Anja-Karina Pahl, Olga Bogatyreva, INTERSECTED, NESTED AND CHAINED
CONTRADICTIONS IN NATURE AND TECHNLOLGY. Proceedings of TRIZCON2003,
Altshuller Institute, Philadelphia, USA , 16-18 March 2003-
NESTED, CHAINED AND INTERSECTING
or ‘COMPLEX’ CONTRADICTIONS
Anja-Karina Pahl and Olga Bogatyreva
Centre for Biomimetics and Natural Technologies,
The University of Bath, BA1 2EU, UK
Tel +44 (0) 1225 826 956 Email ensakp@bath.ac.uk
Abstract
The mathematical structure and representation of chains, nesting and intersection is described in relation to
contradiction in TRIZ. Each of these types of contradiction is related to the others, in 3D, in a conic representation.
Complex contradictions are introduced as those involving two sets of distinctly different original contradictions or
cones. Study of ant behaviour provides some insight into solving complex contradiction –in particular,
contradictions need not be solved in the same sphere in which the problem is caused. Penultimately, the solution to
complex contradiction exists in a concept space, unoccupied by previous solutions. And by involving other spheres
of information to solve complex contradiction, all contingent and relatively simple contradictions should also
automatically be solved.
1.0 INTRODUCTION
In complex systems, it is quite common that contradictions, which exist at one
level of operation, or in one sphere of activity, can also contradict each other at
another level. Then we have a complex situation of ‘contradiction between
contradictions’. What is more, our engineering problem may require us to find
solutions to all of them.
Ideally, all relevant, potentially multiple contradictions should be resolved
simultaneously. In other words, rather than identifying a separate but apparently ‘real’
space-time for each situation, we identify that they occupy the same conceptual spacetime and solve all ‘paired’ contradictions together.
There are several types of complex contradictions. While ‘chained contradictions’
have been introduced by Mann (2000), and the concept of nested solutions has been
introduced by Pahl (2002), TRIZ has as yet no directions for solving these or any
other type of intersecting contradiction.
In this paper therefore, we will review the mathematical structure and
representation of chains, nesting and intersection, and how this is relevant to
describing problems and solutions in TRIZ. Then we will investigate how complex
contradictions have been solved in a natural situation, greatly aiding our process of
doing the same in TRIZ.
1.1 Line of Contradiction
Figure 1 The simplest description of contradiction
is a line that does not involve the resolution.
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Anja-Karina Pahl, Olga Bogatyreva, INTERSECTED, NESTED AND CHAINED
CONTRADICTIONS IN NATURE AND TECHNLOLGY. Proceedings of TRIZCON2003,
Altshuller Institute, Philadelphia, USA , 16-18 March 2003-
To describe the simplest case of a contradiction, we might say that it is graphically
represented by a line or curve with two end points –which symbolize the extreme
targets, goals or conflicting parameters that must be integrated (Fig.1). In this
representation, the line of contradiction is independent of the resolution.
1.2 Chained Contradiction
The next simplest representation, as discussed by Mann (2000), is a ‘chain’ (Fig.
2), where each successive solution moves closer to the optimum.
Said chain evolves since solutions to technical contradictions are not always
discrete and do not always immediately eliminate the entire contradiction –instead
they often merely improve a situation involving conflict. Then, because we have not
achieved the ideal solution, but only a partial one, we must continue to either solve
new problems for the old constraints (Mann, ibid), or, as we argue in this paper,
discover yet more contradictions with different constraints.
In Mann’s (ibid) chain, the contradictions are not obviously related to each other
(cf. Figure 1.) –they are only related to the solution. Thus the chain is, in fact, not a
chain of continuous contradiction, but a scenario of continual resolution.
Further, contrary to Mann’s (ibid) assertion, a paradigm shift can never really
happen, because the entire journey takes place for the same initial set of conflicting
parameters.
In order to initiate a true paradigm shift, we need to involve a second, third or nth
chain. This means we discover, or are presented with, new sets of contradictions or
entirely new end-member parameters, which do not fit into the existing system. When
this additional information or new criteria is made relevant to the original complexity,
it also becomes possible to achieve an overview of our situation. This then, is where it
becomes possible to initiate a shift in our thinking (Fig.3). This concept is extended in
the next section.
