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MyWikiBiz, Author Your Legacy — Tuesday October 07, 2008

This article develops a differential extension of propositional calculus and applies it to a context of problems arising in dynamic systems. The work pursued here is coordinated with a parallel application that focuses 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.

Review and Transition

This note continues a previous discussion on the problem of dealing with change and diversity in logic-based intelligent systems. For ease of reference, I begin by summarizing essential material from previous reports.

Table 1 outlines the notation that I use for propositional calculus. Explained as briefly as possible, I am using only two basic kinds of truth-functional connectives, both of variable k-ary scope.

1. For the first, I use concatenation as a connective to indicate the logical conjunction of k arguments.
2. For the other, I use a bracket of the form ( , , , ) as a connective which says that exactly one of its k arguments is false.

All other truth-functional connectives can be obtained in a very efficient style of representation through combinations of these two forms.

This treatment of propositional logic is derived from the work of C.S. Peirce [P1, P2], who gave this approach an extensive development in his graphical systems of predicate, relational, and modal logic [Rob]. More recently, these ideas were revived and supplemented in an alternative interpretation by G. Spencer-Brown [SpB]. Both of these authors used other forms of enclosure where I use parentheses, but the structural topologies of expression and the functional varieties of interpretation are fundamentally the same.

While working with expressions solely in propositional calculus, the use of plain parentheses to represent logical connectives is simplest to write and easiest to read for both human and machine parsers. In the present text I preserve this form of expression in tables and set-off displays, but in contexts where parentheses are needed for functional notation I will use a distinctive font for logical operators.

The briefest expression for logical truth is the empty word, usually denoted by ε or λ in formal languages, where it forms the identity element for concatenation. To make it visible in this text, I denote it by the equivalent expression "(())", or, especially if operating in an algebraic context, by a simple "1". Also when working in an algebraic mode, I use the plus sign "+" for exclusive disjunction. Thus, we may express the following paraphrases of algebraic forms:

A + B	=	(A, B)
A + B + C	=	((A, B), C)	=	(A, (B, C))

One should be careful to observe that these last two expressions are not equivalent to the form (A, B, C).

NB. The usage that one often sees, of a plus sign "+" to represent inclusive disjunction, and the reference to this operation as boolean addition, is a misnomer on at least two counts. Boole used the plus sign to represent exclusive disjunction (at any rate, an operation of aggregation restricted in its logical interpretation to cases where the represented sets are disjoint (Boole, 32)), as any mathematician with a sensitivity to the ring and field properties of algebra would do:

The expression x + y seems indeed uninterpretable, unless it be assumed that the things represented by x and the things represented by y are entirely separate; that they embrace no individuals in common. (Boole, 66).

It was only later that Peirce and Jevons treated inclusive disjunction as a fundamental operation, but these authors, with a respect for the algebraic properties that were already associated with the plus sign, used a variety of other symbols for inclusive disjunction (Sty, 177, 189). It seems to have been Schroeder who later reassigned the plus sign to inclusive disjunction (Sty, 208). Additional information, discussion, and references can be found in (Boole) and (Sty, 177-263). Aside from these historical points, which never really count against a current practice that has gained a life of its own, this usage does have a further disadvantage of cutting or confounding the lines of communication between algebra and logic. For this reason, I am forced to avoid it here.

A Functional Conception of Propositional Calculus

In the general case, we start with a set of logical features {a₁, …, a_n} that represent properties of objects or propositions about the world. In concrete examples the features {a_i} commonly appear as capital letters from an alphabet like {A, B, C, …} or as meaningful words from a linguistic vocabulary of codes. This language can be drawn from any sources, whether natural, technical, or artificial in character and interpretation. In the application to dynamic systems we tend to use the letters {x₁, …, x_n} as our coordinate propositions, and to interpret them as denoting properties of a system's state, that is, as propositions about its location in configuration space. Because I have to consider non-deterministic systems from the outset, I often use the word state in a loose sense, to denote the position or configuration component of a contemplated state vector, whether or not it ever gets a deterministic completion.

The set of logical features {a₁, …, a_n} provides a basis for generating an n-dimensional universe of discourse that I denote as [a₁, …, a_n]. It is useful to consider each universe of discourse as a unified categorical object that incorporates both the set of points 〈a₁, …, a_n〉 and the set of propositions f : 〈a₁, …, a_n〉 → B that are implicit with the ordinary picture of a venn diagram on n features. Thus, we may regard the universe of discourse [a₁, …, a_n] as an ordered pair having the type (Bⁿ, (Bⁿ → B), and we may abbreviate this last type designation as Bⁿ +→ B, or even more succinctly as [Bⁿ]. (Used this way, the angle brackets 〈…〉 are referred to as generator brackets.)

Table 2 exhibits the scheme of notation I use to formalize the domain of propositional calculus, corresponding to the logical content of truth tables and venn diagrams. Although it overworks the square brackets a bit, I also use either one of the equivalent notations [n] or n to denote the data type of a finite set on n elements.

Qualitative Logic and Quantitative Analogy

These concepts and notations can now be explained in greater detail. In order to begin as simply as possible, I distinguish two levels of analysis and set out initially on the easier path. On the first level of analysis, I take spaces like B, Bⁿ, and (Bⁿ → B) at face value and treat them as the primary objects of interest. On the second level of analysis, I use these spaces as coordinate charts for talking about points and functions in more fundamental spaces.

