Kragujevac Journal of Mathematics
Volume 34 (2010), Pages 131–137.
COMPLETENESS THEOREM FOR PROBABILITY MODELS
WITH FINITELY MANY VALUED MEASURE
IN LOGIC WITH INTEGRALS
VLADIMIR RISTIĆ
Abstract. The aim of the paper is to prove the completeness theorem for probability models, in logic with integrals, whose measures have any finite ranges.
1. Introduction
The probability logics are created as logics appropriate for study of structures arising in Probability Theory. They have been introduced by Jerome Keisler who studied
first order structures in which probability is defined on the domen, i.e. structures
hM, µi where M is a first order structure and µ is probability. He began his study
by probability logic LAP in which ordinary quantifiers ∀x and ∃x are not allowed and
instead of them, quantifier (P x > r) is incorporated, where A is a countable admissible set such that A ⊆ HC and ω ∈ A (see [1]). The formula (P ~x ≥ r)ϕ(~x) means
that the set {~x | ϕ(~x)} has a probability greater then or equal to r.
The development of probability model theory has engendered the need for the study
of logics with a stronger expressive power than that of the logic LAP . Properties of
random variables are usually easier to express using integrals rather than probability
quantifiers. The logic LA∫ , introduced by Keisler in [6] as an equivalent of the logic
LAP , allows us to express many properties of random variables in an easier way. In this
R
logic the quantifiers . . . dx are incorporated instead of the quantifiers (P x ≥ r). The
Key words and phrases. Probability models, Finitely valued measure, Logic with integrals.
2010 Mathematics Subject Classification. 03C70.
Received: February 20, 2010.
Revised: July 26, 2010.
131
132
VLADIMIR RISTIĆ
logics LAP and LA∫ correspond to two alternative approaches to integration theory:
Lebesgue measure theory and the Daniel integral.
In [9], the quite similar investigation has been made for propositional probabilistic
logic which are conceived as the logics for reasoning about uncertainity. The logic
developed here is more expressive and more powerful.
2. Background
In order to prove the main results of the paper, we shall use the following theorem
(see [2]).
Theorem 2.1. Let F be a field of subset of 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.
Since, it is, in a way, measure in a probability models, in the logic LAP , expressed
more explicitly, this feature of a measure can be obtained by axiom
_
³
^
c∈Q+
ϕ∈Φn
´
(P ~x > 0)ϕ(~x) → (P ~x > c)ϕ(~x) ,
where Φn ∈ A and Φn = {ϕ | ϕ has n free variables} (see also [9]).
We will use the next result to create further interest in axiom of finitely many
valued measure in the logic Lfin
A∫ . By induction on the complexity of terms we can
prove that, for each term τ (~x) of LA∫ and for each r ∈ R ∩ A there is a formula ϕτ,r
of LAP such that in every probability structure hA, µi for each ~a ∈ An , the following
is true: τ A (~a) > r iff A |= ϕτ,r [~a] (see [8]).
3. Lfin
A∫ logic and completeness theorem
Let A be a countable admissible set such that A ⊆ HC and ω ∈ A and let L be an
Σ–definable set of finitary relation and constant symbols which has countably many
variables v1 , v2 , . . . , n–ary connective F for each continuous function F : Rn → R,
R
n ≥ 0 such that F ¹Qn ∈ A, and quantifiers . . . dx which build terms with bound
variables. The logic Lfin
A∫ is similar to the logic LA∫ , the only difference is that the list
of axioms of Lfin
A∫ has the following axiom of finitely many valued measure added:
_
"
^ ^ _ Z
c∈Q+ τ ∈D r∈Q k
³
´
Fk,r τ (~x) d~x > 0 →
Z
³
´
#
Fk,r τ (~x) d~x > c ,
COMPLETENESS THEOREM FOR PROBABILITY MODELS
133
where