Figure 2 Contradiction chain or resolution
chain? Each resolution leads to discovery of
a new but somehow related contradiction.
This is a simple bifurcating or binary choice
path! In step-by-step movement, there can
be no paradigm shift, or movement ‘out of
the box’, since all concepts are easily related
to each other.
Figure 3 For the penultimate solution, at the top of
the diagram, the resolution is achieved by bringing
together at least two sets of contradictions which
are initially separate from the other, or related only
distantly. Here it is possible to initiate a paradigm
shift in thinking, since the separate chains signify
different domains or spheres of activity, with
additional information. As new information is
brought to the original system, a super-system of
relationship must ensue.
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Anja-Karina Pahl, Olga Bogatyreva, INTERSECTED, NESTED AND CHAINED
CONTRADICTIONS IN NATURE AND TECHNLOLGY. Proceedings of TRIZCON2003,
Altshuller Institute, Philadelphia, USA , 16-18 March 2003-
1.3 Nested Contradiction
The next simplest representation of contradiction is as a triangle.
As Pahl (2002) pointed out, conflicting or contradictory parameters for a given
problem are not necessarily anchored or anchorable, but can exist anywhere along a
line of contradiction. In the same way, the actual solution is not originally, or usually,
a fixed point or target, but can occur anywhere along a sliding scale that is also
represented by a curve or line –this time at right angles to the first.
A triangle will be created by linking all extremes –that is, linking the (line of)
conflicting parameters with the (chain to ideal) Final Result (Fig. 4).
Of course, potentially infinite triangles may be inscribed within this ideal triangle,
as a result of having one, two or three sides shorter than the ideal.
When these have regular sides and occur in a sequence, as the contradiction set
and/or solution is systematically varied, they will appear to the eye to be ‘nested’.
There are three types of regular nesting explained in figure 4, the most obvious of
which is in the manner of a matriushka doll.
Strictly speaking, irregular triangles created within the meta-triangle, are also
nested. These will be overlapping or intersecting, depending on the degree to which
they are anchored on (or share) a contradiction pair or solution (fig.5).
A
C
B
Figure 4 Several types of regular nested solution
exist. In each of these three types, different solutions
and contradiction sets create theoretical subsystems, systems and supersystems. The horizontal
axis is a sliding scale for degree of contradiction of
the situation. Vertical axis is a sliding scale for
solution, with ideality at the apex. Type A is a
Matriushka-style nesting of contradiction and
solution. Solutions are unique for a given
contradiction set. Each system and subsystem also
has a unique contradiction set. Type B is
contradiction-anchored nesting, where solutions
differ for a given contradiction set (after Pahl, 2002).
All systems and subsystems share a contradiction
set. Type C is solution-anchored nesting, where the
solution is not unique. Different types of
contradiction can lead to an identical solution.
Systems and subsystems do not, therefore, share a
contradiction set.
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Anja-Karina Pahl, Olga Bogatyreva, INTERSECTED, NESTED AND CHAINED
CONTRADICTIONS IN NATURE AND TECHNLOLGY. Proceedings of TRIZCON2003,
Altshuller Institute, Philadelphia, USA , 16-18 March 2003-
This inclusive view of a complex situation actually extends the concept of chained
contradiction presented above, in that all chains or bifurcated lines within a metatriangle must eventually be related (fig.6).
Actually, all three methods of representing contradiction, which are discussed in
this paper, are related. For, as Pahl (2002a,b) has also shown, triangles are a 2D
representation of the path of conical spirals or volume of cones, whose mouth is a
circle or ellipse. Thus we can also imagine contradiction not in exact lines or
triangles, but as circles enclosing more diffuse sets of potential solutions (fig.7).
Intersecting contradictions are for this reason discussed following.
A
C
B
Figure 5 Several types of irregular nested
contradiction exist. Solutions and contradiction
sets create systems etc., as described above. The
horizontal and vertical axes are sliding scales for
contradiction and solution, respectively, as
described above. In Type A, solutions are unique
for a given contradiction set. Each system and
subsystem also has a unique contradiction set.