A pair of spaces, of types Bⁿ and (Bⁿ → B), give typical expression to everything that we commonly associate with the ordinary picture of a venn diagram. The dimension, n, counts the number of "circles" or simple closed curves that are inscribed in the universe of discourse, corresponding to its relevant logical features or basic propositions. Elements of type Bⁿ correspond to what are often called propositional interpretations in logic, that is, the different assignments of truth values to sentence letters. Relative to a given universe of discourse, these interpretations are visualized as its cells, in other words, the smallest enclosed areas or undivided regions of the venn diagram. The functions f : Bⁿ → B correspond to the different ways of shading the venn diagram to indicate arbitrary propositions, regions, or sets. Regions included under a shading indicate the models, and regions excluded represent the non-models of a proposition. To recognize and formalize the natural cohesion of these two layers of concepts into a single universe of discourse, I introduce the type notations [Bⁿ] = Bⁿ +→ B to stand for the pair of types (Bⁿ, (Bⁿ → B)). The resulting "stereotype" serves to frame the universe of discourse as a unified categorical object, and makes it subject to prescribed sets of evaluations and transformations (categorical morphisms or arrows) that affect the universe of discourse as an integrated whole.

Most of the time we can serve the algebraic, geometric, and logical interests of our study without worrying about their occasional conflicts and incidental divergences. The conventions and definitions already set down will continue to cover most of the algebraic and functional aspects of our discussion, but to handle the logical and qualitative aspects we will need to add a few more. In general, abstract sets may be denoted by gothic, greek, or script capital variants of A, B, C, and so on, with elements denoted by a corresponding set of subscripted letters in plain lower case, for example, A = {a_i}. Most of the time, a set such as A = {a_i} will be employed as the alphabet of a formal language. These alphabet letters serve to name the logical features (properties or propositions) that generate a particular universe of discourse. When we want to discuss the particular features of a universe of discourse, beyond the abstract designation of a type like (Bⁿ +→ B), then we may use the following notations. If A = {a₁, …, a_n} is an alphabet of logical features, then A = 〈A〉 = 〈a₁, …, a_n〉 is the set of interpretations, A^ = (A → B) is the set of propositions, and A ^• = [A] = [a₁, …, a_n] is the combination of these interpretations and propositions into the universe of discourse that is based on the features {a₁, …, a_n}.

As always, especially in concrete examples, these rules may be dropped whenever necessary, reverting to a free assortment of feature labels. However, when we need to talk about the logical aspects of a space that is already named as a vector space, it will be necessary to make special provisions. At any rate, these elaborations can be deferred until actually needed.

Philosophy of Notation : Formal Terms and Flexible Types

For much of our discussion propositions and boolean functions are treated as the same formal objects, or as different interpretations of the same formal calculus. This rule of interpretation has exceptions, though. There is a distinctively logical interest in the use of propositional calculus that is not exhausted by its functional interpretation. It is part of our task in this study to deal with these uniquely logical characteristics as they present themselves both in our subject matter and in our formal calculus. Just to provide a hint of what's at stake: In logic, as opposed to the more imaginative realms of mathematics, we consider it a good thing to always know what we are talking about. Where mathematics encourages tolerance for uninterpreted symbols as intermediate terms, logic exerts a keener effort to interpret directly each oblique carrier of meaning, no matter how slight, and to unfold the complicities of every indirection in the flow of information. Translated into functional terms, this means that we want to maintain a continual, immediate, and persistent sense of both the converse relation f^–1 ⊆ B × Bⁿ, or what is the same thing, f^–1 : B → Pow(Bⁿ), and the fibers or inverse images f^–1(0) and f^–1(1), associated with each boolean function f : Bⁿ → B that we use. In practical terms, the desired implementation of a propositional interpreter should incorporate our intuitive recognition that the induced partition of the functional domain into level sets f^–1(b), for b ∈ B, is part and parcel of understanding the denotative uses of each propositional function f.

Special Classes of Propositions

It is important to remember that the coordinate propositions {a_i}, besides being projection maps a_i : Bⁿ → B, are propositions on an equal footing with all others, even though employed as a basis in a particular moment. This set of n propositions may sometimes be referred to as the basic or simple propositions that found the universe of discourse. As typical and collective notations, we may use the forms {a_i : Bⁿ → B} = (Bⁿ ¸> B) to indicate the adoption of a set of a_i as a basis for discourse.

Among the propositions or functions in (Bⁿ → B) are several fundamental sets of 2ⁿ propositions each that take on special forms with respect to a given basis A = {a_i}. Three of these forms are especially common, the linear, the positive, and the singular propositions. Each set is naturally parameterized by the coordinate vectors in Bⁿ and falls into n+1 ranks, with a binomial coefficient giving the number of propositions that have rank or weight k.

The linear propositions, {hom : Bⁿ → B} = (Bⁿ +> B), may be expressed as sums of the following form:

The positive propositions, {pos : Bⁿ → B} = (Bⁿ ¥> B), may be expressed as products of the following form:

The singular propositions, {x : Bⁿ → B} = (Bⁿ ××> B), may be expressed as products of the following form:

In each case the rank k ranges from 0 to n and counts the number of positive appearances of coordinate propositions a_i in the resulting expression.

