it is possible to project a suggestion of what the differential features dp and dq are about on the 2‑dimensional base space X = P×Q by drawing arrows that cross the boundaries of the basic circles in the venn diagram for X, reading an arrow as dp if it crosses the boundary between p and (p) in either direction and reading an arrow as dq if it crosses the boundary between q and (q) in either direction, as indicated in the following figure. Venn Diagram p q dp dq • https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagram-p-q-dp-… Propositions are formed on differential variables, or any combination of ordinary logical variables and differential logical variables, in the same ways propositions are formed on ordinary logical variables alone. For example, the proposition (dp (dq)) says the same thing as dp ⇒ dq, in other words, there is no change in p without a change in q. Given the proposition f(p, q) over the space X = P×Q, the “(first order) enlargement of f” is the proposition Ef over the differential extension EX defined by the following formula. • Ef(p, q, dp, dq) = f(p + dp, q + dq) = f(p xor dp, q xor dq) In the example f(p, q) = pq, the enlargement Ef is computed as follows. • Ef(p, q, dp, dq) = (p + dp)(q + dq) = (p xor dp)(q xor dq) The corresponding cactus graph is shown below. Cactus Graph Ef = (p,dp)(q,dq) • https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-ef-pdpq… Given the proposition f(p, q) over X = P×Q, the “(first order) difference of f” is the proposition Df over EX defined by the formula Df = Ef - f, or, written out in full: • Df(p, q, dp, dq) = f(p + dp, q + dq) - f(p, q) = f(p xor dp, q xor dq) xor f(p, q) In the example f(p, q) = pq, the difference Df is computed as follows. • Df(p, q, dp, dq) = (p + dp)(q + dq) - pq = (p xor dp)(q xor dq) xor pq The corresponding cactus graph is shown below. Cactus Graph Df = ((p,dp)(q,dq),pq) • https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-df-pdpq… This brings us by the road meticulous to the point we reached at the end of the previous post. Differential Logic • 4 • https://inquiryintoinquiry.com/2024/11/03/differential-logic-4-a/ There we evaluated the above proposition, the “first order difference of conjunction” Df, at a single location in the universe of discourse, namely, at the point picked out by the singular proposition pq, in terms of coordinates, at the place where p = 1 and q = 1. This evaluation is written in the form Df|{pq} or Df|{(1, 1)}, and we arrived at the locally applicable law which may be stated and illustrated as follows. • f(p, q) = pq = p and q ⇒ Df|{pq} = ((dp)(dq)) = dp or dq Venn Diagram Difference pq @ pq • https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagram-differe… Cactus Graph Difference pq @ pq • https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-differe… The venn diagram shows the analysis of the inclusive disjunction “dp or dq” into the following exclusive disjunction. • (dp and not dq) xor (dq and not dp) xor (dp and dq) The resultant differential proposition may be read to say “change p or change q or both”. And this can be recognized as just what you need to do if you happen to find yourself in the center cell and require a complete and detailed description of ways to escape it. 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/lnYWY0 cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_an…

