Constructions of Models in Fuzzy Logic with Evaluated Syntax

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1 Constructions of Models in Fuzzy Logic with Evaluated Syntax Petra Murinová University of Ostrava IRAFM 30. dubna Ostrava Czech Republic Abstract This paper is a contribution to the development of model theory of fuzzy logic in narrow sense. It is known that there are many formal systems of fuzzy logic, most of them having traditional syntax and many-valued semantics. Our logic is further generalization, where also syntax is evaluated, i.e. we may consider fuzzy sets of axioms. In the paper we show the possibilities how to construct the model in fuzzy logic with evaluated syntax. Keywords: fuzzy logic in narrow sense with evaluated syntax, Lukasiewicz MV-algebra, model theory in predicate fuzzy logic with evaluated syntax. 1 Introduction This paper is a contributions to the model theory of the predicate fuzzy logics. The first is fuzzy logic with evaluated syntax (Ev L from now on) which is in detail presented in [9] where also its model theory has been founded. It is specific for Ev L that the set of truth values must be the Lukasiewicz MV-algebra whose support set is the interval of reals [0, 1]. Model theory of fuzzy logic with evaluated syntax has been initiated by V. Novák in [9, 7] and further this theory has been elaborated by Murinová and Novák in [4, 5, 6]. In this paper we introduce the possible constructions of the models in Ev L. The constructions of models in fuzzy predicate logics can be separate into few groups. The first possibility is to construct the model due to Henkin theory from constants which will be not discussed in this paper and it is in detail presented in [9]. Other possibility how to construct the models in Ev L is to used omitting types theorem and their consequences (see [6]). At the end, we can not forget the constructions of the models in Ev L using elementary extension, elementary chains (see [2, 4]). 2 Preliminaries The set of truth values is supposed to form Lukasiewicz MV-algebra L L = [0, 1],,,, 0, 1 where is the operation of Lukasiewicz conjunction defined by a b = 0 (a + b 1), is the operation of Lukasiewicz disjunction defined by a b = 1 (a + b) and is the negation operation defined by a = 1 a for all a, b [0, 1]. We may introduce lattice operations by a b = (a b) b, a b = (a b) b. The following two subsections contains a brief overview of the main concepts and notation of Ev L. 2.1 Syntax The language of Ev L is denoted by J and it consists of a set of object variables x, y,..., a set of object constants u 1,..., a set of n-ary functional symbols f, g..., a set of n-ary predicate symbols P, Q... 1, implication connective, logi- 1 The arity n of each functional symbol as well as of predicate symbol, of course, may vary depending on the given symbol. If unnecessary, we will not explicitly stress this in the sequel. 961

2 cal constants a being names of all the truth values a L and the general quantifier. We write, instead of the logical constant 1, 0, respectively. Other connectives (,,,,&, ) are defined in [9]. If L is uncountable then J is uncountable as well. Remark 1. By J we mean a cardinality of the language J. The set of all the well-formed formulas for the language J is denoted by F J. A couple (a A) where a L and A F J is called an evaluated formula. A fuzzy theory T is a fuzzy set of formulas T F J given by the triple T = LAx, SAx, R where LAx F J is a fuzzy set of logical axioms, SAx F J is a fuzzy set of special axioms, and R is a set of sound inference rules which contains the rules of modus ponens (r MP ), generalization (r G ) and the rule of logical constant introduction (r LC ). For the precise definitions see [9]. Given a fuzzy theory T and a formula A. A proof (denoted by w A ) of a formula A is a finite sequence of evaluated formulas which are axioms, or are derived using the inference rules. If w A is a proof with the value Val T (w A ) then T a A means that A is provable in the fuzzy theory T in the degree a = {Val(w) w is a proof of A in T }. If there exists a proof w A such that Val T (w A ) = a then we say that A is effectively provable in T in the degree a (note that this may not always be the case). Definition 1. A fuzzy theory T is contradictory