Towards a Conceptual System Theory
Karl Erich Wolff
Department of Mathematics, University of Applied Sciences
D-64295 Darmstadt, Schoefferstr. 3, Germany
What is a good general definition of a system?
Lin (11],1999) states in his book 'General Systems
Theory: A Mathematical Approach', p. 347:
'There might not exist an ideal definition for general
systems, upon which a general systems theory could be
developed so that this theory would serve as the
theoretical foundation for all approaches of systems
analysis, developed in various disciplines.'
ABSTRACT
Fundamental problems in Mathematical System Theory
concerning the notion of 'state' and 'time' are solved through
the development of a 'Conceptual System Theory' based on
Formal Concept Analysis introduced by Wille [21]. The
basic tool in Formal Concept Analysis, namely the concept
of 'concept', is used to define 'states' and 'phases' in
'conceptual time systems'. The notion of a 'conceptual time
system' generalizes many classical time systems mentioned
in Kalman, Falb, Arbib [9]; Klir [10]; Mesarovic, Takahara
[12]; Pichler [15]; Lin [11].
An important new idea is the formal description of two
aspects of 'time': the first aspect is formalized by 'time
granules' like 'morning' interpreted as 'time objects' used for
the performance of measurements, and the second aspect is
described by 'conceptual time scales' used for the conceptual
representation of 'time theories'. That allows for choosing
discrete, continuous, or multidimensional times.
Another new idea is the combination of the table of
measurements of a 'real system' with a conceptual
granularity tool. This combination is described by 'scaled
many-valued contexts'. The concept lattice of the 'event part'
of a conceptual time system is the 'general state space', its
object concepts are the 'states'.
For a short demonstration of this method an application
to a quality control problem in the chemical industry is
described.
Keywords: Systems, States,
Analysis, Concept Lattices
Time,
Formal
What is a good description of time?
A third problem in System Theory is the notion of
'time'. It is clear that the usual idea of a 'continuous' time,
its description by real numbers, mainly by its linear order
(R, ), leads directly to the problem of the pragmatic
meaning of real numbers as 'time points'. A statement
like 'My son Florian stopped growing at his 18 th birthday
at noon' sounds a little bit crazy. The introduction of a
suitable time interval yields the meaningful statement
'My son Florian stopped growing during his 18th year.' If
we wish to work with time intervals instead of time
points we have another well-known problem: it is
impossible to partition a closed interval into two closed
intervals - which conflicts with our experience of cutting
a usual string into two parts.
Time intervals as 'discrete time points' led to the
general notion of arbitrary time chains (T, ) where a
chain is an ordered set in which any two elements are
comparable. Many authors write that time chains should
not be generalized to arbitrary ordered sets, but we
usually work with several time chains if we look at our
ticket for a flight through different time zones. And if we
say something about our holidays we usually do not
know and often do not need the relation among the home
and the holiday time zones. Clearly, also, the inclusion
relation among time intervals is not a chain.
Therefore a general description of time is needed in
System Theory. It should contain all the classical time
descriptions as special cases. And it should describe the
usual containment relation among 'time granules' like
'morning' and 'early morning'.
This time description should be related to the
description of 'states', in the sense that a system should
be during each 'time granule' in exactly one 'state'.
Concept
1. PROBLEMS IN SYSTEM THEORY
What is a state?
The purpose of this paper is to continue a recent
development in System Theory (Wolff [27]) which started
from the problem that the current concept of 'state' in
Mathematical System Theory could not be described in a
satisfactory way; Zadeh ([31], p.40) wrote in his paper 'The
Concept of State in System Theory':
To define the notion of state in a way which would make
it applicable to all systems is a difficult, perhaps impossible,
task. In this chapter, our modest objective is to sketch an
approach that seems to be more natural as well as more
general than those employed heretofore, but still falls short
of complete generality.
It is clear, that such a general definition of 'state' needs a
general definition of a system.
