Representing emergence with rules

Cover Page

Representing
Emergence with Rules

Author: Jeffrey G. Long (jefflong@aol.com)

Date: August 17, 1994

Forum: Talk presented at the 7th International Conference on Systems Research,
Information and Cybernetics. Sponsored by The International Institute for
Advanced Studies in Systems Research and Cybernetics, and the Society for
Applied Systems Research. Paper published in conference proceedings, available
at http://iias.info/pdf_general/Booklisting.pdf

Contents
Pages 1‐6: Abstract and Preprint of paper

Pages 7‐24: Slides but no text of oral presentation

License
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Uploaded June 22, 2011

Jeffrey G. Long [8/17/1994]

Representing Emergence with Rules


Jeffrey G. Long
San Francisco, CA, USA
jefflong@aol.com

Abstract

Emergence may be defined as the point at which an entity is subject to a new and different class of
rules.  Given that we can describe entities at one level (e.g. the properties of hydrogen and oxygen) in
terms of their probable rules of behavior, and can describe entities at a higher level (e.g. the properties
of water) in the same manner, the essential question of emergence becomes "How does an entity
become subject to a completely new and different set of rules?"  This paper describes the notion of
emergence operationally, by means of a very simple "emergence rule" that declares the existence of
new entities whenever existing entities achieve certain defined statuses.  Conversely, any time an entity
becomes subject to a new and different class of rules, it operationally becomes a new entity.   Entities
change statuses only as a by‐product of processes, which processes can perform only "addition" or
"subtraction" in the broadest senses of the words.  Under this approach, there exist two broad classes of
phenomena: those that follow the classical rules of arithmetic (called resultants), and those that don't
(called emergents).  Both of these may be described by means of qualitative, conditional rules.  The
paper illustrates these concepts with examples of emergence from intentional systems (the U.S.
Constitution) and natural systems (basic chemistry).

Keywords

emergence; processes; limits of mathematics; rules; notation; law

Introduction

The paradoxes of complexity, and in particular the phenomena of emergence, have forced me to
reconsider how we represent the basic and ubiquitous transactions of addition and subtraction.  My
conclusion to date is that we must create another "grammar" that distinguishes resultant from
emergent1 transactions.

The class of transactions that I will call resultant transactions causes no real difficulties for modern
quantitative notation.  These transactions arise when two or more entities can be "added together" in
any order, and their interaction is zero or negligible.  In these transactions, one plus one equals two,
now and forever; in other words, they are additive simpliciter.  In pure arithmetic, this denotes the
union of two sets; in the real world, it may refer to two entities being mixed, or placed nearby each
other, or in some other sense "added".  The notational systems that have been built around numbers
presume the following:

 entities in toto can be added or subtracted simpliciter in all interesting cases
 properties of entities can be added or subtracted simpliciter in all interesting cases


1
-- See G. H. Lewes, who proposed the terminology.

Page 1 of 24



 such transactions are commutative (i.e., that adding A to B is identical in effect to adding B to A)
 they are associative (i.e., that adding A to B and then to C is identical to adding B to C and then to A)
 they are monotonic (i.e. if A < B, then A + X < B + X)
 they are reversible (i.e. that if A has been added to B, then A can be subtracted from B to derive
separate entities again).

These transactional presumptions hold true of all vector spaces, i.e. all sets "of objects or elements that
can be added together and multiplied by numbers  (the result being an element of the set), in such a
way that the usual rules of calculation hold" (Gellert et al, page 362).  They work quite well for the
natural numbers, for many other kinds of mathematical entity (e.g. angles), and for many entities in the
real world (e.g. unbalanced forces); but many transactions in the real world do not fit these criteria.
Such transactions ‐‐ for which I will hijack the phrase emergent transactions ‐‐ occur when two or more
entities are "added together" or "subtracted", and their interaction is significant. There are several signs
that indicate when an emergent transaction has occurred:

 the resulting number of entities cannot be inferred from the number of the components  (the quantity
of the sum cannot be computed from the quantities of the summands)(i.e., a unit increase in summands
produces a non‐unit increase in the sum2)

 the resulting properties cannot be inferred from the properties of the components (the properties of
the sum cannot be computed from the properties of the summands)

 the order of addition (or subtraction) is significant and is not reversible.

The transactions that have created matter, life, society, consciousness ‐‐ and perhaps even notations like
number3 ‐‐  are emergent transactions.  The classic example is the combination of two flammable gasses
‐‐ hydrogen and oxygen ‐‐ to form a non‐flammable liquid: water.  An example of non‐commutativity is
the addition of water to acid versus the addition of acid to water.

