PROBABILITY LOGICS WITH VECTOR VALUED MEASURES. Vladimir Ristić

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1 47 Kragujevac J. Math ) PROBABILITY LOGICS WITH VECTOR VALUED MEASURES Vladimir Ristić Faculty of Teacher Education, Milana Mijalkovića 14, Jagodina, Serbia Received August 01, 2008) Abstract. Probability logic introduced by this paper is based on probability logic L AP. Measure ranges in probability models will not be linearly ordered, more precisely, measures will be vector valued, having ranges Q n [0, 1] n. Axioms and rules of inference are adjusted to determine these types of measures. The completeness theorem for the introduced logic is proved. INTRODUCTION The development of the model theory of first-order logic has brought up the need for the study of logics with a stronger expressive power. This allows us to incorporate into the realm of logic certain common mathematical structures and concepts that have been left out of first-order logic due to its limited scope. Probability logics are logics adequate for the study of structures arising in Probability Theory. Probability logics see [2, 3, 6]) are formed by adding probability quantifiers P x r to first order logic, where P x r)ϕ x) means { x ϕ x)} has probability r. Logic in this paper is extension of logic L AP, is formed by using

2 48 n-typles of reals and probability quantifiers P x r 1,..., r n ). Measures are vector valued and their ranges are [0, 1] n Q n, are not linearly ordered. Loeb s construction is expanded to B-valued measures where it is B Banach space see [4, 5, 8]). The axiom system was given and the completeness theorem for the logic whose models have vector valued measures was proved, which resulted in positive solution of Keisler s problem for the vector space R n see [3]). 1. LOEB COMPLETION OF INTERNAL VECTOR VALUED MEASURES First, we will show that the important Loeb s construction can also be expended to internal spaces of forms X, A, µ, B, where B, is a Banach space and µ is an internal finitely additive, B-valued measure defined on A. Definition 1.1. Let X, A, µ be an internal measure space with internal, finitelyadditive, finite measure µ. A subset A X is Loeb measurable if for every standard ɛ > 0 there exist B, C A such that B A C and st µc\b) ) < ɛ. The family of all Loeb measurable sets is denoted by LA), and let Lµ) denotes the natural extension of µ = st µ on LA). Proposition 1.2. X, LA), Lµ) is a complete measure space with the σ additive measure Lµ). Let B be a Banach space and X, A, µ, B an internal space with finitely additive B-valued measure µ. Let us suppose that the total variation ϑµ, ) defined by ϑµ, A) := sup P { µd) D P }, where P ranges over set of all -finite, A-measurable partitions of X, is a finite internal positive measure on X. Recall that an element x B particularly x R) is finite x fin B)) if x m for some m R. The measure µ := varµ, ) is positive, hence let X, LA), L µ ) be the completion of X, A, µ. Our aim is to define

3 49 Loeb completion X, LA), Lµ),? of the measure space X, A, µ, B. The natural candidate for? is of course B, the nonstandard hull of Banach space B. Recall that B := fin B)/, where x y means that x y is of infinitesimal norm. To simplify notation, x will denote the norm of x for both x B and x B. Analogously to the case B = R the quotient map fin B) B will be called the standard part map and denoted by st. We need the following well-known proposition. Proposition 1.3. Let X, LA), Lµ) be the Loeb completion of the space X, A, µ with finitely-additive, positive, finite measure µ. Then, if A LA), then there exists B A such that Lµ)A B) = 0. Applying the last proposition to the measure space X, A, µ, where µ := µ, ), we get that A LA), iff there exists B A such that L µ )A B) = 0. Also, if B 1, B 2 A are two sets which, in this sence, approximate A, then st µ B 1 B 2 ) L µ ) A B 1 ) + L µ ) A B 2 ) = 0, which means that µb 1 ) µb 2 ) m0), because µb 1 ) µb 2 ) µ B 1 B 2 ). This allows us to give the following definition. Definition 1.4. Let X, A, µ, B be an internal measure space with a finitelyadditive B-valued measure µ so that total variation µ := var µ, ) is finite. Let ) X, LA), L µ be the Loeb space associated with the internal space X, A, µ. For A LA) and B A, with the property L µ )A B) = 0, let Lµ)A) := st µb) ), where st: fin B) B is the mapping defined above. Lµ) is obviously an additive B-valued measure. To prove its σ-additivity we need the following simple inequality: For all A LA) holds Lµ)A) L µ )A).

