A Development of Set Theory in Fuzzy Logic

A Development of Set Theory in Fuzzy Logic Petr Hajek and Zuzana Hanikova Institute of Computer Science Praha Czech Republic Abstract This pap er presents an axiomatic set theory FST Fuzzy Set Theory as a rstorder theory within the framework of fuzzy logic in the style of In the classical ZFC we use a construction similar to that of a Bo oleanvalued universeover an algebra of truth values of the logic we useto show the nontriviali ty of FST We give the axioms of FST Finally we show that FST interprets ZF Intro duction If anything comes to p eoples minds on the term fuzzy set theory b eing used it usually is the theory or theories handling fuzzy sets as realvalued functions on a xed universe within the classical settheoretic universe However as fuzzy or manyvalued logic has b een evolving into a formal axiomatic theory a few exceptions from this general exp ectation have emerged these are formal set theories within the resp ective manyvalued logics ie theories whose underlying logic is governed by a manyvalued semantics We rely esp ecially on those works which develop a theory in the language and style of the classical ZermeloFraenkel set theory ZF We have b een inspired by a series of pap ers developing a theory generalizing ZF in a formally weaker logicintuitionistic and later its strengthening commonly referred to as Godel logic some results and pro ofs carry over to our system For an imp ortant example the axiom of foundation together with a very weak fragment of ZF implies the law of the excluded middle which yields the full classical logic b oth in Godel logic and in the logic we use in this pap er and thus the theory develop ed b ecomes crisp For this reason we start with building a noncrisp universe in which we verify our axioms However all these pap ers fall short in tackling one of the distinguishing traits of manyvalued logic which is the general nonidemp otence of the conjunction conjunction is idemp otent in Godel logic The nonidemp otence of conjunction aects the resulting theory considerably cf in coping with some of the diculties we appreciated an elegant solution found in The author works over the socalled phase spaces as algebras of truth values and builds using an analogy of the construction of a Bo oleanvalued universe over a phasespace instead of a Bo olean algebra a class with class op erations evaluating formulas in the language f g in which he veries the chosen axioms of his set theory Having observed that the standard Luk asiewicz algebra enriched with the op erator is a particular phase space we have studied this pap er thinking of a more general approach employing linearly ordered BLalgebras with this should form a common generalization of the approach of and and that of This pap er is an extension of it brings in a simplied denition of the initial universe and an inner mo del of ZF in FST Regarding the nature of this pap er we omit most of the logical background which is to b e found mainly in as well as some technical details and some pro ofs The authors acknowledge gratefully a partial supp ort from the grant IAA Grant Agency of the Academy of Sciences Czech Republic Prerequisites Denition BL is a formal logical system with basic connectives and dened connectives and where is is is and is and with two quantiers and The axioms are as follows A A A A Aa Ab A A xx t t substitutable for x in t xx t substitutable for x in x x x not free in x x x not free in x x x not free in The deduction rules of BL are mo dus p onens generalization and fg Denition Let C b e an arbitrary schematic extension of BL A C algebra is a BLalgebra L in which all the axioms of C are Ltautologies Theorem Strong completeness Let C be a schematic extension of BL let T be a theory over C and let be a formula of the language of T Then T proves i holds in any safe model M of T over any linearly ordered C algebra The Initial Universe Consider the classical ZFC Fix C as a schematic extension of BL and x a constant L for an arbitrary linearly ordered complete C algebra write L L as usual L denotes b oth the algebra and its supp ort In ZFC let us make the following construction in analogy to the construction of a Bo oleanvalued universe over a complete Bo olean algebra we build the class L V by ordinal induction Dene L L fg L V fg L L V ff Fnc f D f V Rf L g for any ordinal and for limit ordinals L L V V Here Fnc x is a unary predicate stating that x is a function and D x and Rx are unary functions assigning to x its domain and range resp ectively Note that functions