1,
a ≥ r,
Fk,r (a) = 0,
a ≤ r − k1 ,


linear, otherwise ,
and D is a set of terms with finitely many free variables and D ∈ A.
Since the conjunction in the axiom above can be made only according to elements
of the admissible set A, that is
V
τ ∈D
where D ∈ A, we will apply the construction of
the middle model (see [7]).
The probability structure for Lfin
A∫ is a structure hA, µi such that A is a first–order
structure and µ is probability measure on A such that each singleton is measurable,
each RiA is µ(ni ) –measurable and µ is a finitely many valued probability measure. (The
measure µ(n) is the restriction of the completion of µn to the σ–algebra generated by
the measurable rectangles and the diagonal sets {x ∈ An | xi = xj }.) Graded model
is defined as in the logic LAP with the exception that the measures µn are with a
finite range in our logic.
We shall prove that this axiomatization is complete for Σ1 –definable theories with
respect to the class of probability models with finitely many valued measure, by
combining a Henkin construction (see [6] and [8]), and a weak–middle–strong model
construction, such as that of Rašković [7] (see also [3]).
We shall introduce two sorts of auxiliary structures.
Definition 3.1. (i) A weak structure for Lfin
A∫ is a structure hA, Ii where A is a first–
order structure for L and I is a positive linear real function defined on the set of
terms with at most one free variable x and parameters from A (I may be called an
A–Daniell integral an A).
(ii) A middle structure for Lfin
A∫ is a weak structure hA, Ii such that the axiom
of finitely many valued measure holds uniformly in any term τ ; that is, there is a
c ∈ Q+ such that for each term τ and ~a ∈ An , and for each r ∈ Q, there is k, if
³
³
´´
I Fk,r τ (x, ~a)
³
³
´´
> 0 then I Fk,r τ (x, ~a)



