Completeness theorem for probability models with finitely many valued measure

Miodrag Rašković; Radosav Djordjević; Nenad Stojanović

doi:10.1515/math-2019-0016

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Completeness theorem for probability models with finitely many valued measure

Miodrag Rašković , Radosav Djordjević and Nenad Stojanović

Published/Copyright: March 26, 2019

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$Open Mathematics$

From the journal Open Mathematics Volume 17 Issue 1

Abstract

The aim of the paper is to prove the completeness theorem for probability models with finitely many valued measure.

Keywords: Probability models – finitely valued measure

MSC 2010: 03C70; 03B48

1 Introduction

Probability logics introduced by H. J. Keisler are logics appropriate for the study of structures of the form (𝔄, μ) arising in Probability Theory, where 𝔄 is a first order structure and μ is a probability measure on A. The reader can find detailed presentations on the host of probability logics in [9] and the monograph [11]. The basic probability logic L_𝓐P is similar to the infinitary logic L_𝓐 ^[1] except that instead of the ordinary quantifiers ∀x and ∃x, the logic L_𝓐P possesses the probability quantifiers (Px > r).

In this paper using the ideas from [2, 5, 10] we introduce logic LAPfin which is complete for Σ₁ definable theories with respect to the class of probability models with finitely many valued measure. Let us note that our work could be seen as the first step towards the widening of application frame for probability logics since in applied mathematics one often deals with (very large but) finite phenomena.

2 LAPfin logic

The main result which enable us to prove the corresponding Completeness Theorem is the following theorem (see [2]).

Theorem 2.1

Let 𝓕 be a field of subsets of a set Ω. Then μ is a finitely many valued probability measure on 𝓕 if and only if there is a real number c > 0 such that μ(A) > c whenever A ∈ 𝓕 and μ(A) > 0.

The logic LAPfin has all the axiom schemas and rules of inference of L_𝓐P (listed in [9, 11]) as well as the following axiom of finitely many valued measure:

⋁c∈Q+⋀φ∈Φn(Px→>0)φ(x→)→(Px→>c)φ(x→),

where Φ_n ∈ 𝓐 and Φ_n = {φ : φ has n free variables}.

Completeness theorem will be proven by combining a consistency property argument, such as that of [9] or [6], and a weak-middle-strong construction, such as that of [10]. We need two sorts of auxiliary structures.

Definition 2.2

A weak structure for LAPfin is a structure (𝔄, μ_n)_n≥1 such that each μ_n is a finitely additive probability measure on Aⁿ with each singleton measurable and the set φ_a⃗ = {b⃗ : (𝔄, μ_n) ⊨ φ[a⃗, b⃗]} is μ_n-measurable for each φ(x⃗, y⃗) ∈ LAPfin and a⃗ ∈ A^m.
A middle structure for LAPfin is a weak structure (𝔄, μ_n) such that the following is true: there is a c > 0 such that for each formula φ(x⃗, y⃗) ∈ LAPfin and each a⃗ ∈ A^m, if μ_n(φ_a⃗) > 0, then μ_n(φ_a⃗) > c.

Using a consistency property similarly as in [6] or [9] we prove that Σ₁ definable theory of LAPfin is consistent if and only if it has a weak model in which each theorem of LAPfin is true. Let C ∈ 𝓐 be a set of new constant symbols introduced in this Henkin construction and let K = L ∪ C.

Theorem 2.3

(Middle Completeness Theorem) A Σ₁ definable theory T of KAPfin is consistent if and only if it has a middle model in which each theorem of KAPfin is true.

Proof

In order to prove that consistent Σ₁ definable theory T of KAPfin has a middle model, we introduce language M with three sorts of variables, such as that of [10]: X, Y, Z, … variable for sets, x, y, z, … variable for urelements and r, s, t, … variables for reals from [0, 1]. The predicates of M are ≤ for reals, E_n(x⃗, X) for n ≥ 1 and x⃗ = x₁, …, x_n (with the canonical meaning x⃗ ∈ X) and μ(X, r) (with the meaning μ(X) = r). The constant symbols are set constant symbols X_φ, for each φ ∈ KAPfin and r for each r ∈ [0, 1] ∩ 𝓐. The functional symbols are + and ⋅ for reals.

