Keeping An Eye On The Scientists: Bart Kosko's Fuzzy Thinking Tries To Save Logical Positivism

by

Paul Rezendes

I. Introduction

Bart Kosko's Fuzzy Thinking (Hyperion, New York, 1993) manages, despite deep contradictions in the declared philosophical underpinnings of his views, to present a useful portrait of the fuzzy logic engineering revolution that put the intelligence of Japanese electronics ahead of the US's. The deepest contradiction is caused by his early distinction between math and fact, the problem of "fit" between mathematics and the world. Despite making the categorical distinction between mathematical statements and fact statements, he spends much of the rest of the book using mathematical models to present key ideas, as if they showed the inner nature of the real-world situations involved. Eventually, he seems to want to undercut the distinction, although his efforts are not particularly persuasive.

II. Kosko As Erstwhile Logical Positivist

Kosko dresses himself and his thoughts up as anti-establishment, iconoclastic and "eastern." He is thoroughly western, however, in his commitment to the basic and defining tenents of logical positivism. His commitment to that epistemology and world view appears naïve and, ironically, is maintained despite his verbal protests against the doctrine. (pp. 89-90). Kosko declares that logical positivism is the working philosophy of scientists, medicine and engineering [FT at 8]. Others appreciate its weaknesses. "Logical positivism is now almost universally thought to have failed, and not only as an argument against objective values and the possibility of metaphysics." John Post, Metaphysics: A Contemporary Introduction, p. 17 (Paragon House, NY, 1991). But Kosko's faith in basic principles of positivism remains firm. At various points he declares the main articles of faith:

(1) the unbridgeable distinction in kind between (a) statements of math or logic, i.e., formal systems, which permit 100% certitude and truth in their conclusions and as a matter of principle are not about the world, and (b) statements of fact which are neither certain nor absolutely true (pp. 81-83);

(2) An analysis of meaningfulness (and the ability to be a vehicle of truth or falsity) in terms of verification and falsification (pp. 257-59);

(3) A foundationalist epistemology which identifies certain knowledge "bits" - generally, sense experience - as the "best" basis for our knowledge (p. 96);

(4) A constructivist/reductionist approach to knowledge: that the knowledge bits are built up (or "constructed") into knowledge structures in some way while knowledge of any complexity or depth must be reduced or deconstructed into those knowledge bits to be properly handled and known. (At p. 97 he says that he constructs a gray model out of black and white bits.);

(5) A non-cognitivist/ emotivist analysis of ethics. (p. 260-61).

Although Kosko declares that fuzzy logic is the culprit, this death of ethics occurs at the hands of positivist assaults, with nothing whatsoever new added to it by fuzzy logic. It is all too reminiscient of Hume's fork and Ayer's principles for determining metaphysical nonsense. See FT at pp. 256-60.

Despite his underlying faith in positivism, Kosko is not consistent. Hume's fork is sufficient for his rejection of ethics, but Kosko nonetheless attempts to tackle the metaphysical question of existence. P. 269. He acknowledges the vulnerability of the god concept to positivistic analysis, pp. 277-78, but that does not stop him from weighing in on the nature of god as if making an existence statement. Pp. 279-81.

III. Kosko's Influences And His Contribution

What Kosko brings to the table that has the appearance of newness is his apparent rejection of the principle of the excluded middle. The leit-motif of the book is his expressed preference for "A and not-A" over "A or not-A" and his rejection of "bivalent" logic. He attempts to draw an analogy between rejection of the principal of the excluded middle (that is, A or not-A, but not both, and there is no other alternative) and accepting eastern, particularly Buddhist, thought over European, Aristotelean thought. Perhaps the biggest question raised by close analysis of Kosko's reasoning and arguments is this: in what sense and to what extent does Kosko really reject the principle of the excluded middle? Before we examine that question, I briefly discuss the background to Kosko's thought, and what that thought really involves.