Space
Another problem is the formal representation of
'space'. We are all very familiar with the usual
description of 'space' by the 3-dimensional euclidean
affine space R3, but we also use 'space granules' like
'room' and 'cave'. 'Space granules' are mainly treated
nominally, if we use only their names, and sometimes
they are described by measurements. The abstraction of a
1
defined sense) by k-ary relations where k  3 (Peirce
[14], Burch [3,4]), then the binary and ternary relations
can be used as the 'building blocks' of a general system
theory. Since the work on the proof of the Peircean
Reduction Thesis seems to be not finished as yet, it is a
good strategy to focus on binary and ternary relations for
a general description of systems.
A future general mathematical system description
should use not only n-ary relations, but also relational
logic in the sense of Peirce [14], Schröder [17], Tarski
[19] and Burch [3,4]. Since relational logic is not yet
sufficiently developed, one should start with a careful
investigation of binary and ternary relations. They are
studied in many theories for quite different purposes.
To relate the concepts of 'state' and 'phase' in system
theory with the concept of 'concept' in the following I
use Formal Concept Analysis (FCA) developed in the
Research Group Concept Analysis at the Technical
University Darmstadt (Wille [21]; Ganter, Wille, Wolff
[8]; Ganter, Wille [7]).
In section 3 following the conceptual theory of
binary relations is described, where formal contexts,
formal concepts and concept lattices are introduced; in
section 4 the conceptual theory of ternary relations is
described using many-valued contexts, conceptual
scales, and scaled many-valued contexts.
very small 'space granule' to a point in R3 leads to similar
problems to those mentioned for time points.
Space-Time
If we observe objects in space and time we should be
aware of Zenon's famous paradox of the 'flying arrow
standing in each time point'. Its modern version is
Heisenberg's
uncertainty principle describing the
impossibility of a simultaneous 'exact' measurement of the
space and velocity coordinates of an object. Clearly a highspeed-film of a flying arrow will have on each photo a very
sharp picture of the arrow such that we are not able to
estimate the velocity of the arrow using such photos.
Therefore, it is our imagination of the infinite, which
leads us to postulate the possibility of measuring the velocity
of an object in a time point; and the mathematical
construction of the derivation as a limit is clearly done in the
formal theory of the real numbers. Is there any meaningful
relation to a corresponding action in practice? If we accept a
film as a 'virtual reality' each photo in the film would
represent a time point of a discrete time in contrast to our
continuous imagination watching the film. Wherefrom do we
know that 'the real time' is continuous? I agree that the
classical continuous description of time is a very useful and
simple model, but by no means the only meaningful one. It is
just a special aspect of the general description of time in
conceptual time systems.
It is not possible to discuss here further problems of
space and time in contemporary physical theory.
3. FORMAL CONCEPT ANALYSIS
The reason for the introduction of Formal Concept
Analysis (FCA) was to relate the mathematically
oriented theory of lattices and orders to practical
problems. The German industrial norms (DIN) contain
two chapters (DIN 2330, [5]; DIN 2331, [6]) on concepts
and their use in industry. In 1979 Wille recognized that
this description could be formalized by the introduction
of 'formal concepts' of a given data table, which consists
of a set G of object, a set M of attributes and a binary
relation I  G  M. Then the triple K = ( G, M, I ) is
called a formal context, representing just a set of
statements of the form 'object g has attribute m', written
'g I m'.
The basic definition of a 'formal concept' of K is
based on two well-known operations: For any subset X
 G we are interested in the set X of all common
attributes of X, defined formally by X  := {m  M |
gX g I m } and dually for any Y  M we are
interested in the set Y of all common objects of Y,
defined formally by
Y := {g  G  mY g I m }. A formal concept of
a formal context K is a pair (A,B) where A  G, B  M
and A = B and B = A.
A is called the extent, B the intent of (A,B).
This definition of a formal concept (of a given formal
context) has its roots in the description of concepts in the
'Logique de Port Royal' by Arnold and Nicole [1] where
a clear differentiation between intention and extention is
made:
'Or dans ces idées universelles il y a deux choses
qu'il est très important de bien distinguer, la
2. OBJECTS AND SYSTEMS
To master the huge number of sensual impressions, we
have to select, abstract, and combine these impressions to
form 'Gestalten'. Time granules like 'morning' and space
granules like 'room' can be understood as 'objects' in the
sense of basic units used to describe more complicated
structures, called 'systems'. Clearly, the objects can be
understood again as 'systems', if we intend to decompose
them into 'sub-systems'. Further decompositions into 'subsub-systems' can be thought to infinity, but of course they
are pragmatically restricted to finitely many steps.
