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Elementary notions

Logical description of a universe of discourse begins with a set of logical signs. For the sake of simplicity in a first approach, assume that these logical signs are collected in the form of a finite alphabet, \(\mathfrak{A} = \lbrace\!\) “\(a_1\!\)” \(, \ldots,\!\) “\(a_n\!\)” \(\rbrace.\!\) Each of these signs is interpreted as denoting a logical feature, for instance, a property that objects in the universe of discourse may have or a proposition about objects in the universe of discourse. Corresponding to the alphabet \(\mathfrak{A}\) there is then a set of logical features, \(\mathcal{A} = \{ a_1, \ldots, a_n \}.\)

A set of logical features, \(\mathcal{A} = \{ a_1, \ldots, a_n \},\) affords a basis for generating an \(n\!\)-dimensional universe of discourse, written \(A^\circ = [ \mathcal{A} ] = [ a_1, \ldots, a_n ].\) It is useful to consider a universe of discourse as a categorical object that incorporates both the set of points \(A = \langle a_1, \ldots, a_n \rangle\) and the set of propositions \(A^\uparrow = \{ f : A \to \mathbb{B} \}\) that are implicit with the ordinary picture of a venn diagram on \(n\!\) features. Accordingly, the universe of discourse \(A^\circ\) may be regarded as an ordered pair \((A, A^\uparrow)\) having the type \((\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})),\) and this last type designation may be abbreviated as \(\mathbb{B}^n\ +\!\to \mathbb{B},\) or even more succinctly as \([ \mathbb{B}^n ].\) For convenience, the data type of a finite set on \(n\!\) elements may be indicated by either one of the equivalent notations, \([n]\!\) or \(\mathbf{n}.\)

Table 2 summarizes the notations that are needed to describe ordinary propositional calculi in a systematic fashion.

Table 4. Propositional Calculus : Basic Notation
Symbol Notation Description Type
\(\mathfrak{A}\) \(\lbrace\!\) “\(a_1\!\)” \(, \ldots,\!\) “\(a_n\!\)” \(\rbrace\!\) Alphabet \([n] = \mathbf{n}\)
\(\mathcal{A}\) \(\{ a_1, \ldots, a_n \}\) Basis \([n] = \mathbf{n}\)
\(A_i\!\) \(\{ \overline{a_i}, a_i \}\!\) Dimension \(i\!\) \(\mathbb{B}\)
\(A\!\) \(\langle \mathcal{A} \rangle\)

\(\langle a_1, \ldots, a_n \rangle\)
\(\{ (a_1, \ldots, a_n) \}\!\) \(A_1 \times \ldots \times A_n\)
\(\textstyle \prod_i A_i\!\)

Set of cells,

coordinate tuples,
points, or vectors
in the universe
of discourse

\(\mathbb{B}^n\)
\(A^*\!\) \((\operatorname{hom} : A \to \mathbb{B})\) Linear functions \((\mathbb{B}^n)^* \cong \mathbb{B}^n\)
\(A^\uparrow\) \((A \to \mathbb{B})\) Boolean functions \(\mathbb{B}^n \to \mathbb{B}\)
\(A^\circ\) \([ \mathcal{A} ]\)

\((A, A^\uparrow)\)
\((A\ +\!\to \mathbb{B})\)
\((A, (A \to \mathbb{B}))\)
\([ a_1, \ldots, a_n ]\)

Universe of discourse

based on the features
\(\{ a_1, \ldots, a_n \}\)

\((\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))\)

\((\mathbb{B}^n\ +\!\to \mathbb{B})\)
\([\mathbb{B}^n]\)


Special classes of propositions

A basic proposition, coordinate proposition, or simple proposition in the universe of discourse \([a_1, \ldots, a_n]\) is one of the propositions in the set \(\{ a_1, \ldots, a_n \}.\)

