states on polyadic mv-algebras

Studia Logica (2010) 94: 231–243DOI: 10.1007/s11225-010-9233-y © Springer 2010

George Georgescu States on Polyadic

MV-algebras

Abstract. This paper is a contribution to the algebraic logic of probabilistic models

of �Lukasiewicz predicate logic. We study the MV-states defined on polyadic MV-algebras

and prove an algebraic many-valued version of Gaifman’s completeness theorem.

Keywords: polyadic MV-state, Gaifman MV-model.

1. Introduction

The study of probabilities in first order classical logic started with Gaifman’spaper [7]. In [7], the probabilities defined on sets of formulas generalize firstorder theories and an appropriate notion of probabilistic model is introducedby means of Gaifman’s condition ([7], p. 3). Among the results of [7],we mention a completeness theorem stated in terms of probability models(see [7], Theorem 2). Then the theory of probability models for classicalpredicate logic was developed by several authors (see e.g. [1], [6], [8], [18]).Particularly, the paper [6] treats the representation of probabilities definedon polyadic Boolean algebras, the algebraic structures associated with theclassical predicate calculus.

A natural problem is to obtain similar notions and results in the frame-work of other logical systems.

This paper is a contribution to the study of probabilities defined onformulas of infinite-valued �Lukasiewicz predicate logic �L∀ ([10]).

The class of MV-algebras, introduced by Chang in [2], constitutes the al-gebraic base of infinite-valued �Lukasiewicz propositional calculus [13].The main reference for MV-algebras is the monograph [3]. Probabilitieson MV-algebras (= MV-states) were studied in [14], [15], [16], etc.

Our approach to probabilities on �L∀ is purely algebraic. The notion ofpolyadic MV-algebra provides the algebraic counterpart of �L∀ (see [17], [5]).Thus the obiective of this paper is to investigate the MV-states on polyadicMV-algebras.

Special Issue: Algebra and Probability in Many-Valued ReasoningEdited by Ioana Leustean and Vincenzo Marra

232 G. Georgescu

In Section 2 we recall some basic notions and properties of MV-algebrasand MV-states. Section 3 is a brief exposition of results from polyadic MV-algebras needed in the next sections.

Section 4 is concerned with with the polyadic MV-states, introduced byan algebraic identity inspired by Gaifman’s condition. In this framework,an MV-state µ : D → [0, 1] defined on an MV-subalgebra D of a polyadicMV-algebra A can be viewed as a probabilistic version of a theory in �L∀. Wedefine the notion of MV-state model of µ and prove a completeness theorem:any MV-state µ : D → [0, 1] has an MV-state model.

In Section 5 we define the probabilities on formulas of �L∀ (= L-states).We introduce Gaifman’s MV-models and apply the results of Section 4 inorder to prove that any L-state has a Gaifman MV-model.

2. Preliminaries on MV-algebras

In this section we shall present some basic notions and results on MV-algebras and states defined on them (see [2], [3], [14], [15]).

Let 〈A,⊕,�,¬, 0, 1〉 be an MV-algebra.If we define the new operations ∨ and ∧ on A:

a ∨ b = a ⊕ (¬a � b), a ∧ b = a � (¬a ⊕ b),

then 〈A,∨,∧, 0, 1〉 is a bounded distributive lattice.

If a, b ∈ A, then a ≤ b iff a � ¬b = 0 iff ¬a⊕ b = 1.

For any a ∈ A and n ∈ ω we define the elements na and an of A byinduction:

0a = 0, (n + 1)a = na ⊕ a and a0 = 1, an+1 = an � a.

Lemma 2.1. In any MV-algebra the following equalities hold:

(i) a ⊕∨i∈I

ai =∨i∈I

(a ⊕ ai); a �∨i∈I

ai =∨i∈I

(a � ai);

(ii) a ⊕∧i∈I

ai =∧i∈I

(a ⊕ ai); a �∧i∈I

ai =∧i∈I

(a � ai),

whenever these suprema and infima exist.