4 50 Proposition 1.5. Lµ) is a σ-additive measure. Considering the fact that we will work with probability measures taking values in Banach space R n, this result produced by Živaljević see [8]) will be applicable in the process of creating probability models. More precisely, probability vector-valued measure will have a finite total variation. Moreover, a somewhat stronger result given by Oswald and Sun is applicable as well see [5]). Oswald proved see [4]) the vector-valued version of the following Keisler s result see also [7]). If A 1, F 1, µ 1 and A 2, F 2, µ 2 are two internal finitely additive measure spaces, we have the internal product space A 1 A 2, F 1 F 2, µ 1 µ 2, where F 1 F 2 is the internal product algebra generated by F 1 and F 2. This space has as a corresponding Loeb space A 1 A 2, LF 1 F 2 ), Lµ 1 µ 2 ). Also we can construct from the Loeb spaces of A 1, F 1, µ 1 and A 2, F 2, µ 2 the standard product space A 1 A 2, LF 1 ) LF 2 ), Lµ 1 ) Lµ 2 ). Theorem 1.6. Keisler-Fubini) Let f : A 1 A 2 R be Lµ 1 µ 2 )-measurable function. Then 1) For Lµ 2 )-almost all a 2 A 2, the function f, a 2 ): A 1 R is Lµ 1 )-measurable; 2) If f is integrable, then a) for µ 2 -almost a 2, the function f, a 2 ) is integrable over A 1, b) the function ga 2 ) = fa 1, a 2 ) dlµ 1 ) is integrable over A 2, c) ga 2 ) dlµ 2 ) = fa 1, a 2 ) dlµ 1 µ 2 ). 2. SYNTAX AND SEMANTICS Let A be a countable admissible set, i.e. a well-behaved transitive model of Kripke- Platek set theory for a definition of admissible sets see [1]). We will now discuss the

5 51 situation when ω is not an element of A so that each formula is finite. We use Φ Φ) to denote disjunction conjunction) of finite number of formulas. We will assume that the rationals are defined is such a way that Q A; by adding additional reals into A as urelements, we can obtain more probability quantifiers. That is, this construction suggest that n-tuples of rationals and reals are in the set A. We use L AP to denote our logic. Let L be a countable, Σ-definable set of finitary relation and constant symbols no function symbols). Logical symbols that will by used are the same as in L AP logic, except for the quantifiers which are defined in L AP logic by using n-tuples of reals, i.e. P x r 1,..., r n )), where r 1,..., r n ) R n A. Also, the set of formulas are defined as in L AP logic except for the part related to quantification, i.e. if ϕ is a formula of the L AP logic, then P x r 1,..., r n ))ϕ is also a formula of L AP. In the case of our logics, short forms cannot be used as in the logic L AP, i.e. we cannot write P x < r 1,..., r n ))ϕ instead of P x r 1,..., r n ))ϕ, because the measure range is not linearly ordered. The relation on set A [0, 1] n, defined by a 1,..., a n ) b 1,..., b n ) a 1 b 1... a n b n, is a partial order on the given set. Relation < is defined by ) i {1,..., n} ai < b i j {1,..., n} ) a j > b j. Similarly we can define and >. Operations + and will be defined on coordinates as follows: a 1,..., a n ) + b 1,..., b n ) = a 1 + b 1,..., a n + b n ) where a i + b i = mina i + b i, 1) a 1,..., a n ) b 1,..., b n ) = a 1 b 1,..., a n b n ). The satisfiability is defined in a usual way. Definition 2.1. A probability model is a structure A, µ k k<ω = A, µ, where