Among the \(2^{2^n}\) propositions in \([a_1, \ldots, a_n]\) are several families of \(2^n\!\) propositions each that take on special forms with respect to the basis \(\{ a_1, \ldots, a_n \}.\) Three of these families are especially prominent in the present context, the linear, the positive, and the singular propositions. Each family is naturally parameterized by the coordinate \(n\!\)-tuples in \(\mathbb{B}^n\) and falls into \(n + 1\!\) ranks, with a binomial coefficient \(\tbinom{n}{k}\) giving the number of propositions that have rank or weight \(k.\!\)

  • The linear propositions, \(\{ \ell : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{\ell} \mathbb{B}),\) may be written as sums:

    \(\sum_{i=1}^n e_i = e_1 + \ldots + e_n\)  where  \(e_i = a_i\!\)  or  \(e_i = 0\!\)  for  \(i = 1\!\)  to  \(n.\!\)

  • The positive propositions, \(\{ p : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{p} \mathbb{B}),\) may be written as products:

    \(\prod_{i=1}^n e_i = e_1 \cdot \ldots \cdot e_n\)  where  \(e_i = a_i\!\)  or  \(e_i = 1\!\)  for  \(i = 1\!\)  to  \(n.\!\)

  • The singular propositions, \(\{ \mathbf{x} : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{s} \mathbb{B}),\) may be written as products:

    \(\prod_{i=1}^n e_i = e_1 \cdot \ldots \cdot e_n\)  where  \(e_i = a_i\!\)  or  \(e_i = (a_i)\!\)  for  \(i = 1\!\)  to  \(n.\!\)

In each case the rank \(k\!\) ranges from \(0\!\) to \(n\!\) and counts the number of positive appearances of the coordinate propositions \(a_1, \ldots, a_n\!\) 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 \((a_1)(a_2)(a_3).\!\)

The basic propositions \(a_i : \mathbb{B}^n \to \mathbb{B}\) are both linear and positive. So these two kinds of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions.

Finally, it is important to note that all of the above distinctions are relative to the choice of a particular logical basis \(\mathcal{A} = \{ a_1, \ldots, a_n \}.\) For example, a singular proposition with respect to the basis \(\mathcal{A}\) will not remain singular if \(\mathcal{A}\) is extended by a number of new and independent features. Even if one keeps to the original set of pairwise options \(\{ a_i \} \cup \{ (a_i) \}\) to pick out a new basis, the sets of linear propositions and positive propositions are both determined by the choice of basic propositions, and this whole determination is tantamount to the purely conventional choice of a cell as origin.

Reality at the Threshold of Logic

Table 5. A Bridge Over Troubled Waters
\(\mbox{Linear Space}\!\) \(\mbox{Liminal Space}\!\) \(\mbox{Logical Space}\!\)

\(\begin{matrix} \mathcal{X} & = & \{x_1, \ldots, x_n\} \\ \end{matrix}\)

\(\begin{matrix} \underline\mathcal{X} & = & \{\underline{x}_1, \ldots, \underline{x}_n\} \\ \end{matrix}\)

\(\begin{matrix} \mathcal{A} & = & \{a_1, \ldots, a_n\} \\ \end{matrix}\)

\(\begin{matrix} X_i & = & \langle x_i \rangle \\ & \cong & \mathbb{K} \\ \end{matrix}\)

\(\begin{matrix} \underline{X}_i & = & \{(\underline{x}_i), \underline{x}_i \} \\ & \cong & \mathbb{B} \\ \end{matrix}\)

\(\begin{matrix} A_i & = & \{(a_i), a_i \} \\ & \cong & \mathbb{B} \\ \end{matrix}\)

\(\begin{matrix} X \\ = & \langle \mathcal{X} \rangle \\ = & \langle x_1, \ldots, x_n \rangle \\ = & X_1 \times \ldots \times X_n \\ = & \prod_{i=1}^n X_i \\ \cong & \mathbb{K}^n \\ \end{matrix}\)

\(\begin{matrix} \underline{X} \\ = & \langle \underline\mathcal{X} \rangle \\ = & \langle \underline{x}_1, \ldots, \underline{x}_n \rangle \\ = & \underline{X}_1 \times \ldots \times \underline{X}_n \\ = & \prod_{i=1}^n \underline{X}_i \\ \cong & \mathbb{B}^n \\ \end{matrix}\)

