Int. J. Man-Machine Studies (1981) 15, 213-220

Towards a reconciliation of fuzzy logic and standard logic

JOHN FOX+

Department of Medical Physics and Clinical Engineering, and Medical Research Council Social and Applied Psychology Unit, University of Sheffield, Sheffield, U.K.

(Received 9 November 1979, and in revised form 5 March 1980)

Haack (1979) has questioned the need for fuzzy logic on methodological and linguistic grounds. However: three possible roles for fuzzy logic should be distinguished; as a requisite apparatus--because the world poses fuzzy problems; as a prescriptive appara- t u s - t h e only proper calculus for the manipulation of fuzzy data; as a descriptive appara tus--some existing inference system demands description in fuzzy terms. Haack does not examine these distinctions. It is argued that recognition of various different roles for fuzzy logics strengthens the pragmatic case for their development but that their formal justification remains somewhat exposed to Haack's arguments. An attempt is made to reconcile pragmatic pressures and theoretical issues by introducing the idea that fuzzy operations should be carried out on subjective statements about the world, leaving standard logic as the proper basis for objective computations.

Introduction

Haack (1979, 1980) has examined the foundations of fuzzy logic with a view to identifying its contribution to theories of inference and reasoning as compared with th ". contribution of the many-valued logics. She considers that there are two main justifications for the introduction of fuzzy logic; linguistic evidence that the predicates " t rue" and "false" are themselves fuzzy rather than absolute, and the methodological argument that a fuzzy calculus simplifies the manipulat ion of certain data. Haack challenges the linguistic evidence for fuzzy truth and falsehood and further argues that any practical implementat ion of a fuzzy calculus quickly reveals its dependence on intuitive elements; any a t tempt to make use of informal intuitions within a formal apparatus, she says, creates problems which can only be resolved by the intrGduction of mechanisms of " f ea r some" complexity. Haack concludes that we do not need fuzzy logic.

Curiously, however, fuzzy logic and related ideas seem to be proving of practical potential (for example in engineering) and theoretical value (for example in psychology). It seems necessary, therefore, to examine in what ways fuzzy logic may be put to work and whether any of these ways escape Haack ' s criticisms.

What might we need fuzzy logic for?

At least three possible roles for fuzzy logic may be envisaged; as a requisite a p p a r a t u s - - because the world poses fuzzy problems; as a prescriptive appa ra tus - - the only proper

t Now at: Research Computer Unit, Imperial Cancer Research Fund, Lincoln's Inn Fields, London WC2A 3PX, U.K.

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calculus for fuzzy data; as a descriptive appa ra tu s - - some existing inference system requires description in fuzzy terms.

F U Z Z Y L O G I C M A Y BE R E Q U I S I T E

The argument here is that the physical world embodies relationships which are intrinsically fuzzy and therefore some sort of calculus of possibilities that takes account of the fuzziness is necessary. It is not just that the predicates "tal l" or "beaut i ful" are linguistic (or, more generally, subjective) approximations to precise ranges of height or beauty, because they refer to attributes which have objective consequences. Tall trees are vulnerable to gale damage, irrespective or whether they are observed. It seems a natural extension of this fact that an assertion that a particular tree is tall (with respect to its vulnerability) has a fuzzy truth value; the truth of the s ta tement is to be evaluated via its relationships with members of the set of damaged trees.