Fuzzy Thinking is a mathematical application of multi-valued logic. The math-like logic developed by Frege and Russell permitted only two truth-values: "true" and "false." Multi-value logics are a variant of this, retaining the formal structure, but permitting more than two truth values. Logician Jan Lucasiewicz pioneered multi-valued logic in the early twentieth century, and others have taken up and varied the theme. Some versions permit three values (0 for false, ½ for either some intermediate value or for indeterminacy, and 1 for true). Others permit a greater number. Still others permit a continuous range of values, and define the truth of a statement and its contradictory with the formula t(a) = 1 - t(~a), where t(a) is the truth-value of statement a, t(~a) is the truth-value of statement not-a, and these two values range between 0 and 1. That is, there is a direct relationship between the quantified value of the truth or falsity of a statement and its contradictory, with one going up in direct relation to the amount the other goes down. Kosko's work is derived from this last version of multivalued logic. This continuous-value version of truth is applied to "fuzzy sets," a concept developed by Lofti Zadeh. In a fuzzy set, the members are members to one degree or another on the continuum of 0 to 1.

A fuzzy system associates two or more concepts by a rule, but uses quantified truth values to average out results. Suppose, for instance, that a piece of temperature data falls within two cognitively adjacent concepts, say "warm" and "hot." It will fall within each to some degree, and that degree will be assigned. Some operation is associated with each concept as its related output. For instance, in the case of an air conditioner, one data category - a group of certain generic temperature classes characterizing a continuum (cool, temperate, hot) -- will be associated with another category -- certain generic fan speeds (fast, medium, slow), with each class in the first paired with a class in the second. Each of the temperature classes is "fuzzy" in that its data contains data also in an adjacent category; but the closer the data is to the defining center point for each class, the closer its truth value will be to "1," while its truth value in the adjacent category will grow further from one. When a piece of temperature data is received, it is analyzed for its position within the first category (90%, 80%, etc.), and the next (10%, 20%, etc). The output class associated with each of the data categories is modified for the percentage of truth assigned to its related data category and the average result defines the output. By way of example, consider the following simplified example of a fuzzy control system for an air conditioner fan (the actual math is somewhat more complicated, and the full example appears and is explained at FT, pp. 161-178):

1. Cold (0 to 60 degrees), associated with slow speed (0 - 40 rpm);

2. Temperate (50 to 80 degrees), associated with medium speed (20 - 60 rpm);

3. Hot (60 to 100 degrees), associated with fast speed (40 rpm to 80 rpm).

When data of 78 degrees is received, the fuzzy system will calculate that this data falls (say) at the 2% true within "temperate" and 98% true within "hot." The computer will then calculate the average output result, which, for sake of simplicity, could here be calculated as (2% "temperate" X 45 rpm for temperate data) + (98% "hot" X 60 rpm for hot data) = 59.7 rpm.

The two critical philosophical notions Kosko wants us to see at work here are (1) "fuzzy concepts," i.e., generic terms which are capable of having examples which are more or less within them, and (2) multi-valued "truth" quantified as a measure of the extent to which data or output fall within the core or penumbra of a fuzzy concept. However, viewed mathematically, all Kosko has done is to permit ranged input data to produce ranged responses instead of fixed-point responses. Rather than turning up to full blast if the temperature rises to 80, this air conditioner will constantly monitor the air and respond with graded results, a little cooling at 73, more at 80 and more at 92, but reacting all the way up. It provides buffered responses at all temperatures. To what extent does any of this require quasi-mystical discussion of partial truth and falsity or the blending of yin and yang?

IV. Internal Contradictions In Kosko's Exegesis

It is difficult to take Kosko's categorical rejections of bivalence seriously. He himself uses a statement of the form A or ~A. At page 151, he argues: "Zadeh seemed to say that either he was right or most of science and western culture was right. One or the other. The choice seemed clear. I think he was right." P. 151.

If we take as categorically true the statement that, always and everywhere, A and not-A, and this is a matter of varying degrees, then what are we to make of the distinction between matters of fact and truths of logic? Kosko seems to take this as an essential part of his case against bivalence in science: that mathematics does not directly apply to the world and, unlike bivalent logic in which statements are 100% true of false, science deals in partial truths. The types of truth each one uses, he says, are categorically different. Therefore, there is at least one case acceptable to Kosko in which "A and not-A" does not apply.