For example, for 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).

The coordinate projections or simple propositions a_i : Bⁿ → B are both linear and positive. So these two kinds of propositions, the linear or the positive, may be viewed as two different ways of generalizing the class of simple projections. The linear and the positive propositions are generated by taking boolean sums and products, respectively, over selected subsets of the basic propositions in {a_i}. Therefore, each set of functions can be parameterized by the subsets J of the basic index set I = {1, …, n}.

Let us define A_J as the subset of A that is given by {a_i : i ∈ J}. Then we may comprehend the action of the linear and the positive propositions in the following terms:

• The linear proposition l_J : Bⁿ → B evaluates each cell x of Bⁿ by looking at the coefficients of x with respect to the features that l_J "likes", namely those in A_J, and then adds them up in B. Thus, l_J(x) computes the parity of the number of features that x has in A_J, yielding one for odd and zero for even. Expressed in this idiom, l_J(x) = 1 says that x seems odd (or oddly true) to A_J, whereas l_J(x) = 0 says that x seems even (or evenly true) to A_J, so long as we recall that zero times is evenly often, too.

• The positive proposition p_J : Bⁿ → B evaluates each cell x of Bⁿ by looking at the coefficients of x with regard to the features that p_J "likes", namely those in A_J, and then takes their product in B. Thus, p_J(x) assesses the unanimity of the multitude of features that x has in A_J, yielding one for all and aught for else. In these consensual or contractual terms, p_J(x) = 1 means that x is AOK or congruent with all of the conditions of A_J, while p_J(x) = 0 means that x defaults or dissents from some condition of A_J.

Basis Relativity and Type Ambiguity

Finally, two things are important to keep in mind with regard to the simplicity, linearity, positivity, and singularity of propositions.

First, all of these properties are relative to a particular basis. For example, a singular proposition with respect to a basis A will not remain singular if A is extended by a number of new and independent features. Even if we stick to the original set of pairwise options {a_i} ∪ {(a_i)} to select a new basis, the sets of linear and positive propositions are determined by the choice of simple propositions, and this determination is tantamount to the conventional choice of a cell as origin.

Second, the singular propositions Bⁿ ××> B, picking out as they do a single cell or a coordinate tuple of Bⁿ, become the carriers or the vehicles of a certain type-ambiguity that vacillates between the dual forms Bⁿ and (Bⁿ ××> B) and infects the whole hierarchy of types built on them. In plainer language, the terms that signify the interpretations x : Bⁿ and the singular propositions x : Bⁿ ××> B are fully equivalent in information, and this means that every token of the type Bⁿ can be reinterpreted as an appearance of the subtype Bⁿ ××> B. And vice versa, the two types can be exchanged with each other everywhere that they turn up. In practical terms, this allows the use of singular propositions as a way of denoting points, forming an alternative to coordinate tuples.

For example, relative to the universe of discourse [a₁, a₂, a₃] the singular proposition a₁ a₂ a₃ : B³ ××> B could be explicitly retyped as a₁ a₂ a₃ : B³ to indicate the point ‹1, 1, 1›, but in most cases the proper interpretation could be gathered from context. Both notations remain dependent on a particular basis, but the code that is generated under the singular option has the advantage in its self-commenting features, in other words, it constantly reminds us of its basis in the process of denoting points. When the time comes to put a multiplicity of different bases into play, and to search for objects and properties that remain invariant under the transformations between them, this infinitesimal potential advantage may well evolve into an overwhelming practical necessity.

The Analogy Between Real and Boolean Types

There are two further reasons why I am spending so much time on a careful treatment of types, and they both have to do with our being able to take full computational advantage of certain dimensions of flexibility in the types that apply to terms. First, the domains of differential geometry and logic programming are connected by analogies between real and boolean types of the same pattern. Second, the types involved in these patterns have important isomorphisms connecting them that apply on both the real and the boolean sides of the picture.

Amazingly enough, these isomorphisms are themselves schematized by the axioms and theorems of propositional logic. This fact is known as the propositions as types analogy or the Curry-Howard isomorphism [How]. In another formulation it says that terms are to types as proofs are to propositions. This principle seems to have more implications for our subject than I can fully comprehend at present, though I sense that they must be crucial. (Cf. [LaS, 42-46] and [SeH] for a good discussion and further references.) To anticipate the bearing of these issues on our immediate topic, Table 3 sketches a partial overview of the Real to Boolean analogy that may serve to illustrate the paradigm that I have in mind.

The Table exhibits a sample of likely parallels between the real and boolean domains. The central column gives a selection of terminology that I borrow from typical usage in differential geometry and extend in its meaning to the logical side of the Table. These are the varieties of spaces that come up when we turn to analyzing the dynamics of processes that followe their courses through the states of an arbitrary space X. Moreover, when it becomes necessary to approach situations of overwhelming dynamic complexity in a succession of qualitative reaches, then the methods of logic that are afforded by the boolean domains, with their declarative means of synthesis and deductive modes of analysis, supply a natural battery of tools for the task.

