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 whose venn diagram and cactus graph are shown below.

A function like this has an abstract type and a concrete type. The abstract type is what we invoke when we write things like or The concrete type takes into account the qualitative dimensions or “units” of the case, which can be explained as follows.

Let be the set of values

Let be the set of values

Then interpret the usual propositions about as functions of the concrete type

We are going to consider various *operators* on these functions. An operator is a function which takes one function into another function

The first couple of operators we need to consider are logical analogues of two which play a founding role in the classical finite difference calculus, namely:

The *difference operator* written here as

The *enlargement operator*, written here as

These days, 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 its *(first order) differential extension* is constructed according to the following specifications:

where:

The interpretations of these new symbols can be diverse, but the easiest option for now is just to say means “change ” and means “change ”.

Drawing a venn diagram for the differential extension requires four logical dimensions, but it is possible to project a suggestion of what the differential features and are about on the 2-dimensional base space by drawing arrows crossing the boundaries of the basic circles in the venn diagram for reading an arrow as if it crosses the boundary between and in either direction and reading an arrow as if it crosses the boundary between and in either direction, as indicated in the following figure.

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 says the same thing as in other words, there is no change in without a change in

Given the proposition over the space the *(first order) enlargement* of is the proposition over the differential extension defined by the following formula:

In the example the enlargement is computed as follows:

Given the proposition over the *(first order) difference* of is the proposition over defined by the formula or, written out in full:

In the example the difference is computed as follows:

At the end of the previous section we evaluated this *first order difference of conjunction* at a single location in the universe of discourse, namely, at the point picked out by the singular proposition in terms of coordinates, at the place where and This evaluation is written in the form or and we arrived at the locally applicable law which may be stated and illustrated as follows:

The venn diagram shows the analysis of the inclusive disjunction into the following exclusive disjunction:

The differential proposition may be read as saying “change or change 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.

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]]>An efficient calculus for the realm of logic represented by boolean functions and elementary propositions makes it feasible to compute the finite differences and the differentials of those functions and propositions.

For example, consider a proposition of the form graphed as two letters attached to a root node:

Written as a string, this is just the concatenation .

The proposition may be taken as a boolean function having the abstract type where is read in such a way that means and means

Imagine yourself standing in a fixed cell of the corresponding venn diagram, say, the cell where the proposition is true, as shown in the following Figure:

Now ask yourself: What is the value of the proposition at a distance of and from the cell where you are standing?

Don’t think about it — just compute:

The cactus formula and its corresponding graph arise by substituting for and for in the boolean product or logical conjunction and writing the result in the two dialects of cactus syntax. This follows from the fact the boolean sum is equivalent to the logical operation of exclusive disjunction, which parses to a cactus graph of the following form:

Next question: What is the difference between the value of the proposition over there, at a distance of and and the value of the proposition where you are standing, all expressed in the form of a general formula, of course? Here is the appropriate formulation:

There is one thing I ought to mention at this point: Computed over plus and minus are identical operations. This will make the relation between the differential and the integral parts of the appropriate calculus slightly stranger than usual, but we will get into that later.

Last question, for now: What is the value of this expression from your current standpoint, that is, evaluated at the point where is true? Well, substituting for and for in the graph amounts to erasing the labels and as shown here:

And this is equivalent to the following graph:

We have just met with the fact that the differential of the ** and** is the

It will be necessary to develop a more refined analysis of that statement directly, but that is roughly the nub of it.

If the form of the above statement reminds you of De Morgan’s rule, it is no accident, as differentiation and negation turn out to be closely related operations. Indeed, one can find discussions of logical difference calculus in the Boole–De Morgan correspondence and Peirce also made use of differential operators in a logical context, but the exploration of these ideas has been hampered by a number of factors, not the least of which has been the lack of a syntax adequate to handle the complexity of expressions evolving in the process.

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]]>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

The simplest expression for logical truth is the empty word, typically denoted by or 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 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.

It is important to note the last expressions are not equivalent to the 3-place form

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]]>The development of differential logic is facilitated by having a moderately efficient calculus in place at the level of boolean-valued functions and elementary logical propositions. One very efficient calculus on both conceptual and computational grounds is based on just two types of logical connectives, both of variable -ary scope. The syntactic formulas of this calculus map into a family of graph-theoretic structures called “painted and rooted cacti” which lend visual representation to the functional structures of propositions and smooth the path to efficient computation.

The first kind of connective takes the form of a parenthesized sequence of propositional expressions, written and meaning exactly one of the propositions is false, in short, their minimal negation is true. An expression of this form maps into a cactus structure called a *lobe*, in this case, “painted” with the colors as shown below.

The second kind of connective is a concatenated sequence of propositional expressions, written and meaning all the propositions are true, in short, their logical conjunction is true. An expression of this form maps into a cactus structure called a *node*, in this case, “painted” with the colors as shown below.

All other propositional connectives can be obtained through combinations of these two forms. As it happens, the parenthesized form is sufficient to define the concatenated form, making the latter formally dispensable, but it’s convenient to maintain it as a concise way of expressing more complicated combinations of parenthesized forms. While working with expressions solely in propositional calculus, it’s easiest to use plain parentheses for logical connectives. In contexts where ordinary parentheses are needed for other purposes an alternate typeface may be used for the logical operators.

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]]>Differential logic is the component of logic whose object is the description of variation — for example, the aspects of change, difference, distribution, and diversity — in universes of discourse subject to logical description. A definition that broad naturally incorporates any study of variation by way of mathematical models, but differential logic is especially charged with the qualitative aspects of variation pervading or preceding quantitative models. To the extent a logical inquiry makes use of a formal system, its differential component treats the principles governing the use of a *differential logical calculus*, that is, a formal system with the expressive capacity to describe change and diversity in logical universes of discourse.