\(\begin{matrix} A \\ = & \langle \mathcal{A} \rangle \\ = & \langle a_1, \ldots, a_n \rangle \\ = & A_1 \times \ldots \times A_n \\ = & \prod_{i=1}^n A_i \\ \cong & \mathbb{B}^n \\ \end{matrix}\)

\(\begin{matrix} X^* & = & (\ell : X \to \mathbb{K}) \\ & \cong & \mathbb{K}^n \\ \end{matrix}\)

\(\begin{matrix} \underline{X}^* & = & (\ell : \underline{X} \to \mathbb{B}) \\ & \cong & \mathbb{B}^n \\ \end{matrix}\)

\(\begin{matrix} A^* & = & (\ell : A \to \mathbb{B}) \\ & \cong & \mathbb{B}^n \\ \end{matrix}\)

\(\begin{matrix} X^\uparrow & = & (X \to \mathbb{K}) \\ & \cong & (\mathbb{K}^n \to \mathbb{K}) \\ \end{matrix}\)

\(\begin{matrix} \underline{X}^\uparrow & = & (\underline{X} \to \mathbb{B}) \\ & \cong & (\mathbb{B}^n \to \mathbb{B}) \\ \end{matrix}\)

\(\begin{matrix} A^\uparrow & = & (A \to \mathbb{B}) \\ & \cong & (\mathbb{B}^n \to \mathbb{B}) \\ \end{matrix}\)

\(\begin{matrix} X^\circ \\ = & [\mathcal{X}] \\ = & [x_1, \ldots, x_n] \\ = & (X, X^\uparrow) \\ = & (X\ +\!\to \mathbb{K}) \\ = & (X, (X \to \mathbb{K})) \\ \cong & (\mathbb{K}^n, (\mathbb{K}^n \to \mathbb{K})) \\ = & (\mathbb{K}^n\ +\!\to \mathbb{K}) \\ = & [\mathbb{K}^n] \\ \end{matrix}\)

\(\begin{matrix} \underline{X}^\circ \\ = & [\underline\mathcal{X}] \\ = & [\underline{x}_1, \ldots, \underline{x}_n] \\ = & (\underline{X}, \underline{X}^\uparrow) \\ = & (\underline{X}\ +\!\to \mathbb{B}) \\ = & (\underline{X}, (\underline{X} \to \mathbb{B})) \\ \cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})) \\ = & (\mathbb{B}^n\ +\!\to \mathbb{B}) \\ = & [\mathbb{B}^n] \\ \end{matrix}\)

\(\begin{matrix} A^\circ \\ = & [\mathcal{A}] \\ = & [a_1, \ldots, a_n] \\ = & (A, A^\uparrow) \\ = & (A\ +\!\to \mathbb{B}) \\ = & (A, (A \to \mathbb{B})) \\ \cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})) \\ = & (\mathbb{B}^n\ +\!\to \mathbb{B}) \\ = & [\mathbb{B}^n] \\ \end{matrix}\)


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, …, xn〉 \(\cong\) Rn. The coordinate system X = {xi} is a set of maps xi : 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 Li on R, that is, a subset of Rn, and then we define the ith threshold map, or limen xi as follows:

xi : RnB such that:
xi(x) = 1 if xLi,
xi(x) = 0 if otherwise.

In other notations that are sometimes used, the operator \(\chi (\ )\) or the corner brackets \(\lceil \ldots \rceil\) 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:

xi(x) = \(\chi (x \in L_i)\) = \(\lceil x \in L_i \rceil\) = Li(x).

Notice that, as defined here, there need be no actual relation between the n-dimensional subsets {Li} and the coordinate axes corresponding to {xi}, 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, Li is bounded by some hyperplane that intersects the ith axis at a unique threshold value riR. Often, the hyperplane is chosen normal to the axis. In recognition of this motive, let us make the following convention. When the set Li has points on the ith axis, that is, points of the form ‹0, …, 0, ri, 0, …, 0› where only the xi 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 = {xi} 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 ith 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 xi asserts that the representative point x resides above the ith threshold.