All general descriptions of systems should have the
possibility of representing arbitrary relations among the
objects. Therefore, the standard definition of a general
system is a relational structure (G, F) where G is a set and F
is a family of relations of arbitrary arity on G. For several
purposes variations of that general definition have been
developed. For example, for the purpose of describing
processes 'time systems' have been defined where the set T
of 'time points' is playing the role of the set of 'objects', and
'events' are defined as functions on T. This approach is
generalized in the following.
How do we work with relations of arbitrary arity? In
practice, we mainly use relations of an arity k  3, namely
relations of arity zero (constants), relations of arity one
(subsets), binary relations and ternary relations.
Is there a theoretical reason that relations of arity k 3
are sufficient to describe arbitrary systems? Here there is
some hope. If the 'Peircean Reduction Thesis' is true, namely
that any n-ary relation on a set G can be described (in some
2
compréhension et l' étendue.'
The set of all formal concepts of K is denoted by B(K).
The conceptual hierarchy among concepts is defined by set
inclusion: For (A1 , B1 ), (A2 , B2 )  B(K) let (A1 , B1 ) 
(A2 , B2 ) :  A1  A2 (which is equivalent to B2  B1 ).
An important role is played by the object concepts (g)
:= ({g} , {g} ) for g  G and dually the attribute concepts
(m) := ({m} , {m} ) for m  M.
The pair (B(K),) is an ordered set, i.e.,  is reflexive,
antisymmetric, and transitive on B(K). It has some important
properties:

(B(K),) is a complete lattice, called the concept lattice
of K, and any complete lattice is isomorphic to a
concept lattice,

(B(K),) contains the entire information of K, i.e., K
can be reconstructed from B(K),

If B(K) is finite it can be drawn as a line diagram in the
plane, such that K can be reconstructed.
extent of the attribute concept of n in the scale Sm.
Hence, the choice of a scale induces a selection of
subsets of Wm - describing the granularity of the
contextual language about the possible values. The set of
all intersections of these subsets constitutes just the
closure system of all extents of the concept lattice of Sm.
The granularity of the language about the possible
values of m induces in a natural way a granularity on the
set G of objects of the given many-valued context, since
each object g is mapped via m onto its value m(g) and
m(g) is mapped via the object concept mapping m of Sm
onto m(m(g)):
g  m(g)  m(m(g)).
Hence the set of all object concepts of Sm plays the
role of a frame within which each object of G can be
embedded.
For two attributes m, m´  M each object g is mapped
onto the corresponding pair:
g  (m(g), m´(g))  ( m(m(g)), m´(m´(g)) )  B(Sm ) 
B(Sm´ ).
The standard scaling procedure, called plain scaling,
constructs from a scaled many-valued context
((G,M,W,I), (Sm | m  M)), consisting of a many-valued
context (G,M,W,I) and a scale family (Sm | m  M) the
derived context, denoted by
K := (G, {(m,n) | m  M, n  Mm }, J), where
g J (m,n) iff m(g) Im n (g  G, m  M, n  Mm ).
The concept lattice B(K) can be (supremum-)
embedded into the direct product of the concept lattices
of the scales (Ganter, Wille [7]). That leads to a very
useful visualization of multidimensional data in so-called
nested line diagrams, which is implemented in the
program TOSCANA (Vogt, Wille [20]).
Scaled many-valued contexts are essentially the same
as information channels in the sense of Barwise,
Seligman [2] which was shown by the author (Wolff
[28]).
Scaled many-valued contexts are 'conceptually
ordered versions' of knowledge bases in the sense of
Rough Set Theory. That was shown by the author (Wolff
[29]).