There is an obvious precedent for introducing some sort of calculus--probabil i ty theory. The events of the physical world are uncertain and probabilistic formalisms (and linguistic informalisms) allow us to compute and communicate our degree of uncertainty. Of course the existence of probabili ty theory may lead us to suspect that fuzzy logic is little more than a minor variant of it, since one could make statements, about the likelihood of gale damage, in our example, that for all practical purposes are equivalent to s tatements about tallness. I think this would be to miss the greater generality of fuzzy concepts. Suppose for instance that I wanted a decision rule with which to assess the priority of tree felling, where the relevant concerns are that the tree should be tall enough to give an economic return but it should not be allowed to grow so tall that there is an unacceptable risk of damage from the elements. In terms of fuzzy logic the following priority rule is entirely meaningful:

priority(T) = maxfvalue(T), risk(T)). (1)

By treating each function of T (a tree) as one having a truth value a fuzzy computat ion may be carried out over a wider range of dimensioned functions than would be allowable for probabil i ty theory. In this case we are able to mix probabilistic functions (risk) and non-probabilist ic functions (value) at will. All that is required is that we supply compatibili ty functions to t ransform values f rom the natural domain of the function into the normalized truth domain.

FUZZY LOGIC MAY BE PRESCRIPTIVE

If one were to suppose that fuzzy logic were requisite as a general purpose calculus then one immediately encounters the question of whether (a particular) fuzzy logic is the proper f ramework for carrying out computat ions on fuzzy data. To pursue the link with probabili ty theory, the evaluation of the s ta tement p(x) & p(y) is straightforward once the values of p(x) and p(y) are available. The realization of the conjunction opera tor as a multiplier is proper in the sense that it is justifiable f rom axioms of set theory. Fur thermore the f ramework of probabil i ty has led to the development of many procedures with provably optimal propert ies for information processing (such as Bayes ' rule for calculation of posterior probabilities). In short, probabili ty theory is p roper in the sense of being justifiable and efficient. The analogous issues for fuzzy logic are whether its fundamental elements are justified [for example, can the realization of x & y

TOWARDS FIJZZY/STANDARD LOGIC RECONCILIATION 2 1 5

in "s tandard" fuzzy logic as min(x, y) be justified'?] and whether the efficiency of the whole apparatus is demonstrable .

Haack complains that the per formance of the logical system is impossible to evaluate because the contribution of any chosen set of primitive operators (max, rain, etc.) cannot be separated from intuitive judgements about set membersh ip functions. For example, suppose that we identify the fuzzy predicate beautiful with the constituents fashionable and shapely, and we know how fashionable and shapely Alex is, then we might wish to assert that:

beautiful(Alex) = fashionable(Alex) & shapely(Alex) (2)

and further that

truth(beautiful(Alex)) = rain(truth(fashionable(Alex)), truth(shapely(Alex))), (3)

or some such. The trouble is that the machinery of fuzzy logic is being applied to predicates whose semantics are intuitive and beyond scrutiny. Under such circum- stances how can we possibly determine whether the computat ions are efficient or even justified? It seems to me that this objection has some force (for instance it is not entirely clear that subjective number systems always display even the e lementary propert ies of formal number systems, such as transitivity). On the other hand it may well prove possible in practice to employ actuarial techniques in the compilation of se t -member - ship functions, in a way which is analogous to the compilation of probabil i ty distributions f rom frequency counts. Alternatively it may be possible to develop pro- cedures by which rationality in the definition of subjective se t -membership functions is enforced, or at least irrationality detected. Finally if we limit ourselves to the mat ter of efficiency we may note that designers are often more concerned to improve on the existence of some sub-optimal inference procedure rather than to achieve perfection (e.g. Mamdani and Assilian, 1975; Shortliffe, 1976). So the criticism, that the foun- dations of fuzzy logic are not clear enough to specify when and how it should be properly used, may be overstated. However , it does seem to be correct that, as presently stated, there is not a clear division between the contribution of membership functions and truth-value computat ions to fuzzy reasoning, and that for all its pragmatic and heuristic virtues fuzzy logic may rest on somewhat insecure foundations if highly subjective data are employed.

