Pattern Logic Series, Part III: Imperfect Limitation & Distinction
In the part I of this exploration into pattern logic we introduced the fundamental relational pattern of simple limitation. In Part II we will considered compound limitation. Here in Part III we pick up on a thread from Part I having to do with patterns that lack inputs. This is the exploration of imperfect limitation.
Imperfect Limitation
It has already been pointed out in Part I that the interpretive rubric of limitation can be applied even in the absence of one or both of the thing or use channel inputs. We can refer to this as “imperfect limitation”. Inserting the concept of “no occasion” for the value of a channel without input, the interpretation proceeds as we can now examine.
- An ADEPT LION disjoint occasion is a primitive occasion having the pattern of only a use channel input [u] from the indeterminate entity that it is disjoint with.
- An ADEPT LION non-predication occasion is a primitive occasion having the pattern of only a thing channel input [u] from the indeterminate entity that it is non-predicative of.
These definitions obviously require further explanation to be grokked.
Frame-Region Diagrams
To understand these definitions, it is helpful to employ what we will call Frame-region diagrams. These are the Euler-like diagrams that are on the right side of the figure shown below. These diagrams are a 2 dimensional representations of the extent of limitation between related concepts. Given a simple frame-region diagram with two ellipses, each will be representative of an ADEPT LION occasion, and their arrangement will convey the limitation that exists between them. There are 9 basic arrangements as cataloged in the following diagram.
- Top Concept — A rectangle with a solid line represents the universe of discourse, the whole world, or the logical top concept.
- Bottom Concept — A rectangle (or ellipse) with a dashed line represents nothing, the void, or the logical bottom concept.
- Alternative — A diagonal line that bisects a shape represents a mutually-distinct and exhaustive partitioning of a concept into two choices. Boolean truth values would assign one concept to the TRUE value and the other to the FALSE value. Another example of alternative would be a logical concept (“dog”) and its logical complement (“not-dog”).
- Restatement — a visual representation that two concepts share the same notion and extent and are thus equivalent.
- Superordinate — one concept subsumes the entirety of a second concept within it.
- Subordinate — the entirety of one concept is subsumed by another concept.
- Similar — two concepts have some common occasion.
- Dissimilar — two concepts share no common occasion.
- Ruled-out — a diagrammatic technique to restrict the possibility of an arrangement when a range of several possibilities needs to be considered.
Yes, it is the case that superordinate is an alternate expression of subordinate with a simple change in the labeling of the respective concepts involved. The description of “Ruled-out” points to the fact that unlike Venn Diagrams, which show all possible logical combinations in one diagram, Frame-region diagrams are used to show a set of all possible arrangements and then from that set of diagrams, some may be excluded, or “ruled out”. This means that pattern logic is often ambiguous — requiring multiple diagrams to capture all of the possible states of arrangement and corresponding logical relationship between some given concepts.
An example of this ambiguity is present within the most basic patterns of limitation itself. Revisiting the trickle diagram from part I and the braid diagrams of part II, if we were to add the corresponding frame-region diagram for the relationship of limitation, it would look like this…
We can see that a limitation can be ambiguously interpreted as both a subordination and a restatement in frame region diagramming. We can think of this as analogous to the idea of a weak inequality operator in mathematics such as “less-than-or-equal-to”. When “x is less-than-or-equal-to 5” we are asserting many possibilities like, “x” is “3”, or “x” is “5”. The point is simply that we are making an ambiguous assertion.
And if, for example, we were to add a second limitation to our pattern (see the next figure) that went in the opposing direction, then we would have a situation of “mutual limitation”. This would eliminate the subordinate arrangement and leave only the restatement which makes sense because this is one form of equality. If I were to claim that “this pizza is a portion of my dinner” and then also that “my dinner is a portion of this pizza”, then you might rightly conclude that “this pizza was my dinner”. So it is with mutual limitation. These two statements rule out the possibilities that I ate anything more than pizza for dinner or that similarly, dinner consisted in anything more than pizza.
The example is clearly trivial but it does show how the frame-region diagramming method uses ambiguity and ruling-out to interpret the situation.
Let’s employ frame-region diagramming in the interpretation of some imperfect limitations.