Finally, Fuzzy Theory, introduced by Zadeh [32],
also developed some notion of a scale, namely the
linguistic variables (Zadeh [33]). It was shown by the
author (Wolff [25,26]) that Fuzzy Theory can be
extended (by replacing the unit interval in the definition
of the membership function by an arbitrary ordered set
(L,)) to so-called L-Fuzzy Theory, which allows for
developing analogously to Formal Concept Analysis a
Fuzzy Scaling Theory which is equivalent to Conceptual
Scaling Theory.
Line diagrams of finite concept lattices can be drawn
automatically by computer programs (Wille [22], Vogt,
Wille [20]) and serve as an important communication tool
for the representation of multidimensional data.
One of the most famous infinite examples is the context
(Q, Q, Q) of the rational numbers Q with the usual rational
ordering Q. The concept lattice (B(Q, Q, Q),) is
isomorphic to the complete lattice of all real numbers
including  and - with the usual ordering on this set.
4. CONCEPTUAL REPRESENTATIONS OF
GRANULARITY
An arbitrary ternary relation on a set G of 'objects' is a
special case of a ternary relation among three sets of objects.
In formal descriptions of measurements by data tables the
following three sets play a fundamental role: A set G of
'objects', a set M of 'measurements' and a set W of values
(German: 'Werte') which are related by a ternary relation
whose elements (g,m,w) are interpreted as 'object g has at
measurement m the value w'. That leads to the following
definition of a many-valued context (G,M,W,I) as a
quadruple of four sets, where the elements of G are called
'objects', the elements of M 'many-valued attributes', the
elements of W 'values', and I is a ternary relation, I 
GMW, such that for any g  G, m M there is at most
one value w satisfying (g,m,w)  I. Therefore, a manyvalued attribute m can be understood as a (partial) function,
and we write m(g) = w iff (g,m,w)  I. A many-valued
attribute m is called complete iff for any g  G there is
(exactly one) w  W such that m(g) = w. (G,M,W,I) is called
complete if each m  M is complete.
The central granularity-choosing process in conceptual
scaling theory is the construction of a formal context Sm =
(Wm, Mm, Im) for each mM such that Wm  mG := {m(g) |
gG }. Such formal contexts, called conceptual scales,
represent a contextual language about the set of values of m.
Usually one chooses Wm as the set of all 'possible' values of
m with respect to some purpose. Each attribute n Mm is
called a scale attribute. The set n = {w | w Im n } is the
5. CONCEPTUAL TIME SYSTEMS
In this section I explain the principal ideas
concerning conceptual time systems as defined by the
author (Wolff [27]).
The first idea is to replace 'time points' by 'time
granules' understood as basic time objects like 'morning'
and 'early morning'. In the same way as we describe
3
'space granules' like 'lecture room' (for the purpose of giving
lectures) not necessarily by geometric length measurement
values, we wish to describe time granules in a general and
flexible way including a granularity suitable for the actual
purpose. Therefore, we choose a scaled many-valued context
T as the description of the time.
The second idea is to preserve the classical separation of
the 'time' and 'space' part of a system so as to be able to
generalize the notion of a phase space to arbitrary conceptual
time systems. Therefore, I choose for the description of
events observed during the time granules of G again a scaled
many-valued context C over the object set G. That is
described in the following definition.
Definition: 'state space and time granule space of a
conceptual time system'
Let (T, C) be a conceptual time system and KT and
KC the derived contexts of T and C. For each time
granule g we define the state s(g) of (T, C ) at time
granule g by s(g) := C(g) := the object concept of g in
KC and the time granule concept t(g) of (T, C ) at time
granule g by t(g) := T(g) := the object concept of g in
KT.
The set S(T, C):= {s(g) | g  G } is called the state
space of (T, C) the set G(T, C):= {t(g) | g  G } is
called the time granule space of (T, C).
This definition yields the 'partition meaning' of
states, namely, that the set G of time granules is
partitioned by the states, or, equivalently, that a system is
at each time granule in exactly one state.
The investigation of subsystems of a conceptual time
system shows that states of a subsystem do not
necessarily correspond to states in the given system, if
we take as 'correspondence' the natural part embedding
of the concept lattice of the subsystem into the concept
lattice of the given system. For the details the reader is
referred to Wolff ([27]).
