Fuzzy Sets and Systems 25 (1988) 357--367 North-Holland

357

T H E O R I E S OF P O S S ~ I L ~ : META-.4~,~OMAT|C$ A N D $ E ~ N T I C $

Ladislav J. KOHOUT Man-Computer Studies Group, Department of Computer Science, Brunel, the Umversity of West London, Uxbridge, Middx. UB8 3PH, England, U.K.

. . . . . Received February 1986

To Professor Lotfi Zadeh

This paper outlines the fundamentals of possibiiistic systems, of which the classical fuzzy sets, probabilistic fuzzy sets and fuzzy sets of the 2-nd type are particular instances, it presems the meta-axiomatics of the possibilistic systems that motivated Lotfi Zadeh for his possibilistic interpretation of fuzzy sets. This paper also develops the basic fundamentals of the semantics of possibilistic systems within a ,ewdotic framework. It concludes with the construction cMled 'contraction of a possibilistic system into its extension', thus providing a formal formulation of the pragmatics of fu~_y membership function. This procedure is a meta°thcoretica~ application of the Checklist Paradigm of Bandler and Kohout. The last section outlines a possibilistic logic of existence and the dynamics of construction.

Keywords: Possibilistic systems, Semantics and pragmadcs of fuzzy membership function, Semiotics, Foundations of fuzzy sets, gelationa| products, Epistemological questions of fuzziness, Meta-~iomatics, Logic of existence.

L On the fo~dation~ studies of f ~ y sets

in a discussion of the natlzre and the character of fuzzy membership function we have to pay attention to the foundations of f u r y sets. In the foundational studies we have ~o distinguish:

(1) mathematical questions, (2) lo~ca! questions, (3) ontological and epistemological questions. In (1) we deal with the structure, in (2) we add to the structure the logical form

and the mode of inference. In (3) we hay, - deal with the problem of reality of existence of the chosen primitive (atomic) ~,~ ~cepts, ~s well as with the questions of seiection and justification of the appropriate meaning of the scientific and technological concepts employed. We have also to add the problematics of methods of enquiry.

It is also important to distinguish between the theory, the applications, a~d the foundational studies of both. In all these three endeavours the questions listed in (1)-(3) above are relevant.

0165-0114/88/$3.50 © 1988, Elsevier Science Publishers B.V. (North-Holland)

358 L.L gohom

2. On p - ~ b ' ~ e in terp~f ion of f m ~ sets

It is known that Zadeh's possibilistic interpretation of fuzzy sets [18] was motivated by an article on 'Possible automata' by Gaines and Kohout [3]. But it is not so commonly recognized that the 'Possible automata ~ paper contains two, vastly different from each other, concepts of possibility. Zadeh himself notes that " . . . the interpretation of the concept of possibility in the theory of possibility is quite different from that of modal logic . . . " (Zadeh, [18~, p. 4, footnote 2). This concept of possibility I used in [6] for constructing algebraic generalizations of Pinkava type many-valued logics. Its application to medical diagnostics was given in [7]. In this paper I present its meta-~iomatics linked to semiotic systems. Zadeh noted that he employed " . . . the theory of fuzzy sets as a basis for the theory of possibility..." (Zadeh, [18], p. 27, Acknowledgment). Here, in the last section, I reverse this process and derive from a possibilistic model a fuzzy set.

Since 1974 I have devoted some effort to the development of formal foundations, methodology and semiotics of possibilistic systems [7, 8, 9, 10]. It evolved into the methodology of Activity Structures [10, 13], that has found its application in the design of Knowledge Based Systems, Brain Modelling and elsewhere [14].

3. P~sie notions md defmitiom of the semiotics o~' ,posab~s~ie systems

This section contains some notions of semiotics that were developed for dealing with the semantics and pragmatics of possibilistic systems, it is based on a p~:¢viously unpublished note of mine [11], which also contains further details.

