Pattern Logic Series Part II: Compound Limitation & Hierarchy

In part I of this series on pattern logic we introduced a fundamental relationship called limitation. In this part we will explore what happens when the simple form of limitation between conceptual entities becomes more complex: the emergence of expressions for equality and hierarchy.

Compound Limitation

A compound limitation is a pattern that employs more than one limitation occasion. In the preceding part we examined the compound limitations for each of four basic species in which there are two related occasions for each kind of limitation. In most of the examples that follow the exploration is limited to the variety of limitation called an inclusion trickle (SEH\\). This is the variety in which both the portion and extent of the limitation are universal, indeterminate entities (also referred to as “concepts”).

The diagrams used so far distinguish explicitly between the thing [t] and use [u] channel inputs and are called rivulet diagrams. This explicit display of channel information can quickly become visually overwhelming as patterns become larger so it is helpful to utilize the visually-simpler braid diagram. A braid diagram displays a relationship of limitation as a directed arrow rather than a node with thing and use inputs as in the trickle diagram. The thing input, or portion, is the node that contacts the tail of the arrow. The use input, or extent, is the node that contacts the head of the arrow, leaving the entire arrow to represent the limitation.

Here is a comparison of the trickle diagram and the braid diagram for the basic pattern of limitation that we started with.

Comparison of the Trickle Diagram method with the Braid Diagram method. The occasion of limitation is represented as a directed arrow.

In the prior discussion of taxonomic inclusion of universals we saw the emergence of “union” and “overlap” patterns and we can now complete this most basic set of compound limitations with a third possibility of “mediation”.

• Union: an extent with two portions (two limitations).
• Mediation: the extent of one limitation is also the portion of a second limitation.
• Overlap : a portion with two extents (two limitations)

The three basic arrangements of patterns consisting of two limitation occasions (the arrows) and three entities (rounded rectangles). The Union and Overlap patterns displayed here as braid diagrams correspond to the rivulet diagrams above where the form of limitation is specifically taxonomic inclusion of universals.

These terms of union, mediation and overlap must be thought of as arrangements or “pattern sets” that do not introduce an atomic concept, but rather a new means of relating atomic concepts which are the occurrences that compose the arrangement. For instance, we should not jump to equate “union” with logical disjunction or “overlap” with logical conjunction as if we were arbitrarily introducing a desired concept in support of some axiomatic approach. In fact, these patterns are introducing three sets of mutually-ambiguous possibilities. For instance, it could be the case that all three nodes in each of the diagrams above are equivalent occasions of the same concept. This may also not be the case, but these patterns in themselves do not answer the question as to whether we are asserting three equivalent concepts or not.

That doesn’t make these three pattern sets the same however. We could work out the possibilities for each of these three arrangements to get a better handle on exactly where the ambiguity is but for now, let us see what happens with resolving this intrinsic ambiguity when we add one more limitation to create a triangular arrangement of three concepts which we can call “completely limiting each other”.

Cyclic Equivalency and Hierarchy

Given three occasions that completely limit each other (each and every pairing of regions has a limitation between them) they can either form a cyclic equivalency or a hierarchy.

Cyclic Equivalency is composed of uni-directional mediations: alpha is a mediation of beta and gamma, beta is a mediation of alpha and gamma, and gamma is a mediation of beta and alpha. This pattern equates the conceptual values of alpha, beta and gamma.

Cyclic Equivalency is a ring of uni-directional mediations that equates a triad of concepts.

A Hierarchical Triad consists of three component occasions with each distinguished as:

• Top: the union of the other two occasions. The “highest” or most “general” concept of the triad.
• Middle: the mediation of the other two occasions. The “intermediate” concept of the triad.
• Bottom: the overlap of the other two occasions. The “lowest” or most “specific” concept of the triad.

Hierarchical Triads determine the relative position of three concepts through the collective meaning of the three compound limitation patterns of union, mediation and overlap.

These triadic combinations can compose a branching conceptual hierarchy of any count of tiers. Unless an equality is intended, cyclic arrangements should be avoided so as to maintain the tiers. The existence of cyclical equality patterns will “condense” nodes within the hierarchy structure into single concepts with multiple equivalent expressions of values.

Example of a Simple Hierarchy

The simple hierarchy of animal concepts on the left is represented in pattern logic relations of taxonomical inclusion on the right. Although a three-concept hierarchical triad requires three relations, concepts do not need to be repeated unnecessarily. So in this example, because both cats and dogs are mammals which are also animals, the inclusion of the mammal concept in the animal concept only need occur once rather than twice.