Type B is contradiction-anchored nesting, where
solutions differ for a given contradiction set. All
systems and subsystems share a contradiction
set. Type C is solution-anchored nesting, where
the solution is not unique. Different types of
contradiction can lead to an identical solution.
Systems and subsystems do not, therefore, share
a contradiction set.
1.4 Intersected Contradictions
As we know, 1D lines (which create chains) are simpler mathematical constructs
than 2D areas (for example, triangles or circles), which are themselves simpler than
3D volumes (created by conical or vortical spirals). Thus the least simple
representation of TRIZ contradiction involves circular sets and 3D cones.
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Anja-Karina Pahl, Olga Bogatyreva, INTERSECTED, NESTED AND CHAINED
CONTRADICTIONS IN NATURE AND TECHNLOLGY. Proceedings of TRIZCON2003,
Altshuller Institute, Philadelphia, USA , 16-18 March 2003-
Mathematically, we can calculate the precise point where an intersection of
two curves or single lines occurs, by using the ‘simultaneous equation’ method of
equating equations. We can do a similar exercise to find the overlap of two areas
(fig.8).
Figure 6 Chains are related in a metatriangle. Also, when individual solutioncontradiction relationships are
Intersecting
Contradiction
retrospectively
extrapolated, the cause
and effect relationship of chained
contradictions becomes nested. For
example, the cause of a 2nd degree
contradiction (see definition in body
text), which appears subsequent to,
and contingent on, its prior 1st degree
contradiction, may extend the original
field or sphere in which we should look
for contradiction.
A
Figure 7 Triangles and circles are
related in a cone.
Figure 8 The intersection of two sets
occurs at two distinct points.
Figure 9 The 3D situation of
complex contradiction involves
relating distinctly different original
criteria or contradicting pairs. This
diagram shows the relationship of
such chains, nests and intersected
sets, resulting in a ‘cones-in-a-cone’
situation. Note that the slope of the
wall of the ideal cone is the
minimum, which will enclose the
two complexly conflicting sets. The
slope of the wall of the cone for any
intermediate solution must be such
as to fall within the ideal (which is
not the case in this diagram!).
Otherwise, it will suggest
that the original contradictory sets
must be expanded. Note that this
‘cones-in-a-cone ‘situation is
reminiscent of that proposed,
but not explained, by Mann et al.
(2002).
The advantage of looking at contradictions in this way, is that we see that we can
combine two or more completely different spheres of activity (the individual metatriangles or chains of previous sections), each of which have their own internal
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Anja-Karina Pahl, Olga Bogatyreva, INTERSECTED, NESTED AND CHAINED
CONTRADICTIONS IN NATURE AND TECHNLOLGY. Proceedings of TRIZCON2003,
Altshuller Institute, Philadelphia, USA , 16-18 March 2003-
constraints and contradictions. Thus the real 3D picture of complex contradiction is
described in Figure 9.
SOLVING INTERSECTED CONTRADICTION
Now we will look at how nature solves intersected contradictions.
2.1 Advice from Ants
The ideal result for any society is ‘cooperation’, whereby the society acts as a
unified organism and survives in equilibrium with the environment. However, at the
same time as satisfying society, there are conflicting demands on individuals. Each
member wants also to expend as little energy as possible. In other words, the colony
demands working elements, but nobody wants to work –a very typical situation for
any society!
For the ant society, we can cite two main pairs of contradictions, which must be
solved together: specialization-generalization and cooperation-competition. We can
formally state that the main contradiction is to ‘minimise personal use of energy and
maximise reliability of the colony’. Because the pairs of contradictions involve
different spheres of activity, the simultaneous solving of both (ie. the pairing of pairs)
is a complex contradiction, not a simple one. Graphically, we have two intersecting
cones or circles, with different constraints (fig.10).
In Table 1., all possible solutions of all simple and complex contradictions for this
situation are presented (Bogatyreva, 2002). The type of response, which we might call
the ‘solution’ or ‘management’ of ants, is divided into four spheres of activity, which
for convenience we call Activation, Controlling, Coordination, Governing. These are
different levels of creating cooperation, which are brought into play as the colony
evolves.