In any MV-algebra A we define the operations → and ↔ by a → b =¬a⊕ b and a ↔ b = (a → b) ∧ (b → a), respectively.

Lemma 2.2. The implication operation → has the following properties:

(i) a → a = a;

States on Polyadic MV-algebras 233

(ii) a ≤ b iff a → b = 1;

(iii) a � b ≤ c iff a ≤ b → c;

(iv) a → (b → c) = a � b → c;

(v) If a ≤ b then b → c ≤ a → c and c → a ≤ c → b;

(vi) a → b = ¬b → ¬a.

Lemma 2.3. In any MV-algebra the following properties hold:

(i) (∨i∈I

ai) → b =∧i∈I

(ai → b); b → (∨i∈I

ai) =∨i∈I

(b → ai);

(ii) (∧i∈I

ai) → b =∨i∈I

(ai → b); b → (∧i∈I

ai) =∧i∈I

(b → ai),

whenever these suprema and infima exist.

An MV-filter is a non-empty subset F of A which is closed under � andsuch that a ∈ F , a ≤ b imply b ∈ F . The MV-filter filt(X) generated byX ⊆ A has the form

filt(X) = {a ∈ A : a1 � · · · � an ≤ a, for some n ∈ ω and a1, . . . , an ∈ X}.

If F is an MV-filter of A then the congruence a ∼F b iff a ↔ b ∈ Fdefines the quotient MV-algebra A/F = A/ ∼F . We denote by a/F thecongruence class of a ∈ A.

Following [14], an MV-state on the MV-algebra A is a function m : A →[0, 1] such that the following two conditions hold:

(S1) m(0) = 0, m(1) = 1;

(S2) If a � b = 0 then m(a ⊕ b) = m(a) + m(b).

Lemma 2.4. If m is an MV-state on A then the following properties hold:

(i) m(¬a) = 1 − m(a);

(ii) m(a ⊕ b) = m(a) + m(b) − m(a � b);

(iii) m(a ∨ b) = m(a) + m(b) − m(a ∧ b);

(iv) If a ≤ b then m(a) ≤ m(b);

(v) m(a) + m(a → b) = m(b) + m(b → a);

(vi) If m(a → b) = 1 then m(a) ≤ m(b).

234 G. Georgescu

Proof. The properties (i) - (iv) are established in [14], (v) in [9] and (vi)follows from (v).

Proposition 2.5. [12] Let A′ be a subalgebra of the MV-algebra A. AnyMV-state m′ : A′ → [0, 1] can be extended to an MV-state m : A → [0, 1].

3. Monadic and polyadic MV-algebras

In this section we shall recall some definitions and properties of monadic andpolyadic MV-algebras (see [17], [5]).

Let A be an MV-algebra. An existential quantifier on A is a function∃ : A → A that satisfies the following conditions:

(A0) ∃0 = 0;

(A1) a ≤ ∃a;

(A2) ∃(a � ∃b) = ∃a� ∃b;

(A3) ∃(a ⊕ ∃b) = ∃a⊕ ∃b;

(A4) ∃(a � a) = ∃a � ∃a;

(A5) ∃(a ⊕ a) = ∃a ⊕ ∃a.

The existential quantifier on an MV-algebra is an algebraic notion thatgeneralizes the existential quantifier on a Boolean algebra ([4], [11]). Theaxioms (A0) - (A5) arise from the properties of the existential quantifier in�Lukasiewicz predicate logic (see e.g. [10]).

The notion of universal quantifier on an MV-algebra is defined in adual way.

A monadic MV-algebra is a pair 〈A,∃〉, where A is an MV-algebra and∃ is an existential quantifier on A.

Lemma 3.1. [17] In any monadic MV-algebra 〈A,∃〉 the following propertieshold:

(i) ∃1 = 1;

(ii) ∃∃ = ∃;

(iii) ∃(¬∃a) = ¬∃a;

(iv) ∃(∃a� ∃b) = ∃a � ∃b;

(v) ∃(∃a⊕ ∃b) = ∃a ⊕ ∃b;


(vi) ∃(a ∧ ∃b) = ∃a ∧ ∃b;

(vii) ∃(a ∨ b) = ∃a ∨ ∃b;

(viii) If a ≤ b then ∃a ≤ ∃b.

