## Differential Logic and Dynamic Systems • Overview

In modeling intelligent systems, whether we are trying to understand a natural system or engineer an artificial system, there has long been a tension or trade‑off between dynamic paradigms and symbolic paradigms.  Dynamic models take their cue from physics, using quantitative measures and differential equations to model the evolution of a system’s state through time.  Symbolic models use logical methods to describe systems and their agents in qualitative terms, deriving logical consequences of a system’s description or an agent’s state of information.  Logic-based systems have tended to be static in character, largely because we have lacked a proper logical analogue of differential calculus.  The work laid out in this report is intended to address that lack.

This article develops a differential extension of propositional calculus and applies it to the analysis of dynamic systems whose states are described in qualitative logical terms.  The work pursued here is coordinated with a parallel application focusing on neural network systems but the dependencies are arranged to make the present article the main and the more self-contained work, to serve as a conceptual frame and a technical background for the network project.

# Document History

## Document History

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## In the Way of Inquiry • Objections to Reflexive Inquiry

Inquiry begins when an automatic routine or normal course of activity is interrupted and agents are thrown into doubt concerning what is best to do next and what is really true of their situation.  If this interruptive aspect of inquiry applies at the level of self-application then occasions for inquiry into inquiry arise when an ongoing inquiry into any subject becomes obstructed and agents are obliged to initiate a new order of inquiry in order to overcome the obstacle.

At such moments agents need the ability to pause and reflect — to accept the interruption of the inquiry in progress, to acknowledge the higher order of uncertainty obstructing the current investigation, and finally to examine accepted conventions and prior convictions regarding the conduct of inquiry in general.  The next order of inquiry requires agents to articulate the assumptions embodied in previous inquiries, to consider their practical effects in light of their objective intents, and to reconstruct forms of conduct which formerly proceeded through their paces untroubled by any articulate concern.

Our agent of inquiry is brought to the threshold of two questions:

• What actions are available to achieve the aims of the present activity?

The inquirer is faced in the object of inquiry with an obstinately oppositional state of affairs, a character marked by the Greek word pragma for object, whose manifold of senses and derivatives includes among its connotations the ideas of purposeful objectives and problematic objections, and not too incidentally both inquiries and expositions.

An episode of inquiry bears the stamp of an interlude — it begins and ends in medias res with respect to actions and circumstances neither fixed nor fully known.  As easy as it may be to overlook the contingent character of the inquiry process it’s just as essential to observe a couple of its consequences:

First, it means genuine inquiry does not touch on the inciting action at points of total doubt or absolute certainty.  An incident of inquiry does not begin or end in absolute totalities but only in the differential and relative measures which actually occasion its departures and resolutions.

Inquiry as a process does not demand absolutely secure foundations from which to set out or any “place to stand” from which to examine the balance of onrushing events.  It needs no more than it does in fact have at the outset — assumptions not in practice doubted just a moment before and a circumstance of conflict that will force the whole situation to be reviewed before returning to the normal course of affairs.

Second, the interruptive character or escapist interpretation of inquiry is especially significant when contemplating programs of inquiry with recursive definitions, as the motivating case of inquiry into inquiry.  It means the termination criterion for an inquiry subprocess is whatever allows continuation of the calling process.

## In the Way of Inquiry • Reconciling Accounts

The Reader may share with the Author a feeling of discontent at this point, attempting to reconcile the formal intentions of this inquiry with the cardinal contentions of experience.  Let me try to express the difficulty in the form of a question:

What is the bond between form and content in experience, between the abstract formal categories and the concrete material contents residing in experience?

Once toward the end of my undergrad years a professor asked me how I’d personally define mathematics and I told him I saw it as “the form of experience and the experience of form”.  This is not the place to argue for the virtues of that formulation but it does afford me one of the handles I have on the bond between form and content in experience.

I have no more than a tentative way of approaching the question.  I take there to be a primitive category of “form‑in‑experience” — I don’t have a handy name for it yet but it looks to have a flexible nature which from the standpoint of a given agent easily passes from the “structure of experience” to the “experience of structure”.

## In the Way of Inquiry • Material Exigency

Our survey of obstacles to inquiry has dealt at length with blocks arising from its formal aspects.  On the other hand, I have cast this project as an empirical inquiry, proposing to represent experimental hypotheses in the form of computer programs.  At the heart of that empirical attitude is a feeling all formal theories should arise from and bear on experience.

