Haack On Fuzzy Logic
L.A. Zadeh who introduced the term "fuzzy logic" reserves it for the result of a second stage of fuzzification, motivated by the idea that "true" and "false" are themselves vague: a family of systems in which the indenumerably many values of truth values of the base logic are superseded by denumerably many fuzzy truth values, true, false, very true, fairly true, not very true, etc. For fuzzy logic, Zadeh tells us, such traditional concerns as approximation, proof procedures, etc. are "peripheral" because fuzzy logic is not just logic of fuzzy concepts, but is logic, which is itself fuzzy. (1) Susan Haack criticizes Zadeh on the grounds that fuzzy logic is not well motivated, since truth does not come in degrees. Inevitably some will protest that fuzzy logic is working, and so that her distaste for it can only be an expression of a Fregean prejudice. But she claims that if fuzzy logic were reconstrued as merely a partially formalized description of the mental processes by means of which human operators adjust air-conditioners, gas flames, brakes and the like, then it would have some connection with fuzzy technology, but would be no threat to classical logic. If however fuzzy logic is taken to be, as Zadeh and his followers claim, a rival theory, fuzzy technology is irrelevant to its philosophical bona fides. (2) The idea is to accommodate vagueness within the framework of classical logic by means of a non-classical semantics in which vague evidence counts as true just in case it would be true for all the ways of making it precise.
After explaining how radically fuzzy logic departs not only from classical logic, but also from the classical conception of what logic is and does, Haack criticizes fuzzy logic for its methodological extravagances and its linguistic inconsistencies. She argues that despite the considerable new complexities it introduces, fuzzy logic does not avoid but actually requires the imposition of artificial precision - the very fault for which Zadeh blames classical logic. She also points out that the linguistic evidence does not support the main contention motivating fuzzy logic that "true" and "false" like "bald" and "tall" are predicates of degree. Linguistic, metaphysical, and methodological considerations all speak against degrees of truth. (3) Haack concludes that fuzzy logic is not a viable competitor of classical logic. A process of weighted averaging "defuzzifies" - Carnap and Haack might say, "precisifies" - this fuzzy input to produce a precise one. If fuzzy logic is construed as an attempt to represent the mental processes thorough which people go when making adjustments to kiln-thermostats, air-conditioners, etc., there is a connection with fuzzy technology. But of course, so construed fuzzy logic is not a theory in the same domain as classical logic. In fact, so construed, it is obviously not properly describable as a "logic" at all. (4)
Already Frege has pointed out that the shortcomings stressed here are rooted in a certain softness and mutability of (ordinary) language, which nevertheless is necessary for its versatility and potential for development. We build for ourselves artificial hands, tools for particular purposes, which work with more accuracy than the hand can provide. And this accuracy possible through the very stiffness and inflexibility of parts the lack of which makes the hand so dexterous. Word-language is inadequate in a similar way. We need a system of symbols from which every ambiguity is banned, which has a strict logical form from which the content cannot escape. (5) F.S.C. Schiller also concludes that formal logic ... is incapacitated by its self-imposed limitations from dealing with the problems of actual theory and from radically interpreting the conception of truth implied in such a theory ... We need, in short, a second (non-formal) logic which will be applicable to life and relevant to actual thought. (6) Haack points out that the term "fuzzy logic" seems to be used in the literature to refer to two related but distinct enterprises. (i) the interpretation of familiar infinitely-many-valued logic in terms of set theory, and (ii) the development on the basis of (i) of a family of new logical systems in which the truth-values are themselves fuzzy sets. (7) Haack's objections do not apply to the base logic of (i) but only to the more radical second enterprise. The rationale for the development of fuzzy logic is somewhat as follows: Informal arguments suffer from vagueness of indeterminacy, so that classical logic is hopelessly inadequate to represent them. The traditional response (e.g. Carnap, Logical Foundations Of Probability 1950, ch. 1) (8) is that informal arguments must be tidied up, or "regimented" so that classical logic will apply. The fuzzy logician proposes instead to loosen up, or "fuzzify" classical logic to obtain a new logic which is directly applicable to unregimented informal arguments.
Fuzzy logic then results from two stages of "fuzzification" of classical logic: (i) a move from 2-valued to indenumerably many-valued logic as a result of treating object-language predicates as denoted fuzzily rather than classical sets yielding the base logics, and (ii) a move to countably many fuzzy truth values as a result of treating the meta-language predicates "true" and "false" as denoting fuzzy subsets of the set of values of the base logic, yielding "fuzzy logic" proper. The indenumerably many-valued logics which result from the first stage of fuzzification are, so to speak, standard non-standard logics, whereas fuzzy logic, the result of the second stage of fuzzification, is a very radical departure from classical logic. Indeed it would be true to say that Zadeh has challenged, not just classical logic, but also the classical conception of what logic is and what it aims to do. (9) Among the distinctive features of fuzzy logic to which Zadeh draws attention are the following: its truth values are fuzzy, local and subjective, its set of truth values is not closed under the usual propositional operations, and linguistic approximations have to be introduced to guarantee closure. In the second stage of fuzzification the truth-values of the base logic, the set of points in the interval (0,1) are replaced by fuzzy subsets of that set, referred to as "fuzzy truth-values."