That leads to the introduction of 'general states' in the
sense of the following definition. The notion of general
states and their conceptual ordering is a mathematical
description of the 'order meaning' of states, which is
intended in phrases like "the 'state of reading' is a substate of 'the state of living' ".
Definition: 'conceptual time system'
Let G be an arbitrary set and T := ((G, M, W, IT), (Sm |
m  M)) and C := ((G, E, V, I), (Se | e  E )) scaled manyvalued contexts (on the same object set G). Then the pair
(T, C) is called a conceptual time system on G.
T is called the time part and C the event part of (T, C).
This definition of a conceptual time system describes the
notion of time at three places: the first is the set G of time
granules, which are understood as arbitrary time objects used
for the time part as well as for the event part of (T, C) ; the
second is the many-valued context (G, M, W, IT),
interpreted as the data table of 'time measurements' taken
during the time granules of G; the third one is the scale
family (Sm | m  M) which can be used for two purposes,
namely the description of a granularity in the conceptual
language about the values of time measurement, as well as
for the description of theoretical constraints on time theories.
Definition: 'general time granules and general states'
Let (T, C) be a conceptual time system and KT and
KC the derived contexts of (T, C).
Each concept (A,B) of KT is called a general time
granule of (T, C).
The concept lattice B(KT) is called the time space of
(T, C).
For (A,B), (C,D)  B(KT) we say that (A,B) is a timesub-concept of (C,D) iff (A,B)  (C,D) in B(KT).
6. STATES
Now we are able to introduce 'states' and the 'state space'
of a conceptual time system.
The word 'state' is mainly used with an observation of a
thing (or 'system') which is stable during some range of time,
'stable' clearly being only relative to some of its properties.
For example, we say, that 'the patient is now again in a
healthy state'.
In a conceptual time system (T, C) we would like to say
'The system is at time granule g in state s(g)'. The central
idea in the introduction of states in conceptual time systems
is that a state s(g) should be described by the event values
e(g) – but these values should be considered in the
conceptual frame described by the event scales Se = (We, Me,
Ie). Two values v, w  We are called e-equivalent iff they
have the same object concept in Se. If the measurements at
two different time granules g and h yield e-equivalent values
for each event e, then the state s(g) should be the same as the
state s(h). That shows that the states of (T, C) should be
defined as the object concepts of the derived context KC. As
to the time part we should look at the object concepts of the
derived context KT.
Each concept (A,B) of KC is called a general state of
(T, C).
The concept lattice B(KC) is called the general state
space of (T, C).
For (A,B), (C,D)  B(KC) we say that (A,B) is a substate of (C,D) iff (A,B)  (C,D) in B(KC).
This definition will be explained in the example in
section 8.
4
scaled many-valued context with four many-valued
attributes and each of the four scales has just one scale
attribute. The derived context KC is shown in the
following Table 1.
7. THE PHASE SPACE
In classical Mathematical System Theory a phase of a
system is a pair (t, s) of a point t of time and a state s. In a
conceptual time system (T, C) the notion of phase space is
defined as follows.
input<625 pressure<118 reflux<140 energy<600
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Definition: 'phase space of a conceptual time system'
Let (T, C) be a conceptual time system on G and KT and
KC the derived contexts of T and C.
The apposition KT|KC of the derived contexts is called the
phase context of (T, C).
The direct product B(KT)  B(KC) of the two concept
lattices of KT and KC is called the general phase space of (T,
C ).
Each element of B(KT)  B(KC) is called a general phase of
(T, C).
The concept lattice B(KT|KC) is called the phase space of (T,
C).
For any general time granule (A,B)  B(KT) and any general
state (C,D)  B(KC) we say that the system (T, C) is at the
general time granule (A,B) in the general state (C,D) iff
A  C (which is equivalent to T(A,B)  C(C,D) in
B(KT|KC), where T and C are the part embeddings of KT
and KC into KT|KC).
For any time granule g  G the pair (t(g), s(g))  B(KT) 
B(KC) is called a phase of (T, C) and we say that (T, C) is in
the phase (t(g), s(g)) or (T, C) is at time granule g in the
state s(g) iff T(t(g))  C(s(g)) in B(KT|KC).