taking the value on any element of their domain are not considered as elements of the universe Finally we put L L V V 2On L L Observe that for V V L We dene two binary functions from V into L assigning to any tuple x y L V the values jjx y jj and jjx y jj representing the truth values of the two predicates and jjx y jj y x if x D y otherwise jjx y jj if x y otherwise We now use induction on the complexity of formulas to dene for any formula x x the free variables in b eing x x a corresp onding nary n n L n function from V into L The induction steps admit the following cases we just write for short is then jjjj is then jjjj jj jj jjjj is then jjjj jj jj jjjj is then jjjj jj jj jjjj is then jjjj jj jj jjjj is then jjjj jj jj V xu is x then jjjj L u2V W is x then jjjj xu L u2V Here xu is the result of substituting u for the variable x in We use the symb ols for b oth logical connectives and op erations in L L Denition Let b e a closed formula We say that is valid in V i jj jj is provable in ZFC As usual we take closures of formulas containing free variables when considering their validity L Lemma ZFC proves for u V ux jjx ujj for x D u jju ujj L ZFC proves for u v w V ijju v jj jjv w jj jju w jj iijju v jj jjv w jj jju w jj iiijju v jj jjv w jj jju w jj Proof immediate i immediate ii if v w then jjv w jj so jju v jj jju v jj jjv w jj jju w jj jjv w jj jju w jj otherwise jjv w jj in which case the statement holds trivially iii analogously ut L Lemma Substitution For any formula ZFC proves u v V jju v jj jjujj jjv jj L Lemma Bounded quantiers ZFC proves x V W xy jjy jj i jjy xy jj jjy y x y jj y 2D x V xy jjy jj ii jjy xy jj jjy y x y jj y 2D x W W xy jjy xjj jjy jj Proof i jjy y x y jj L y 2D x y 2V jjy jj since jjy xjj is nonzero only if y D x and in that case it is xy ii analogously ut Corollary jjx y jj jju xu y jj jjuu x u y jj The Theory FST We intro duce the theory FST in the language fg we take to b e a logical symb ol with the usual axioms imp osing reexivity symmetry transitivity and congruence wrt on the corresp onding relation The underlying logic of FST is a schematic extension C p ossibly void of BL when proving theorems within FST we only rely on the logical axioms of BL thus any schematic extension will for example interpret ZF On the other hand for a given extension C the L universe V for L a C algebrawill yield an interpretation of FST over C The reader will have noticed that ours is a crisp equality This was imp osed by the following fact Lemma A theory with comprehension for open formulas and pairing or singletons over a logic which proves the propositional formula proves x y x y x y Proof Given x y let z b e s t u z u fxgu x i e u z u x Since x x we have x z If y x then y z by congruence but then y x thus we have proved y x y x thus by assumption on the logic x y x y ut Thus eg in L ukasiewicz logic and in pro duct logic we get a crisp Also under the usual formulation of extensionality crispness of implies crispness of in a theory with pairings and unions cf We b orrow the elegant solution from a mo dication of extensionality uses the op erator which invalidates the pro of of crispness of from the crispness of Denition FST is a rst order theory in the language fg with the follow ing axioms i extensionality xy x y x y y x ii empty set xy y x iii pair xy z uu z u x u y iv union xz uu z y u y y x v weak p ower xz uu z u x vi innityz z x z x fxg z vii separation xz uu z u x u x for any formula not con taining z as a free variable viii collection xz u xv u v u xv z u v for any for mula not containing z as a free variable ix induction xy xy x xx for any formula x supp ort xz Crispz x z The s in the axioms of weak p ower and induction are intro duced to weaken the statements the s after an existential quantiers is used to guar antee that an element with the p ostulated prop erty exists in every mo del of the theory which do es not follow from the semantics of the quantier As usual in the formulation of some axioms we use the functions of empty set singleton and union which can b e intro duced using the appropriate axioms as in classical ZF and also the usual denition of For a detailed treatment of intro ducing functions in BL see The function intro duced by the weak p ower set axiom will b e denoted WP x for a set x We use the following denition of crispness of a set Denition Crisp x uu x u x Theorem Let be a closed formula provable in FST Then ZFC proves L jjjj in V L Having proved this theorem we shall take the lib erty to call V

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