It is usually expedient to take these spaces two at a time, in dual pairs of the form X and (X → K). In general, one creates pairs of type schemas by replacing any space X with its dual (X → K), for example, pairing the type X → Y with the type (X → K) → (Y → K), and X × Y with (X → K) × (Y → K). Here, I am using the word dual in its larger sense to mean all of the functionals, not just the linear ones. Given any function f : X → K, the converse or inverse relation corresponding to f is denoted as f^–1, and the subsets of X that are defined by f^–1(k), taken over k in K, are called the fibers or the level sets of the function f.

Theory of Control and Control of Theory

In the boolean context, a function f : X → B is tantamount to a proposition about elements of X, and the elements of X constitute the interpretations of that proposition. The fiber f^–1(1) comprises the set of models of f, or examples of elements in X satisfying the proposition f. The fiber f^–1(0) collects the complementary set of anti-models, or the exceptions to the proposition f that exist in X. Of course, the space of functions (X → B) is isomorphic to the set of all subsets of X, called the power set of X and often denoted as Pow(X) or 2^X.

The operation of replacing X by (X → B) in a type schema corresponds to a certain shift of attitude towards the space X, in which one passes from a focus on the ostensibly individual elements of X to a concern with the states of information and uncertainty that one possesses about objects and situations in X. The conceptual obstacles in the path of this transition can be smoothed over by using singular functions (X ××> B) as stepping stones. First of all, it's an easy step from an element x of type Bⁿ to the equivalent information of a singular proposition x : X ××> B, and then only a small jump of generalization remains to reach the type of an arbitrary proposition f : X → B, perhaps understood to indicate a relaxed constraint on the singularity of points or a neighborhood circumscribing the original x. I have frequently discovered this to be a useful transformation, communicating between the objective and the intentional perspectives, in spite perhaps of the open objection that this distinction is transient in the mean time and ultimately superficial.

It is hoped that this measure of flexibility, allowing us to stretch a point into a proposition, can be useful in the examination of inquiry driven systems, where the differences between empirical, intentional, and theoretical propositions constitute the discrepancies and the distributions that drive experimental activity. I can give this model of inquiry a cybernetic cast by realizing that theory change and theory evolution, as well as the choice and the evaluation of experiments, are actions that are taken by a system or its agent in response to the differences that are detected between observational contents and theoretical coverage.

All of the above notwithstanding, there are several points that distinguish these two tasks, namely, the theory of control and the control of theory, features that are often obscured by too much precipitation in the quickness with which we understand their similarities. In the control of uncertainty through inquiry, some of the actuators that we need to be concerned with are axiom changers and theory modifiers, operators with the power to compile and to revise the theories that generate expectations and predictions, effectors that form and edit our grammars for the languages of observational data, and agencies that rework the proposed model to fit the actual sequences of events and the realized relationships of values that are observed in the environment. Moreover, when steps must be taken to carry out an experimental action, there must be something about the particular shape of our uncertainty that guides us in choosing what directions to explore, and this impression is more than likely influenced by previous accumulations of experience. Thus it must be anticipated that much of what goes into scientific progress, or any sustainable effort toward a goal of knowledge, is necessarily predicated on long term observation and modal expectations, not only on the more local or short term prediction and correction.

Propositions as Types and Higher Order Types

The arrangement of types collected in Table 3 (repeated below) can serve as a good introduction to several ideas about higher order propositional expressions (HOPE's) and also about the propositions as types (PAT) isomorphism.

First, observe that the type of a tangent vector at a point, also known as a directional derivative at that point, has the form (Kⁿ → K) → K, where K is the chosen ground field, in the present case either R or B. At a point in a space of type Kⁿ, a directional derivative operator takes a function on that space, an f of type (Kⁿ → K), and maps it to a ground field value of type K. This value is known as the derivative of f in the direction [Che46, 76-77]. In the boolean case,  : (Bⁿ → B) → B has the form of a proposition about propositions, in other words, a proposition of the next higher type.

Next, by way of illustrating the propositions as types theme, consider a proposition of the form X ⇒ (Y ⇒ Z). One knows from propositional calculus that this is logically equivalent to a proposition of the form (X ∧ Y) ⇒ Z. But this equivalence should remind us of the functional isomorphism that exists between a construction of the type X → (Y → Z) and a construction of the type (X × Y) → Z. The propositions as types analogy permits us to take a functional type like this and, under the right conditions, replace the functional arrows "→" and products "×" with the respective logical arrows "⇒" and products "∧". Accordingly, viewing the result as a proposition, we can employ axioms and theorems of propositional calculus to suggest appropriate isomorphisms among the categorical and functional constructions.

Finally, examine the middle four rows of Table 3. These display a series of isomorphic types that stretch from the categories that are labeled Vector Field to those that are labeled Derivation. A vector field, also known as an infinitesimal transformation, associates a tangent vector at a point with each point of a space. In symbols, a vector field is a function of the form  : X →  that assigns to each point x of the space X a tangent vector to X at that point, namely, the tangent vector [Che46, 82-83]. If X is of type Kⁿ, then is of type Kⁿ → ((Kⁿ → K) → K). This has the pattern X → (Y → Z), with X = Kⁿ, Y = (Kⁿ → K), and Z = K.