Differential Logic • 7 • https://inquiryintoinquiry.com/2024/11/06/differential-logic-7-a/ Differential Expansions of Propositions — Panoptic View • Enlargement Maps — The “enlargement” or “shift” operator E exhibits a wealth of interesting and useful properties in its own right, so it pays to examine a few of the more salient features playing out on the surface of our initial example, f(p, q) = pq. A suitably generic definition of the extended universe of discourse is afforded by the following set‑up. • Let X = X₁ × … × Xₖ. • Let dX = dX₁ × … × dXₖ. • Then EX = X × dX = X₁ × … × Xₖ × dX₁ × … × dXₖ For a proposition of the form f : X₁ × … × Xₖ → B, the “(first order) enlargement” of f is the proposition Ef : EX→B defined by the following equation. • Ef(x₁, …, xₖ, dx₁, …, dxₖ) = f(x₁ + dx₁, …, xₖ + dxₖ) = f(x₁ xor dx₁, …, xₖ xor dxₖ) The “differential variables” dx_j are boolean variables of the same type as the ordinary variables x_j. Although it is conventional to distinguish the (first order) differential variables with the operational prefix “d”, that way of notating differential variables is entirely optional. It is their existence in particular relations to the initial variables, not their names, which defines them as differential variables. In the example of logical conjunction, f(p, q) = pq, the enlargement Ef is formulated as follows. • Ef(p, q, dp, dq) = (p + dp)(q + dq) = (p xor dp)(q xor dq) Given that the above expression uses nothing more than the boolean ring operations of addition and multiplication, it is permissible to “multiply things out” in the usual manner to arrive at the following result. • Ef(p, q, dp, dq) = p·q + p·dq + q·dp + dp·dq To understand what the “enlarged” or “shifted” proposition means in logical terms, it serves to go back and analyze the above expression for Ef in the same way we did for Df. To that end, the value of Efₓ at each x in X may be computed in graphical fashion as shown below. Cactus Graph Ef = (p,dp)(q,dq) • https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-ef-pdpq… Cactus Graph Enlargement pq @ pq = (dp)(dq) • https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-enlarge… Cactus Graph Enlargement pq @ p(q) = (dp)dq • https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-enlarge… Cactus Graph Enlargement pq @ (p)q = dp(dq) • https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-enlarge… Cactus Graph Enlargement pq @ (p)(q) = dp dq • https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-enlarge… Collating the data of that analysis yields a boolean expansion or “disjunctive normal form” (DNF) equivalent to the enlarged proposition Ef. • Ef = p q · Ef| p q + p (q) · Ef| p (q) + (p) q · Ef|(p) q} + (p)(q) · Ef|{p)(q) Here is a summary of the result, illustrated by means of a digraph picture, where the “no change” element (dp)(dq) is drawn as a loop at the point pq. Directed Graph Enlargement pq • https://inquiryintoinquiry.files.wordpress.com/2020/04/directed-graph-enlar… Logical Formula Enlargement pq • https://inquiryintoinquiry.files.wordpress.com/2024/11/logical-formula-enla… We may understand the enlarged proposition Ef as telling us all the ways of reaching a model of the proposition f from the points of the universe X. 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/5NAZ9q cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_an…

Differential Logic • 9 • https://inquiryintoinquiry.com/2024/11/08/differential-logic-9-a/ Propositional Forms on Two Variables — Table A2 arranges the propositional forms on two variables according to another plan, sorting propositions with similar shapes into seven subclasses. Thereby hangs many a tale, to be told in time. Table A2. Propositional Forms on Two Variables • https://inquiryintoinquiry.files.wordpress.com/2020/04/table-a2.-propositio… 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/LYpp4B cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_an…

Differential Logic • 11 • https://inquiryintoinquiry.com/2024/11/10/differential-logic-11-a/ Transforms Expanded over Ordinary and Differential Variables — As promised last time, in the next several posts we'll extend our scope to the full set of boolean functions on two variables and examine how the differential operators E and D act on that set. There being some advantage to singling out the enlargement or shift operator E in its own right, we'll begin by computing Ef for each of the functions f : B×B → B. Enlargement Map Expanded over Ordinary Variables — We first encountered the shift operator when we imagined ourselves being in a state described by the truth of a certain proposition and contemplated the value of that proposition in various other states, as determined by a collection of differential propositions describing the steps we might take to change our state. Restated in terms of our initial example, we imagined ourselves being in a state described by the truth of the proposition pq and contemplated the value of that proposition in various other states, as determined by the differential propositions dp and dq describing the steps we might take to change our state. Those thoughts led us from the boolean function of two variables f₈(p, q) = pq to the boolean function of four variables Ef₈(p, q, dp, dq) = (p , dp)(q , dq), as shown in the entry for f₈ in the first three columns of Table A3. Table A3. Ef Expanded over Ordinary Variables {p, q} • https://inquiryintoinquiry.files.wordpress.com/2020/04/ef-expanded-over-ord… Let's catch a breath here and discuss the full Table next time. 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/5AqoEK cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_an…