However, Zadeh thinks that to allow all fuzzy subsets of the unit interval would result in "unimaginable complexity," and so instead a countable structured set of "linguistic truth-values" is introduced. (10) At stage (i) object language predicates determine, not classical sets, to which objects either definitely do or definitely do not belong, but fuzzy sets, in which objects have degrees of membership. The truth-values of fuzzy logic are not only fuzzy but also "subjective" and "local." By calling values subjective, Zadeh means that it is simply arbitrarily laid down which values of the base logic belong to which linguistic truth-values to what degree. There are rules for calculating which rules belong to what degree to very true, or not very true etc., but the upshot depends on critical subjective assignment of the primary terms. By calling the values local, Zadeh means that the assignments to the primary term are defined only for the specified set of properties and may have to be differently defined for another set. Zadeh's linguistic truth-values represent only a countable subset of the set of all fuzzy subsets of the indenumerably many values of the base logic, and consequently the set of linguistic truth-values is not closed under the usual propositional operations. It may be that while 'p' and 'q' has linguistic truth-values, 'not-p' and 'p v q' though they denote fuzzy subsets of the values of the base logic, do not denote fuzzy subsets which happen to be assigned to linguistic truth-values. This motivates the introduction of the concept of "linguistic approximation." In consequence of the introduction of linguistic approximation, Zadeh says, one has in fuzzy logic not exact but merely approximate inference. In speaking of "inference" as approximate Zadeh is referring to the relation of logical consequence, and claiming that this relation is inexact or comes in degrees. The consequence of a given set of premises depends in an essential way on the meaning attached to fuzzy sets which appear in the premises. (11) The claim is that inference must be characterized semantically rather than syntactically, and this seems to be a consequence of the approximate character of validity rather than the logical character of truth. (12) If, as Zadeh claims, validity in fuzzy logic can only be characterized semantically, it is obvious that questions of axiomatization and proof practice procedure (which characterize a system syntactically) or of completeness and soundness (which characterize a system syntactically) or of completeness and soundness (which relate syntax and semantics) will drop out of consideration.
Zadeh offers us not only a radically non-standard logic, but also a radically non-standard conception of the nature of logic. It would scarcely be an exaggeration to say that fuzzy logic lacks every feature that the pioneers of modern logic wanted logic for. It sacrifices what have traditionally been regarded as the crucial advantages of formalism-precise formal rules of inference, the security offered by consistency and completeness results. While traditionally logic has corrected or avoided vagueness, fuzzy logic compromises with it. It is not just logic of vagueness; it is-what from Frege's point of view would have been a contradiction in terms-a vague logic. Indeed Zadeh's approach has stronger affinities with attitudes of those critics of formalism-such as Schiller and Strawson-who urge the inadequacy of any formal system to the subtleties of ordinary language than with the traditional stance of formal logicians. (13) Haack considers fuzzy logic to be methodologically extravagant and linguistically incorrect. Zadeh seems to rely on two main reasons for adopting fuzzy logic: a methodological reason, that it avoids complexities inevitably introduced by regimentation of informal arguments, and a linguistic reason that it is the proper way to acknowledge that "true" and "false" are not precise but fuzzy predicates. Zadeh emphasizes that we should prefer to change our logic to cope with vagueness rather than regiment informal discourse and continue to rely on classical logic. He claims that regimentation introduces excessive complexity, but that its replacement by fuzzy logic produces a net gain in simplicity. Haack concludes that it is pretty clear that it does not: resort to fuzzy logic by no means avoids all the complications of regimentation, for it introduces enormous complexity of its own, and it still requires the imposition of artificial precision.