Disjoint Occasions
Recall the species of limitation called inclusion that allowed us to compose a taxonomical inclusion relation between two universal concepts in Part II. These universal concepts are modeled as ADEPT LION shadow entities and are indeterminate in their logical effects within patterns. It is important to understand the meaning of “indeterminate” in this usage. It does not mean that the participation of these concepts is without purpose, only that the meaning is entirely conceptual and not material in its effect. If we do actually omit these universal concepts/indeterminate entities from the pattern, then the pattern will interpret “no occasion” in their place. The laws of pattern logic that govern the interpretation of patterns in the first consideration can be found elsewhere but we do require them to ground the meaning of these imperfect limitations.
In the presence of a use channel indeterminate entity but the absence of any thing channel input, the composition of the pattern is evaluated as,
α : β ⊒ _
which we can read as:
“the alpha occasion is composed of the beta occasion predicating (or including) no occasion”
The Law of Non-Counter-Predication (V) speaks explicitly to this pattern and states that:
α : β ⊇_ ≡ α ⊈ β ≡ β ⊈ α ≡ α φ β
This statement of the law relies on an implicit application of the Law of Commutation (II) to arrive at the conclusion that the concepts of alpha and beta are actually disjoint (symbolized above as φ)from each other.
The Law of Intersection (VII) states:
α : β ⊇ γ ≡ γ = α ∩ β
and applying this to our original composition of the pattern by supplying “no occasion” for the value of gamma we get:
α : β ⊇ _ ≡ _ = α ∩ β
which we can read as:
“no occasion equals alpha intersecting beta”
Now, looking over the possible arrangements of frame-region diagramming with these assertions:
alpha does not predicate (include or equal) beta
- rules our superordinate and restatement
- rules out alpha as the top concept and beta as the bottom concept
beta does not predicate (include or equal)alpha
- rules out subordinate and restatement
- rules out beta as the top concept and alpha as the bottom concept
no occasion equals alpha intersecting beta
- rules out similar
This leaves us with the two arrangements of alternative and dissimilar. Both are forms of disjoint: alternative would be the interpretation that alpha and beta are logical complements of each other and dissimilar would simply assert that they have nothing in common but are not in combination exhaustive of the universe of discourse. So we conclude that the pattern we have been discussing, specifically the SEH-\ trickle means that there is some concept beta that is providing a use channel to a concept alpha and that these two concepts are thereby either complements of each other or simply disjoint. The SEH-\ trickle is therefore called the else trickle.
Non-Predication Occasions
If we were to repeat this examination with the pattern of a sole input through the thing channel and no occasion for the use channel we would arrive at an interpretation of the succession trickle (SEH\-).
The potential states of some succession trickle alpha and its use-channel-providing neighbor beta are alternative, dissimilar, similar or subordinate. Thus it could be considered an even more ambiguous pattern than the else trickle. The only thing the succession trickle really does do is to rule out the possibility that alpha subsumes or equals beta.
So is this Complement and/or Negation?
If we were to consider two concepts having nothing to do with each other, like “red” and “pickle” it would be natural to say “red is not-pickle” or “pickle is not-red”. The “not” part of these assertions requires careful handling though. If “red” and “pickle” were truly logical complements then anything we could think of would be one or the other. This is clearly not the case. All they are is the dissimilar arrangement. If we took “red” and “not red” we would have a pair of logical complements and in terms of frame-region diagrams we would have just the alternative frame-region arrangement. The “disjoint” of the Else trickle is more like the relationship between “red” and “pickle” than between “red” and “not red” but this pattern could be interpreted as either of these examples because of its ambiguity.
I can’t resist but to point out for any lovers of Plato out there, that these subtle distinctions regarding what the word “not” can ambiguously mean in natural language was one seemingly endless source of entertaining philosophical bewilderment that featured prominently in Platonic dialogues such as Parmenides and Sophist.
Non-predication is even more messy in its ambiguous meaning. Take the concepts of “tire” and “car”. An assertion of non-predication might go something like “a tire is not a car”. If this were the pattern of the concept “tire” non-predicating the concept “car”, then these two concepts could be alternative, similar, dissimilar or subordinate arrangement. They just can’t be a restatement or a superordinate arrangement. In other words, the concept “tire” will never include the concept “car”. That is all that is being asserted by the non-predication “a tire is not a car”. And what good is that? On its own, perhaps not much, unless you are seeking to obfuscate, but when non-predication becomes part of a larger pattern, some more clearly useful expressions begin to emerge.
But for now, if we want an unambiguous pattern for logical complement or negation we will have to continue searching.