Simple examples of differential logical calculi are furnished by *differential propositional calculi*. A differential propositional calculus is a propositional calculus extended by a set of terms for describing aspects of change and difference, for example, processes taking place in a universe of discourse or transformations mapping a source universe to a target universe. Such a calculus augments ordinary propositional calculus in the same way the differential calculus of Leibniz and Newton augments the analytic geometry of Descartes.

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]]>One of my readers on Facebook told me “venn diagrams are obsolete” and of course we all know they become unwieldy as our universes of discourse expand beyond four or five dimensions. Indeed, one of the first lessons I learned when I set about implementing Peirce’s graphs and Spencer Brown’s forms on the computer is that 2-dimensional representations of logic are a death trap on numerous conceptual and computational counts. Still, venn diagrams do us good service in visualizing the relationships among extensional, functional, and intensional aspects of logic. A facility with those connections is critical to the computational applications and statistical generalizations of propositional logic commonly used in mathematical and empirical practice.

At any rate, intrepid readers will have provisioned their visual imaginations fully enough at this point to pick their way through the cactus patch ahead. The outline below links to my last, best introduction to Differential Logic, which I’ll be working to improve as I serialize it to this blog.

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]]>*The most fundamental concept in cybernetics is that of “difference”, either that two things are recognisably different or that one thing has changed with time.*

W. Ross Ashby • *An Introduction to Cybernetics*

The times are rife with distraction, so let’s pause and retrace how we got to this place.

Our last reading in *Cybernetics* brought us in sight of a convergence or complementarity between the triadic relations in Peirce’s semiotics and those in Ashby’s regulator games. There’s a lot more to explore in that direction and I plan to get back to it soon.

The two threads intertwined here, Cybernetics and Differential Logic, both spun off a thread on Pragmatic Truth, asking what theories of truth are compatible with Peircean disciplines of pragmatic thinking. That’s a topic with a tangled history but the latest local tangle is documented in the following posts and excerpts.

Pragmatic inquiry into a candidate concept of truth would begin by applying the pragmatic maxim to clarify the concept as far as possible and a pragmatic definition of truth, should any result, would find its place within Peirce’s theory of logic as formal semiotics, in other words, stated in terms of a formal theory of triadic sign relations.

There are many conceptions of truth — linguistic, model-theoretic, proof-theoretic — for the moment I’m focused on cybernetics, systems, and experimental sciences and this is where the pragmatic conception of truth fits what we naturally do in those sciences remarkably well.

The main thing in those activities is the relationship among symbol systems, the world, and our actions, whether in thought, among ourselves, or between ourselves and the world. So the notion of truth we want here is predicated on three dimensions: the patch of the world we are dealing with in a given application, the systems of signs we are using to describe that domain, and the transformations of signs we find of good service in bearing information about that piece of the world.

We do not live in axiom systems. We do not live encased in languages, formal or natural. There is no reason to think we will ever have exact and exhaustive theories of what’s out there, and the truth, as we know, is “out there”. Peirce understood there are more truths in mathematics than are dreamt of in logic and Gödel’s realism should have put the last nail in the coffin of logicism, but some ways of thinking just never get a clue.

That brings us to the question —

- What are formalisms and all their embodiments in brains and computers good for?

For that I’ll turn to cybernetics …

The Survey linked above recaps the reading of Ashby’s *Cybernetics* up to the present date.

Meanwhile, the inquiry into Pragmatic Truth branched off at another point when a question from Stephen Paul King demanded an answer in terms of Differential Logic. That point of departure is documented in the following post.

This updates the state of the threads linking pragmatic truth, cybernetics, and differential logic. Disentangling them to any large extent has always been difficult if not impossible, at least for me.

- Differential Propositional Calculus • Part 1 • Part 2
- Differential Logic • Part 1 • Part 2 • Part 3
- Differential Logic and Dynamic Systems

• Part 1 • Part 2 • Part 3 • Part 4 • Part 5

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]]>Plato • *Alcibiades* • 135 A

This is a Survey of blog and wiki posts relating to Cybernetics.

- Chapter 10 • Regulation In Biological Systems

Questions about Abduction in AI and Computer Science raised in the Ontolog Forum prompted me to look up previous discussions tracing the integral relationship among information, inquiry, and the three types of inference. Here’s a sample of links.

- Survey of Abduction, Deduction, Induction, Analogy, Inquiry
- Survey of Pragmatic Semiotic Information

- Abductive Inference, Concept Formation, Hypothesis Formation
- Information = Comprehension × Extension • Revisited
- Pragmatic Semiotic Information (Ψ)

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]]>*The most fundamental concept in cybernetics is that of “difference”, either that two things are recognisably different or that one thing has changed with time.*

W. Ross Ashby • *An Introduction to Cybernetics*

Re: Cybernetics Communications • Klaus Krippendorff

- KK:
- To me, differences are the result of drawing distinctions. They don’t exist unless you actively draw them. So, the act of drawing distinctions is more fundamental than the differences thereby created.

I often return to that line from Ashby. This time I thought it made an apt segue from the scene of propositional calculus, where universes of discourse are ruled by collections of distinctive features, to the differential extension of propositional calculus, which enables us to describe trajectories within and transformations between our logical universes.

So I agree with Klaus Krippendorff about “which came first”, the distinctions drawn or the states distinguished in space or time. The primitive character of distinctions is especially salient in this setting since our formalism for propositional calculus is built on the forms of distinction pioneered by C.S. Peirce and augmented by George Spencer Brown.

- Differential Propositional Calculus • Part 1 • Part 2
- Differential Logic • Part 1 • Part 2 • Part 3
- Differential Logic and Dynamic Systems

• Part 1 • Part 2 • Part 3 • Part 4 • Part 5

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