Differential Logic • 13 • https://inquiryintoinquiry.com/2024/11/14/differential-logic-13-a/ Transforms Expanded over Ordinary and Differential Variables — Two views of how the difference operator D acts on the set of sixteen functions f : B×B → B are shown below. Table A5 shows the expansion of Df over the set {p, q} of ordinary variables and Table A6 shows the expansion of Df over the set {dp, dq} of differential variables. Difference Map Expanded over Ordinary Variables — Table A5. Df Expanded over Ordinary Variables {p, q} • https://inquiryintoinquiry.files.wordpress.com/2024/11/df-expanded-over-ord… Difference Map Expanded over Differential Variables — Table A6. Df Expanded over Differential Variables {dp, dq} • https://inquiryintoinquiry.files.wordpress.com/2024/11/df-expanded-over-dif… 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/546XW3 cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_an…

Differential Logic • 15 • https://inquiryintoinquiry.com/2024/11/18/differential-logic-15-a/ Differential Fields — The structure of a differential field may be described as follows. With each point of X there is associated an object of the following type: a proposition about changes in X, that is, a proposition g : dX → B. In that frame of reference, if Xº is the universe generated by the set of coordinate propositions {p, q} then dXº is the differential universe generated by the set of differential propositions {dp, dq}. The differential propositions dp and dq may thus be interpreted as indicating “change in p” and “change in q”, respectively. A differential operator W, of the first order type we are currently considering, takes a proposition f : X → B and gives back a differential proposition Wf : EX → B. In the field view of the scene, we see the proposition f : X → B as a scalar field and we see the differential proposition Wf : EX → B as a vector field, specifically, a field of propositions about contemplated changes in X. The field of changes produced by E on pq is shown in the following venn diagram. Enlargement E(pq) : EX → B • https://inquiryintoinquiry.files.wordpress.com/2024/11/field-picture-pq-enl… E(pq) = p ∙ q ∙ (dp)(dq) + p ∙ (q) ∙ (dp) dq + (p) ∙ q ∙ dp (dq) + (p) ∙ (q) ∙ dp dq The differential field E(pq) specifies the changes which need to be made from each point of X in order to reach one of the models of the proposition pq, that is, in order to satisfy the proposition pq. The field of changes produced by D on pq is shown in the following venn diagram. Differential D(pq) : EX → B • https://inquiryintoinquiry.files.wordpress.com/2024/11/field-picture-pq-dif… D(pq) = p ∙ q ∙ ((dp)(dq)) + p ∙ (q) ∙ (dp) dq + (p) ∙ q ∙ dp (dq) + (p) ∙ (q) ∙ dp dq The differential field D(pq) specifies the changes which need to be made from each point of X in order to feel a change in the felt value of the field pq. 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/LGv6Q0 cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_an…

Differential Logic • 17 • https://inquiryintoinquiry.com/2024/11/22/differential-logic-17-a/ Enlargement and Difference Maps — Continuing with the example pq : X → B, the following venn diagram shows the enlargement or shift map E(pq) : EX → B in the same style of field picture we drew for the tacit extension ε(pq) : EX → B. Enlargement E(pq) : EX → B • https://inquiryintoinquiry.files.wordpress.com/2024/11/field-picture-pq-enl… E(pq) = p ∙ q ∙ (dp)(dq) + p ∙ (q) ∙ (dp) dq + (p) ∙ q ∙ dp (dq) + (p) ∙ (q) ∙ dp dq A very important conceptual transition has just occurred here, almost tacitly, as it were. Generally speaking, having a set of mathematical objects of compatible types, in this case the two differential fields εf and Ef, both of the type EX → B, is very useful, because it allows us to consider those fields as integral mathematical objects which can be operated on and combined in the ways we usually associate with algebras. In the present case one notices the tacit extension εf and the enlargement Ef are in a sense dual to each other. The tacit extension εf indicates all the arrows out of the region where f is true and the enlargement Ef indicates all the arrows into the region where f is true. The only arc they have in common is the no‑change loop (dp)(dq) at pq. If we add the two sets of arcs in mod 2 fashion then the loop of multiplicity 2 zeroes out, leaving the 6 arrows of D(pq) = ε(pq) + E(pq) shown in the following venn diagram. Differential D(pq) : EX → B • https://inquiryintoinquiry.files.wordpress.com/2024/11/field-picture-pq-dif… D(pq) = p ∙ q ∙ ((dp)(dq)) + p ∙ (q) ∙ (dp) dq + (p) ∙ q ∙ dp (dq) + (p) ∙ (q) ∙ dp 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/lP1a2M cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_an…