From fuzzy logic to fuzzy mathematics: a methodological manifesto
|Title:||From fuzzy logic to fuzzy mathematics: a methodological manifesto|
|Journal:||Fuzzy Sets and Systems|
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The scope of the programme in fuzzy mathematics
This paper assumed that deductive fuzzy logics could provide foundations for the whole of traditional fuzzy mathematics. While this is true to some extent, the best-suited area of applicability of the approach was later clarified in the paper On the difference between traditional and deductive fuzzy logic. -- LBehounek 16:40, 2 September 2008 (CEST)
- Although some may be tempted to found new mathematics on many-valued logics (the Manifesto is cited here), this grand purpose still looks out of reach if not delusive. It sounds like a paradox of its own since we use classical mathematics to formally model many-valued logic notions. What could be named "many-valued mathematics" essentially looks like an elegant way of expressing properties of many-valued extensions of Boolean concepts in a Boolean-like syntax. For instance, the transitivity property of similarity relations is valid in Lukasiewicz logic, and, at the syntactic level, exactly looks like the transitivity of equivalence relations, but should be interpreted as the triangular inequality of distances measures.
To answer the criticism, the following clarification should be given first. Formal fuzzy mathematics based on the methodology of the Manifesto can essentially be understood in one of the following two ways:
- In a "traditionalist view", logic-based fuzzy mathematics is just a methodologically advantageous treatment of traditional fuzzy mathematics, where in exchange for voluntary restrictions on the language and methods we get a formalism that enables us to derive theorems of certain forms more easily (cf. Section 3.4 of Fuzzy class theory or the paper Relational compositions in Fuzzy Class Theory). Under this explanation, the formulae of higher-order fuzzy logic describe the behavior of membership functions valued in the real unit interval (or more generally in an appropriate semilinear residuated lattice). One of the benefits of this approach comes from the fact that many notions of traditional fuzzy mathematics turn out to be expressed by formulae of exactly the same form as analogous notions of traditional mathematics---e.g., T-transitivity by a formula expressing classical transitivity, only reinterpreted in many-valued logic. This enables a treatment of fuzzy notions similar to that of classical notions: e.g., the proofs of theorems often just copy classical proofs. Moreover, it allows us to extrapolate this observation and find new important notions of fuzzy mathematics by reinterpreting classical definitions in fuzzy logic. Finally, by employing many-valued logic, all defined notions are naturally graded, which radically facilitates the study of graded properties (in the sense of Siegfried Gottwald's monograph A Treatise on Many-Valued Logics) of fuzzy relations and other fuzzy notions.
- Alternatively, under a "foundationalist view", logic-based fuzzy mathematics presents a fundamental treatment of fuzzy mathematics (a "new mathematics" indeed, as labeled by Dubois), based on non-classical logics. This interpretation understands fuzzy sets as a primitive notion, axiomatized (or governed) by the axioms and rules of the non-classical logic, similarly as crisp sets are governed (and can be axiomatized) by the axioms and rules of classical logic. In this approach, fuzzy sets are not represented or modeled by their membership functions, but are primitive objects sui generis. Pre-theoretical considerations on examples of vague propositions suggest that they are governed by the laws of the fuzzy logic MTL (or some of its variations). Importantly, the justification of the laws is pre-theoretical and independent of any model of fuzzy sets in classical mathematics. Based on the axioms and rules of fuzzy logic, a formal theory of fuzzy sets can be developed, with the intended informal semantics of actual fuzzy sets, i.e., vaguely delimited collections of objects---similarly as the intended informal semantics of classical sets is that of sharply delimited collections of objects. The formal semantics of fuzzy logic is formed by fuzzy sets described by (a fragment of) the very same theory itself---similarly as the semantics of classical logic is formed by sets described by (a fragment of) classical set theory (it is the same form of circularity). It turns out that, incidentally, the theory of fuzzy sets can be formally interpreted in classical mathematics: this formal interpretation is what is more usually called "the many-valued semantics", in which fuzzy sets become interpreted by membership functions. Although classical mathematics is thus, by means of the formal interpretation, capable of faithful modeling fuzzy mathematics, it does not establish its priority over fuzzy mathematics, as both theories can be founded independently of each other and are faithfully interpretable in each other (a reverse formal interpretation of classical mathematics in fuzzy mathematics can be done by means of the propositional connective Δ---which is no wonder as the connective is intended to represent crisp propositions among fuzzy ones and is axiomatized by the laws valid for crisp sets). Both classical and fuzzy mathematics are therefore of equal standing as foundational theories, and precedence can be given to one of them only based on some pragmatic criteria: classical mathematics may be preferred because of our long experience with it or because of its simplicity (as it only considers crisp sets); fuzzy mathematics, on the other hand, can be preferred in vague contexts because it renders vaguely delimited sets more directly, and because of the advantages of its apparatus in proving theorems on fuzzy sets as described under the traditionalistic view above.
It can thus be seen that the three points of the above criticism of many-valued mathematics from Dubois' paper, namely that
- ... we use classical mathematics to formally model many-valued logic notions,
- What could be named "many-valued mathematics" essentially looks like an elegant way of expressing properties of many-valued extensions of Boolean concepts in a Boolean-like syntax, and that
- the transitivity property of similarity relations ... should be interpreted as the triangular inequality of distances measures,
only apply to the traditionalistic view of the non-classical theory. The second statement is explicitly admitted in the Manifesto (p. 643):
- Admittedly, a formal theory over fuzzy logic is just a notational abbreviation of classical reasoning about the class of all models of the theory.
Still, the advantages of the logic-based approach fully justify the development of logic-based fuzzy mathematics even under the traditionalistic explanation. The possibility of the foundationalist attitude, however, shows that the non-classical theory need not be regarded as formally modeling many-valued notions by using classical mathematics. Rather, the non-classical notions can be regarded as primitive and independent of classical mathematics: since the theory is syntactical, it does not need to presuppose that classical mathematics has been developed first. And under the foundationalist approach, transitivity of similarities is not interpreted as the triangular inequality of distance measures, but indeed as transitivity of unsharply delimited relations (as primitive entities). The name "transitivity" is justified here by the fact that from Trans(R) one can infer, by the rules of fuzzy logic, any instance of Rxy & Ryz → Rxy, where & represents cumulation of premises and → transmission of truth in vague contexts (cf. Section 4 of On the difference between traditional and deductive fuzzy logic); only accidentally it coincides, when fully true, with the notion of T-transitivity that is known from traditional fuzzy mathematics and that expresses the triangle inequality of distance measures.
In sum, the criticism only applies to the traditionalist understanding of logic-based fuzzy mathematics, and not to the foundationalist one. But even under the traditionalistic view, logic-based fuzzy mathematics has undisputable advantages described above, which fully justify its development.
-- LBehounek 17:39, 2 September 2008 (CEST)
References for this page
- Dubois D.: "On ignorance and contradiction considered as truth values", Logic Journal of IGPL 16 (2008): 195--216.
- Běhounek L.: "An alternative justification of the axioms of fuzzy logics" (abstract). The Bulletin of Symbolic Logic 13 (2007): 267. 
- P. Hájek, L. Běhounek: "Fuzzy logics among substructural logics" (tutorial). Second World School on Universal Logic, Xi'an 2008.