Kosko flirts briefly with the rejection of this distinction. Pp. 275-76. But the distinction is not, in the end, rejected. Rather, Kosko purports to point out what he believes to be a parallelism between math and science. That does not, of course, undercut the distinction so much as raise the problem of coordination across the divide. Kosko can at best suggest an analogy to two separately growing bushes, each of which approach each other in shape while growing from different roots. (pp. 276-77) The analogy, however, is bad. What are the "branches" or "leaves" of each bush and how do we match them one-to-one? Math theorems are not theorems about world events or their explanations. And scientific discoveries are not discoveries about mathematical theory. We cannot make one-to-one correlations between a math theory and a scientific theory. Rather, math is the language used to express scientific theory. Further, Kosko is probably just wrong about science following math. It seems far more likely that the two are in a symbiotic relationship. When observed fact does not conform to mathematical model, alternative mathematical systems (e.g., curved space geometries) which previously were little more than intellectual curiosities are looked to for model-building. Or, alternative maths are developed to deal with problems presented by the sciences - as with the development of differential calculus. The suggestion that math is "prior" in any sense is unsupported. But, further, the distinction in kind Kosko draws had not been undercut, and bivalence applies unmodified in the realm of fact and mathematical statements in Kosko's universe.

Given the importance of the science/math distinction to Kosko's case for non-bivalence in science, one has to wonder why he trots out what he describes as paradoxes lurking at the heart of mathematics: Russell's paradoxes concerning classes. (pp. 25-26; 97ff.) In fact, at one point he seems to indicate that his task is to prove even logic and mathematics is, in some sense, fuzzy notwithstanding the sharp distinction between fact and math/logic. (p. 97) This brings us to the real problem with Kosko's entire approach: despite paying lip service to "fuzziness," he ultimately depends on bivalence at three levels: "foundational," "input" and "output."

Consider the air conditioner example. While the input categories ("hot", etc.) and the output categories ("fast," etc.) are treated as fuzzy and having partial truth, not so the input and output themselves. Is there any question but that the fuzzy system depends on being able to say truthfully "it is 78 degrees right now" or "the fan speed is now 100 rpm"? The model does not calculate relative truth or likelihood here. There is simple bivalence: either it is or it isn't 78 degrees, and either the fan is turning at 100 rpm or not. If Kosko wants his system to work, he has to avoid an infinite regress of efforts to find the weighted average.

V. How Far Is Kosko Committed To The Rejection Of Bivalence?

Kosko does not need to wax philosophical to get his result. He does not need to think of overlapping A's and not-A's in our everyday life. He only needs to permit varying responses at varying input data without having to digitize each and overcomplicate the machine, and that is what his model does.

Kosko wants to treat the application to the world of quantified concepts described in a mathematical model as being a matter of fact that is subject to non-bivalence: "hot," the class of temperatures between 60 and 100 degrees, may be truly predicated of temperatures in that range, but only to some measured degree. Here Kosko stumbles on one of the principles with which he starts the book: there is the problem of fit between math and the world. When he defines the fuzzy set "hot," he creates the internal logic of the formal mathematized system - 78 will always produce a truth value relative to "hot" of (say) 98% hot, no matter what (subject to redefinition by a learning system, of course), because that is how the system is designed. The statement "78 degrees is 98% hot" (more precisely, "it is 98% true that it is hot when it is 78 degrees") is not a factual statement at all. It is analytically true; formally true as a result of the internal logic of the system. It has the same sort of truth as Kosko set aside for mathematics generally when he admitted that bivalent logic is sufficient for mathematical statements such as 1+1=2. The only function of the "truth" value in the fuzzy system is to provide a method for calculating varying responses from varying data. The same function could have been served by a factor referred to as the "buffer factor" rather than "truth." Further, the term "hot" that we apply in day-to-day usage is not the "hot" as defined in the formalized system. On this analysis, to say that a fuzzy system requires that we surrender bivalence in everyday life is unnecessary. When I say it is hot today, it well could be 100% true in our ordinary sense of true, even if the fuzzy system has assigned 82% truth, with respect to the formally defined set of "hot" temperatures, to today's temperature.

But Kosko spends a good deal of time trying to argue for non-bivalence, for "A and not-A". It is worth spending some time examining what arguments he uses, why he uses them and what they might prove.