We remark that ∃(A) = {a ∈ A : ∃a = a} is an MV-subalgebra of A.

Lemma 3.2. [5] In any monadic MV-algebra 〈A,∃〉, we have ∃(a → b) ≤a → ∃b, for all a ∈ ∃(A) and b ∈ A.

Let I be a non-empty set. An I-polyadic MV-algebra is a structure

〈A, I, {S(σ)}σ∈II , {∃(J)}J⊆I〉,

where A is an MV-algebra, S(σ) is an endomorphism of the MV-algebra Aand ∃(J) is an existential quantifier on A, such that, for all p ∈ A, σ, τ ∈ II

and J, J ′ ⊆ I, the following properties hold:

(P1) S(δI)p = p, where δI is the identity of I;

(P2) S(στ) = S(σ)S(τ);

(P3) ∃(J ∪ J ′) = ∃(J)∃(J ′);

(P4) ∃(∅) = δA;

(P5) If σ|I−J = τ |I−J then S(σ)∃(J) = S(τ)∃(J);

(P6) If σ|σ−1(J) is injective then ∃(J)S(σ) = S(σ)∃(σ−1(J)).

This I-polyadic MV-algebra will be denoted by 〈A, I, S,∃〉. The cardinalnumber of I will be called the degree of 〈A, I, S,∃〉.

The first example of polyadic MV-algebra is the Lindenbaum-Tarski al-gebra of �Lukasiewicz predicate logic. If X, I are non-empty sets, then the setS(XI , [0, 1]) of functions p : XI → [0, 1] becomes an I-polyadic MV-algebraby putting (S(σ)p)(x) = p(xσ) and (∃(J)p)(x) =

∨{p(y) : y|I−J = x|I−J},

for all p ∈ S(XI , [0, 1]), σ ∈ II , J ⊆ I and x ∈ XI .

We fix an I-polyadic MV-algebra 〈A, I, S,∃〉. If p ∈ A then J ⊆ I is asupport of p if ∃(I − J)p = p. We say that p is independent from J if I − Jis a support of p.

Lemma 3.3. If card(I) ≥ 2, p ∈ A and J ⊆ I, then the following areequivalent:

(i) J is a support of p;

236 G. Georgescu

(ii) S(σ)p = S(τ)p, for all σ, τ ∈ II such that σ|J = τ |J ;

(ii) S(σ)p = p, for all σ ∈ II such that σ|J = δ|J .

The polyadic MV-algebra 〈A, I, S,∃〉 is locally-finite if each element p ∈ Ahas a finite support. In this case, Jp will denote the minimal support of p,i.e. the intersection of all supports of p.

Throughout this paper we shall assume that the polyadic MV-algebra〈A, I, S,∃〉 is locally-finite, of infinite degree.

Lemma 3.4. If J is a support of p ∈ A and K ⊆ I then ∃(K)p = ∃(K ∩ J)pand J − K is a support of ∃(K)p.

Lemma 3.5. If J is a support of p and σ ∈ II , then σ(J) is a support ofS(σ)p.

Proposition 3.6. For all p ∈ A, σ ∈ II and J ⊆ I the following equalityholds:

S(σ)∃(J)p =∨{S(τ)p : τ |I−J = σ|I−J}.

Notation: If i1, . . . , in, k1, . . . , kn ∈ I then the function(k1, . . . , kn/i1, . . . , in) : I → I will be defined by

(k1, . . . , kn/i1, . . . , in)(j) =

{kt, if j = it (t = 1, . . . , n)j, otherwise.

Corollary 3.7. For any i ∈ I, ∃(i)p =∨{S(j/i)p : j ∈ I}.