The example here demonstrates how a simple hierarchy can be expressed in pattern logic. The top of the hierarchy is the concept “animal” and it contains two branches, “mammal” and “reptile”. Under these middle concepts we find three concepts at the bottom level: “cat”, “dog” and “snake”. Given these arrangements we see that [animal] is a union, [mammal, reptile] are mediations, and [cat, dog, reptile] are overlaps.

This example helps us to visualize the concept of Hierarchical Siblings. Siblings in a hierarchy are either:

1. sibling branches — mediations that share the same union (like mammal and reptile) or

2. sibling leaves — overlaps that share the same mediation (like cat and dog).

It also becomes clear that Hierarchical Parents are either:

1. parent trunks — unions having one or more mediations (like animal) or

2. parent branches — mediations having one or more overlaps (like mammal and reptile)

Finally, we can identify Hierarchical Children as either:

1. child branches — mediations (like mammal and reptile)

2. child leaves — overlaps (like cat, dog and snake)

A Braid Diagram of an example simple hierarchy.

This next figure is the same example shown as a braid diagram. It looks similar to the traditional way that a hierarchy is drawn except for the addition of the dashed lines. It is understandable to think that these dashed lines are superfluous from the perspective of communicating the idea of a hierarchy visually. But in pattern logic these relationships are not superfluous at all but entirely necessary to unambiguously express the meaning of this hierarchical arrangement of concepts. In one sense it makes the implicit concept of transitivity explicit. For example, we reason transitively that because “mammal includes cat” and “animal includes mammal” that “animal includes cat”. This intuitive step of deduction through transitive relationships of inclusion is explicitly stated in pattern logic by the dashed arrow going from cat to animal. In a way then, what the typical hierarchical diagram implicitly states is an activity of deductive reasoning through transitive inclusion and pattern logic is capturing not only the involved concepts but also an explicit representation of this additional activity of deductive reasoning.

What pattern logic is therefore doing is providing a means of knowledge representation that is explicit as to the meaning of what would otherwise be merely implicit. It is the basis of a universal data model expressing both concepts and logical relations.

So what does this simple hierarchy look like as a “universal data model” in ADEPT LION? The following self-referencing table of integers would suffice and is an abbreviated snippet of a typical ADEPT instance.

An ADEPT LION instance of a simple hierarchy. The integer values in the thing and use channels point to spark values which identify the occasion, that is each row of the table. The word channel captures the value of that occasion. This tabular arrangement can also be conceived of as a directed graph in which the rows (uniquely identified by the “spark” value) are vertices and the thing and use channels are two different relationship types that reference another vertex as the supplier of an incoming edge of that particular channel type.

While it may be tempting to think that we have composed an ontology with this example, it is more precise to speak of this merely as a taxonomy. The distinction would be that the relations in an ontology are concerned with terms that are quantified. In other words, while it is natural for us to implicitly equate taxonomical statements like “animal includes mammal” with logical propositions like “all mammals are animals”, pattern logic sees these as distinctively different patterns. A propositional statement in an ontology requires quantifiers like “all” whereas taxonomical inclusion does not require this quantification. See these links for more on ontology, quantification and reasoning in pattern logic.

The ADEPT LION instance of an animal taxonomy in a graphical visualization. The blue vertices correspond to the labels for the 14 rows of data in the table above. There are additional triangular patterns associated with these blue nodes that instantiate these symbolic values prior to their participation in the relations of limitation. This is why the count of occasions will appear to have increased.

Summary of Compound Limitation

Our exploration into pattern has produced some additional useful discoveries in this Part II. Namely, the expression of equivalent concepts using cyclic limitations and the expression of tree-like hierarchies using non-cyclic triads of mutually-limiting occasions. An important feature of each of these discoveries is that the patterns expressing the universal concepts themselves have taken on no commitment in pattern as to their “intrinsic identity”. In other words, these remain mere concepts with no dependency on other occasions to explain their meaning and yet they have freely contributed to arrangements that do capture higher-order arrangements like equality and hierarchy. For more on what these concepts could represent in pattern if they were to commit to an intrinsic identity, the mappings of pattern logic primitive occasions to the Kantian “judgements” is a fascinating place to begin that journey. And for a similar exploration of the meaning of patterns reserved for logical individuals, the Kantian or Aristotelian “categories” will lead us off in that direction.

Another important point here is that although we have called these acyclic triads “hierarchies” they are also expressing a concept of “sequence” as well and from a mathematical point of view, we could speak of these as “weak inequalities”. But rather than heading off into mathematics, our next destination will be the Imperfect Limitation. There we will find a nuanced and critical means of understanding what it means to “not” be something else.