The principal ideas underlying this definition are
explained in the following example.









































Table 1: The derived context of the event part
The general state space is drawn in Figure 1.
8. APPLICATIONS
For a short demonstration of the practical and theoretical
value of this method I choose a quality control problem in
the chemical industry. For proprietary reasons I describe the
problem in a modified version.
Quality control in a distillation column
To control the quality of a chemical product the process
in a distillation column was observed during approximately
one month. Actually, only 20 days were used to take
measurements. At each day for each of 13 'variables' (like
'pressure') only one value was measured. The resulting
many-valued context with the 20 days as time granules and
the 13 'variables' as many-valued attributes has real values
(and also some missing values). For proprietary reasons the
corresponding 2013 data table is not shown here. But the
construction of scales and the resulting derived context can
be demonstrated.
For the purpose of understanding the 'behavior' of the
distillation column with respect to the four many-valued
attributes 'input', 'pressure', 'reflux' and 'energy' we decided
(together with the experts on the process) to use a very
simple threshold scaling, namely, choosing just one scale
attribute selecting the 'small' values of the corresponding
many-valued attribute. Therefore, the conceptual time
system (T,C) we are going to construct has as event part a
Figure 1: A four-dimensional general state space
This figure shows the general state space of (T,C),
that is the concept lattice of KC embedded into the
Boolean lattice of all 16 = 24 "possible states" of the four
attributes of KC. For more details about this supremumembedding of the general state space the reader is
5
referred to Ganter, Wille [7, and Wolff [27]. We now explain
how to read this line diagram. The top point denotes the
formal concept (G,) with extent G={1, ...,20} and intent .
The four points just under the top concept describe the
attribute concepts denoted by their attribute names. The
object concepts are also denoted by their object names.
These object concepts are exactly the states of (T,C), and
their points are drawn black. An object g has an attribute m
if and only if their is an upward-leading path from the point
of g to the point of m. Therefore, time granule 1 has the
attributes 'input < 625' and 'reflux < 140' and no others. The
object concept (1) = ({1,3,4,5,6,7,8}, { 'input < 625', 'reflux
< 140' }) is the state with maximal frequency. The two gray
points are also formal concept, hence general states. The
attribute concept of 'reflux < 140' is indeed a state in the
subsystem obtained by deleting the attribute 'energy < 600',
since it is the object concept of the time granule 10 in this
subsystem. Similarly the top concept is a state (of 18) in this
subsystem.
The white points in Figure 1 do not belong to the general
state space, but they have an important meaning: each of
them represents an implication valid in the general state
space. For the well-developed theory of implications in
formal contexts the reader is referred to Ganter, Wille [7].
We here read some examples from Figure 1.
The white point 'having' the attributes {'pressure < 118',
'reflux < 140'} describes the implication that each time
granule g which has these attributes also has the attribute
'energy < 600'.
The following 'clause' holds in this context, since the
attribute concept of 'reflux < 140' is not a state: if 'reflux <
140' then 'input < 625' or 'energy < 600'.
In the following we construct the time part T. In general
we can take an arbitrary scaled many-valued context
representing 'time measurements', for example values
describing the time intervals during the event measurements
at this time granule. In this example we are interested in
understanding the dynamics of the system, that is the time
dependency of the states. The only information about the
time in this example is coded in the names '1' to '20' of the
days (the time granules): their ordering as natural numbers
represents the time order of the days. Formally expressed:
day 'x' was before day 'y' if and only if x < y. The names do
not mean in this example that 'x+1' denotes the next day after
day 'x'. Therefore, we only use the ordinal structure of the
time granules.
The purpose of the following construction of the time
part T is to generate a very short description of the process
which can serve to inform the director about his distillation
column.
To describe the time part T and the role of the scales in
this example some notation is necessary for the following
simple scaling procedure. From Figure 1 we see that the
process can be partitioned meaningfully into three parts. To
do this we choose two many-valued attributes "t>8" and
"t>12" where t is understood as a variable for the time
granules and "t>x" has value 1 iff t>x, and value 0 otherwise
(xG). The two-valued context is shown in abbreviated form
in Table 2.