6 52 1) A = A, R i, c j i I,j J is a model in the sense of first order logic; 2) Each µ k, k < ω, is a σ-additive) probability measure an A k, each µ k is vectorvalued with values in [0, 1] n Q and the sequence of measures µ k k < ω satisfies the Fubini property, that is: i) For all m, k µ m+k is an extension of the product measure µ m µ k ; ii) Each µ k is invariant under permutations. That is, whenever π is a permutation of {1, 2,..., k}, and S dom µ k ), if πs = { ) a π1),..., a πk) a1,..., a k ) S }, then πs dom µ k ) and µ k πs) = µ k S); If S dom µ m+k ), then iii) For all b A k, { a a, b) S } domµ m ); iv) For all r 1,..., r n ) R n, { b µ m {a a, b) S} > r 1,..., r n ) } dom µ k ) ; v) µ m+k S) = µ S mdx) ) µ k dy). 3) Each atomic formula with k free variables is measurable with respect to µ k. 3. A COMPLETE AXIOMATIZATION Axioms and rules of inferences of logic L AP are: 1) All axioms of logic L A without quantifiers 2) Monotonicity of the quantifier: i) P x r 1,..., r n ) ) ϕ x) P x s 1,..., s n ) ) ϕ x), where s 1,..., s n ) r 1,..., r n ); ii) P x > r 1,..., r n ) ) ϕ x) P x r 1,..., r n ) ) ϕ x); iii) P x r 1,..., r n ) ) ϕ x) P y r 1,..., r n ) ) ϕ y); 3) Axioms and rules about probability:

7 53 i) From ψ ϕ x) infer ψ P x 1,..., 1) ) ϕ x); ii) P x 0,..., 0) ) x x, µ 1 ) 0,..., 0) ) ; iii) Additivity axioms. P x 1,..., 1) )[ ϕ x) ψ x) )] P x r1,..., r n ) ) ϕ x) P x s 1,..., s n ) ) ψ x) P x r 1 + s 1,..., r n + s n ) ) ϕ x) ψ x) ) ; and P x r1,..., r n ) ) ϕ x) P x s 1,..., s n ) ) ψ x) P x r 1 + s 1,..., r n + s n ) ) ϕ x) ψ x) ) ; iv) Monotonicity of probability measure P x 1,..., 1) ) ϕ x) ψ x) ) P x r1,..., r n ) ) ϕ x) P x r 1,..., r n ) ) ) ψ x) ; v) Probability measure is continuous at 0. We use the following rule scheme: For any k < ω, k > 0, from ϕ P y < 1,..., 1 k k) ) [ r1 P x 1,..., r m n m) 1, r1,..., r n )) ) ψ, [ r1 ) m = 1, 2,... infer ϕ, where 1,..., r m n m) 1, r1,..., r n ) in the last formula represents a set of all n-tuples a 1,..., a n ) so that a 1,..., a n ) r1 1,..., r ) m n 1 m and a1,..., a n ) < r 1,..., r n ). 4) Fubini property axiom: i) Permutation axiom: If π is a permutation of {1,..., n}, P x1,..., x n r 1,..., r n ) ) ϕ P x π1,..., x πn r 1,..., r n ) ) ϕ;

8 54 ii) Product independence: P x r1,..., r n ) ) P y s 1,..., s n ) ) ϕ x, y) and P x y r 1 s 1,..., r n s n ) ) ϕ x, y), P x r1,..., r n ) ) P y s 1,..., s n ) ) ϕ x, y) P x y r 1 s 1,..., r n s n ) ) ϕ x, y), where all variables in x, y are distinct. 5) In addition, we will add a rule that secures the situation in which measure will take values only in the set [0, 1] n Q. Namely: if ϕ P x q 1,..., q n ) ) ψ, for every q 1,..., q n ) [0, 1] n Q, then we infer ϕ; Of course, the list of rules of inference includes also modus ponens and conjunction rule, as in the logic L AP. Remark 3.1. The adapted Hoover s Iterated integration axioms can be derived from the axioms Product independence: Let P is partition of intervals [0, 1] n consisted from S 1, S 2,..., S k where S i = [s i 1, s i 2) [s i 3, s i 4) [s i 2n 1, s i 2n), where are all specified intervals are subset of [0, 1]; 1 l k P x r l 1,..., r l n) ) P y S l )ϕ x, y) P x y ) r1, l..., rn) l s l 1, s l 3,..., s l 2n 1) ϕ x, y), where r l 1,..., r l n) [0, 1] n, 1 l k; and let P is partition of intervals [0, 1] n consisted from S 1, S 2,..., S k where S i = [s i 1, s i 2] [s i 3, s i 4] [s i 2n 1, s i 2n],