Figure 10 Complex contradiction in an ant nest involves
two sets or pairs of criteria which must be
simultaneously resolved. As shown in Figure 8., there
are at least two distinct solutions, mathematically
defined by the points of intersection, or overlap.
In a young colony, or when an individual is young, each ant can theoretically
perform any task, though it is (somehow) activated to choose and do only one task at a
given time. At such an early stage, expertise is so low and competition between
individuals so intense, that there are no stable interactions. However, the more active
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Anja-Karina Pahl, Olga Bogatyreva, INTERSECTED, NESTED AND CHAINED
CONTRADICTIONS IN NATURE AND TECHNLOLGY. Proceedings of TRIZCON2003,
Altshuller Institute, Philadelphia, USA , 16-18 March 2003-
ants guide others in a task. They are the activators. Once an ant has chosen its task,
this determines which group it can join –ants band together to perform a similar or
identical function, and can help each other, or cooperate in such teams. Because the
level of function is still quite simple within a team, this requires little effort in
coordination between individuals –it is fairly automatic. Yet at the same time, there
may be an ant who has control over ‘membership’ of a team, selecting only fit and
able individuals. Next, as the number of ants in a colony increases, the number of
functions that can be performed increases –which means the number of teams that
form to carry out these functions increases. This now requires coordination of action
between teams, a higher level of complexity – thus one ant will take on the task of
making sure the team can cooperate with others in every necessary way. Last, there is
coordination of all the teams in the nest, carried out by an individual who has in its
lifetime performed many functions, and can now sit ‘outside the nest’ and regulate it
or govern.
Table 1: Types of “management” employed by ants to solve simple and complex
contradictions in four spheres of activity
Contradictions
Complex
Simple
Space
(fixed
material
boundaries)
1a Specialization
1b Generalization
2a Cooperation
2b Competition
Any combination of IV
1 and 2
Sphere of activity
Time
Structure/
(fixed seasonal Function
and diurnal
(dynamic
constraints)
individual
response)
Interaction
(dynamic
communal
response)
III
II
I
Bogatyreva’s (2002) paper indicates that contradictions such as ‘cooperationcompetition’ can be solved simply by controlling (II), and in the complex sense by
activation (I), both of which call forth specialised behaviour –meaning that someone
who likes doing something will attract and activate friends for this particular job.
Likewise, the contradiction ‘specialization-generalization’ can be solved simply
by coordination (III) – inducing a strong behavioural hierarchy. Competition between
individuals is hereby effectively ‘frozen’ in the interaction structure –there is just no
point in fighting every time, if all leaders are recognisable. Governing (IV) is a way of
solving a complex contradiction between individual and social spheres, by locating a
central government for outdoor work in a nest.
There are additional considerations, which determine how a contradiction is
solved. For example, biologically, colony “management” depends both on size of
population and complexity of society. Ants with a small population demonstrate
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Anja-Karina Pahl, Olga Bogatyreva, INTERSECTED, NESTED AND CHAINED
CONTRADICTIONS IN NATURE AND TECHNLOLGY. Proceedings of TRIZCON2003,
Altshuller Institute, Philadelphia, USA , 16-18 March 2003-
Activation (I) as the way of solving complex contradiction. Ants with a complex
social structure solve the same complex contradiction by Governing (IV).
2.2 Degree of contradiction
We know from earlier in this paper, that we can geometrically relate cones and
circles with triangles and lines. We thus also know we can eventually relate all chains,
nests and sets of contradiction.
That means there is no absolute measure of the degree of contradiction.
Essentially, whatever the starting point is, where our initial contradictions are
defined, must be our first level or degree of contradiction. Subsequent contradictions
identified at the next stage of problem-solving will be second degree or level 2,
thereafter third degree, and so on. Each is countable only in a chain as consequent, or
contingent on, an intial or prior pair of contradictions and within the same overall set.