"t>8"
"t>12"
0
0
1
1
1
1
1
0
0
0
0
1
1
1
1
.
9
.
13
.
20
Table 2: The many-valued context of the time part T
For each of the two many-valued attributes we
choose the same scale S described in Table 3.
S
0
1
'w = 1'

Table 3: A scale for the transformation of a twovalued context into a formal context
The derived context KT of this time part T is
described by Table 2 where each '1' is replaced by ''.
The time space, which is the concept lattice B(KT), is a
chain with three formal concepts. (Remark: I am aware
that this simple example can not demonstrate the rich
possibilities of the scaling procedure, but it is suitable for
demonstrating the general phase space.)
The general phase space
The following diagram shows the general phase
space B(KT)  B(KC) of (T,C) embedded in the direct
product of the time space and the four-dimensional
boolean lattice containing the state space. The direct
product is drawn as a 'nested' line diagram where the
time space is taken as the 'rough structure' and the fourdimensional boolean lattice as the 'fine structure' of the
diagram. For each time granule g the phase of g is
visualized by the point labeled 'g'. By definition, the
phase of g is (t(g), s(g)), where t(g) = T(g) and s(g) is
the state C(g). For example the phase of time granule 1
can be seen in Figure 2 as follows: look at the time
concept t(1) = T(1), which is 'top concept' in the 'rough
structure' and then look into the 'fine structure' at the
state s(1) = C(1) which is labeled '1' in Figure 2.
The arrows indicate transitions between 'successive'
phases, such as from the phase of g (g < 20) to the phase
of g+1 (if these phases are different). Here we use the
linear ordering of the time granules in this example. A
general conceptual theory of transitions will be published
later.
In each of the three 'big' points in the 'rough
structure' there are two special arrows, one pointing to
the start and one coming from the end of the process
represented in this 'big' point. For example, the transition
from the phase of 12 to the phase of 13 is drawn as the
end-arrow in the 'middle big point' and as the start-arrow
in the 'bottom big point'.
6
A conceptual film
We can understand the general phase space of Figure
2 as a conceptual film with just three photos. To discuss
the meaning of these photos with respect to 'sharpness'
and 'uncertainty' we compare this short film with the
corresponding long film with 20 photos, each showing
exactly one phase of the system, for example photo 1
shows the phase of 1, (T(1), C(1)), graphically
represented as one black point with label 1 (for C(1)) in
the lattice of the state space, which itself is drawn in the
'big' point of T(1), the top point in the time chain of 20
time concepts.
In contrast to the 20-photo film our 3-photo film
shows in each photo many phases. That is essentially the
same as on a usual photo of a quick motion, showing for
example the not sharply represented wings of a flying
bird. In our short film the knowledge of the ordinal
meaning of the labeling of the time granules together
with the names of the time granules in the diagram gives
us the certainty about each single phase of the process.
But on a customary photo we do not have the knowledge
as to which part of the photo was at first exposed to light.
This uncertainty occurs in our 3-photo film on each
photo if we forget the ordinal meaning of the labeling of
the time granules, which is indeed not represented in the
conceptual time system of this example. In this sense all
three photos are not sharp, while a photo taken during
the time granules 3 to 8 would be perfectly sharp, in the
sense that all the phases (t(g), s(g)) are equal for g 
{3,...8}.
Clearly, a change to a coarser granularity in the state
space can lead from not sharp to sharp photos (in another
conceptual time system); for example omitting the
attribute 'reflux < 140' would lead from the not sharp
photo in top of Figure 2 to a sharp photo taken during the
first 8 time granules, since it would not represent the
change of the reflux value from time granule 1 to 3.
9. CONCLUSION AND PERSPECTIVES
This paper offers some contributions to the solution
of the problems described in the introduction.
Figure 2: A general phase space with transitions
The role of conceptual time systems
Though the main problem, namely how to define a
general system, is not solved, it seems that there is some
hope that it can be reduced to the problem of how to
define systems using only k-ary relations where k3.