9 55 where are all specified intervals are subset of [0, 1]; 1 l k P x r l 1,..., r l n) ) P y S l )ϕ x, y) P x y ) r1, l..., rn) l s l 2, s l 4,..., s l 2n) ϕ x, y), where r l 1,..., r l n) [0, 1] n, 1 l k. The proof is rather technical, and axioms Product independence are used k times. Definition 3.2. A weak model for L AP is a structure A = M, R A i, c A j, µ k i I,j J,k N, such that M, Ri A, c A j, is a classical model, each µ i I,j J k is a finitely additive probability measure on A k with each singleton measurable and takes values in the set [0, 1] n Q n and with the set { c A k A ϕ[ a, c ] } µ k -measurable for each ϕ x, y) L AP and each a A. In this case, we will define ϕ like in the logic L AP, except for the part for quantification. Namely, since a negation in the logic L AP cannot go trough the formula containing quantifiers; under P x r 1,..., r n )) ϕ ) we will assume simply the negation of formula P x r 1,..., r n ) ) ϕ. Let C be a countable set of new constant symbols and let K = L C. Then we form the logic K AP property. corresponding to K and we introduce a notion of a consistency Definition 3.3. A consistency property for L AP is a set S of countable sets s of sentences of K AP which satisfies the following conditions for each s S: C 1 ) Triviality rule) S; C 2 ) Consistency rule) Either ϕ / s or ϕ / s; C 3 ) -rule) If ϕ s, then s {ϕ } S; C 4 ) -rule) If Φ s, then for all ϕ Φ,s {ϕ} S;

10 56 C 5 ) -rule) If Φ s, then for some ϕ Φ, s {ϕ} S; C 6 ) P-rule) If P x > 0,..., 0) ) ϕ x) s, then for some c C, s { ϕ c) } S; C 7 ) If ϕ x) K AP is an axiom, then { P x ) } i) s 1,..., 1) ϕ x) S; ii) s { ϕ c ) } S, where c C; C 8 ) Continuity rule) For each k < ω, r 1,..., r n ) [0, 1] n and formula ϕ x, y) of K AP with only finitely many free variables there is some m < ω such that P s { y < 1,..., 1 k k)) [ r1 P x 1,..., r m n m) 1, r1,..., r n )) ) } ϕ x, y) S; C 9 ) For any formula ϕ x) of logic K AP with only finitely many free variables, for some q 1,..., q n ) [0, 1] n Q, there is s { P x = q 1,..., q n ) ) ϕ x) } S. Theorem 3.4. If S is a consistency property, then any s 0 S has a probability model. The completeness theorem follows from this theorem because the set of all countable, consistent sets of formulas K AP is a consistency property. Proof. This consist of two lemmas. First, we will find for s 0 a weak model, and then by applying Loeb s construction on internal weak model we will make a probability model for s 0. Lemma 3.5. If S is consistency property, then any s 0 S, has a weak model. Proof. We define a complete sequence s k k < ω of elements of S as follows: Let ϕ k k < ω be an enumeration of the sentences of K AP. s 0 is given. Given s k choose s k+1 to satisfy the following conditions: 1) s k s k+1 ;

11 57 2) If s k {ϕ k } S, then ϕ k s k+1 ; 3) If s k {ϕ k } S, ϕ k = Φ, then for some θ Φ, θ s k+1 ; 4) If s k {ϕ k } S and ϕ k = P x > 0,..., 0) ) ψ x), then for some c C, ψ c ) s k+1 ; 5) Continuity rule) Let ) ψ k x, y), r k k < ω be an enumeration of the pairs of formulas of KAP which have only finitely many free variables, and n-tuples of reals, listed so that each pair occurs infinitely often. Then for some m, P y < 1 k,..., 1 k )) P x [ r k 1 m,..., 1 m ), r k) ) ψ k x, y) s k+1 ; 6) Let ψ k x ), q k ) k < ω be an enumeration od the pairs of formulas of KAP which have only finitely many free variables, and n-tuples of rationales, then P x = q1,..., q n ) k ) ψk x) s k+1. Now let s ω = n s n. Let T be the set of constants of K AP. For c, d T, let c d iff c = d s ω. Then, is an equivalence relation. Let [c] denote the equivalence class of the constant c. Let A have the universe set A = { [c] c T }. If R is an n-placed relation symbol and c 1,..., c n C, then A R [c 1 ],..., [c n ] ) iff R c 1,..., c n ) s ω. Define µ k on the subset of A k definable by formulas of L AP with parameters from A, by µ k { a A k A ϕ[ a, c ] } = q 1,..., q n ) iff P x = q1,..., q n ) ) ϕ x, c ) s ω. The axioms about probability guarantee that the µ k are finitely additive probability measures with ranges in [0, 1] n Q. It is routine to check that A ϕ [c 1 ],..., [c n ] ) whenever ϕ c 1,..., c n ) s ω. Therefore A is a weak model of s ω, and hence a model of s 0. Lemma 3.6. Let A, µ be a weak model for L AP. Then the model A, Lµ), obtained by applying the Loeb process to A, µ, is a probability model, and for each ϕ x) L AP, a A A, µ ϕ[ a] iff A, Lµ n ) ϕ [ a].