A. The Sorites Paradox

The sorites paradoxes are simple and follow the same pattern. Here is "the heap": One grain of sand does not make a heap. Adding one does not make a heap; and so on. The same argument can be developed for the gradual changes from red to yellow, or night to day.

The sorities paradoxes are unconvincing. We may nod assent to them when recited but, as with Zeno's paradoxes, we wouldn't bet on them. The runner overtakes the tortoise; sooner or later, there are enough grains for a heap; and night does become day, even if we cannot say precisely when. Somewhere there is a problem in the reasoning or in the argument. We might not know where, but it must be there because we are capable of distinguishing cases.

It is not necessary to directly resolve the sorites paradox to dull its sting. The paradox relies on certain features: an attribute continuous through time, and gradual change. Our everyday experience, however, does not present itself as a continuum. We see discontinuity and difference, demarcations. We recognize boundaries and contradistinctions. We also recognize clear cases. Kosko also recognizes clear cases. The construction of the fuzzy system requires the development of curves to assign data truth value within a concept. The curves in his air conditioner example or in my simplified example are triangles. The curve for hot in my example, starts with 60 degrees and ends at 100. At 60, the truth value is 0, rising to 100 at 80, then dropping to 0 again at 100. (The system in my example would fail. It would need an additional category for temperatures in the 80 and up range) Centering such triangular approximations on a spike where the truth value is 100% is to admit that there is at least one clear case, one paradigm. Such paradigms or clear cases permit us to break up the continuity of the continuum necessary for the sorites paradox to work. Perhaps we have not yet identified the fault in the reasoning; but we know it is wrong, and can at least show that it is, if not how.

Kosko's purpose in citing the sorites paradox was to demonstrate "A and not-A." The uncertainty we feel in the "middle position" between clear cases on the continuum, however, does not show that A and not-A. It might be that at that point neither A nor not-A, or it might mean something else. Meanwhile, at the clear cases (in fuzzy systems, at the 100% truth points) A is not not-A. While Kosko would reduce this to a "special case of grey," two things need to be acknowledged. First, recognition of "clear cases" breaks up the continuum necessary for the sorites paradox to work. Second, Kosko does not make an important distinction which he needs to observe to make some of his later arguments. I will discuss this later in section VI.

B. Russell's Paradoxes

Russell's paradoxes are cited, it appears, to show that fuzziness, a state of "A and not-A" obtains in math as well. But Russell's paradoxes do not directly condemn math. They only condemn math if both (1) the fixes offered by Russell do not work, and (2) Russell's program of reducing math to logic through set theory is considered necessary. If either of these is not the case, math is not condemned to Kosko's version of fuzziness.

It is possible to reject Russell's program and simply accept mathematics as equally basic with logic. This complicates things, to be sure, but it does not seem totally out of bounds.

Putting this aside, let's look more closely at Russell's paradoxes. They deal with problems of the logic of set-inclusion and self-reference. The barber problem is one example: a barber shaves all and only those men who do not shave themselves. If he shaves himself, then he does not, and vice-versa. This appears to produce a situation in which the statement "the barber shaves himself" ("TBSH") is both true and false, and Kosko would make this a case of "A and not-A." It is, instead, a problem of formal unsolveability. Given the statements of the problem, one cannot formally resolve the truth of TBSH. One wants instead to visit the barber and watch what he does. Of course, that changes the sense of "truth" with which we are working. If we decide to resolve the matter by looking, we are talking about correspondence truth under Kosko's scheme, which is different from the coherence sense that makes this a paradox.

The problem of this shift in the different senses of truth is obvious in what is now known as Russell's paradox: the set of all sets which do not have themselves as members. We cannot here simply go out and "look at the barber"; all we have is the formal logic of the system. And within the system, it appears that if this set is a member of itself, it is not, and vice-versa. But the net result is the same as with the barber problem. It is not that it is equally true that the set of all sets which do not have themselves as members is and is not a member of itself. It is undecideable without a "fix," some additional axiom that avoids the problem. Whatever the limitations of Russell's theory of types, it at least offers the possibility of providing a fix, a criterion of decideability. Kosko rejects Russell's efforts at fixes as ad hoc and unsupported. Perhaps this is a function of his sharp distinction between correspondence truth, the truth of science, and coherence, the truth of logic. Given that these are the only two ways that something can be true in Kosko's scheme, the fact that a proposition fixes a paradox without itself being either coherent with a system or correspondent to a fact means that it cannot be true. But why are these two criteria exhaustive for "truth"? Treating them as jointly exhaustive of the field of truth presents the same core problem faced by verificationist theories: their criterion of truth does not fall into either camp, and a number of things we accept as true will likewise fail. Under a broader conception of truth, a fix can be true simply because it is a fix to a system which otherwise works.