One of the seminal thinkers about emergent evolutionism, C. Lloyd Morgan, suggested that emergent
transactions produce qualitative changes; but I observe that they can also create quantitative changes.
Conversely, resultant transactions also produce qualitative changes, such as the addition of blue and
yellow watercolors to make green watercolor.  Figure 1 illustrates the key distinctions between a
resultant grammar of interactions and an emergent grammar:

Resultant Resultant Emergent Emergent
Grammar Grammar Grammar Grammar
Addition Subtraction Addition Subtraction
Quantitative 1 + 1 = "11" = 2 "111" ‐ 1 = "11" 1 + 1 = ¬2 2 ‐1 = ¬1
Qualitative A + B = "AB" "ABC"‐B= "AC" A + B = C C ‐ B = A, B

Figure 1: Two Grammars of Interaction


2
-- Thus non-linear transactions are a subset of emergent transactions.
3
-- In January 1994 I gave a talk exploring the idea that notations are real in the Platonic sense of being pre-existing
rather than emergent. I confess I am presently confused about my beliefs in this area.

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Thus the key distinction between resultant and emergent transactions is not whether they are
quantitative or qualitative, but whether they are predictable.  Emergent transactions do seem to violate
the traditional and common‐sense beliefs that every effect has a cause, and that full knowledge of
causes permits one to deduce or predict all of their effects.  Bertrand Russell, in support of these beliefs,
asserted that "Emergent properties represent merely scientific incompleteness, which would not exist in
the ideal physics" (Russell, 1929).  Whatever the causality issues may be, if we can simply describe our
empirical experience accurately we will have a far greater understanding of the world we live in.

Representation Issues

If we look at the universe as changing at all, then we must acknowledge the existence of processes and
the need to clearly represent processes.  Processes in the real world can only add or subtract, in the
sense described above; all other mathematical operations are shorthand notational conveniences.  But
simply describing the effects of processes is uninteresting: we could take a photograph of the result of a
process, or we could otherwise quantify its result (say, by tracking total population in an ecosystem); but
these brute facts do not give us any real understanding of what is going on within the process. To
represent the internal relations of processes we must use rules. One might say (as Lloyd Morgan did)
that the "effective relatedness" of things, both internally and externally, changes in emergent
transactions; and the effective relatedness is that which is represented by rules.

After defining rules that accurately replicate key features of observed processes, then using Occam's
razor and principles of algorithmic complexity we can select our favorites.   The goal is to select the
simplest (i.e. fewest bytes) set of rules that defines all behavior of interest.

In spite of their unpredictability (whether it be inherent or temporary), emergent transactions are lawful
and can be studied just like any other transactions.  Given that we can describe entities at one level A
(e.g. the properties of hydrogen and oxygen) in terms of their probable rules of behavior, and can
describe entities at a higher level B (e.g. the properties of water) in the same manner, we are left "only"
with the need to define or discover rules that describe how an entity becomes subject to a completely
new and different set of rules, i.e. how it becomes a distinct new entity.

As we cannot (by the definition of emergence) use any algebra or form of logical inference to determine
from one set of facts what another, related set might look like, we can only describe the conditions
under which one set of facts becomes another, different set of facts.   This requires the use of
conditional statements.  My work thus far indicates that there is a way to canonically represent rules at
a higher level of abstraction than the mere individual rule; I call this approach "Ultra‐Structure" (see
Long & Denning, 1994).

An operating rule in Ultra‐Structure has one or more factors (IF conditions) and one or more
considerations (THEN conditions), which may be thought of as attributes in a relational table.  The
factors determine whether the rule and its considerations will be inspected and possibly executed.
Once a rule has been selected for inspection, the considerations are used to determine in the context of
other rules whether that rule will be executed.  Abstractly, a rule with, say, three factors A, B, and C, and
three considerations X, Y, and Z is a conditional statement which is interpreted as follows:

if A and B and C then consider X, Y, Z.

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The factors of a rule are always ANDed together.  Logical OR is represented by multiple rules.  Each
record is unique.  Groups of such conditional rules that share the same factors and considerations are
called a ruleform.

A General Class of Emergence Rules

In modeling the U.S. Constitution using Ultra‐Structure, I've found (not surprisingly) a fairly precise set of
steps defining how a normal human becomes a U.S. Senator.4  The powers (properties) of Senators are
quite different than the qualitative properties of ordinary citizens.  For example, Senators cannot be
arrested during their attendance at a session of Congress, or while going to or from a session of
Congress, except for a treason, felony, or breach of the peace; nor can a Senator be appointed, during
the time for which he was elected, to any office which was created or had its pay increased during his
term of office; and ‐‐ most importantly ‐‐ their vote counts on matters before the Senate, unlike (say)
mine.  Likewise, the Constitution specifies the conditions under which a bill may become a law; and we
all know that laws have very different properties than bills!