FUZZY LOGIC MAY B E DESCRIPTIVE

It seems possible that inference systems exist whose behaviour may be naturally described in fuzzy terms. It seems to the present author that the very existence of such a system would imply a need for fuzzy concepts irrespective of whether there are more "serious" justifications that this particular logical system is required to cope with a fuzzy environment, or that the logic confers greater efficiency than some alternative inference mechanism. Haack ignores this empirical matter. However , evidence is accumulating that psychological theory could benefit f rom the use of fuzzy logic or some variant of it; the first hints came from linguistics and Lakott ' s (1972) theory of hedges as well as Zadeh ' s writings, but more recent empirical research suggests that membersh ip of a class may subjectively be a continuous kind of relationship (Rips, Shoben & Smith, 1973; Rosch, 1973, 1975). It seems, fur thermore, that we are able to process fuzzy data

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in a consistent and systematic manner (Oden, 1977) and that cognitive processes in medical problem-solving may be accurately modelled using a close relative of a fuzzy logical scheme (Fox, 1981). The history of at tempts to apply standard logic to human reasoning, on the other hand, is littered with failures (see, for example, Braine, 1978; Joffe-Falmagne, 1975) and if fuzzy logic were to clarify this field at all it would be justification enough for its introduction.

It appears then that there may be good pragmatic reasons for the introduction of some sort of calculus that unifies computat ions of an imprecise kind, and that human information processing may exploit just such a calculus. There are precedents for the development . The multivalued logics a t tempted to deal with some of the same linguistic and formal problems of classical logics, though without the radical reinterpretat ion of truth proposed by Zadeh. On the other hand, probabili ty and variants on it such as Shortliffe & Buchanan 's (1975) theory of imprecise reasoning respond by translating imprecision into terms of uncertainty. But rather than compromising with vagueness or imprecision by creating special categories (e.g. not true but not false) or by use of translation, fuzzy logic assimilates it entirely. The price, of course, is that truth loses its traditional precise and objective quality in favour of an intuitive semantic base. Haack argues that this step is object ionable and that its methodological advantages are illusory.

T o w a r d s a reconci l iat ion

Is the solution to ignore the logicians? Should we say that the performance of a logical system (whether the system is natural or contrived) should be the sole criterion for its admissibility, irrespective of whether it is logically secure? I think not, this approach is likely only to be a palliative as the system is turned on to more and more demanding problems. It would be preferable if we could find some sort of reconciliation between the views of Haack and Zadeh; interestingly the seeds of such a reconciliation are to be found in their own writings since it appears to be possible to preserve the concept of fuzzy se t -membership without challenging the classical meaning of truth.

To illustrate, we should return to the linguistic evidence for fuzzy set -membership . A lot of weight has been placed, of course, on the observation that many linguistic predicates, such as tall or beautiful, have poorly specified semantics. Zadeh and others have argued that the sense of predicates like tall concerns the extension to the set of tall things and that, crucially, membersh ip of that set can profitably be mapped onto the range of tallness values of the individuals in the set. Fur thermore if we say "X is very tall" we map it into a particular sub-range of values, and if we say " X is extremely tall" we map it into a different sub-range of values, and so on. Thus fuzzy concepts are legitimized by an appeal to the linguistic usage of predicates and various modifiers, and, in a sense, truth is relegated to the role of an underlying metric, should we wish to carry out any sort of computat ion on the propositions.

But Haack is dubious about the legitimacy of exploiting truth in this way and denies that linguistic usage is in its favour. For example she points out that if truth is a predicate of degree then it should be possible to apply the same range of adverbial modifiers that may be used with other predicates of degree. But while the s ta tement "John is somewhat aggressive" seems acceptable enough she feels that locutions like "somewhat t rue" are not, and that this must be because " somewha t " is restricted to modifying

TOWARDS F U Z Z Y / S T A N D A R D LOGIC R E C O N C I L I A T I O N 217

predicates of degree while " t ru th" is not among predicates of the type. Elsewhere (Haack, 1980) she elaborates this argument:

"Of the following modifiers of 'tall' (I'll call them "degree" modifiers): 'extremely', 'rather', 'fairly', 'pretty', 'relatively', 'unusually', none applies to 'true'. And of the following modifiers of 'true' (I'll call them "success" modifiers): 'absolutely', 'perfectly', 'wholly', 'almost', none applies to 'tall'."