Once again, it is not necessary to solve Russell's paradox directly to take the sting out of it, because it does not show what Kosko thinks it shows. First, it does not demonstrate a statement that is both true and false; it demonstrates a statement that is formally undecideable. More importantly, it does not show that every statement within the system of math is itself both true and false. Math statements are formally decideable. Buddha ("A and not-A") does not reign supreme in math, and math has not been shown to be fuzzy.

VI. The Bipolarity of Kosko's Thinking, And Other Responses To Bivalence

There is much that is muddled in Kosko's book. He often confuses whether something is or is not known with certainty with the question of whether or not its truth is of the bivalent sort. See, e.g., pp. 8, 85-6, 90. Of course, showing that we can never prove scientific statements truth with 100% certainty is not the same as saying that they are not either wholly true or wholly false. I may believe, to a fair degree of certainty, but not 100%, that my son is doing homework in his room. But it either is or is not the case that homework is being done.

As noted at the outset, one of the deepest muddles in the book is the rejection of math as descriptive of reality while at the same time using math models at critical junctures. I will here note two examples, criticizing their logic, but pointing them out primarily to show that Kosko has not fully absorbed the message that math has only a rough fit on the world.

At p. 101 and following, Kosko discusses his view that paradox is the point where the truth of two contradictories is equally balanced. But he puts the point in purely mathematical terms, using the mathematical model of a one-dimensional continuum. He states "[a] simple argument proves that self-reference paradoxes correspond to the mid-point of the truth-line from zero to one." He then asserts that paradox means that t(a) = t(~a). But, he says, the truth of t(a) and t(~a) are related by the mathematical formula t(~a) = 1 - t(a). Combining the two statements, and using simple algebra, t(a) = t(~a) = ½. On this reasoning, paradox is the mid-point of the line from 0 to 1, i.e., at ½. But this reasoning does not prove that "self-reference paradoxes correspond to the mid-point of the truth-line from zero to one" unless we already buy into the notion that truth values can be quantified from zero to one, that their relationship is linear, and that the relationship is bipolar as between two contradictory statements. In other words, we must already buy into a math-like model of truth for this proof to work; and the conclusion it presumed in the construction of premises in that model. But why should we accept that truth, especially correspondence truth, can be modeled in this way? As we have already seen, the success of fuzzy systems is no proof because what is referred to there as "truth value" could as easily have been called "buffering."

Another example of Kosko's commitment to math modeling despite his protestations to the contrary is his proof of why there is something rather than nothing. In that argument he suggests that there can not be nothing because if there is, math is destroyed. Putting aside why this matters, his reasoning is faulty. He argues that the measure of fuzzy entropy in a system cannot be measured if there is nothing in the system because it results in dividing zero by zero. Pp. 272-76. Of course, the problem with this argument is that it attempts to apply a formula that measures something ("fuzzy entropy" in a system) to measure nothing. Kosko's use of the fuzzy entropy formula to prove there must be something not only begs the question (by assuming that math must survive even if nothing exists), it just simply misapplies it. I cannot divide up "no apples" among "no people" because there is nothing to divide and nothing to divide it among. I cannot measure the fuzzy entropy of nothing because there is nothing to measure. In addition to being a bad argument, however, this misapplication of a mathematical formula to address a metaphysical problem demonstrates how thoroughly immersed in a mathematics-is-reality world view Kosko is, whatever distinctions he might make.