if there is a formula A and proofs w A and w A of A and A, respectively, such that Val T (w A ) Val T (w A ) > 0. It is consistent in the opposite case. 2.2 Semantics The semantics is defined by generalization of the classical semantics of predicate logic. A model for the language J is V = V, P V,..., f V,..., u 1,... where V is a set, P V V n are n-ary fuzzy relations assigned to each n-ary predicate symbol P (n depends on P ),..., f V are ordinary n-ary functions on V assigned to each n-ary functional symbol f, and u 1,... V are designated elements which are assigned to each object constant u 1... J. Let V be a model for the language J. An cardinal of the model V is the cardinal V. V is said to be finite, countable or uncountable if V is finite, countable or uncountable. Let V be a model for the language J. We extend J to the language J V by new constants being names for all the elements from V, i.e. constants will be denoted by the corresponding bold-face letter, namely J V = J {v v V }. (1) For interpretation of closed terms, formulas and the derived connectives (see [9]). Let Var(J) be a set of all variables of the language J and V be a model for J. Interpretation of a general formula A is using an evaluation e : V ar(j) V. Definition 2. A formula A(x 1,..., x n ) is satisfied in V by the evaluation e, e(x 1 ) = v 1,... e(x n ) = v n in the degree a if [v 1,..., v n ]) = a. Definition 3. A formula A(x 1,..., x n ) is true in V in the degree a if a = V(A) = { [v 1,..., v n ])} for all evaluation e. Definition 4. Let T be a fuzzy theory over Ev L and V be a model for J. We say that the model V is a model of the fuzzy theory T and write V = T, if SAx(A) V(A) holds for all formulas A F J. 962

3 Theorem 1. (compactness) If each finite fuzzy subtheory T of T has a model then T has a model. 3 Model-Theoretic Constructions in Ev L 3.1 Elementary extension Definition 5. Let V and W be models for the language J. Then we say that V is a submodel of W, in symbols V W, if V W and for every atomic formula A F J [v 1,..., v n ]) = W(A x1,...,x n [v 1,..., v n ]) (2) holds where v 1,..., v n V. Definition 6. Let V and W be models for the J. We say that V is a strong submodel of W, in symbols V W, if V W and for every formula A F J [v 1,..., v n ]) W(A x1,...,x n [v 1,..., v n ]) (3) holds where v 1,..., v n V. Remark 2. If (3) holds only for every atomic formula A F J then we say that V is a weak submodel of W, in symbols V W. Definition 7. Let V and W be models for the language J. Then we say, that V is an elementary submodel of W, in symbols V W, if V W, V W and for every formula A F J [v 1,..., v n ]) = W(A x1,...,x n [v 1,..., v n ]) (4) holds where v 1,..., v n V. When V is an elementary submodel of W then we also say that W is an elementary extension. Lemma 1. Let V and W be models for the same language J. Let V W and for every w W V there is v V such that W(A[w]) W(A[v]) holds for every formula from F J. Then the following are equivalent: a) V W b) For every formula ( x)a(x, x 1,..., x n ) and every v 1,..., v n V the following holds: If W(( x)a[v 1,..., v n ]) = a then there is an v V such that W(A[v, v 1,..., v n ]) = a. 3.2 Elementary diagram Let V be a model for J. Let J V be a language which was defined in (1) as extension of J by new constants. We may then expand V to the model V V = (V, v) v V for J V by interpreting each new constat v by the element v. Definition 8. The elementary diagram of V is the fuzzy theory Th(V V ) such that Th(V V ) = {a A(x 1,..., x n ) V V (A(x 1,..., x n )) = a} holds for every closed formula A(x 1,..., x n ) of J V. Sometimes instead of Th(V V ) we will write Γ V. Lemma 2. Let Γ V be the elementary diagram of V. If V W, then V W if and only if (W, v) v V = Γ V. 3.3 Downward Löwenheim-Skolem-Tarski theorem Theorem 2. Let V be a model of the cardinality α and let J β α. Then V has an elementary submodel of the cardinality β. Furthermore, given any set X V of the cardinality β, V has an elementary submodel of the cardinality β which contains X. 4 Models constructed from constants in Ev L The one possibility how to construct models in Ev L is to used omitting types theorem if we work with the fuzzy set of formulas in the free variables among x 1,..., x n (see [6]). This theorem makes possible to extend the power of classical logic by characterizing properties that are too complicated to be expressed by one formula but can be expressed using a set of formulas and to construct models for such situations. 963