Assume that I+ ⊇ I and 〈A+, I+, S+,∃+〉 is an I+-polyadic MV-algebra.Consider the set A = {p ∈ A+ : I is a support of p}. For every p ∈ A,

σ ∈ II and J ⊆ I, we define S(σ)p = S+(σ+)p, were σ+ = σ ∪ δI+−I and∃(J)p = ∃+(J)p. Then 〈A, I, S,∃〉 is an I-polyadic MV-algebra and A+ willbe called an I+-dilation of A.

Proposition 3.8. Every I-polyadic MV-algebra A has an I+-dilation A+

such that card(A) = card(A+).

Proposition 3.9. If A+ is an I+-dilation of an I-polyadic MV-algebra A,then every element q ∈ A+ has the form q = S+(k1, . . . , kn/i1, . . . , in)p,where p ∈ A, k1, . . . , kn ∈ I+ − I and q is independent from {i1, . . . , in}.

A constant of a polyadic MV-algebra 〈A, I, S,∃〉 is a function c from P(I)to the endomorphisms of the MV-algebra A, such that for all J,K ⊆ I andσ ∈ II , the following properties hold:


(C1) c(∅) = δA;

(C2) c(J ∪ K) = c(J)c(K)

(C3) c(J)∃(K) = ∃(K)c(J − K);

(C4) ∃(J)c(K) = c(K)∃(J − K);

(C5) c(J)S(σ) = S(σ)c(σ−1(J)).

Lemma 3.10. If c is a constant of A, J is a support of p ∈ A and K ⊆ I,then c(K)p = c(K ∩ J)p and J − K is a support of c(K)p.

Let A+ be an I+-dilation of A. Then A+ becomes an I-polyadic MV-algebra, by putting, for all p ∈ A+, σ ∈ II and J ⊆ I:

S(σ)p = S+(σ+)p, ∃(J)p = ∃+(J)p.

This I-polyadic MV-algebra will be denoted by A(K), where K = I+−I.Of course, A is embedded in A(K). Every k ∈ K induces a constant of A(K):k(J) = S+(k/J), denoted by the same symbol k, where

(k/J)(i) =

{k, if i ∈ Ji, if i �∈ J .

Every element q ∈ A(K) has the form q = k1(i1) · · · kn(in)p, where p ∈ A,k1, . . . , kn ∈ K and i1, . . . , in ∈ I. We say that A(K) is obtained from A+

by fixing the variables of K.

The previous notions and results will be used without mention in thenext section.

4. Polyadic MV-states

This section is concerned with polyadic MV-states, a class of MV-statesdefined on a polyadic MV-algebra. They are introduced by a Gaifman-likecondition that expresses the behaviour of these MV-states w.r.t. the algebraicexistential quantifier. Regarding the MV-states defined on MV-subalgebrasof a polyadic algebra as “probabilistic MV-theories”, we define their MV-state models and prove that any such MV-state has an MV-state model.

We fix an I-polyadic MV-algebra 〈A, I, S,∃〉. Let U be a non-empty setsuch that U ∩ I = ∅ and denote I+ = I ∪ U . Let us consider an I+-dilation〈A+, I+, S+,∃+〉 of A and the I-polyadic MV-algebra A(U) obtained fromA+ by fixing the variables of U .

238 G. Georgescu

Definition 4.1. Let m : A(U) → [0, 1] be an MV-state. We say that m is apolyadic MV-state defined on A(U) if for any p ∈ A(U) and i ∈ I such thatJp ⊆ {i}, the following condition holds:

(1) m(∃(i)p) = sup{m(n∨

t=1kt(i)p) : n ∈ ω, k1, . . . , kn ∈ U}.

According to the definition of the constants k1, . . . , kn associated withthe elements k1, . . . , kn ∈ U (see the end of Section 3), we have kt(i)p =S+(kt/i)p, for all t = 1, . . . , n. Therefore condition (1) can be written in thefollowing form:

(1’) m(∃(i)p) = sup{m(n∨

t=1S+(kt/i)p) : n ∈ ω, k1, . . . , kn ∈ U}.