Applying the propositions as types analogy, one can follow this pattern through a series of metamorphoses from the type of a vector field to the type of a derivation, as traced out in Table 4. Observe how the function f : X → K, associated with the place of Y in the pattern, moves through its paces from the second to the first position. In this way, the vector field , initially viewed as attaching each tangent vector to the site x where it acts in X, now comes to be seen as acting on each scalar potential f : X → K like a generalized species of differentiation, producing another function f : X → K of the same type.

Reality at the Threshold of Logic

Table 5 accumulates an array of notation that I hope will not be too distracting. Some of it is rarely needed, but has been filled in for the sake of completeness. Its purpose is simple, to give literal expression to the visual intuitions that come with venn diagrams, and to help build a bridge between our qualitative and quantitative outlooks on dynamic systems.

The left side of the Table collects mostly standard notation for an n-dimensional vector space over a field K. The right side of the table repeats the first elements of a notation that I sketched above, to be used in further developments of propositional calculus. (I plan to use this notation in the logical analysis of neural network systems.) The middle column of the table is designed as a transitional step from the case of an arbitrary field K, with a special interest in the continuous line R, to the qualitative and discrete situations that are instanced and typified by B.

I now proceed to explain these concepts in more detail. The two most important ideas developed in the table are:

• The idea of a universe of discourse, which includes both a space of points and a space of maps on those points.

• The idea of passing from a more complex universe to a simpler universe by a process of thresholding each dimension of variation down to a single bit of information.

For the sake of concreteness, let us suppose that we start with a continuous n-dimensional vector space like X = 〈x₁, …, x_n〉 Rⁿ. The coordinate system X = {x_i} is a set of maps x_i : Rⁿ → R, also known as the coordinate projections. Given a "dataset" of points x in Rⁿ, there are numerous ways of sensibly reducing the data down to one bit for each dimension. One strategy that is general enough for our present purposes is as follows. For each i we choose an n-ary relation L_i on R, that is, a subset of Rⁿ, and then we define the i^th threshold map, or limen x_i as follows:

x_i : Rⁿ → B such that:

x_i(x) = 1 if x ∈ L_i,

x_i(x) = 0 if otherwise.

In other notations that are sometimes used, the operator or the corner brackets can be used to denote a characteristic function, that is, a mapping from statements to their truth values, given as elements of B. Finally, it is not uncommon to use the name of the relation itself as a predicate that maps n-tuples into truth values. In each of these notations, the above definition could be expressed as follows:

x_i(x) = = = L_i(x).

Notice that, as defined here, there need be no actual relation between the n-dimensional subsets {L_i} and the coordinate axes corresponding to {x_i}, aside from the circumstance that the two sets have the same cardinality. In concrete cases, though, one usually has some reason for associating these "volumes" with these "lines", for instance, L_i is bounded by some hyperplane that intersects the i^th axis at a unique threshold value r_i ∈ R. Often, the hyperplane is chosen normal to the axis. In recognition of this motive, let us make the following convention. When the set L_i has points on the i^th axis, that is, points of the form ‹0, …, 0, r_i, 0, …, 0› where only the x_i coordinate is possibly non-zero, we may pick any one of these coordinate values as a parametric index of the relation. In this case we say that the indexing is real, otherwise the indexing is imaginary. For a knowledge based system X, this should serve once again to mark the distinction between acquaintance and opinion.

States of knowledge about the location of a system or about the distribution of a population of systems in a state space X = Rⁿ can now be expressed by taking the set X = {x_i} as a basis of logical features. In picturesque terms, one may think of the underscore and the subscript as combining to form a subtextual spelling for the i^th threshold map. This can help to remind us that the threshold operator ( )_i acts on x by setting up a kind of a "hurdle" for it. In this interpretation, the coordinate proposition x_i asserts that the representative point x resides above the i^th threshold.

Primitive assertions of the form x_i(x) can then be negated and joined by means of propositional connectives in the usual ways to provide information about the state x of a contemplated system or a statistical ensemble of systems. Parentheses "( )" may be used to indicate negation. Eventually one discovers the usefulness of the k-ary just one false operators of the form "( , , , )", as treated in earlier reports. This much tackle generates a space of points (cells, interpretations), X = 〈X〉  Bⁿ, and a space of functions (regions, propositions), X^  (Bⁿ → B). Together these form a new universe of discourse X ^• = [X] of the type (Bⁿ, (Bⁿ → B)), which we may abbreviate as Bⁿ +→ B, or most succinctly as [Bⁿ].

The square brackets have been chosen to recall the rectangular frame of a venn diagram. In thinking about a universe of discourse it is a good idea to keep this picture in mind, where we constantly think of the elementary cells x, the defining features x_i, and the potential shadings f : X → B, all at the same time, remaining aware of the arbitrariness of the way that we choose to inscribe our distinctions in the medium of a continuous space.

Finally, let X* denote the space of linear functions, (hom : X → K), which in the finite case has the same dimensionality as X, and let the same notation be extended across the table.

We have just gone through a lot of work, apparently doing nothing more substantial than spinning a complex spell of notational devices through a labyrinth of baffled spaces and baffling maps. The reason for doing this was to bind together and to constitute the intuitive concept of a universe of discourse into a coherent categorical object, the kind of thing, once grasped, which can be turned over in the mind and considered in all its manifold changes and facets. The effort invested in these preliminary measures is intended to pay off later, when we need to consider the state transformations and the time evolution of neural network systems.