Let S be the first-order theory of M_𝓐 which has the following list of formulas:

Axiom of well-definedness

(∀X) ⋀_n<m ¬ (∃ x⃗, y⃗) (E_m(x⃗, y⃗, X) ∧ E_n(x⃗, X)), where {x⃗} ∩ {y⃗} = ∅;
Axiom of extensionality

(∀ x⃗) (E_n(x⃗, X) ↔ E_n(x⃗, Y)) ↔ X = Y;
Axioms of satisfaction
1. (∀x⃗) (E_n(x⃗, X_∧Φ) ↔ ⋀_φ∈Φ E_n(x⃗, X_φ))
2. (∀x⃗) (E_n(x⃗, X_¬φ) ↔ ¬ E_n(x⃗, X_φ))
3. (∀x⃗) (E_n(x⃗, X_{(Px⃗≥r)φ} ↔
  
  ↔ ((∃₁ X) (μ(X) ≥ r ∧ (∀ y⃗) (E_n+m(x⃗, y⃗, X_φ) ↔ E_m(y⃗, X))))),
  
  where μ(X) ≥ r is the formula (∃ s)(s ≥ r ∧ μ(X, s));
4. (∀x⃗) ((∀y⃗)E_n+m(y⃗, x⃗, X_R) ↔ E_n+m(c⃗, x⃗, X_R)),
  
  for each predicate R(c⃗, x⃗) ∈ 𝓐;
Axioms of measure
1. (∀X) (∃₁ r)μ(X, r);
2. (∀X) (∀ Y) ((μ(X, r) ∧ μ(Y, s) ∧ ¬ ⋁_n≥1 (∃x⃗) (E_n(x⃗, X) ∧ E_n(x⃗, Y))). →
  
  → (∃ Z) (⋀_n≥1 (∀x⃗) (E_n(x⃗, Z) ↔ (E_n(x⃗, X) ∨ E_n(x⃗, Y))) ∧ μ(Z, r + s)));
Axiom of finitely many valued measure
1. (∃c>0)(∀X)(μ(X)>0¯→μ(X)>c),
where μ(X) > r is the formula (∃ s)(s > r ∧ μ(X, s));
Axioms for an Archimedean field (for real numbers);
Axioms which are transformations of axioms of KAPfin

(∀x⃗)E_n(x⃗, X_φ), where φ is an axiom of KAPfin ;
Axiom of realizability of T

(∀x)E₁(x,X_φ), for each sentence φ in T.

A standard structure for M_𝓐 is the structure

B=B,P,EnB,μB,+,⋅,≤,XφB,rn≥1,φ∈K′,r∈F,

where P ⊆ ⋃_n≥1 𝓟(B), EnB ⊆ Bⁿ × P, F = F′ ∩ [0, 1], F′ ⊆ ℝ is a field, μ^𝔅 : P → F, +, ⋅ : F² → F, ≤ ⊆ F², XφB ∈ P and K′ ⊆ KAPfin .

The theory S is Σ₁ definable over 𝓐. To prove that S is consistent it is enough, by Barwise Compactness Theorem (see [1]), to show that S₀ ⊆ S, S₀ ∈ 𝓐 has a standard model. First, note that a weak structure (𝔄, μ_n) for KAPfin can be transformed into a standard structure by taking: XφB = {a⃗ : (𝔄, μ_n) ⊨ φ[a⃗]} and P={XφB:φ∈KAPfin}. Since the axiom

⋁c∈Q+⋀φ∈(S0′)n(Px→>0)φ(x→)→(Px→>c)φ(x→)

holds in the weak model (𝔄, μ_n), where S0⊆S0′,S0′ ∈ 𝓐 is the closure for the substitution of constant symbols from C and disjunction and (S0′)n={φ∈S0′:φ has n free variables}, it follows that

A,P,EnA,μA,+,⋅,≤,{a→∈An;(A,μn)⊨φ[a→,c→]},r}n≥1,φ∈S0,r∈[0,1]∩A

where P = {{a⃗ ∈ Aⁿ ; (𝔄, μ_n) ⊨ φ[a⃗, c⃗]} : φ ∈ S₀}, is the standard model for S0′ and S₀, too.