DefuMfiou 3.1. A collection of some elements together with their interrelatioii-o ships is called a space-aggregate.

Detinlflou 3.2. A semiotic system ~ = (~ , ~, ~/,) is a triple of space-aggregates. It contains the following as its components:

(a) .~ ~pace, called a linguistic space with the universe of discourse consisting of languages. It contains abstract signs as its proper part.

(b) ~'-space, called the co~cept-space, its universe of discourse being abstract concepts.

(c) ~-space, called the physical space. It has for its universe of discourse the collections of the concrete entities of the real world.

Definition 3.3~ The assignment relations of a semiotic space: (a) denogation :

~elation D:

(b) significaaon:

relation S:

Ceim × Pelm,

Leim × Celm,

Theoeie~ of possibility 359

(C) reference:

relation R: Lelm x Pelm

with three kinds of elements Celm, Lelm and Pelm such that Lehn ~ ~-space, Celm e CO-space and Pelm ~ ~-space.

Dele t ion 3.4. The three types of assignment spaces are defined as follows: (a) denota~on space: :~(Celm~ Pelm)~ (b) signification space: ~(Le lm~Celm) , (c) reference space: ~(Lelm ~ Pelm),

where ~(A-~ B) is the lattice of all fuzzy relations from A to B, A and B being elemems of types as shewn in (3.4a), (3.4b) and (3.4c) above.

If we restrict ourselves to the Boolean lattice of all the crisp relations, i.e. ~(A ~ B), ie talk about a crisp assignment space.

Defer|on 9.5. Syntax, semantics and pragmatics of a possibilistic system. (a) A space-aggregate together with some dependence subrelationship between

its subaggregates form the syntax of a space. So we talk about Le-syntax, C~-syntax and ~-syntax.

(b) A pair of space aggregates together with their corresponding assignment relation, form the semantics.

(c) The system depicting how and why a particular semantics is formed determines the pragmaacs of a possibilistic system.

4. Abstract fo~dafiom o~' ~ i b ~ t i c systems a.cd its meta-a~omaties

4. I. From objects with properties w objects with membership .func~on

We have a universe of definite objects Obj w~i~ the associated collection of properties Prop, Each object a ~ Obj possesses a subco~.lection of properties from the collection Prop. By 'definite' I mean making a distinction in the sense of Spencer-Brown (Cf. Kohout and Pinkava, [12]).

This universe 0~ of the definite objects with the associated properties can be modelled in the theory of classes by a relation

0~:Obj × ~(Prop)

where ~(Prop) is the crisp or fuzzy power set in its usual sense and Obj is a class. 0~ is a possibilistic object-universe. Any subrelation U such that U = ~ is covering and univalent is a definite object-universe. Any other proper subrelation of ~ we call constrained possibilistic universe.

A definite group (i.e. collection) of observers may single out some objects according to some definite selection criteria and describe them i~ some language. Given a family of definite selection criteria for each observer, there is a definite possibility of marking various objects according to some definite selectk~ criterion c ¢ Cfit. Of selections we have at least two kinds: descriptions aad

360 L.Z Koho~a

constructions. If we take two objects with corresponding associated properties and relate them by some logical relation, we may summarize the attached properties by some measure over the property space-this gives the Checklist Paradigm [1, 23].

4.2. Meta-axiomatics

M e ~ o ~ o m L Ontologically neutral existence of the object universe. There exist shr~ultaneously: a definite family of objects Obj, a family of properties Prop and a definite relation ~ (a possibilistic object-universe)

~ :Obj × ~(Prop).

M e ~ - ~ o m 2. Pragmatic commitment. A particular universe 0~ can belong to nothing else but to the space-aggregates of the semiotic space, i.e.

M e ~ - ~ o m 3, Epistemological unicity. • does not belong to more than one of the space-aggregates of the sen~otic space.