Distinction
The perfect and imperfect limitation patterns can be combined to craft patterns that are expressive of distinction between concepts. We can further divide what we mean by “distinction” into two forms: proper distinction and general distinction.
One important impact of the distinction pattern which we will look at here is that it provides a means for developing pattern sets that can help clear up the ambiguous possibilities intrinsic to limitation. Specifically, we can express when some part is a “proper part” of some whole because the distinction expresses the existence of “something else” which is part of the whole but not part of the proper part.
Proper Distinction: the portion of a union that is an overlap between the union and a disjoint occasion of the other portion.
A proper distinction pattern set thus consists of four occasions:
- A union (extent) having two portions,
- One portion has,
- A disjoint occasion which is also the extent of,
- The second portion, which thus is an overlap between the union and the disjoint occasion.
We can introduce some additional terms in the realm of “universal mereology” (in contrast to the “individual mereology” found inthe discussion of parthood from Part I)
Whole: The extent of the union is thus a ‘whole’ (alpha in the diagram).
Proper Parts (of a proper distinction): a union having a proper distinction (gamma) between two of its portions (beta and delta).
Proper Part: Each portion (beta or delta) is thus a ‘proper part’ of the ‘whole’ (alpha).
General Distinction: the portion of a union that is an overlap between the union and a non-predication occasion of the other portion
A general distinction pattern set thus consists of four occasions:
- A union(extent) having two portions,
- One portion has,
- A non-predication occasion which is also the extent of,
- The second portion, which thus is an overlap between the union and the non-predication occasion.
We can now add a term for general part and identify a second kind of related proper part to our emerging system of universal mereology. By “universal mereology” we are talking about the part and whole relationships between universal concepts such as the “figment” that is part of “my imagination” as opposed to that “dream” over there…
General Part: the portion of the whole (beta in the diagram) which has the non-predication (gamma) which is an extent of the other portion(delta) of the whole (alpha).
Proper Part (of a general distinction): the portion of the whole which is also an overlap (delta in the diagram) of the whole (alpha) and the non-predication (gamma) of the general part (beta).
Ambiguous Equality
Ambiguous Equality: two occasions are said to be ‘ambiguously equal’ or ‘ambiguously equivalent’ under the following patterns:
- Ambiguous Equality of Indistinct Mediation: they are the portion and extent of a mediation and the portion has no distinction involving the mediation, and the mediation has no distinction involving the extent. The mediation is the occasion of equality.
- Ambiguous Equality of Indistinct Overlap: they are the extents of an overlap and there is no distinction involving the overlap for either extent. The overlap portion is the occasion of equality.
- Ambiguous Equality of Indistinct Union: they are the portions of a union and there is no distinction involving the union for either portion. The union extent is the occasion of equality.
Inequality
Inequality: two or more occasions are said to be ‘unequal’ or ‘inequivalent’ when either:
- Inequality of Distinct Mediation: they are the portion and extent of a mediation and the mediation has some distinction involving the portion or the extent has some distinction involving the mediation. The mediation is the occasion of inequality.
- Inequality of Distinct Overlap: they are the extents of an overlap and there is a distinction involving the overlap for either extent. The overlap portion is the occasion of inequality.
- Inequality of Distinct Union: they are the portions of a union and there is a distinction involving the union for either portion.
Proper Intersection
Proper Intersection: an overlap of two extents in which both of the extents have some distinction involving the overlap portion.
- The overlap portion of a proper intersection is called the ‘proper intersect’.
- By definition, an inclusion cannot be a proper intersection, but given any intersection, the overlap portion of the intersection will be included in each of the extents of that intersection (see that beta is included in alpha and beta is included in gamma in the frame-region diagram below).
Summary of Imperfect Limitation
In this Part III of our exploration into pattern logic we have again hit upon some very interesting and useful logical expressions. Namely distinction, ambiguous equality, inequality and proper intersection. It would likely be a fruitful exercise to further consider how these expressions in pattern align, or distinguish themselves from the standard understandings of logical operations such as implication, intersection, and complement. Greater precision in handling the meaning of the word “not” is an important part of disambiguating natural language and the additional granularity of expression found in the patterns for disjoint and non-predicating concepts provides a novel basis for approaching this challenge systematically. Other interesting tangents arise here for mathematical topics such as the ordered sequence and the operation of subtraction.
But our journey into pattern logic will next visit a long-standing debate in the history of philosophy that echoes even today in our approaches to understanding the mind-body connection and the role of language in our models of experience and reality, that is “the problem of universals”.