That demonstrates from a theoretical point of view the
importance of binary and ternary relations. And they are
the main tools on which the notion of a 'conceptual time
system' is based which is emphasized here as a quite
general definition of a system. Conceptual time systems
are defined using the notion of a 'scaled many-valued
context', which is the central tool in Conceptual Scaling
Theory. Scaled many-valued contexts play a very
important role in general knowledge processing, as
evidenced by the fact that they are equivalent to
Figure 2 visualizes three clear 'motions' of the process;
they can be verbalized as follows: the first motion from time
granule 1 to 8 is just a short 'side-leap' from the phase of 1 to
the phase of 2 and back to the phase of 1, which is the same
as the phases of 3,...8; the second motion from time granule
8 to 12 leads down to the phase of 12 where all
measurement values are low; and the third motion from time
granule 12 to 20 goes on a 'high' way (where many values
are not low) to the final phase of 20.
Beside the three main motions of the process from this
diagram we also see some dependencies: If 'pressure < 118'
then 't>8', that is, all time granules satisfying 'pressure < 118'
also satisfy 't>8'. Much more detail in the process can be
seen from this diagram, but now I wish to discuss the
meaning of this diagram as a conceptual film.
7
information channels in the sense of Barwise and Seligman
[2] and they are 'ordered versions' of the knowledge bases in
Rough Set Theory (Pawlak, [13]; Wolff [29]). Further, they
are the basic tool for the generalization of linguistic variables
in classical Fuzzy Theory (Zadeh, [33]) to 'realized linguistic
variables' in L-Fuzzy Scaling Theory (Wolff [25,26]).
attributes.
Perspectives
Clearly, the introduction of conceptual time systems
and the formal definitions of the notions of 'state', 'state
space' and 'phase space' are just the starting point for the
development of a conceptual system theory. It should be
related to 'Conceptual Graphs' in the sense of Sowa [18]
and to 'Concept Graphs' of 'Power Context Families' in
the sense of Prediger and Wille [16], which are
introduced to describe conceptual graphs in the
framework of Formal Concept Analysis. A power
context family is a contextual description of a relational
structure. It should be related in the future to the general
description of 'time' in conceptual time systems.
There are many other promising fields of research
where Conceptual System Theory can be applied. For
example, it is necessary to develop a conceptual
transition theory, to study deterministic systems, for
example automata theory, to relate the conceptual theory
of implications to rules and laws, to investigate
dependencies arising from the object distribution as
studied in classical statistics, and to unfold the
conceptual meaning of the use of probability theory, for
example in classical physics, including the role of
metrics, energy, and continuous and discrete spectra
which now can be studied from the conceptual point of
view.
The role of time granules and time scales
The time representation in a conceptual time system by a
scaled many-valued context is general enough to contain the
usual continuous time as well as discrete times, as well as
arbitrary ordered times. The combination of the time
granules (as objects of the many-valued time context and the
event context) and the scales for the time and event
measurements is what is really new. That gives us not only
the possibility of describing the 'time-meaning' of the time
granules, but also using an appropriate granulation of the
conceptual language about the time granules. Furthermore,
the scales serve as a theoretical framework for the
formulation of rules or laws. For example, the scale of a time
chain represents the rule, that any two time granules are
comparable (in the concept lattice of this scale).
Therefore it is now possible to study in a single
conceptual time system finitely many time granules and
infinitely many scale attributes of a large time scale, for
example the 'rational' scale (Q,Q,), which has a concept
lattice isomorphic to the ordering of the real numbers
including  and -. Hence, in conceptual time systems a
pragmatically meaningful restriction to finitely many time
granules does not contradict the application of a continuous
time theory.
The separation of time granules and time scales leads to a
pragmatically meaningful and simple description of
'simultaneous measurements' as measurements performed
during the same time granule – in contrast to 'simultaneous
measurements' at the same time point, defined as a real
number. Clearly, we could take the set R of real numbers as
the set of time granules of a conceptual time system, but
what should be the pragmatic meaning of performing a
measurement at such a time granule or taking two different
measurements at the same time granule?
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The notion of a 'formal concept' is the central tool for the
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8
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9