Tables of Propositional Forms

To prepare for the next phase of discussion, Tables 6 and 7 collect and summarize all of the propositional forms on one and two variables. These propositional forms are represented over bases of boolean variables as complete sets of boolean-valued functions. Adjacent to their names and specifications are listed what are roughly the simplest expressions in the cactus language, the particular syntax for propositional calculus that I use in formal and computational contexts. For the sake of orientation, the English paraphrases and the more common notations are listed in the last two columns. As simple and circumscribed as these low-dimensional universes may appear to be, a careful exploration of their differential extensions will involve us in complexities sufficient to demand our attention for some time to come.

Propositional forms on one variable correspond to boolean functions f : B¹ → B. In Table 6 these functions are listed in a variant form of truth table, one which rotates the axes of the usual arrangement. Each function f_i is indexed by the string of values that it takes on the points of the universe X ^• = [x]  B¹. The binary index generated in this way is converted to its decimal equivalent, and these are used as conventional names for the f_i , as shown in the first column of the Table. In their own right the 2¹ points of the universe X ^• are coordinated as a space of type B¹, this in light of the universe X ^• being a functional domain where the coordinate projection x takes on its values in B.

Propositional forms on two variables correspond to boolean functions f : B² → B. In Table 7 each function f_i is indexed by the values that it takes on the points of the universe X ^• = [x, y]  B². Converting the binary index thus generated to a decimal equivalent, we obtain the functional nicknames that are listed in the first column. The 2² points of the universe X ^• are coordinated as a space of type B², as indicated under the heading of the Table, where the coordinate projections x and y run through the various combinations of their values in B.

A 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 some later stage. I am trying to develop a differential theory of qualitative equations that parallels the application of differential geometry to dynamic systems. The idea of a tangent vector is key to this 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 these constructions that can be discovered within the realm of propositional calculus, so long as they serve to fill out the general theme.

Differential Propositions : The Qualitative Analogues of Differential Equations

In order to define the differential extension of a universe of discourse [A], the initial alphabet A must be extended to include a collection of symbols for differential features, or basic changes that are capable of occurring in [A]. Intuitively, these symbols may be construed as denoting primitive features of change, qualitative attributes of motion, or propositions about how things or points in [A] may change or move with respect to the features that are noted in the initial alphabet.

Hence, let us define the corresponding differential alphabet or tangent alphabet as dA = {da₁, …, da_n}, in principle just an arbitrary alphabet of symbols, disjoint from the initial alphabet A = {a₁, …, a_n}, that is intended to be interpreted in the way just indicated. It only remains to be understood that the precise interpretation of the symbols in dA is often conceived to be changeable from point to point of the underlying space A. (For all we know, the state space A might well be the state space of a language interpreter, one that is concerned, among other things, with the idiomatic meanings of the dialect generated by A and dA.)

In the above terms, a typical tangent space of A at a point x, frequently denoted as T_x(A), can be characterized as having the generic construction dA = 〈dA〉 = 〈da₁, …, da_n〉. Strictly speaking, the name cotangent space is probably more correct for this construction, but the fact that we take up spaces and their duals in pairs to form our universes of discourse allows our language to be pliable here.

Proceeding as we did before with the base space A, we can analyze the individual tangent space at a point of A as a product of distinct and independent factors:

dA = ∏_i dA_i = dA₁ × … × dA_n.

Here, dA_i is an alphabet of two symbols, dA_i = {(da_i), da_i}, where (da_i) is a symbol with the logical value of "not da_i". Each component dA_i has the type B, under the correspondence {(da_i), da_i} {0, 1}. However, clarity is often served by acknowledging this differential usage with a superficially distinct type D, whose intension may be indicated as follows:

D = {(da_i), da_i} = {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 this 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.

An Interlude on the Path

It may help to get a sense of the relation between B and D by considering the path classifier (or equivalence class of curves) approach to tangent vectors. As if by reflex, the thought of physical motion makes us cross over to a universe marked by the nominal character [X]. Given the boolean value system, a path in the space X = 〈X〉 is a map q : B → X. In this case, the set of paths (B → X) is isomorphic to the cartesian square X² = X × X, or the set of ordered pairs from X.

We may analyze X² = {‹u, v› : u, v ∈ X} into two parts, specifically, the pairs that lie on and off the diagonal:

X² = {‹u, v› : u = v} ∪ {‹u, v› : u ≠ v}

In symbolic terms, this partition may be expressed as:

X² Diag(X) + 2 * Comb(X, 2),

where:

Diag(X) = {‹x, x› : x ∈ X},

and where:

Comb(X, k) = "X choose k" = {k-sets from X},

so that:

Comb(X, 2) = {{u, v} : u, v ∈ X}.

We can now use the features in dX = {dx_i} = {dx₁, …, dx_n} to classify the paths of (B → X) by way of the pairs in X². If X Bⁿ then a path in X has the form q : (B → Bⁿ) Bⁿ × Bⁿ B²ⁿ (B²)ⁿ. Intuitively, we want to map this (B²)ⁿ onto Dⁿ by mapping each component B² onto a copy of D. But in our current situation "D" is just a name we give, or an accidental quality we attribute, to coefficient values in B when they are attached to features in dX.