Lastly, note that a standard model 𝔅 of S can be transformed into a middle model 𝔅 of T by taking:

x⃗ ∈ R^𝔅 iff EnB (x⃗, X_R) for an n-ary relational symbol R ∈ L,
μnB¯ (X) = r iff μ^𝔅(X, r) for X ∈ 𝓟(B).

It follows from the Loeb-Hoover-Keisler construction (see [6, 9, 11]) that the axiom of finitely many valued measures implies that (⋆) holds for all internal sets in the nonstandard superstructure. The property (⋆) also holds for all Loeb measurable sets because these can be approximated by internal ones. Thus, it follows from Theorem 2.1. that each middle model in which all theorems of LAPfin hold is elementary equivalent to a probability model for LAPfin . As a consequence of the preceding we obtain the following theorem.

Theorem 2.4

(Completeness Theorem for LAPfin ) A Σ₁ definable theory T of LAPfin is consistent if and only if T has a probability model with finitely many valued measure.

Finally, let us note that structure (𝔄, μ), where μ is a finitely many valued probability measure, cannot be axiomatized so that extended completeness theorem holds. The following example of a countable consistent theory T in LAPfin does not have a probability model with finitely many valued measure.

Example. Let L = {R₁(x), R₂(x), … } be a Δ₁ definable set which is not a subset of an element of 𝓐, and let φ₁,φ₂, … be an enumeration of all formulas from LAPfin . Then there exists the first predicate, denoted by R_{φ_n}(x), not occurring in φ₁, …, φ_n; otherwise L ⊆ TC(φ₁) ∪ … ∪ TC(φ_n) ∈ 𝓐, which would imply that L ∈ 𝓐 as a Δ₁ definable set.

It is obvious that the countable theory

T={(Pxy≥1)x≠y}∪{(Px>0)(Rφ1(x)∧…∧Rφn(x)):n∈ω}∪{(Px<1/2n)Rφn(x):n∈ω}

does not have any probability model with finitely many valued measure. We prove that T is consistent in LAPfin .

Let I be a unit interval [0, 1] and let μ be a Lebegue measure on I. For B_n = [0, 1/2ⁿ⁺¹) we have 0 < μ(B_n) < 1/2ⁿ. Let An1…nki1…ik=Bn1i1∩…∩Bnkik be a Boolean atom, where Bni = B_n for i = 1 and Bni = I ∖ B_n for i = −1.

We interpret the predicates by taking Rn(A,μ)=Bn,if Rn=Rφm for some mB1,otherwise.

Since only finitely many predicates R_{φ_n} can occur in an element of 𝓐, it follows that the set

An1…nki1…ik:μAn1…nki1…ik>0

is finite. The theory T and all axioms of LAPfin are satisfied except perhaps the axiom of finitely many valued measure. But, for

c=minμAn1…nki1…ik:μAn1…nki1…ik>0,

T is consistent in LAPfin .

3 Conclusion

In this paper we have used very fruitful technique introduced by Raskovic in [10]. The technique is developed for solving Keisler’s problem about probalilistic logics with two measures. More precisely, Keisler proposed research problem of developing a model theory and particulary proving the completeness theorem for the class of biprobability models, and specially for structures with two measures μ₁, μ₂ such that μ₁ is absolutely continuous with respect to μ₂. After paper [10], the method which was used in the proof becomes a very powerful technique for producing new results. In the first instance, the method has given completeness for many logics that appear in Probability Theory. As an example (for more see [11]) we point out a corresponding result for the extension L(∫₁, ∫₂)_𝓐 of logic with integrals related to analytic models, [3]. Furthermore, using the same ideas, the completeness theorem is proven for the logic appropriate for the study of topologies on proper classes [4]. The idea for future research is to systematize many applications of this method. Future research should be dedicated to finding appropriate logics for classes of structures equipped with two monotone collections connected in the different ways.

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Received: 2018-04-26

Accepted: 2019-01-29

Published Online: 2019-03-26

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https://doi.org/10.1515/math-2019-0016

Keywords for this article

Probability models – finitely valued measure

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