Met~-~om 4. Ontologically neutral existence of possibilisitc universe. Given a class of criteria Crit for making distinctions, a class of objects Obj and a class of properties Prop, the possibilistic universe H is defined as:

~ : Cdt x 5(Obj ~- @(Prep)).

Me~-~vdom 5. Possibility of exi8'~ence of multiplicity of recta-descriptions of different semiotic types. H belongs to at least one space-aggregate of the semiotic s~ ace 2'.

4.3. Observation in possibilistic systems

Delet ion 4.3.]l. Any possibilistic universe H such that H is either a part of an £¢-,,~pace or of a ~-space is called a descriptor space.

M e ~ - ~ e m 6. Possibi!istic universe of observational descriptions. The possibi!is- tie universe of observational descriptions, so called £~-universe is ,:onstructed as follows:

g2: $(Obs-~ ~(H))

where Obs is a class of observers and ~(H) is a fuzzy descriptor space. A relation D ~ $:(/7) such that it is a projection of Q is cal~ed possible Juzzy observation. A relational pair (i.e. extensivel, y given element (a, D) of Q) where a ~ Obs is a particular observer, is called an observation subjective to the observer a.

Theories of possi~tity 361

$. ~ p t b n s and pr-~ednres for identification of membership ~nc~on

The purpose of the membership function is to make crisp or fuzzy distinctions, but this cannot be done without naming. Hence, in order to be able to provide meta-axiomaties of fuzzy membership function in the object-universe ~ , we have to associate a name-universe with the possibilistic universe fir.

5.1. A recta-algorithm for iden~'fication

The identification procedure is based on making distinctions and performing constructi~ons as follows:

(IM) Single out an object-universe ~. (2M) By choosing the class of criteria for making distinctions fix the cor-

responding possibilistic universe/7. (3C) Associate an appropriate name-~ni~erse fd with the corresponding

possibilistic universe If. (4C) Form the membership function space by associating an appropriate

valuation algebraic structure (e.g. a Pinkava algebra, [14, 15, 16]; or a semiring [3, 4]; or a lattice).

(5M, C) Identify the correct value of the membership function of every object of ~t relative to the selected criterion of distinction c ~ Crit.

Note that the letter M in the above denotes making a distinction (i.e. perception) while the |etter C denotes a construction (i.e. action).

6. Con,~lcfion of ~ sets by means of possibi~tic systems

6. I. Gn ~i~c~s?,,a and extensive definition of a set

Elements of a set (or a relation) can ~ther be given as a list, by enumerating them explicitly, this is the definition by extensicn; or they can b~ ~pecified ~nstead by giving their properties, without listing the elements explicitly. Thf, s is called definition by intension. In the fi~t instance of extensive definition, we check belonging of an element by looking it up in the list. In the second instance, the membership is tested by checking the properties. The former definition, by extension is customary in most mathematics.

The notion of intension and extension can be illustrated by the following example due to Troelstra, which is concerned with an equational definition of a set of numbers.

The binary functions + and + ' on natural numbers defined by the recursion equation (S for successor, ~'~ y numerical variables)

x + O = x , ~O+'y=y , x + Sy = s ( x + y), t s x + 'y = s(x + 'y),

represent extensionaUy the same fu,~c~ion, but they are given by distinct pairs of equations con esponding to distinct competationai procedures; hence they are not

362 L.L Kohout

intensionally equal. However, extensionally their elements (objects) satisfy both pairs of conditions which specify different properties each.

The role of the membership function of an element of a fuzzy set then acquires the precise meaning in this framework. It is the degree of possibil/ty to which an element a belongs to a specific aggregate of objects (i. e. elements) declared by their extension. In symbols ~(a ) := ~r(a ¢ A), where A is which the element a explicitly appears. It can be seen that for i0enfification of membership function can generate an although each operates on a different possibilistic system.

the name of the list in two distinct algorithms identical extensive set,

6. 2. Contraction of a possibilistic system into an extensively defined set of ob/ects

Extensively defined fuzzy or crisp set can be constructed from the possibilistic systems of Section 4 by contraction of a possibilistic system into its extension.