Haack emphasizes that one needs to distinguish between two kinds of vagueness, unidimensional versus multidimensional vagueness which reflects the distinction between predicates like "old" or "tall" on the one hand, and "beautiful" or "capable" on the other. Regimentation of unidimensional vagueness (as e.g. by means of the rule that all and only persons over 6 feet are to count as tall) result in artificiality, but not in complexity. These complexities arrive in the regimentation of multi-dimensional vagueness-one can assign a fuzzy set to "tall" instead of settling on an arbitrary cut off point. But this does not avoid the need to regiment multidimensional vagueness, and yet it is here that regimentation introduces complexities. Haack points out that in Zadeh's "A Fuzzy-Algorithm Approach to the Definition of Complex or Imprecise Concepts" (14) the formal apparatus is redundant, since "beautiful" is to be defined ostensibly. (15) This is how fuzzy logic introduces enormous complexities in her opinion. Perhaps it will be claimed that it is better to have complexities in the formalism than at the level of translation from informal discourse into the symbolism. This might be a reasonable principle for, in general, formal manipulations are subject to definite, precise rules, unlike the process of paraphrase from informal argument to symbolism which is a matter of discretion, and the whole point of formalization is to make what was previously a matter of judgment into a mere application of routine. Haack however points out that this argument cannot be used in the present case, for the complexities introduced by fuzzy logic are such as to nullify the usual definite, mechanical, routine character of formal rules (e.g. Zadeh concludes that the choice of suitable linguistic approximation is a matter of discretion). Fuzzifying logic only postpones, it does not eliminate the need to introduce arbitrary boundaries. The very radical character of fuzzy logic results from the second stage of fuzzification, and the rationale for the second stage of fuzzification is that the metalanguage predicates "true" and "false" are fuzzy. The view of truth on which the motivation for fuzzy logic relies however, is mistaken, according to Haack. It is only at the second stage of fuzzification that one is obliged to admit degrees of truth. (16) Haack's objections focus on Zadeh's willingness to allow not only "very true" and "more or less true" but also such bizarre locutions as "rather true" and "not very true." Haack therefore emphasizes that the linguistic arguments for fuzzy logic are confused, and since neither of the main arguments that are offered in its favor are acceptable, she concludes that we do not need fuzzy logic. (17)
Much of the progress in modern logic beyond Aristotle is due to the invention of a precise and powerful formalism, and this is why Haack is reluctant to weaken it. What motivates her to regard deviant and fuzzy logics as extensions rather than rivals of classical logic is its demonstrated capacity for refinement and progress. Thus she sharply distinguishes between a logic dealing with fuzzy concepts, (she accepts), and one which is itself fuzzy, i.e. where "true" and "false" cease to be precise concepts (she rejects). She shows that Quine is inconsistent in his treatment of deviant logic, but in spite of her misgivings she agrees with him that there are no clear cases where it makes sense to discard classical logic on pragmatic grounds, For Quine, to accept a deviant logic is to change the subject, and Haack has no conclusive argument to refute this. Whereas for Frege a vague logic is a contradiction in terms, for Zadeh fuzziness in logic is a matter of degree. Logicists like Frege and Russell stress the pervasive vagueness of ordinary language and maintain that logic does not apply to it, yet they take logic to be more basic (i.e. comprehensive) than mathematical reasoning. It was Wittgenstein who in his later work realized that consistency is only one possible language game we play, and his "family resemblances" demonstrated the incapacity of classical logic to do justice to overlapping similarities. In due course this led to the rejection of "logocentrism" by Derrida, and to treating logic a tool of male domination by some radical feminists. While it is often more convenient to retain as much as possible of classical logic because of its simplicity and familiarity, there is nothing in the hermeneutical view of logic to render it immune from revision. Yet to treat logic as a canon of interpretation conflicts with Haack's idea of what logic is and does.
(1) Susan Haack, Deviant Logic, Fuzzy Logic, Beyond the Formalism, The University of Chicago Press, Chicago and London 1996, p. ix
(2) ibid. p. xii
(3) ibid. p. 229.
(4) ibid. p. 231.
(5) G. Frege, "On the Scientific Justification of a Conceptual Notation," 1882/translation by Bynum T. Ward in Gottlob Frege: Conceptual Notation and Related Articles, Oxford: Claredon Press 1982, p. 86.
(6) F.S.C. Schiller, Formal Logic, A Scientific and Social Problem, London: Macmillan 1912, p. 8.
(7) Haack, p. 233.
(8) R. Carnap, Logical Foundations of Probability, Chicago: Chicago University Press 1950.
(9) Haack, p. 233.
(10) ibid. p. 234.
(11) L.A. Zadeh and R.E. Bellman, "Local and Fuzzy Logics," in M. Dunn and G. Epstein, Eds, Modern Uses of Multiple-Value Logics, Dordrecht: Reidel 1977, pp. 106-107.
(12) Haack, p. 236.
(13) ibid. p. 237.
(14) L.A. Zadeh, "A Fuzzy-Algorithm Approach to the Definition of Complex or Imprecise Concepts," International Journal of Man-Machine Studies 1976 vol.8 p.269n.
(15) Haack, p.238.
(16) ibid. p. 240.
(17) ibid. p. 242.