Every season of growth in empirical knowledge begins with a rush to the sources of experience.  Every fresh-thinking reed of intellect is raised to pipe up and chime in with the still-viable canons of inquiry in one glorious paean to the personal encounter with natural experience.

But real progress in the community of inquiry depends on observers being able to orient themselves to objects of common experience — the uncontrolled exaltation of individual phenomenologies leads as a rule to the disappointment and disillusionment which befalls the lot of unshared enthusiasms and fragmented impressions.

Look again at the end of the season and see it faltering to a close, with every novice scribe rapped on the knuckles for departing from that uninspired identification with impersonal authority which expresses itself in third-person passive accounts of one’s own experience.

A turn of events so persistent must have a cause, a force of reason to explain the dynamics of its recurring moment in the history of ideas.  The nub of it’s not born on the sleeve of its first and last stages, where the initial explosion and the final collapse march along their stubborn course in lockstep fashion, but is embodied more naturally in the middle of the above narrative.

Experience exposes and explodes expectations.  How can experiences impact expectations unless the two types of entities are both reflected in one medium, for instance and perhaps without loss of generality, in the form of representation constituting the domain of signs?

However complex its world may be, internal or external to itself or on the boundaries of its being, a finite creature’s description of it rests in a finite number of finite terms or a finite sketch of finite lines.  Finite terms and lines are signs.  What they indicate need not be finite but what they are, must be.

### Fragments

The common sensorium.

The common sense and the senses of common.

This is the point where the empirical and the rational meet.

I describe as empirical any method which exposes theoretical descriptions of an object to further experiences with that object.

## In the Way of Inquiry • Formal Apology

Using form in the sense of abstract structure, the focus of my interest in this investigation is limited to the formal properties of the inquiry process.  Among its chief constituents are numbered all the thinking and unthinking processes supporting the ability to learn and to reason.  This formal apology, the apologetics of declaring a decidedly formal intent, will be used on numerous occasions to beg off a host of material difficulties and thus avoid the perceived necessity of meeting a multitude of conventional controversies.

### Category Double-Takes

The first use of the formal apology is to rehabilitate certain classes of associations between concepts otherwise marked as category mistakes.  The conversion is achieved by flipping from one side of the concept’s dual aspect to the other as the context demands.  Thus it is possible in selected cases to reform the characters of category mistakes in the manner of categorical retakes or double-takes.

### Conceptual Extensions

The second use of the formal apology is to permit the tentative extension of concepts to novel areas, giving them experimental trial beyond the cases and domains where their use is already established in the precedents of accustomed habit and successful application.

This works to dispel the “in principle” objection that any category distinction puts a prior constraint on the recognition of similar structure between materially dissimilar domains.  It leaves the issue a matter to be settled by post hoc judgment, a matter of what fits best “in practice”.

### Explosional Recombinations

Another obstacle to inquiry is posed by the combinatorial explosion of questions arising in complex cases.  The embarrassment of riches found here is deceptively deadly to the ends of inquiry in the very measure it appears so productive at first.  An eye to form provides a way to manage the wealth of material diversity by identifying formal similarities among materially distinct domains.  It allows the same formal answer to unify a host of concrete questions under a single roof, overall reducing the number of distinct topics that need to be covered.

### Interpretive Frameworks

Iterations of the recombinatorial process generate alternative hierarchies of categories for controlling the explosion of parts in the domain under inquiry.  If by some piece of luck an alternative framework is uniquely suited to the natural ontology of the domain in question, it becomes advisable to reorganize the inquiry along the lines of the new topic headings.

But a complex domain seldom falls out that neatly.  The new interpretive framework will not preserve all the information in the object domain but typically capture only another aspect of it.  To take the maximal advantage of all the different frameworks that might be devised it is best to quit depending on any one of them exclusively.  Thus, a rigid reliance on a single hierarchy to define the ontology of a given domain passes over into a flexible application of interpretive frameworks to make contact with particular aspects of one’s object domain.

## In the Way of Inquiry • Justification Trap

There is a particular type of “justification trap” a person can fall into, of trying to prove the scientific method by solely deductive means, that is, of trying to show the scientific method is a good method by starting from the simplest possible axioms, principles everyone would accept, about what is good.