At first sight these examples seem pretty damning evidence but they benefit from refinement. Take the first example "somewhat t rue". Leaving aside Haack's linguistic intuition that the locution is unacceptable, which I do not entirely share, one might also comment that the locution "somewhat certain" is unacceptable and it should follow that "cer ta in" cannot be a predicate of degree. But in common usage one is allowed to express degrees of certainty (though perhaps not with this particular form of words) and indeed the notion of mathematical probability seems to depend upon some such concept. Perhaps one should doubt the veracity of these intuitive linguistic arguments, and the clue to why lies in Haack's elaboration of the argument. She observes that. certain modifiers apply to "tall" but not to " t rue" , and vice versa, but the difference here is surely that "tall" is an open-ended dimension, tallness can increase without limit, and so "perfectly tall" or "wholly tall" are meaningless. In short there are semantic restrictions on the application of modifiers to predicates which are separate from whether or not they must be applied to predicates of degree.

Fur thermore it seems to me that locutions like " ra ther t rue" and "extremely t rue", while clumsy, are a great deal more acceptable than "wholly tall". They are not usual or common constructions but I find them perfectly meaningful in a way that "wholly tall" is not. But what do they mean? Zadeh defines truth to be a predicate of degree, normalized on the unit interval, so that " ra ther" and "ext remely" have straightforward senses as ranges of truth values within the interval. As Haack points out, these ranges, and the functions mapping them on to truth values, cannot be specified in a principled way so we have at some point got to introduce intuitive judgements and consequently lose the hard-won concept of logical truth. One way to preserve the concept of logical truth and the machinery of fuzzy logic is to make a distinction between truth and what I shall call " t ru th-prominence" . To illustrate the distinction consider the proposition

Goliath was tall. (4)

Now if Goliath was, say, six feet tall then we would say that the proposition is true, but if Goliath was nine feet tall we would not only approve the proposition but fur thermore would be willing to say it is "very t rue", "strikingly t rue", "particularly t rue" and possibly even "extremely t rue". In other words we not only recognize that the proposition has the property of being true, a binary attribute covering the necessary condition for the validity of the proposition, but also that it has an ancillary attribute of degree.

I propose that this ancillary attribute of t ruth-prominence is intrinsically subjective, infinitely variable and the proper base for fuzzy logic and fuzzy sets, while truth is regarded as objective (whether or not it is verifiably so), binary-valued, and the domain of classical logic.

To use Haack's (1974) concepts, if fuzzy logic is applied to t ruth-prominence values then fuzzy logic ceases to be a "rival" logic and becomes an "extension" of classical

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logic. It is not designed to substitute for classical logic, rather it is compatible with its precursor and is applicable to a different family of (subjective) computations.

Separat ion of truth f rom t ru th-prominence seems to have a number of advantages. First of all we can give an answer to the question of why locutions like " ra ther t rue" and "ext remely t rue" are not wholly unacceptable, although they are admittedly awkward. I suggest that this is because of a conflict of selection restrictions associated with different readings of the assertion. " I t is extremely true that Goliath was tall" has at least the following readings:

" 'Gol ia th was tall ' is true to a high degree" ; (5) " 'Gol ia th was tall ' is strikingly t rue" ; (6) " 'Gol ia th was tall ' is true by anybody 's s tandards". (7)

Sta tement (5) a t tempts to apply a graded evaluation term inappropriately to a pro- position about history, while s tatements (6) and (7) involve quite reasonable inferences or presumptions which may be made by a reader given a fact of history. Since the reader has these subjective alternative points of view on the s ta tement he does not find it as object ionable as "Gol ia th is wholly tall" for which there is no obvious viewpoint f rom which it would be meaningful.