Therefore, define dx_i : X² → B such that:

dxi(‹u, v›)	=	( xi(u) , xi(v) )
	=	xi(u) + xi(v)
	=	xi(v) – xi(u).

In the above transcription, the operator bracket of the form "( … , … )" is a cactus lobe, signifying just one false, in this case among two boolean variables, while "+" is boolean addition in the proper sense of addition in GF(2), and is thus equivalent to "–", in the sense of adding the additive inverse.

The above definition is equivalent to defining dx_i : (B → X) → B such that:

dxi(q)	=	( xi(q0) , xi(q1) )
	=	xi(q0) + xi(q1)
	=	xi(q1) – xi(q0),

where q_b = q(b), for each b in B. Thus, the proposition dx_i is true of the path q = ‹u, v› exactly if the terms of q, the endpoints u and v, lie on different sides of the question x_i.

Now we can use the language of features in 〈dX〉, indeed the whole calculus of propositions in [dX], to classify paths and sets of paths. In other words, the paths can be taken as models of the propositions g : dX → B. For example, the paths corresponding to Diag(X) fall under the description (dx₁)…(dx_n), which says that nothing changes among the set of features {x₁, …, x_n}.

Finally, a few words of explanation may be in order. If this concept of a path appears to be described in a roundabout fashion, it is because I am trying to avoid using any assumption of vector space properties for the space X which contains its range. In many ways the treatment is still unsatisfactory, but improvements will have to wait for the introduction of substitution operators acting on singular propositions.

The Extended Universe of Discourse

Next, we define the so-called extended alphabet or bundled alphabet EA as:

EA = A ∪ dA = {a₁, …, a_n, da₁, …, da_n}

This supplies enough material to construct the differential extension EA, or the tangent bundle over the initial space A, in the following fashion:

EA	=	A × dA
	=	〈EA〉
	=	〈A ∪ dA〉
	=	〈a1, …, an, da1, …, dan〉,

thus giving EA the type Bⁿ × Dⁿ.

Finally, the tangent universe EA ^• = [EA] is constituted from the totality of points and maps, or interpretations and propositions, that are based on the extended set of features EA:

EA ^• = [EA] = [a₁, …, a_n, da₁, …, da_n],

thus giving the tangent universe EA ^• the type (Bⁿ × Dⁿ +→ B) = (Bⁿ × Dⁿ, (Bⁿ × Dⁿ → B)).

A proposition in the tangent universe [EA] is called a differential proposition and forms the analogue of a system of differential equations, constraints, or relations in ordinary calculus.

With these constructions, to be specific, the differential extension EA and the differential proposition h : EA → B, we have arrived, in concept at least, at one of the major subgoals of this study. At this juncture, I pause by way of summary to set another Table with the current crop of mathematical produce (Table 8).

The adjectives differential or tangent are systematically attached to every construct based on the differential alphabet dA, taken by itself. Strictly speaking, we probably ought to call dA the set of cotangent features derived from A, but the only time this distinction really seems to matter is when we need to distinguish the tangent vectors as maps of type (Bⁿ → B) → B from cotangent vectors as elements of type Dⁿ. In like fashion, having defined EA = A ∪ dA, we can systematically attach the adjective extended or the substantive bundle to all of the constructs associated with this full complement of 2n features.

Eventually we may want to extend our basic alphabet even further, to allow for discussion of higher order differential expressions. For those who want to run ahead, and would like to play through, I submit the following gamut of notation (Table 9).

Intentional Propositions

In order to analyze the behavior of a system at successive moments in time, while staying within the limitations of propositional logic, it is necessary to create independent alphabets of logical features for each moment of time that we contemplate using in our discussion. These moments have reference to typical instances and relative intervals, not actual or absolute times. For example, to discuss velocities (first order rates of change) we need to consider points of time in pairs. There are a number of natural ways of doing this. Given an initial alphabet, we could use its symbols as a lexical basis to generate successive alphabets of compound symbols, say, with temporal markers appended as suffixes.

As a standard way of dealing with these situations, I produce the following scheme of notation, which extends any alphabet of logical features through as many temporal moments as a particular order of analysis may demand. The lexical operators p^k and Q^k are convenient in many contexts where the accumulation of prime symbols and union symbols would otherwise be cumbersome.

The resulting augmentations of our logical basis found a series of discursive universes that may be called the intentional extension of propositional calculus. The pattern of this extension is analogous to that of the differential extension, which was developed in terms of the operators d^k and E^k, and there is an obvious and natural relation between these two extensions that falls within our purview to explore. In contexts displaying this regular pattern, where a series of domains stretches up from an anchoring domain X through an indefinite number of higher reaches, I refer to a particular collection of domains based on X as a realm of X, and when the succession exhibits a temporal aspect, as a reign of X.

For the purposes of this discussion, let us define an intentional proposition as a proposition in the universe of discourse QX ^• = [QX], in other words, a map q : QX → B. The sense of this definition may be seen if we consider the following facts. First, the equivalence QX = X × X′ motivates the following chain of isomorphisms between spaces:

(QX → B)		(X × X′ → B)
		(X → (X′ → B))
		(X′ → (X → B)).