Often this happens, in spite of the fact one really knows better, simply in the process of arranging one’s thoughts in a rational order, say, from the most elementary and independent to the most complex and derivative, as if for the sake of a logical and summary exposition.  But when does that rearrangement cease to be a rational reconstruction and start to become a destructive rationalization, a distortion of the genuine article, and a falsification of the authentic inquiry it attempts to recount?

Sometimes people express their recognition of this trap and their appreciation of the factor it takes to escape it by saying there is really no such thing as the scientific method, that the very term “scientific method” is a misnomer and does not refer to any kind of method at all, in short, the development of knowledge cannot be reduced to any fixed method because it involves in an essential way such a large component of non-methodical activities.  If one’s idea of what counts as method is fixed on the ideal of a deductive procedure then it’s no surprise one draws that conclusion.

## In the Way of Inquiry • Initial Unpleasantness

Clouds and thunder:
The image of Difficulty at the Beginning.
Thus the superior man
Brings order out of confusion.

I Ching Hexagram 3

Inquiry begins in doubt, a debit of certainty and a drought of information, never a pleasant condition to acknowledge, and one of the primary obstacles to inquiry may be reckoned as owing to the onus one naturally feels on owning up to that debt.  Human nature far prefers to revel in the positive features of whatever scientific knowledge it already possesses and the mind defers as long as possible the revolt it feels arising on facing the uncertainties that still persist, the “nots” and “not yets” it cannot as yet and ought not deny.

### Reference

• The I Ching, or Book of Changes, R. Wilhelm and C.F. Baynes (trans.), Foreword by C.G. Jung, Bollingen Series 19, Princeton University Press, Princeton, NJ.  1st edition 1950, 2nd edition 1961, 3rd edition 1967.

## In the Way of Inquiry • Obstacles

Upon this first, and in one sense this sole, rule of reason, that in order to learn you must desire to learn, and in so desiring not be satisfied with what you already incline to think, there follows one corollary which itself deserves to be inscribed upon every wall of the city of philosophy:

Do not block the way of inquiry.

C.S. Peirce, Collected Papers, CP 1.135–136.
From an unpaginated ms. “F.R.L.”, c. 1899.

The discussion in this Chapter addresses a set of conceptual and methodological obstacles standing in the way of the current inquiry, threatening to undermine a reasonable level of confidence in the viability of its proceeding, all of which problems I think can be overcome.

Often the biggest obstacle to learning more is the need to feel one already knows.  And yet there are some things a person knows, at least, in comparison to other things, and it makes sense to use what one already knows best in order to learn what one needs to know better.  The question is, how does one know which is which?  What test can tell what is known so well it can be trusted in learning what is not?

One way to test a supposed knowledge is to try to formulate it in such a way that it can be taught to other people.  A related test, harder in some ways but easier in others, is to try to formalize it so completely that even a computer could go through the motions that are supposed to be definitive of its practice.

Both ways of testing a supposition of knowledge depend on putting knowledge in forms which can be communicated or transported from one medium or system of interpretation to another. Knowledge already in a concrete form takes no more than a simple reformation or transformation, otherwise it takes a more radical metamorphosis, from a wholly disorganized condition to the first inklings of a portable or sharable form.

## In the Way of Inquiry • Recircus

I must lie down where all the ladders start
In the foul rag and bone shop of the heart.

W.B. Yeats

I have in mind circling back to a point in my project on Inquiry Driven Systems, namely, the chapter addressing various Obstacles to the Project.

## Riffs and Rotes • Happy New Year 2023

$\text{Let} ~ p_n = \text{the} ~ n^\text{th} ~ \text{prime}.$

$\text{Then} ~ 2023 = 7 \cdot {17}^2 = p_{4} p_{7}^{2} = p_{{p_1}^{p_1}} p_{{p_4}}^{p_1} = p_{{p_1}^{p_1}} p_{{p_{{p_1}^{p_1}}}}^{p_1}$

No information is lost by dropping the terminal 1s.  Thus we may write the following form.

$2023 = p_{p^p} p_{{p_{p^p}}}^p$

The article referenced below tells how forms like these correspond to a family of digraphs called riffs and a family of graphs called rotes.  The riff and rote for $2023$ are shown in the next two displays.