By Lemma 2.4, (1) and the usual computation rules in an MV-algebraone can prove the following lemma:

Lemma 4.2. An MV-state m : A(U) → [0, 1] is a polyadic MV-state iff forany p ∈ A(U) and i ∈ I such that Jp ⊆ {i}, the following equality holds:

(2) m(∀(i)p) = inf{m(n∧

t=1S+(kt/i)p) : n ∈ ω, k1, . . . , kn ∈ U}.

Definition 4.3. Let D be an MV-subalgebra of A and µ : D → [0, 1] anMV-state. An MV-state model of µ is a pair (U,m), where U is a non-emptyset and m : A(U) → [0, 1] is a polyadic MV-state such that m|D = µ.

Theorem 4.4. Let D be an MV-subalgebra of the polyadic MV-algebra A.Any MV-state µ : D → [0, 1] has an MV-state model.

Proof. For any element ∃(i)p of A with Jp ⊆ {i} we consider a newelement k∃(i)p such that ∃(i)p �= ∃(j)q implies k∃(i)p �= k∃(j)q (we assumethat k∃(i)p �∈ I). Let us denote by K1 the set of elements k∃(i)p obtainedin this way. We remark that I ∩ K1 = ∅. Consider the I-polyadic MV-algebra A1 = A(K1) obtained from an I ∪ K1-dilation of A by fixing thevariables of K1. Applying a similar procedure to A1, we get a new set K2

with I∩K2 = K1∩K2 = ∅ and the I-polyadic MV-algebra A2 = A1(K2). Byinduction, we obtain a sequence (Kn)n≥1 of nonempty sets and a sequenceA = A0 ⊆ A1 ⊆ A2 ⊆ . . . of I-polyadic MV-algebras such that the followingproperties hold:

• Kn ∩ Km = ∅, for all distinct n,m ∈ ω − {0};• Kn ∩ I = ∅, for all n ∈ ω − {0};


• An+1 = An(Kn+1), for all n ∈ ω.

Denote U =∞⋃

n=1Kn and consider the I-polyadic MV-algebra A(U) ob-

tained from an I ∪ U -dilation of A by fixing the variables of U (we observethat I and U are disjoint). Thus A(U) is isomorphic to the inductive limitlim−−→n∈ω

An.

Now we take the following subset of A(U):

E = {∃(i)p → S+(k∃(i)p/i)p : p ∈ A(U), i ∈ I, Jp ⊆ {i}}.

Let B be the MV-subalgebra of A(U) generated by D ∪ E. Then anyelement b ∈ B has the form

(3) b = τA(U)(d1, . . . , dn, e1, . . . , em),

where τ(x1, . . . , xn, y1, . . . , ym) is an MV-term, d1, . . . , dn ∈ D ande1, . . . , em ∈ E.

Fact 1: If F is the MV-filter of A(U) generated by E then F ∩A = {1}.

In order to prove Fact 1, let us consider a ∈ F ∩ A. In accordance toF = filt(E), there exist l1, . . . , ln ∈ ω − {0}, t1, . . . , tn ∈ ω, p1, . . . , pn ∈ A,i1, . . . , in ∈ I and k1 ∈ Kl1 , . . . , kn ∈ Kln such that

(4) [∃(i1)p1 → S+(k1/i1)p1]t1 � · · · � [∃(in)pn → S+(kn/in)pn]tn ≤ a.

Denoting w = [∃(i2)p2 → S+(k2/i2)p2]t2 � · · · � [∃(in)pn → S+(kn/in)pn]tn ,the inequality (4) becomes [∃(i1)p1 → S+(k1/i1)p1]t1 � w ≤ a, i.e.

(5) [∃(i1)p1 → S+(k1/i1)p1]t1 ≤ w → a.

If r is a natural number such that t1 ≤ 2r, then, by (5) we get

(6) [∃(i1)p1 → S+(k1/i1)p1]2r

≤ w → a.

We observe that k1 does not belong to the minimal support of w → a,hence ∃(k1)(w → a) = w → a, so, by (6) we obtain the inequality

(7) ∃(k1)([∃(i1)p1 → S+(k1/i1)p1]2r

) ≤ w → a.