With such intentional systems, as opposed to natural systems, we do not "predict" what happens if we
add A to B, nor do we explore what happens if we reverse the prescribed procedure for things: we
simply collectively, as a society, declare what happens.  And in cases of doubt, we have a mechanism ‐‐
the Supreme Court, in the Constitution ‐‐ that is empowered to declare clarifications of rules. .  Other
examples of intentional emergence  are marriage, corporations, and nations.    Such emergence‐by‐
declaration is obviously a phenomenon that can only occur in cases where institutional facts can be
generated by human intentionality.  But what about natural emergence?

We commonly represent changing states of affairs by modifying the attributes (predicates) possessed by
entities.  Thus if Tom is a person‐entity and Tom gains 5 pounds, it is still Tom in some fundamental
sense, but now his weight attribute is N+5 pounds.  If Tom grows a mustache, he is still Tom and we can
still represent him by modifying selected attributes.  Tom is still the same entity, and subject to the
exact same regulative rules as he always was.  But suppose Tom turns out to be "swampman", the
creature created spontaneously from stuff in a swamp?  He still looks and acts like a person (I say), but
now he may not be covered by the laws governing treatment of humans, for he is not in fact human.
How do we represent this state of affairs?  More generally, under what conditions (rules) can we say
that entity A ceases to exist and becomes entity B, subject to wholly new and different rules, and having
in some cases wholly new attributes?

In natural systems, this identity question arises when chemical compounds change their state.  We tend
to think of ice, water and steam as different states of the same entity; and yet each state acts very
differently, i.e. it follows different rules.  Thus, from an Ultra‐Structure point of view, liquid water,
steam, and ice must be defined and treated as three separate and distinct entities, because each is
subject to different classes of rules. The fact that they are inter‐translatable by the addition or
subtraction of heat energy is represented by "emergence rules", stating that ice plus heat makes water,

4
-- In modeling the Constitution using Ultra-Structure, I've concluded provisionally that the essence (deep structure)
of any regulative legal system is the probabilistic assignment of new legal statuses (new predicates or qualities) to
some legal entities by other legal entities.

Page 4 of 24



and water plus heat makes steam.  Thus we may model an entity (e.g. water) plus attributes (e.g. state);
but because the resultant acts so differently, we treat certain cases as distinct entities each subject to
their own rules.  It does not matter whether the new entity is apparent or real; only that it behaves
differently.

In a strictly descriptive sense, we can imagine a notation having a grammar whereby adding or
subtracting Entity X and Entity Y does not follow the classic (resultant) grammar of arithmetic. By
definition we cannot specify what any two (or more) entities will add up to; thus the properties that
emerge must be declared; they cannot be deduced.  And they must be stated contingently, as
conditional rules.  Furthermore, "adding" cannot be treated as an abstract operation, because in fact
what is being added matters: we must define more concrete operations such as "added to X", "added to
Y", "added to Z", etc.  Each such situation can then be treated as a predicate of an entity: i.e., a status.
Thus, in the general case,

If Entity X acquires status C, then it (becomes) (is treated henceforth as) New Entity Y.

Examples are:

 X = bill, C = approved by House, Senate, and President, Y = law
 X = ordinary citizen, C = wins election, Y = U.S. Senator
 X = U.S. Senator, C = loses election, Y = ordinary citizen
 X = hydrogen, C = added to oxygen, Y = water
 X = water, C = heat, Y = steam
 X = pawn, C = reaches 1st rank, Y = queen.

I call these "emergence rules", and the class of such rules I call an "emergence ruleform", for by
effectively redefining (renaming) the type of entity one is dealing with in an Ultra‐Structure model, that
entity becomes immediately subject to completely new and different rules. This and like facts cannot be
expressed in mathematics, which is the notation of resultants; it can only be expressed by a contingent‐
rule‐expressing notation such as Ultra‐Structure.

In the most general case, for every permutation of statuses that an entity may have, it may be subject to
new and different classes of rules.  Therefore if its status changes for any reason (including the mere
passage of time), it (or the new entities it becomes) may behave in unexpected ways.  Perhaps it is for
this reason that the general semanticists like to index words: so we don't always presume that an entity
at t=1 is the same entity at t=2.  It does not matter whether the changes are the result of human
intention or not: the application of a rule that declares the existence of new entities whenever existing
entities achieve a defined status permits us to replicate, in a computer model, the characteristics that
we see in real‐world emergent transactions: they are novel; they are sudden; and they are not
predictable by an understanding of their causes.

Conclusion

We may speculate, like Sellars, that emergence occurs because "at specific degrees of complexity of
organization, new  properties are formed in order to establish a fresh and simpler point of departure"

Page 5 of 24



(Blitz, page 180). Although we may not know why any given emergent transaction occurs as it does,
"merely" documenting its behavior and rules is still real science, not a yielding to irrationalism5.