Cons~efion 6.2.1. Contraction into extension. Given a collection of some selection criteria and some collection of objects by two relations R and S such that

R 'Obj x Gprop, S" Scrit x Gprop

we can consz~uct a collection of subsets by a relationa~ product (ef. [1,2]) as follows:

!

where - are the cort~iectives of some base many-valued logic; Q, R, S are relations in the matrix form, R T iS the conver~e of R.

This construction is an instance of the contraction by means of a square relational product S E]R r. The basic technical prerequisites necessary for understanding our relational products are summarized in the Appendix for reader's convenience. Here, it suffices to point oat that in the resultant relation Qt.k, the index i determines an afterset generated by a particular selection criterion s ¢ Scrit and the index k a particular element of that afterset.

The semantic description of the meaning of the product can be elucidated as follows. The relation $, relating (Stilt x Prop) composed with the relation R T relating (Prop× Obj), gives the relation Q relating (Scritx Obj). We are matching the properties of our descriptive frame with the properties of the observed objects. If we choose the arithmetic sum for the ^ operator and the usual two-valued equivalence for ~ , and also assume that Prop are crisply related in the relations R and S, we obtain fuzzy sets of the 1-st type. However, if the relations R and S are fuzzy and - is a many-valued equivalence operator, we obtain an interval approximation of the f~-cy sets of the 2-nd type. Exact representation of the fuzzy sets of the second type can be obtained by exchanging the fuzzy points for the possibilisfic distributions. "i~is is done simply taking ~(Gprop) instead of (Gprop) in the relations R and S.

Theories of possibility 363

Numerically, the value of Q~.k represents the degree of the match of the properties of our frame of description with the properties of objects.

6. 3. Matching the potentialities of description and observation

The contraction based on the square product assumes that the identification procedure is symmetric, finding the match of the properties by means of some many-valued logic equivalence operator. But an equivalence can be generated by the intersection of the effects of some non-symmetric implications. For any two components of the product, say S 0 E R~ can also be expressed as (S0---~ R~)^ (S 0 ~---R~). This leads to the necessity of distinguishing between S <] R" and S~>R r relational PrOducts. Hence, we should examine the meaning the two distinct triangular products in this context. This leads to the following construction.

Constmdion 6.3.1. Let us have S (a descriptive framework), observational framework) as defined in 6.2.1. Then

and R (an

(a) (s - A (s,j,-- !

selects the objects compatible with S (i.e. admissible objects), and

(b) (S < - A (so--, i

lists the selections compatible with the framework of observation (admissible selections).

Construction 6.3.1 is very general and can be viewed as a blueprint for a large variety of distinct species of possibilistic fuzzy set theories. However, I have not yet linked the constructions of this section with my possibilistic system of Section 4 explicitly. This can be done by relating Gprop (in Constructions 6.2.1 and 6.3ol) to ~(Prop) of Meta-Axiom 1 of Section 4.2 by a one-to-one and onto mapping. This provides an explicit possibilistic interpretation of the above constructions.

It is also possible to contract into intension, describing the objects by means of some formulas over their properties. It can be done by forming the square products of the type (SUI R r) U! R combined in a suitable syntactic logical normal f3rm, or directly by formulas of GUHA-type. However, this is outside the :~cope of this article and will be described elsewhere.

7. Possibnisti¢ lo~c of e~Jtence

We have assumed that the ~ relation does not change with time, and that the objects and properties retain their existence all the time. We can relax this assumption and assume instead, that either the properties or the objects can be created and anih/lated. Theft means that the foresets and aftersets of 0~ change their size.

L.Z Ka~m

When we considered the stable situation where all the objects and properties existed all the time:, it was permissible to talk about objects, without referring to properties and vice versa. This involved making the explicit assumption that both t ~ s of entities had stable existence and were related in the way specified by ~.