Viewed in this light, an intentional proposition q may be rephrased as a map q : X × X′ → B, which judges the juxtaposition of states in X from one moment to the next. Alternatively, q may be parsed in two stages in two different ways, as q : X → (X′ → B) and as q : X′ → (X → B), which associate to each point of X or X′ a proposition about states in X′ or X, respectively. In this way, an intentional proposition embodies a type of value system, in effect, a proposal that places a value on a collection of ends-in-view, or a project that evaluates a set of goals as regarded from each point of view in the state space of a system.

In sum, the intentional proposition q indicates a method for the systematic selection of local goals. As a general form of description, we may refer to a map of the type q : QⁱX → B as an "i^th order intentional proposition". Naturally, when we speak of intentional propositions without qualification, we usually mean first order intentions.

Many different realms of discourse have the same structure as the extensions that have been indicated here. From a strictly logical point of view, each new layer of terms is composed of independent logical variables that are no different in kind from those that go before, and each further course of logical atoms is treated like so many individual, but otherwise indifferent bricks by the prototype computer program that I use as a propositional interpreter. Thus, the names that I use to single out the differential and the intentional extensions, and the lexical paradigms that I follow to construct them, are meant to suggest the interpretations that I have in mind, but they can only hint at the extra meanings that human communicators may pack into their terms and inflections.

As applied here, the word intentional is drawn from common use and may have little bearing on its technical use in other, more properly philosophical, contexts. I am merely using the complex of intentional concepts - aims, ends, goals, objectives, purposes, and so on - metaphorically to flesh out and vividly to represent any situation where one needs to contemplate a system in multiple aspects of state and destination, that is, its being in certain states and at the same time acting as if headed through certain states. If confusion arises, more neutral words like conative, contingent, discretionary, experimental, kinetic, progressive, tentative, or trial would probably serve as well.

Life on Easy Street

The finite character of the extended universe [EA] makes the problem of solving differential propositions relatively straightforward, at least, in principle. The solution set of the differential proposition q : EA → B is the set of models q^–1(1) in EA. Finding all of the models of q, the extended interpretations in EA that satisfy q, can be carried out by a finite search. Being in possession of complete algorithms for propositional calculus theorem proving makes the analytic task fairly simple in principle, though the question of efficiency in the face of arbitrary complexity may always remain another matter entirely. While the fact that propositional satisfiability is NP-complete may be discouraging for the prospects of a single efficient algorithm that covers the whole space of [EA] with equal facility, there appears to be much room for improvement in classifying special forms and in developing algorithms that are tailored to their practical processing.

In view of these constraints and contingencies, my focus shifts to the tasks of approximation and interpretation that support intuition, especially in dealing with the natural kinds of differential propositions that arise in applications, and in the effort to understand, in succinct and adaptive forms, their dynamic implications. In the absence of direct insights, these tasks are partially carried out by forging analogies with the familiar situations and customary routines of ordinary calculus. But the indirect approach, going by way of specious analogy and intuitive habit, forces us to remain on guard against the circumstance that occurs when the word forging takes on its shadier nuance, indicting the constant risk of a counterfeit in the proportion.

Back to the Beginning : Exemplary Universes

To anchor our understanding of differential logic, let us 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 obvious simplicity of these cases, it is possible to observe how central difficulties of the subject begin to arise already at this stage.

A One-Dimensional Universe

Let X = {x₁} = {A} be an alphabet that represents one boolean variable or a single logical feature. In this example I am using the capital letter "A" in a more usual informal way, to name a feature and not a space, at variance with my formerly stated formal conventions. At any rate, the basis element A = x₁ may be interpreted as a simple proposition or a coordinate projection A = x₁ : B₁ ¸> B. 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 {x : B ××> B}. The space of linear propositions X* = {hom : 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 {pos : B ¥> B} = {A, 1}, of rank 1 and 0, respectively, where "1" is understood as denoting the constant function 1 : B → B. In sum, there are propositions altogether in the universe of discourse, comprising the set X^ = {f : X → B} = {0, (A), A, 1} (B → B).

The first order differential extension of X is EX = {x₁, dx₁} = {A, dA}. If the feature "A" is understood as applying to some object or state, then the feature "dA" may be interpreted as an attribute of the same object or state that says that it is changing significantly with respect to the property A, or that it has an escape velocity with respect to the state A. In practice, differential features acquire their logical meaning through a class of temporal inference rules.

For example, relative to a frame of observation that is left implicit for now, one is permitted to make the following sorts of inference: From the fact that A and dA are true at a given moment one may infer that (A) will be true in the next moment of observation. Altogether in the present instance, there is the fourfold scheme of inference that is shown below:

It might be thought that we need to bring in an independent time variable at this point, but an insight of fundamental importance appears to be that the idea of process is more basic than the notion of time. A time variable is actually a reference to a clock, that is, a canonical or a convenient process that is established or accepted 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 these inquiries only wrap up puzzles in further riddles, and are obviously too involved to be handled at our current level of approximation.

Observe that the secular inference rules, used by themselves, involve a loss of information, since nothing in them can tell us whether the momenta {(dA), dA} are preserved or changed in the next instance. In order to know this, we would have to determine d²A, and so on, pursuing an infinite regress. Ultimately, in order to rest with a finitely determinate system, it is necessary to make an infinite assumption, for example, that d^kA = 0 for all k greater than some fixed value M. Another way to escape the regress is through the provision of a