Primitive assertions of the form xi(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〉 \(\cong\) Bn, and a space of functions (regions, propositions), X^ \(\cong\) (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 xi, 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.

The Extended Universe of Discourse

Table 8

Table 8. Differential Extension : Basic Notation
Symbol Notation Description Type
\(\operatorname{d}\mathfrak{A}\) \(\lbrace\!\) “\(\operatorname{d}a_1\)” \(, \ldots,\!\) “\(\operatorname{d}a_n\)” \(\rbrace\!\) Alphabet of

differential
symbols

\([n] = \mathbf{n}\)
\(\operatorname{d}\mathcal{A}\) \(\{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}\) Basis of

differential
features

\([n] = \mathbf{n}\)
\(\operatorname{d}A_i\) \(\{ (\operatorname{d}a_i), \operatorname{d}a_i \}\) Differential

dimension \(i\!\)

\(\mathbb{D}\)
\(\operatorname{d}A\) \(\langle \operatorname{d}\mathcal{A} \rangle\)

\(\langle \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle\)
\(\{ (\operatorname{d}a_1, \ldots, \operatorname{d}a_n) \}\)
\(\operatorname{d}A_1 \times \ldots \times \operatorname{d}A_n\)
\(\textstyle \prod_i \operatorname{d}A_i\)

Tangent space

at a point:
Set of changes,
motions, steps,
tangent vectors
at a point

\(\mathbb{D}^n\)
\(\operatorname{d}A^*\) \((\operatorname{hom} : \operatorname{d}A \to \mathbb{B})\) Linear functions

on \(\operatorname{d}A\)

\((\mathbb{D}^n)^* \cong \mathbb{D}^n\)
\(\operatorname{d}A^\uparrow\) \((\operatorname{d}A \to \mathbb{B})\) Boolean functions

on \(\operatorname{d}A\)

\(\mathbb{D}^n \to \mathbb{B}\)
\(\operatorname{d}A^\circ\) \([\operatorname{d}\mathcal{A}]\)

\((\operatorname{d}A, \operatorname{d}A^\uparrow)\)
\((\operatorname{d}A\ +\!\to \mathbb{B})\)
\((\operatorname{d}A, (\operatorname{d}A \to \mathbb{B}))\)
\([\operatorname{d}a_1, \ldots, \operatorname{d}a_n]\)

Tangent universe

at a point of \(A^\circ,\)
based on the
tangent features
\(\{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}\)

\((\mathbb{D}^n, (\mathbb{D}^n \to \mathbb{B}))\)

\((\mathbb{D}^n\ +\!\to \mathbb{B})\)
\([\mathbb{D}^n]\)


Table 9

Table 9. Higher Order Differential Features

\(\begin{array}{lllll} \operatorname{d}^0 \mathcal{A} & = & \{a_1, \ldots, a_n\} & = & \mathcal{A} \\ \operatorname{d}^1 \mathcal{A} & = & \{\operatorname{d}a_1, \ldots, \operatorname{d}a_n\} & = & \operatorname{d}\mathcal{A} \\ \end{array}\)

\(\begin{array}{lll} \operatorname{d}^k \mathcal{A} & = & \{\operatorname{d}^k a_1, \ldots, \operatorname{d}^k a_n\} \\ \operatorname{d}^* \mathcal{A} & = & \{\operatorname{d}^0 \mathcal{A}, \ldots, \operatorname{d}^k \mathcal{A}, \ldots \} \\ \end{array}\)

\(\begin{array}{lll} \operatorname{E}^0 \mathcal{A} & = & \operatorname{d}^0 \mathcal{A} \\ \operatorname{E}^1 \mathcal{A} & = & \operatorname{d}^0 \mathcal{A}\ \cup\ \operatorname{d}^1 \mathcal{A} \\ \operatorname{E}^k \mathcal{A} & = & \operatorname{d}^0 \mathcal{A}\ \cup\ \ldots\ \cup\ \operatorname{d}^k \mathcal{A} \\ \operatorname{E}^\infty \mathcal{A} & = & \bigcup\ \operatorname{d}^* \mathcal{A} \\ \end{array}\)