If at the end we are left to look in wonder at a world where we expect one plus one to equal two, much
as we once presumed parallel lines could never intersect, we need not only "Consider and bow the
head", as Morgan suggested: we may and indeed must respond with creative new technologies of
representation.  And then consider and bow the head.

References

Blitz, D., Emergent Evolution: Qualitative Novelty and the Levels of Reality.  Boston: Kluwer Academic
Publishers, 1992.

Gellert, W., Kustner, H., Hellwich, M., and Kastner, H., The VNR Concise Encyclopedia of Mathematics.
New York: Van Nostrand Reinhold Company, 1975.

Lewes, G.H., Problems of Life and Mind.  London, 1874.

Long, J., and Denning, D., "Ultra‐Structure: A Design Theory for Complex Systems and Processes", in
Communications of the ACM (in press)

Miller, D.L., Emergent Evolution and the Scientific Method.  Doctoral dissertation, University of Chicago
Department of Philosophy, 1932.

Morgan, C.L., Emergent Evolution: The Gifford Lectures.  London: Williams and Norgate, 1927.

Russell, B., The Analysis of Matter.  London: Allen and Unwin, 1927. Quoted in Blitz, D., Emergent
Evolution: Qualitative Novelty and the Levels of Reality.  Boston: Kluwer Academic Publishers, 1992.


5
-- See D. L. Miller's dissertation

Page 6 of 24



Emergence

Nonlinearity

Anomalies of Complex Systems

Page 7 of 24



Old Goal: Understand the WHY of emergence & nonlinearity

New Goal: Accept brute facts. Develop the ability to accurately represent (model)
all kinds of addition and subtraction, including transactions that demonstrate
apparent emergence and nonlinearity

Why is Not the Question

Page 8 of 24



A B
Attributes Attributes
or Quantity or Quantity

Summands

C
Attributes
or Quantity
Sum

Latin addere, to add

derived from ad = to or towards

and dere = to put

 to put towards

Contrast with abdere, to put away

Page 9 of 24



Emergence:

When an entity is suddenly subject

to a new and different class of rules,

as observed qualitatively or quantitatively.

Operational Definition Includes Nonlinearity & Emergence

Page 10 of 24



value color
width shape

height purpose
mass age

velocity strength
acceleration substance

(et cetera)

Entity Attributes

Distinction Arises from the Aristotelian Legacy

Page 11 of 24



Entities Statuses

Entity
Status
Emergence
Rules

Status
Assignment
Rules

Deep Structure of Addition & Subtraction

Page 12 of 24



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Page 17 of 24



Both quantities of entities and properties of entities are additive and subtractive
simpliciter:

 commutative: a+b = b+a

 associative: (a+b)+c = a + (b+c)

 monotonic: if a < b, then (a+x) < (b+x)

 reversible: if c = a+b, then c‐a = b

Resultant Transactions Have Certain Assumptions

Page 18 of 24



Unconditional Conditional

Dimensionless math

Dimensional
physics

Dimensional-Plus chemistry,
accounting

Science has Known that
Not All Things can be Meaningfully Added Together

Page 19 of 24



Unconditional Conditional

continuous
Low Interaction functions

Medium Interaction chaos

High Interaction statistics

And Science is Learning How to Represent Complex Interactions...
Adapted from Sally Goerner, Chaos and the Evolving Ecological Universe (1994)

Page 20 of 24



Quantities of entities and properties of entities are not necessarily additive and
subtractive simpliciter; most transactions in the real world are:

 non‐commutative: a+b  b+a

 non‐associative: (a+b)+c  a + (b+c)

 non‐monotonic: if a < b, then (a+x) >, =, or < (b+x)

 non‐reversible: if c = a+b, then c‐a  b

1+1=2

1 + 1 = ???

Which are a Special (Limited) Case of the Assumptions for Resultant Transactions

Page 21 of 24



Factors Considerations

F1 F2 F3 F4 C1 C2 C3 C4 C5 C6
R1 A B C D U V W X Y Z

R2 E F G H I J K L M N

R3
Rules
R4

R5

R6

Universals

But we Still Need to Show Conditions (IFs) and Other Kinds of Considerations
(THENs) (Besides Units of Measure)

Page 22 of 24



Summary

We currently create "Numbers‐Plus" by:

 adding dimensions to represent qualities

 adding extra operational procedures to represent special handling rules

But we must also:

 explicitly add environmental conditions ("factors") onto the operations

 add other "considerations" besides UM that affect how a rule is to be executed.

This will permit us to better represent known facts about:

 emergent behavior (i.e. properties of sum not predictable from properties of
summands)

 nonlinear behavior (i.e. quantity of sum not predictable from quantities of
summands; output not commensurate with input)

Page 23 of 24



1+1=2

Proceed With Caution!

Page 24 of 24

Representing emergence with rules

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