1,
> c where
a ≥ r,
Fk,r (a) = 0,
a ≤ r − k1 ,



linear, otherwise .
134
VLADIMIR RISTIĆ
In both cases, for a term τ , we define τ A inductively as for probability models,
except that at the integral step we define
ÃZ
!A
τ (x, ~y ) dx
³
´
(~a) = I τ, (x, ~a) .
By means of a Henkin construction argument similarly as in [8] we prove that a
Σ1 –definable theory of Lfin
A∫ is consistent if and only if it has a weak model in which
each theorem of Lfin
A∫ is true. Let C ∈ A be a set of new constant symbols introduced
in this Henkin construction and let K = L ∪ C.
The construction of the middle model from the weak model is the key part of the
proof of the main result and we present it with the following theorem.
fin
Theorem 3.1 (Middle Completeness Theorem). A Σ1 –definable theory T of KA∫
is
fin
consistent if and only if it has a middle model in which each theorem of KA∫
is true.
fin
has a middle
Proof. In order to prove that consistent Σ1 –definable theory T of KA∫
model, we introduce language M with four kind of variables: X, Y, Z, . . . variables
for sets, x, y, z, . . . variables for urelements, r, s, t, . . . for reals and P, Q, R, S, T, . . .
variables for terms which represent functions An → R, n ≥ 0. Predicates are ≤ for
t
reals, Ens (~x, X) (n ≥ 1 and ~x = x1 , . . . , xn ) for sets, En+1
(~x, r, T ) (n ≥ 0) (with the
meaning T (x1 , . . . , xn ) = r), and I(T, r) (T : A0 → R or T : A1 → R and I(T ) = r iff
I(T, r)). Function symbols are +, · and set of function symbols F for terms and reals,
fin
such that F ∈ CA (Rn ). Constant symbols are Xϕ for each formula ϕ of KA∫
, Tτ for
each term τ and r̄ for each real number r ∈ R ∩ A.
Let S be the following theory of MA which has the following list of formulas:
1. Axioms of well–definedness
V
(a) (∀X)
(b) (∀T )
V
´
³
s
(~x, ~y , X) ∧ Ens (~x, X) where {~x} ∩ {~y } = ∅,
x, ~y ) Em
n<m ¬ (∃~
´
³
t
t
(~x, s, T ) ,
(~x, ~y , r, T ) ∧ En+1
x, ~y , r, s) Em+1
n<m ¬(∃~
´
³
t
t
(c) (∀T ) (∀~x, r, s) (En+1
(~x, s, T )) → r = s .
(~x, r, T ) ∧ En+1
2. Axioms of extensionality
´
³
(a) (∀~x) Ens (~x, X) ↔ Ens (~x, Y ) ↔ X = Y ,
³
t
t
(b) (∀~x, r) (En+1
(~x, r, S)) ↔ T = S .
(~x, r, T ) ↔ En+1
3. Axioms of terms
³
h
³
´
t
t
x, y, 1, T1l(x=y)
x, y, 0, T1l(x=y) ∨ E2+1
(a) (∀x, y) E2+1
(b) (∀~x)
h
t
En+1
³
´
~x, 0, T1l(R(~x)) ∨
t
En+1
³
~x, 1, T1l(R(~x))
´i
,
´i
,
COMPLETENESS THEOREM FOR PROBABILITY MODELS
t
(c) E0+1
(r̄, Tτ ) where τ has the form r ,
h
³
135
³
t
t
(d) (∀~x) En+1
(~x, r, Tτ ) ↔ (∃P ) (∀y, s) E1+1
(y, s, P )
´
t
↔ En+1+1
(~x, y, s, Tσ ) ∧ I (P, r)
h
³
´
t
~x, r, TF(τ1 ,...,τk ) ↔
(e) (∀~x) En+1
´i
³V
where τ has the form
k
i=1
R
σ (~x, y) dy ,
´i
t
En+1
(~x, ri , Tτi )∧r = F (r1 , . . . , rk )
4. Axioms of satisfaction
³
´
t
(a) (∀~x) Ens (~x, Xτ ≥0 ) ↔ En+1
(~x, r, Tτ ) ∧ r ≥ 0 ,
³
´
(b) (∀~x) Ens (~x, X¬ϕ ) ↔ ¬Ens (~x, Xϕ ) ,
³
(c) (∀~x) Ens (~x, X∧ϕ ) ↔
´
V
ϕ
Ens (~x, Xϕ ) .
5. Axioms of integral operators I
(a) (∀T )
³V
n≥2
´
t
(~x, r, T ) ↔ (∃1 s) I (T, s) ,
¬ (∃~x, r) En+1
(b) (∀r) I (Tr , r) ,
³
´
(c) (∀S, T, r, s) I(r · T + s · S) = r · I(T ) + s · I(S) ,
³
´
t
(d) (∀T ) (∀x) (∃r ≥ 0) E1+1
(x, r, T ) → (∃s ≥ 0) I (T, s) .
6. Axiom of finitely many
µ valued measure
(∃c) (∀T )
V
³
W
r∈Q
k
´
³
¶
I Fk,r (T ) > 0 → I Fk,r (T ) > c