In addition to avoiding some of the linguistic difficulties that are raised by the concept of fuzzy truth, the proposed notion of t ru th-prominence also avoids the methodological problems identified by Haack in her complaints about the introduction of intuitive elements into a formal system. I have commented that fuzzy logic applied to truth- prominence values is an extension of standard logic, not a rival to it. This means that theorems within the fuzzy domain are compatible with, and hence constrained by, all the standard theorems. Suppose we encounter this assertion:

Goliath is tall and loves children. (8)

If both the proposit ions are classically true then (8) is true, otherwise it is false. If we make use of t ru th-prominence values in place of truth values this does not abrogate the theorems which apply in standard logic for the reason that t ru th-prominence rests upon truth. A s ta tement cannot have a positive t ru th-prominence if the s ta tement is false. By implication normal proposit ional analyses still apply and so (8) cannot have a truth- prominence of its own independent of the t ru th-prominence values of its component propositions. We may now choose some machinery for dealing with conjunction, disjunction and so forth, but applied to t ru th-prominence values rather than truth or fuzzy truth values, without encountering any unexpected inconsistencies or contradictions.

Use of t ru th-prominence in conjunction with truth also minimizes the difficulties associated with the concept of error. If we allow any truth function at all, as in fuzzy logic, there is nothing to prevent us giving a false s ta tement a fuzzy truth value, even if that value is small. Hence if it were known that Goliath was habitually cruel to women and children there is still nothing to prevent us assigning a positive truth value to the assertion "Gol ia th loves children". By distinguishing truth and truth prominence, however, and fur thermore requiring that a t ru th-prominence cannot be positive if the truth value is not positive, we prevent such abuses. This constraint underlines the way in which t ru th-prominence offers the subjective counterpar t to objective truth which continues to fulfill its traditional role as a standard of "correctness".

T O W A R D S F U Z Z Y / S T A N D A R D L O G I C R E C O N C I I . I A T I O N 219

Although t ru th-prominence may not violate truth it can be used to qualify truth in important ways. For example the s ta tement

Richard Nixon is an important figure (9)

was certainly true in 1974, but would we wish to assert it now? If we try to deny it then we are getting close to suggesting that truth is linked to time, an idea fraught with difficulty. An alternative is to suggest that (8) is true, but that its prominence declines with time. (Indeed it may decline with many factors including individual, cultural and geographical factors.)

Tru th-prominence is a concept with practical application. In designing information processing systems for computers we often require some scheme for sorting out priority among competing processes. Prominence offers an important unifying basis for employing diverse priority heuristics (Fox, 1981). For example suppose we consider a system of production rules of the form

If A & B & C then-do X, Y, Z, (10)

then for any sizeable set of rules, where a number of rules may be candidates for execution at any one time, we have to face the "conflict resolut ion" problem. If in choosing we only consider the truth values of the premises of the rules then this is no help because the truth values for all the candidates are the same, " t rue" . If, on the other hand, we consider the t ru th-prominence of the premise of each rule this provides a parameter which can be used for selection. Returning to the idea that t ru th-prominence offers a common metric for computat ion over an unlimited range of subjective factors (risk, value, beauty, tallness, recency, etc.) we can see that it also provides a common medium for control of information processing. Elsewhere we have confirmed that the idea does indeed provide a viable basis for controlling this kind of processing (Fox, 1981; Fox, Barber & Bardhan, 1980).

Conclusion

In this paper we have discussed Haack ' s objections to the need for fuzzy logic. I have a t tempted to clarify what might be meant by "need " and it is my view that some sort of general imprecise or fuzzy calculus is required, and indeed that there is some evidence that human thinking reflects this requirement. Nevertheless Haack ' s criticisms of the linguistic and methodological justifications for Zadeh ' s concept of fuzzy sets and fuzzy logic have considerable force. I have presented the concept of t ru th-prominence as a substitute for fuzzy truth in the belief that it may offer a basis for reconciling the pragmatic need for a fuzzy logic and the theoretical weaknesses in the existing formulation.

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