We did not ~ u m e that the ~bjects had existed without having any property. The minimal condition mvolve~ then was the as s~pt ion that an object has at least one p~operty, that is its existence. Invariably, it was assumed that the properties were fixed. On the other hand. if we assume that o ~ and q/p change with time, we have to capture this by a, possibly infinite, family of relations v~t indexed by time. It is useful to assume that the time index t ranges over a partially ordered semiring. We can easily .restrict it to a lattice, or even to a strict order, such as reals or integers. The semiring generally determines the structure of time.

Let us describe this time variable structure more formally. Where O is the set of all possible objects, 0 ' ~ ~ ( 0 ) is the set of the objects existing at the time instant t. Simihrly, for the properties, we have p t¢ ~(p). O r and p r are the sequences of all O t and pt subsets appearing during the interval of existence of the system. On these complex objects, we can define various necessities and possibilities over the objects and properties, as below:

Possibility: o lTJo, o~ , v ~ p .

(For the notation see the Appendix.) Then the possibilistic reali~ability of a system being constructed can be

constructively defined by the following sequence of activities: (1) Collecting non-contradictory necessary properties of the construction, we

fol~ possible semi-objects (i.e. incompletely specified objects) or objects. (2) Collecting from the semi-objects or objects obt ' ~ e d in the previous step

those that satisfy some additional properties, we restrict the family of the possible objects.

(3) Repeating the activity of (2) above until we obtain the objects of the desired properties (actualisation of objects) or until the additional properties show incompatible with the earlier chosen properties (impossibility of a construc- tion), we complete this process.

$o Conclusion

In Section 6 I have given an explicit construction of a fuzzy subset with the support taken from Obj. This elucidates the meaning of fuzzy membership ~nction and aleviates the criticism that a membership function cannot be logically and objectively constructed. If we use as the 'second order objects', the properties of the possibilJstic systems that are attached to the original Obj, we can constract 'two-layered' systems that capture all tile features of ~ z ~ sets of the

Theories of pouibil~ - 365

~ n d type and of probabilistic sets, in an appropriate pragmatic system that can justify the constructions des~bed in Sections 5 and 6.

In order to deal with the normative aspects of the systems developed by Hisdal [5], i.e. to 'depsychologize' them we would have to use the who|e semiotic system, i.e. the whole triple of ~-space, cO-space and ~ . s p a c e . ~ i s fact indicates that her systems capture some features of the field of possibifistic systems that remain neglected by other approaches.

A p p e ~ : F ~ , ~ ~ and pe~l~e~ of ee~fio~

The theory of crisp and fuzzy relation~ products developed by W. Ban~er and L..L Kohout, started with the obse~ation that the relational composition (products) can be expressed by means of logical connectives and that apart fIom the AND and oa conn~tives, also the both IMPLICATIONS and EQUIVALENCE ,V~n be used to form useful relational products with interesth:g and useful properties. For the early resets, see our 'Essex' report of 1977 [22]. The most s~Jccha~ but useful overview is [2], originally numbered as FRP-23 (August 19~) in our 'E~ex' FuT~ Research Project ~eri~,~'~ of ~',eports. It was submi~ed to the publishe~ ~ 1982 and unfortunately still re~alns in the pubfishers press. For the app~cafions of the products see [1]. Reference [2] deals also with n-ary relations.

In what follows R is a fuzzy relation from X to Y that belongs ~o the collection of all fuzzy relations ~ ( X ::> Y) from X to Y. The matrix rep:esentation of a fuzzy relation R will ~ denoted by the symbol R~. ^ , v, --->, ~--, ~ are logical connectives of a many-valued base lo~c. For a useful fist see ~23]o

Standard (A~.~ALIS~C) foresets and aflersets

The afterset x~R is for any element x~ in X the subset of Y consisting of those elements yj to which x~ is related via R.