The commitment to a mathematical model of truth creates a distortion in Kosko's analysis. His truth is always bi-polar, along a single axis. That, however, is not the thrust of many of his examples. Consider, for instance, his example of an apple. When it is a whole apple, Kosko agrees that it is so with 100% truth. As more bites get taken from it, whether or not it is an apple becomes a matter of degree. So, he suggests, there is a continuum of values from 0 to 1 corresponding to the truth of whether or not the partially-eaten thing is an apple. Suppose our question were "is that an apple or is that a pear"? Half-eaten or not, the response "it is an apple" is true, and completely so to the extent our language permits truth. If, on the other hand, a young boy is asked to go to the store and come back with an apple for grandma, and he returns with a half-eaten apple he has not returned with what he was asked for. The application of our generic terms is not a matter of a single bi-polar continuum; there are networks of uses and meaning, purposes and goals. To put the value of truth on a single continuum is a distortion.

Underlying Kosko's approach is the theme of the blending of "opposites," a notion with roots not only in Buddhism and the yin-yang symbol, but also in western philosophy. Heraclitus pronounced "And as the same thing there exists in us living and dead and the waking and the sleeping and young and old; for these things having changed round are these." (Fr. 88) But the blending of opposites does not mean that A is not-A. Plato made this point in the Parmenides (129a ff) as well as in other dialogues, especially the Phaedo and the Republic. Kosko himself admits the distinction between "A" and "not-A" on the one hand and the things which are both "A" and "not-A" when he rejects the possibility of nothingness because of the collapse of the bipolar continuum of truth resulting when one tries to measure the fuzzy entropy of nothingness. Pp. 274.

We must distinguish "A or not-A" from the possibility of something being capable of being characterized by opposites. Where "A" is the characteristic of being a raven, "not-A" is not "the opposite of raven." It is everything else which is not raven. To put the distinction between raven and not-raven on a bipolar continuum is therefore misleading. There are many ways that being a raven can shade off into something else. While that shading might matter in one context, it also might not matter in another, as the apple example, above, shows. If I want a plaster statue of a raven, it does not matter that it is not covered with feathers. It might even be made of white plaster.

The bipolarity of opposites is not the right analogy for the multivalence of truth. There are other responses to bivalence besides the type of uni-dimensional multivalence Kosko uses for fuzzy systems. Kosko's fuzzy systems are based on single and quantifiable factors. Some concepts are based on the assessment of multiple factors. Consider "motherhood." This notion has a number of dimensions, including affection to kin, nuturing behavior, biological relationship, and reliance by dependents. We pick our examples of "mothers" by assessing the various factors, and our assessment might not always be the same, depending on our purposes. In that case, the truth of an ascription of motherhood might vary in different contexts, and we might be willing to reject a strong bivalence in favor of truth as more-or-less; but it will not be a simple matter of position on a one-dimensional continuum.

VII. The Measurement of Truth

The comments of section VI highlight the point made in section V, that the notion of "truth-value" appearing in the fuzzy system calculations is not the same as the "truth" of ascribing a description to a particular. We can see this by considering the air conditioner example yet again. 80 degrees is the "clear case" of "hot," where the fuzzy system assigns 100% truth value for the purpose of the calculations. 79.5 degrees is not 100% hot; it is assigned something less. But here the sorites paradox turns on Kosko. We do not consider it any less "true" that it is hot at 79.5 degrees, but the system suggests that we do. And its functioning requires that we do so.

VIII. Concluding Comments

Fuzzy systems do not offer us a new way to look at knowledge and truth, nor do they provide us with a methodology for determining good metaphysics from bad. Kosko's philosophical version erroneously combines the intellectually bankrupt doctrines of logical positivism that bedeviled mid-twentieth century philosophy with the erroneous conflation of mathematics and reality that bedeviled classical physics and mechanics in the nineteenth century, covered with the façade of quasi-eastern thinking. On the whole it is a philosophical failure.

Fuzzy systems and fuzzy logic make for smart machines. It is a mistake, however, to conclude that they are the future of intelligence or the right way to think about thinking. They are a useful tool. Nothing more.

Paul Rezendes is a practicing lawyer and amateur philosopher who received his BA in philosophy in 1978. His interests in law and philosophy came together in his article "Professionals As Attorneys: Are Lawyers Ethically Unique?" published in Volume 3, Numbers 3 & 4, Fall/Winter 1994 of The Journal Professional Ethics.

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