To some extent it will be self-evident that the degree or level of complexity of
contradiction is equivalent to the degree to which a given structure (ie. set of
contradictions) is different from a second or third set, since this will require more
connections to be made between chains of information or contradiction, before sets
can be adequately related. However, it is unlikely that this will be obvious at the
starting point.
In other words, we define a complex contradiction as having constraints in two
different sets of spheres of action, which are not contingent on each other, at the level
of our observation.
In any case, at each stage of solving a problem, we must ask these sorts of
questions: How is one set of contradictions related to another set of contradictions?
Do we have (two or more) completely different cones/triangles/ circles/end-member
criteria, or not? In other words, are sets of contradictions truly independent at each
new starting point, or are they contingent on each other?
As corollary, from Bogatyreva’s (2002) research, we can add that a simple
contradiction will be solved close to the original sphere of influence. Thus, a simple
contradiction in relationship is solved in the sphere of the dynamic response of the
individual (Structure/Function). A simple contradiction in individual function can be
solved purely by dynamic community response (Interactions sphere).
On the other hand, solutions of complex contradictions lie outside the original
areas of influence. For ants these solutions lie in what Bogatyreva (2002) calls the
spheres of ‘Space’ and ‘Time’. These are domains of action, which may originally
have formed in response to demands of the society –for example, the spatial limit of
the nest is a function of the size of the colony. Similarly, the division of a workforce
into simultaneously labouring teams (creating ‘work divided in time’), also depends
on the size of the colony. However, the constraints these spheres later impose on both
the individual and the community become anchored.
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Anja-Karina Pahl, Olga Bogatyreva, INTERSECTED, NESTED AND CHAINED
CONTRADICTIONS IN NATURE AND TECHNLOLGY. Proceedings of TRIZCON2003,
Altshuller Institute, Philadelphia, USA , 16-18 March 2003-
3.0 DISCUSSION
Now we will discuss whether intersected contradictions are a peculiarity of
complex living systems, or whether we can find them in thinking techniques too.
Let’s have a look at typical TRIZ task, described in many Altshuller books. The
task is about hot liquid slag in a bucket, which needs to be transported from the blast
furnace to the place where it will be used.
While transporting it, a hard crust appears on the slag surface. To remove the slag
from the bucket, it will later be necessary to break the crust with a special device. But
the crust is thick and the shape of bucket is conical, which means that even when the
liquid is poured out, lots of hard slag will be left rimming the bucket. An additional
procedure for breaking and removing the hard slag is required. The problem is how to
avoid loss of slag at the same time as minimizing additional procedures.
We test the solutions made available by understanding ant colony organization
above.
Basically, there are two functions to be improved – time of slag transportation and
slag insulation. These create two simple contradictions – a time contradiction
(transport / do not transport) and a structure contradiction (insulate / do not insulate).
The simplest decisions, and the most desirable ones, are to transport and let slag
harden (do not insulate) and they form a complex contradiction with the goal (ideal
result) of the system, which is to empty the bucket easily and without loss of slag.
We may wish to introduce a metal bucket lid, as a new structure in this system.
This will help stop spillage during transport. But will it help or hinder the slag
hardening?
Table 2: Solution table for applying complex contradiction
in ant behaviour to a problem with slag
Decisions
Contradiction
Space
Time
Structure/
Interaction
Function
SIMPLE
1. Transport
Convert quickly
2. Do not transport
Convert
after and move slag
converting devices
transporting
to blast furnace
SIMPLE
1.To
delay
Harden (slag)
hardening, lid
should exist.
2. To let slag
Do not harden
harden,
lid
should
not
exist
COMPLEX
Add ice as
Transport
new structure
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Anja-Karina Pahl, Olga Bogatyreva, INTERSECTED, NESTED AND CHAINED
CONTRADICTIONS IN NATURE AND TECHNLOLGY. Proceedings of TRIZCON2003,
Altshuller Institute, Philadelphia, USA , 16-18 March 2003-
Do not insulate (let
slag harden)
in system, to
make
an
insulating
foam
crust,
that is easily
removed
If we consider the solutions from ant organization, proposed in this paper, a
simple time contradiction should be solved in space while a simple structure/function
contradiction is best solved in time. We should also try to make decisions or find
solutions in the structure/function and interactions spheres. The best decision
regarding the complex contradiction ‘transport /do not insulate’ is in
Structure/Function sphere.