Using Corollary 3.7, it is easy to see that ∃(i1)p1 = ∃(k1)S+(k1/i1)p1.Thus, by applying the axiom (A4) and Lemma 3.2, the following equalitieshold:

∃(k1)([∃(i1)p1 → S+(k1/i1)p1]2r

) = [∃(k1)(∃(i1)p1 → S+(k1/i1)p1)]2r

=[∃(i1)p1 → ∃(k1)S+(k1/i1)p1]2

r

= 1.

240 G. Georgescu

Hence, by (7) we get w → a = 1, so w ≤ a. Applying many times theprevious argument we obtain a = 1, hence F ∩ A = {1}.

Fact 2: If b ∈ B then b/F = d/F , for some d ∈ D.In order to prove Fact 2, let b ∈ B, hence b = τA(U)(d1, . . . , dn, e1, . . . , em) forsome MV-term τ(x1, . . . , xn, y1, . . . , ym), d1, . . . , dn ∈ D and e1, . . . , em ∈ E.Then

b/F = (τA(U)(d1, . . . , dn, e1, . . . , em))/F= τA(U)/F (d1/F, . . . , dn/F, e1/F, . . . , em/F )= τA(U)/F (d1/F, . . . , dn/F, 1/F, . . . , 1/F )= (τA(U)(d1, . . . , dn, 1, . . . , 1))/F .

Denoting d = τA(U)(d1, . . . , dn, 1, . . . , 1), we have d ∈ D and b/F = d/F .Let us consider the function µ′ : B/F → [0, 1] defined by µ′(b/F ) = µ(d),

where b ∈ B, d ∈ D and b/F = d/F (cf. Fact 2, such an element d exists).In order to prove that µ′ is well defined, we consider two elements d, d′ ∈ Dsuch that b/F = d/F = d′/F . Thus d ↔ d′ ∈ F ∩A = {1}, hence d = d′ andµ(d) = µ(d′). It is clear that µ′ is an MV-state on the MV-algebra B/F .

Since B/F is an MV-subalgebra of A(U)/F , by Proposition 2.5, µ′ can beextended to an MV-state µ∗ : A(U)/F → [0, 1]. The function m : A(U) →[0, 1] defined by m(a) = µ∗(a/F ) for any a ∈ A(U) is an MV-state on A(U).We shall prove that m is a polyadic MV-state.

If p ∈ A(U) such that Jp ⊆ {i}, then ∃(i)p → S+(k∃(i)p/i)p ∈ F , so

m(∃(i)p → S+(k∃(i)p/i)p) = µ∗((∃(i)p → S+(k∃(i)p/i)p)/F ) = 1.

By Lemma 2.4, (vi), it follows that m(∃(i)p) ≤ m(S+(k∃(i)p/i)p).Since S+(k∃(i)p/i)p ≤ ∃(i)p, we get

m(∃(i)p) = m(S+(k∃(i)p/i)p) = sup{(n∨

t=1S+(kt/i)p) : k1, . . . , kn ∈ U}.

Thus m is a polyadic MV-state on A(U). If a ∈ D then m(a) =µ∗(a/F ) = µ′(a/F ) = µ(a), hence m|D = µ. Thus (U,m) is a model of µ.

5. Application to probabilistic model theory of �L∀

This section is concerned with probabilistic models of �L∀. We define thenotions of Gaifman MV-state and Gaifman MV-model and use Theorem 4.4in order to obtain an MV-version of Gaifman’s completeness theorem ([7],Theorem 2).

A language L of the infinite-valued �Lukasiewicz predicate logic �L∀ con-sists of the following primitive symbols: variable symbols, constant symbols,


predicate symbols, the connectives → and ¬, the existential symbol ∃ and theparantheses (, ), [, ]. We shall assume that the set V of variable symbols isinfinite. The set Form of formulas of L is introduced as usual (see [10]). Weshall use the following abbreviations: ϕ⊕ψ = ¬ϕ → ψ, ϕ�ψ = ¬(ϕ → ¬ψ),ϕ ∨ ψ = (ϕ → ψ) → ψ, ϕ ∧ ψ = ¬(¬ϕ ∨ ¬ψ), ϕ ↔ ψ = (ϕ → ψ) ∧ (ψ → ϕ).We shall consider the axiomatization of L given in [10], p.111. If ϕ is aformal theorem of L then we write � ϕ.

