Inquiry Into Inquiry

Commentary On Small Models

The consequence of dealing with “practically infinite extensions” becomes crucial in building neural network systems capable of learning and adapting, since the adaptive competence of any intelligent system is limited to the objects and domains it is able to represent.  If we seek to design systems which operate intelligently with the full deck of propositions dealt by intact universes of discourse then we must supply those systems with succinct representations and efficient transformations in that domain.

Beyond the ability to learn and adapt, which taken at the ebb so often devolves into bare conformity and confirmation bias, the ability to inquire and reason makes even more demands on propositional representation.  The project of constructing inquiry driven systems forces us to contemplate the level of generality embodied in logical propositions — we can see this because the progress of inquiry is driven by evident discrepancies among expectations, intentions, and observations, and each of those components of systematic knowledge takes on the fully generic character of an empirical summary or an axiomatic theory.

A compression scheme by any other name is a symbolic representation — and this is what the differential extension of propositional calculus is intended to supply.  But why is this particular program of mental calisthenics worth carrying out in general?

The provision of a uniform logical framework for describing time‑evolving systems makes the task of understanding complex systems easier than it would otherwise be when we try to tackle each new system de novo, “from scratch” as we say.  Having a uniform medium ready to hand helps both in looking for invariant representations of individual cases and also in finding points of comparison among diverse structures otherwise appearing to be isolated systems.  All this goes to facilitate the search for compact knowledge, to apply what is learned from individual cases to the general realm.

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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.

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• Differential Logic and Dynamic Systems
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Example 1. A Square Rigging (concl.)

If we eliminate from view the regions of ruled out by the dynamic law then what remains is the quotient structure shown in the following Figure.  The picture makes it easy to see how the dynamically allowable portion of the universe is partitioned between the respective holdings of and   As it happens, the fact might have been expressed “right off the bat” by an equivalent formulation of the differential law, one which uses the exclusive disjunction to state the law as

What we have achieved in this example is to give a differential description of a simple dynamic process.  We did this by embedding a directed graph, representing the state transitions of a finite automaton, in the share of a boolean lattice or n‑cube cut out by nullifying all the regions the dynamics outlaws.

With growth in the dimensions of our contemplated universes it becomes essential, both for human comprehension and for computer implementation, that dynamic structures of interest be represented not actually, by acquaintance, but virtually, by description.  In our present study we are using the language of propositional calculus to express the relevant descriptions, and to grasp the structures embodied in subsets of n‑cubes without being forced to actualize all their points.

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• Differential Logic and Dynamic Systems
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Example 1. A Square Rigging (cont.)

Because the initial space is one‑dimensional we can easily fit the second order extension within the compass of a single venn diagram, charting the pair of converging trajectories as shown in the following Figure.

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• Differential Logic and Dynamic Systems
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Urge and urge and urge,
Always the procreant urge of the world.

— Walt Whitman • Leaves of Grass

Example 1. A Square Rigging

Returning to the universe of discourse based on a single feature suppose we are given the initial condition and the second order differential law   Since the equation is logically equivalent to the disjunction we may infer two possible trajectories, as shown in the following Table.

In either case the state is a stable attractor or terminal condition for both starting points.

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The clock indicates the moment . . . . but what does
     eternity indicate?

— Walt Whitman • Leaves of Grass

A One-Dimensional Universe (concl.)

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 are changed or unchanged in the next moment.  To know that one would have to determine 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 for all greater than some fixed value   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.

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• Differential Logic and Dynamic Systems
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A One-Dimensional Universe (cont.)

The first order differential extension of is   If the feature is interpreted as applying to some object or state then the feature may be taken as an attribute of the same object or state which tells it is changing significantly with respect to the property as if it bore an “escape velocity” with respect to the state   In practice, differential features acquire their meaning through a class of temporal inference rules.

For example, relative to a frame of observation to be left implicit for now, if and are true at a given moment, it would be reasonable to assume will be true in the next moment of observation.  Taken all together we have the fourfold scheme of inference shown below.

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• Differential Logic and Dynamic Systems
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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

A One‑Dimensional Universe

Let be a logical basis containing one boolean variable or logical feature   The basis element may be regarded as a simple proposition or coordinate projection   Corresponding to the basis is the alphabet which serves whenever we need to make explicit mention of the symbols used in our formulas and representations.

The space of points (cells, vectors, interpretations) has cardinality and is isomorphic to   Moreover, may be identified with the set of singular propositions

The space of linear propositions is algebraically dual to and also has cardinality   Here, is interpreted as denoting the constant function amounting to the linear proposition of rank while is the linear proposition of rank

Last but not least we have the positive propositions of rank and respectively, where is understood as denoting the constant function

All told there are propositions in the universe of discourse collectively forming the set

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I would have preferred to be enveloped in words, borne way beyond all possible beginnings.

— Michel Foucault • The Discourse on Language

Back to the Beginning • Exemplary Universes

To anchor our understanding of differential logic let’s look at how the various concepts apply in the simplest possible concrete cases, where the initial dimension is only 1 or 2.  In spite of the simplicity of these cases it is possible to observe how central difficulties of the subject begin to arise already at this stage.

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• Differential Logic and Dynamic Systems
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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 makes the task of solving differential propositions relatively straightforward.  The solution set of the differential proposition is the set of models in   Finding all models of the extended interpretations in which satisfy 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 with equal facility but there appears to be much room for improvement in classifying special forms and developing algorithms tailored to their practical processing.

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• Differential Logic and Dynamic Systems
• Differential Logic • Life on Easy Street

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