Thus the ant’s complex contradiction rule works in our technical system too. As
the system described is very simple, there is no second way of solving the complex
contradiction.
4.0 CONCLUSION
In this paper we have proposed that, when solving simple and complex problems,
different types of contradictions will become evident. Depending on the degree of
complexity of problem, chained, nested or intersected contradictions can be identified.
Ultimately, these are related to each other –geometrically the situation is represented
in 3D cones.
This situation has previously not been discussed in TRIZ theory.
We have also shown that ant behaviour can give us some useful insight into
solving contradictions –in particular, that contradictions need not be solved in the
same sphere in which the problem is caused. Instead, we can look for resolution
within another sphere of action. Thus, a contradiction in individual function can be
solved in other spheres (space, time or group interaction) but not in its own field.
Similarly, a space contradiction may be more easily solved in time, and so on.
We conclude that:
(i)
the resolution of any complex contradiction (that is between pairs of
contradictions originating in different spheres of activity), necessarily
involves chains or nests of other, previously solved contradictions –
whether we can see the links or not
(ii)
the penultimate solution exists in a concept space, which previous
solutions do not occupy
(iii)
by solving any complex contradiction via resolution within other spheres,
a resolution of any contingent and relatively simple contradictions will
also automatically be achieved.
References
Mann, D. (2000) Contradiction Chains, TRIZ-Journal, (online), August
Pahl, A-K. (2002a) Contradiction and analogy as the basis for inventive thinking,
TRIZ-Journal (online), August.
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Anja-Karina Pahl, Olga Bogatyreva, INTERSECTED, NESTED AND CHAINED
CONTRADICTIONS IN NATURE AND TECHNLOLGY. Proceedings of TRIZCON2003,
Altshuller Institute, Philadelphia, USA , 16-18 March 2003-
Pahl, A-K. (2002b) What S-curves Really Are… ETRIA Conference Proceedings,
Strasbourg, Nov 6-9.
Bogatyreva, O. (2002) Management in ant colonies: Natural analogue of the
resolution of social contradictions using TRIZ tools, ETRIA Conference Proceedings,
Strasbourg, Nov 6-9.
Mann, D., and De Wulff, S (2002) Evolutionary Potential in Technical and Business
Systems –The Next Stage. ETRIA Conference Proceedings, Strasbourg, Nov 6-9.
Biographies
Anja-Karina Pahl BSc MSc
Anja-Karina Pahl is Research Officer in the Department of Mechanical Engineering at the University
of Bath. Previously, she was a Lecturer in Engineering Geology at the University of Western Australia
and worked internationally with the Mining and Petroleum Industries. She holds a Masters degree (for
work on the structural evolution of macro- and micro- extensional fault systems) from Monash
University, Melbourne and continues to be interested in methods for 3D structural modelling in the
Earth Sciences. In 2000, she received formal training in TRIZ from Dr. Iouri Belski (RMIT) while
developing short-courses on inventive and technical ‘thinking tools’, associated with the Swinburne
School of Management MBA and Executive Development Programme in Melbourne. Her ongoing
research interest is multidisciplinary, in the context of ‘Creativity as an Exact Science’.
Olga Bogatyreva DSc
Olga Bogatyreva is Research Officer in the Department of Mechanical Engineering at the University of
Bath. She holds a doctorate in Biology, dedicated to Ecology and Ethology of ants, and another
doctorate in Philosophy, on General Systems Theory. Before coming to Bath University, Olga was
leading scientific researcher at the Institute of Philosophy and Law, Siberian Branch of Russian
Academy of Sciences, Novosibirsk, and Professor in the Siberian Institute of International Affairs
(lecturing on General Systems Theory and TRIZ). Her research interests are in the field of animal
behaviour science, especially in evolution of societies. She has published 41 articles and 2 books
covering bumblebee and ant ecology, ethology and biology –foraging activity and communal
behaviour, human sociology and ethology and general system theory.
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