4 Convention 1 By Σ(x 1,..., x n ) we denote a set of evaluated formulas from the language J such that each formula A(x 1,..., x n ) a Σ(x 1,..., x n ) has all its free variables among x 1,..., x n. If we work with predicate fuzzy logic with rational logical constants (see [8]) then we will write Σ r (x 1,..., x n ) instead of Σ(x 1,..., x n ). All the membership degrees in Σ r are supposed to be rational. Definition 9. (Realization of Σ) Let Σ be a fuzzy set introduced above and let V be a model for the language J. We say that Σ is realized in the model V in a degree c > 0 if there is an n-tuple v 1,..., v n V of elements such that V(A(v 1 x1,..., v n xn )) c Σ(A) holds for each evaluated formula A a Σ. We say that an n-tuple of elements v 1,..., v n V realizes Σ in V in some non-zero degree. Definition 10. (Omitting Σ) Let Σ be a fuzzy set introduced above and V be a model for the language J. We say that Σ is b-omitted in V if to each n-tuple v 1,... v n V of elements there is a formula A a Σ such that 0 < b a and V(A(v 1 x1,..., v n xn )) < b. (5) Definition 11. (Isolation of Σ in T ) Let T be a fuzzy theory and let Σ(x 1,..., x n ) be a fuzzy set of formulas. Then we say that Σ is isolated in T if there is a fuzzy set of formulas Φ(x 1,..., x n ) which is consistent with T such that holds for every V = T Φ. V = Σ (6) Definition 12. We say that Σ is non isolated if for every fuzzy set of formulas Φ(x 1,..., x n ) consistent with T, if V = T, v 1,..., v n V and V(A(v 1 x1,..., v n xn )) a holds for every A a Φ then there is B b Σ such that V( B(v 1 x1,..., v n xn )) > b. Theorem 3. (Omitting Σ) Let T be a consistent fuzzy theory with the language J and Σ(x 1,..., x n ) be a set of evaluated formulas which is not isolated in T. Then there exists a model of T which omits Σ(x 1,..., x n ) in some non-zero degree. In this subsection, we will confine to the case when the language J contains only rational logical constants, i.e. unlike the case above, it can be countable. Theorem 4. (Omitting Σ r ) Let T be a consistent fuzzy theory in the countable language with the rational logical constants J r and Σ r (x 1,..., x n ) be a set of evaluated formulas which is a non isolated in T. Then there exists a countable model of T which omits Σ r (x 1,..., x n ) in some non-zero degree. 5 n 0 -Logic Omitting types theory in the classical logic has a lot of consequences (see [1]) and there is possibility to generalize a lot of them. In this paper we give one application of omitting types theorem which is n 0 -logic. In the special case n 0 -logic gives the way how to construct non-standard model in Ev L. Definition 13. Let J be a predicate language containing the binary predicate =. The equality axioms (will be denoted by EAx) for a binary predicate = in J are (E1)x = x, (F 4)x 1 = y 1... (x n = y n f(x 1,..., x n ) = f(y 1,..., y n )...), (P 5)x 1 = y 1... (x n = y n P (x 1,..., x n ) P (y 1,..., y n )...). Elementary arithmetic Let J P A = {+,, =, S, 0} where = is binary predicate for crisp equality, +, are binary function symbols (addition, multiplication), S is unary functional symbols S (called the successor function) and 0 is a constant symbol. Elementary arithmetic or Peano arithmetic (will be denoted by PA) has the equality axioms plus the following list of axioms (will be denoted by PAx): (P A1) S(x) 0 964

5 (P A2) S(x) = S(y) x = y (P A3) x + 0 = x (P A4) x + S(y) = S(x + y) (P A5) x 0 = 0 (P A6) x S(y) = (x y) + x (P A7)(A(x0) ( x)(a(x) A(xS(x)))) ( x)a(x). Axioms (PA3) and (PA4) are the usual recursive definition of + in terms of 0 and S, the axioms (PA5) and (PA6) are the recursive definition of in terms of 0, S and +. The whole list of axioms (P A7) is called the additional axiom scheme of induction. Peano arithmetic is not finitely axiomatizable but if the induction scheme (PA7) is replaced by the single axiom (P A8)( x)(x 0 ( y)(x = S(y))), we obtain a finitely axiomatizable subtheory of PA. Terms 0, S(0), SS(0) we will call numerals and denote by 0, 1, 2,.... Generally n denotes the term S n (0) where S