1,



a≥r
1

k


linear, otherwise
where Fk,r (a) = 0,
a≤r−
7. Axiom for an Archimedean field and all theorems of the theory of real closed
field
fin
8. Axioms which are transformations of axiom of KA∫
(∀~x) Ens (~x, Xϕ ) .
9. Axioms of realizability of all sentences ϕ in T
(∀x) E1s (x, Xϕ ) .
A weak structure hA, Ii can be transformed to a standard structure B = hB, P, Q,
B
E
B
t
F, Ens , Em+1
, I B , ≤, +, ·, Fn , XϕB , TτB , r̄ for MA where B = A, P ⊆
S
n≥0
S
n≥1
P(An ), Q ⊆
P(An × R), F ⊆ R a field, XϕB ∈ P for a formula ϕ and TτB ∈ Q for a term τ .
n
o
n
fin
}, TτB (~a) = τ A (~a) for
We can take XϕB = ~a ∈ An | A |= ϕ[~a] , P = XϕB | ϕ ∈ KA∫
n
fin
} and IkB (TτB ) = I(τ ) for a term τ with at most
~a ∈ An , Q = TτB | τ is term of KA∫
one free variable.
The theory S is Σ1 definable over A and S0 ⊆ S, S0 ∈ A has a standard model
because the axiom of finitely many valued measure holds in the weak model. It follows
.
136
VLADIMIR RISTIĆ
by means of the Barwise Compactness Theorem (see [1]) that S has a standard model
B which can be transformed to a middle model B̄ of T by taking:
τ B̄ (~x) = r
³
B
t
iff Em+1
(~x, 1, Tτ ) RB̄ (~a)
holds iff
B
³
t
Em+1
~a, r, T1l(R(~x))
´´
I B̄ (τ (~a, y)) = I B (Tτ (~a,y) )
for ~a ∈ B n .
This completes the proof.
¤
Now we shall prove the completeness theorem for the logic Lfin
A∫ . The method of
proof is similar to the completeness proof of Keisler [6]. It uses Loeb’s construction
in [8] which corresponds to the Daniell integral. We first state a simple case of Loeb’s
results as a lemma.
Lemma 3.1 (Loeb). In an ω1 –saturated nonstandard universe, let L be an internal
vector lattice of functions from an internal set M into ∗R (the set of hyper real numbers), and let I be an internal positive linear functional on L, such that 1 ∈ L and
I(1) = 1. Then there is a complete probability measure µ on M such that for each
finitely bounded φ ∈ L, the standard part of φ is integrable with respect to µ and its
integral is equal to the standard part of I(φ).
fin
Theorem 3.2 (Completeness Theorem for Lfin
A∫ ). A Σ1 –definable theory T of LA∫ is
consistent if and only if T has a probability model with finitely many valued measure.
Proof. The soundness part is easy. To prove hard part let C be countable set of new
constant symbols. We use the Henkin construction to obtain a maximal consistent
fin
set Φ of sentences of KA∫
where K = L ∪ C, such that:
(1) if Γ ⊆ Φ and
R
V
fin
Γ ∈ KA∫
, then
V
Γ ∈ Φ,
(2) if ( τ (x)dx > 0) ∈ Φ, then (τ (c) > 0) ∈ Φ, for some c ∈ C.
fin
, the Henkin theory Φ
Since Φ is complete and contains all the axioms of KA∫
induces a weak structure hA, Ii with universe A = C, such that every sentence in Φ
holds in hA, Ii.
The next step is the construction of the middle model for theory T of Lfin
A∫ . Based
on the Theorem 3.1. we get the middle model in which the axiom of finitely many
valued measure holds uniformly in any term τ . Then we pass to an ω1 –saturated
nonstandard universe and form internal middle integral structure h∗A, ∗Ii. The Loeb
construction of Daniell integral produces a probability measure µ on ∗A such that for
COMPLETENESS THEOREM FOR PROBABILITY MODELS
137
each term τ (x) with parameters from ∗A, the standard part of ∗I(τ ) is the integral of
the standard part of τ
∗
∗A
with respect to µ. Similarly, we can construct a probability
n
measure µn on A using iterated integrals and in that way we get a graded model
h∗A, µn i. From Loeb’s construction, the axiom of finitely many valued measure and
the Theorem 2.1. we conclude that the measures µn , n ∈ N will have finite ranges.
Finally, the axiom product measurability, which is essentially a translation of the
axiom (B4 ) for LAP (see [6]), is satisfied almost every where in the graded model
h∗A, µn i. This yields a probability model hB, µi of T, where B = ∗A and µ = µ1 . ¤
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Faculty of Education,
University of Kragujevac,
M. Mijalkovića 14, 35000 Jagodina,
Serbia
E-mail address: [email protected]