12 58 Proof. Let V S) be a superstructure over S and R A S. We suppose that a formula ϕ x, a) with parameters from A, a weak model A, and the relation are represented by sets in V S). Then ϕ x, a) and A are sets in the nonstandard universe V S), and is an internal relation. If the context is clear we write simply. At the beginning of this paper we proved that in this case it is possible to infer Loeb s process. A, L µ) satisfies the Fubini property because A, µ satisfies the Fubini property axioms form which it follows that A, µ satisfies them in the nonstandard sense), and because of the very mode of constructing measure on the model. The main step in our proof is to show that for each ϕ x) L AP and a A A, µ ϕ[ a] iff A, Lµ) ϕ [ a]. 1) To prove 1) we prove by induction on formulas that for ϕ x, y) L AP, a A { c L µ k ) A A, Lµ) ϕ [ c, a] } { c A A, µ ϕ[ c, a ] }) = 0. Since in our logic L AP we have only finite formulas, the only nontrivial case for the last equality is ϕ y) = P x r 1,..., r n ) ) ψ x, y). From now on we shall suppress parameters from A. Let ϕ y) = P x r 1,..., r n ) ) ψ x, y). Then we have P x r1,..., r n ) ) ψ x)) µ k { x ψ x)} r 1,..., r n ) in and P x r1,..., r n ) ) { ψ x) k N) µ k x ψ x) } r 1 1,..., r ) k n 1 k. R n On the basis of the rule of inference measure continuity, we have this is just the limited case) { P x 0 = Lµ k ) c A A, µ r1,..., r n ) ) ) ψ x, c) P x r 1,..., r n ) ) } ψ x, c).

13 59 By the triangle argument L µ k ) { c A, µ ϕ c ) ϕ c ) } { P x Lµ k ) c A, µ r1,..., r n ) ) ) ψ x, c ) P x r 1,..., r n ) ) } ψ x, c ) + { + Lµ k ) c A, µ P x r 1,..., r n ) ) } ψ x, c ) { c A, Lµ) P x r 1,..., r n ) ) }) ψ x, c ). The first term is 0. By applying the induction hypothesis: Lµ m+k ) { d, c ) A, µ ψ[ d, c ] } { d, c ) A, Lµ) ψ[ d, c ] }) = 0. So, for all c s but a set of Lµ k )-measure 0 we have: Lµ m ) { d A, µ ψ[ d, c ] } { d A, Lµ) ψ[ d, c ] }) = 0. So, for all c s but a set of Lµ k )-measure 0 we have: Lµ m ) { d A, µ ψ[ d, c ] } r 1,..., r n ) iff Lµ m ) { d A, Lµ) ψ[ d, c ] } r 1,..., r n ). Hence the second term in the inequality is 0. If some definable set S in model A, µ had measure q 0, then, as a consequence of 1), we have that set S will keep the same measure also in model A, Lµ), and vice versa, for any formula ϕ a) L AP. This concludes the proof of the lemma and theorem. References [1] J. Barwise, Admissible Sets and Structures, Springer-Verlag, Berlin, [2] D. N. Hoover, Probability logic, Annals of mathematical logic 14, 1978),

14 60 [3] H. J. Keisler, Probability quantifiers, in: Model-theoretic logics. etds. J. Barwise, S. Feferman, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1985), [4] H. Osswald, Bochner and Pettis integrable functions an Loeb spaces, to appear. [5] H. Osswald and Y. Sun, On the extensions of vector valued Loeb measures, Proceedings of the American Mathematical Society ) [6] M. Rašković and R. Djordjević, Probability quantifiers and operators, Vesta, Beograd, [7] A. V. Uglanov, Fubini s theorem for vector-valued measures, Math. USSR SB, 1991), 692), [8] R. T. Živaljević, Loeb completion of internal vector valued measures, Math. Scand ),