n (0) is S(S(... (S(0))...)), n copies of S for natural number n N. The standard model of PA is the model N =< N, 0 N, S N, + N, N > where N is the set of natural numbers, 0 N is zero, + N and N are addition and multiplication of natural numbers, S N (n) = n+1 for each n and = is interpreted absolutely. All other (non-isomorphic) models are called nonstandard. Let J be a language of the first-order predicate fuzzy logic with evaluated syntax which consists of the symbols of the language of PA and contains fuzzy equality. Let T be a fuzzy theory of the predicate fuzzy logic with evaluated syntax which contains logical axioms of Ev L, some special axioms and Peano axioms which are crisp and provable in the degree 1. Lemma 3. Let T be a fuzzy theory with the the language J. Then T x = y y = x, T x = y (y = z x = z). Definition 14. (n 0 -model) Let J be a language as above and 1 = S(0), 2 = SS(0),.... By n 0 - model we mean a model V in which V = {0, 1, 2,...}, that is V omits the fuzzy set Σ(x) = {a n x n an > 0, for all n < n 0, n N}, (7) If n 0 = ω then we suppose that lim n a n > 0 where either n 0 N or n 0 = ω. Definition 15. (n 0 -consistent) Let T be a fuzzy theory, J(T ) be a language. A fuzzy theory T is said to be n 0 -consistent if there is no formula A(x) of F J(T ) such that T = e(n) A(n) and T = d ( x) A(x) (8) where e(n) > 0, d = n<n 0 e(n) such that e(n) d > 0 holds true for all n < n 0 where n N. Definition 16. (n 0 -complete) Let T be a fuzzy theory J(T ) be a language. A fuzzy theory T is said to be n 0 -complete if for every formula A(x) of F J(T ) and for all n < n 0, n N T = e(n) A(n), e(n) > 0 (9) implies T = c ( x)a(x) where c n<n 0 e(n). Note that (9) implies T = f(n) A(n) for some f(n) e(n). The equality f(n) = e(n) holds true only for complete theories. We will use this fact below. The following lemma follows from the omitting types theorem. Lemma 4. Let T be consistent fuzzy theory and J(T ) be a language. Let T be a n 0 -complete. Then T has an n 0 -model. Lemma 5. Let T be consistent fuzzy theory and J(T ) be a language. Let V be a n 0 -model of T. Then T is n 0 -consistent. The n 0 -inference rule is the following: r n0 = e(0) A(0), e(1) A(1),... e ( x)a(x) (10) 965

6 where A(x) is any formula from J and e = n<n 0 e(n). If n 0 = ω then the n 0 -inference rule is infinitary. n 0 -logic is formed by adding the n 0 - rule to the axiom and rules of inference of the first-order fuzzy logic with evaluated syntax and allowing infinitely long proofs. Theorem 5. (n 0 -Completeness Theorem) Let T be consistent fuzzy theory in the language J(T ). A theory T in J(T ) is consistent in n 0 -logic if and only if T has n 0 -model. 6 Conclusion In this paper, we continued developing the model theory of fuzzy logic. In this paper, we have focused of fuzzy logic with evaluated syntax that is based of the Lukasiewicz algebra of truth values. It is specific for this logic, that it deals with fuzzy theories, i.e. fuzzy sets of formulas that can be determined by fuzzy sets of axioms. This enables to consider axioms that are not fully convincing. The main result of this paper was to present the possibilities how to construct the models in fuzzy logic with evaluated syntax. [5] P. Murinová-Landecká, Omitting types Theory in Fuzzy Logic with Evaluated Syntax, Journal of Electrical Engineering, vol. 55, no. 12s, 2004, [6] P. Murinová and V. Novák, Omitting Types in Fuzzy Logic with Evaluated syntax, unpublished. [7] V. Novák, Joint consistency of fuzzy theories, Mathematical Logic Quaterly, 48, 4, 2002, [8] V. Novák, Fuzzy Logic With Countable Evaluated Syntax, Proc. World Congress IFSA 2005, Beijing, China, [9] V. Novák, I. Perfilieva, and J. Močkoř, Mathematical Principles of Fuzzy Logic, Kluwer, BostonDordrecht, Acknowledgments This investigation has been partially supported by project MSM of the MŠMT ČR. References [1] C.C. Chang and H.J. Keisler, Model Theory, North-Holland Publishing Company, Amsterdam, London, [2] P. Cintula and P. Hájek, On theories and models in fuzzy predicate logics, unpublished. [3] P. Hájek, Metamathematics of fuzzy logic, Kluwer, Dordrecht, [4] P. Murinová-Landecká, Model theory in Fuzzy Logic with Evaluated Syntax Extended by Product, Journal of Electrical Engineering, vol. 54, no. 12s, 2003,