The afterset of relation R and element x~ from X with R" X ~-- Y is

x,R := {yj I x, Ryj ^ yj Y}.

The afterset can be completely described by the i-th row of the m a t ~ R~. Similarly, for any element Yi in If, the foreset Ry i is the subset of X which

consists of those elements xi related by R to yj. The foreset is then

:= (x, i x, ^ x, X } .

The foreset is completely described by the ]-th column of Ru.

Nonstandard (Possu3Ius~c) forese~ ~d aflerse:$

Also the both IMPLICATIONS and the EQUIVALENCE of any manyova~ued base logic are useful in defin~g the foresets and after~ets. Laci< of space prevents me to give their semantics here,

/~erset : x~R := ~j ]xiRyj*yj e Y}, Foreset: Ryj := {xi ] xiRyj, xj a X}~

366 LY. Kohout

whe~ * is a connective chosen from the set {~--,-% - } of connectives of any many-valued base logic.

A d ~ o ~ e a W

I would like to thank, for stimulating discussions concerning the nature of ~ e membership function, Ellen Hisdal, Robin Giles, George Klir and Milan Zeleny. To Farouk Dowlatshahi I am indebted for many discussions concerning his Perspeetivism that is interestingly related to possibilistic systems. Brian Qaines I want to thank for bringing me into this field and for years of stim~|ating collaboration during 'the old Essex days'.

Postscript

A new important paper was published while I was completing this contribution to the special issue, a paper by Brian Gaines and Mildred Shaw, touching d;,rectly upon the issues discussed here. I quote:

"Kelly's psychology and Brown's logic both take as primitive an axiom of unrestricted predication. In set-theoretic terms they take comprehension as a priori and derive all other concepts from t h i s . . . i t is misleading to translate.°, into logic and set theory. . . . . They are defining a bas is . . , which is prior to both and fTom which logic and set theory must be derived." [19] Further, in 1980 Gaines suggested that the notion of distinction defines a

system. It appears that there is a vast and potentially unexplored framework of possibil/stic systems which could unify the approaches of Gaines and Shaw [19], of H/sdal [5,21], the checklist paradigm of Bundler and Kohout [23], and the topic of this paper. "l~is awaits further research.

Referene~

[I] W. Bandler and L.J. Kohout, Semantics of implication operators and fuzzy re'-ational products, Intemat. J. Man-Macl~ne Stud. 12 (1980) 89-116. Reprinted in: A. Mamdani and B.R. Gaines, Eds., Fuzzy Reasoning and its Applications (Academic Press, London, 1981) 219-246.

[2] W. Bandler and L.L Kohout, Mathematical relations, Fuzzy Research Project Technical Report FRPo23, Dept. of Mathematics, Umversity of Essex, August 1982. Reprinted in: M.G. Singh et al., Bds., International Encyclopedia of System~ and Control (Pergamon Press, New York- Oxford, 1987), in press.

L3jr 1 ~n .... o G~.~,~ . . . . . . . . . o,~d ~. J Kohout, Possible automata, in: Proceedings of 1975 lnternat. Symposium on Multiple-Valued Logics (IEEE~ New York, 1975) 183-196.

~4] B.R. Gaines and L.L Kohout, The lo~c of automata, lnternat. J. General Systems 2 (!975) 191-208.

[5] E. H/sdal, A theory of logic based on probability, Part !, The TEE model, Remarch Report (lastit,,~te of Informatics, Uni~,ers/ty of Os~o, Oslo, 1984).

[6] L.K. Kohout, The Pinkava many-valued complete logic syste~ns and their application to the design of many-~ah~ed switching circuits, in: D.C. Rine, Ed., Proceedings of 1974 International Symposium on Multiple-Valued Logic (IEEE, New York, !974) 261-284.

~ ~ ~f poss~ba~y 367

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[131

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{ 11

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