Let us consider the following equivalence relation ∼ on Form: ϕ ∼ ψiff � ϕ ↔ ψ. We denote ϕ the equivalence class of the formula ϕ ∈ Form.Then Form/∼ = {ϕ : ϕ ∈ Form} becomes a V -polyadic MV-algebra (theLindenbaum-Tarski algebra of L).

If C is a set of new constants then Form(C) will denote the set of for-mulas of the language L(C) obtained from L by adding the constants of C.

Let D be a set of formulas of L such that• D contains the formal theorems of L;• D is closed under the connectives → and ¬.Then D is also closed under the connectives ⊕, �, ∨, ∧, ↔ and D/∼ =

{ϕ : ϕ ∈ D} is an MV-subalgebra of Form/∼.

Definition 5.1. A function µ : D → [0, 1] is an L-state if the followingconditions hold for all ϕ,ψ ∈ D:

(i) If � ϕ then m(ϕ) = 1;

(ii) If � ¬(ϕ � ψ) then m(ϕ ⊕ ψ) = m(ϕ) + m(ψ).

Lemma 5.2. Let µ : D → [0, 1] be an L-state and ϕ,ψ ∈ D. Then

(a) µ(¬ϕ) = 1 − µ(ϕ);

(b) If � ϕ ↔ ψ then µ(ϕ) = µ(ψ).

Proof.

(a) Since � ϕ⊕¬ϕ and � ¬(ϕ�¬ϕ) we get 1 = µ(ϕ⊕¬ϕ) = µ(ϕ)+µ(¬ϕ).(b) Assume � ϕ ↔ ψ, hence � ϕ → ψ and � ψ → ϕ. Then � ¬ϕ ⊕ ψ

and � ¬(¬ϕ�ψ), so 1 = µ(¬ϕ⊕ψ) = µ(¬ϕ) + µ(ψ) = 1− µ(ϕ) + µ(ψ), i.e.µ(ϕ) = µ(ψ).

Let µ : D → [0, 1] be an L-state. By Lemma 5.2, one can consider thefunction µ : D/∼ → [0, 1], defined by µ(ϕ) = µ(ϕ), for all ϕ ∈ D. Thus µ isan MV-state on the MV-algebra D/∼.

Definition 5.3. Let U be a set of new constant symbols and m : Form(U)→ [0, 1] an L(U)-state. m is said to be a Gaifman state on L(U) if for anyformula ϕ(x) of L(U), the following equality holds

242 G. Georgescu

m(∃xϕ(x)) = sup{m(n∨

i=1ϕ(ci)) : n ∈ ω, c1, . . . , cn ∈ U}.

Definition 5.4. A Gaifman MV-model of an L-state µ : D → [0, 1] is a pair(U,m), where U is a set of new constant symbols and m is a Gaifman stateon L(U) such that m|D = µ.

Let (U,m) be a Gaifman MV-model of µ : D → [0, 1]. Then the function

m : Form(U)/∼ → [0, 1],m(ϕ) = m(ϕ), for all ϕ ∈ Form(U),

is a polyadic MV-state. We remark that the restriction of m to D/∼ is µ,hence (U, m) is an MV-state model of µ.

Theorem 5.5. Any L-state µ : D → [0, 1] has a Gaifman MV-model.

Proof. By Theorem 4.4 and the previous remarks.

Remark 5.6. Theorem 5.5 is the MV-version of the completeness theoremproved by Gaifman for probabilistic models of classical predicate logic [7].

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George Georgescu

Faculty of Mathematics and Computer ScienceUniversity of BucharestAcademiei 14, RO 010014Bucharest, [email protected]

states on polyadic mv-algebras

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