Non-standard probability, coherence and conditional probability on many-valued events

International Journal of Approximate Reasoning 54 (2013) 573–589

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International Journal of Approximate Reasoning journal homepage: www.elsevier.com/locate/ijar

Non-standard probability, coherence and conditional probability on many-valued events Franco Montagna ∗ , Martina Fedel, Giuseppe Scianna Department of Mathematics and Computer Science ‘R. Magari’, Pian dei Mantellini 44, 53100 Siena, Italy

ARTICLE INFO

ABSTRACT

Article history: Received 16 July 2012 Received in revised form 4 February 2013 Accepted 4 February 2013 Available online 19 February 2013

The usual coherence criterion by de Finetti is extended both to many-valued events and to conditional probability. Special attention is paid to assessments in which the betting odds for conditioning events are zero. This case is treated by means of infinitesimal probabilities. We propose a rationality criterion, called stable coherence, which is stronger than coherence in the sense of no sure loss. © 2013 Elsevier Inc. All rights reserved.

Keywords: States Hyperstates Coherence Conditional probability MV-algebras Non-standard analysis

1. Introduction This paper is about conditional probability over many-valued events, and takes also care of the case where the conditioning event has zero probability. We tried to make this introduction accessible also to non-specialists of many-valued logic. In any case, in Section 2 the reader may find the background material needed to understand the whole paper. According to de Finetti [8], the probability of an event φ is the amount of money (betting odd in the sequel) α that a fair bookmaker B would accept for the following game:

• A gambler G chooses a real number λ and pays λα to B (possibly, λ < 0; we stipulate that paying β < 0 is the same as receiving −β ). • If φ is true, then G gets back λ. • If φ is false, then G gets back nothing. The only rationality criterion proposed by de Finetti is the following: suppose B accepts bets on the events φ1 , . . . , φn with betting odds α1 , . . . , αn , respectively. Then the assessment  : φ1  → α1 , . . . , φn  → αn is said to be coherent if there is no system of bets λ1 , . . . , λn which causes to B a sure loss. In other words,  is coherent if for all λ1 , . . . , λn there is a valuation connectives, formally, v (roughly speaking, a map from the set of events into {0, 1} respecting the truth tables of classical  a homomorphism from the algebra of events into the two-element boolean algebra) such that ni=1 λi (αi − v(φi )) ≥ 0. A famous result by de Finetti establishes a relation between coherence and probability operators: Theorem 1.1 [8]. An assessment  i = 1, . . . , n, Pr(φi ) = αi .

: φ1 → α1 , . . . , φn → αn is coherent iff there is a probability distribution Pr such that, for

∗ Corresponding author. E-mail addresses: [email protected] (F. Montagna), [email protected] (M. Fedel), [email protected] (G. Scianna). 0888-613X/$ - see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ijar.2013.02.003

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De Finetti’s characterization was also extended to non-classical logics e.g. in [29,24,1]. In particular, Mundici [24] extended this framework to many-valued events, which are represented as elements of an MV-algebra. In this many-valued context, events may take values in [0, 1], not necessarily in {0, 1}. MV-algebras allow us to treat not only events with intermediate truth values, but also continuous random variables. Indeed, a rather general example of MV-algebra is constituted by the algebra C(X )[0, 1] of all continuous functions from a compact Hausdorff space X into [0, 1], with operations ⊕ and ∼ defined, for all f , g ∈ C(X ) and for all x ∈ X, by (f ⊕ g )(x) = min{f (x) + g (x), 1}, and (∼ f (x)) = 1 − f (x). Now any continuous random variable h on X can be represented as h = Mf + K for some f ∈ C(X )[0, 1] and for some constants M , K. Hence, continuous random variables can be represented by means of MV-events. The role of a probability distribution is played by states on MV-algebras (see [23]). A state on an MV-algebra A is a map s from A into [0, 1] such that:

• s(1) = 1. • If φ  ψ = 0, then s(φ ⊕ ψ) = s(φ) + s(ψ), where φ  ψ =∼ (∼ φ⊕ ∼ ψ). A valuation on an MV-algebra A is a map v from the universe of A into [0, 1] satisfying v(0) = 0, v(∼ φ) = 1 − v(φ) and v(φ ⊕ ψ) = min{v(φ) + v(ψ), 1}. An MV-assessment (in the sequel, asssessment) is a map  : φ1  → α1 , . . . , φn  → αn from a finite subset {φ1 , . . . , φn } of an MV-algebra A into [0, 1].  is said to be coherent iff for any sequence λ1 , . . . , λn of  real numbers there is a valuation v such that ni=1 λi (αi − v(φi )) ≥ 0. In other words,  is coherent if there is no system of bets causing a sure loss to B. The MV-analogue of de Finetti’s theorem reads: Theorem 1.2 [24]. Let  : φ1  → α1 , . . . , φn state s on A such that, for i = 1, . . . , n, s(φi ) =

→ αn be an assessment on an MV-algebra A. Then  is coherent iff there is a αi .

By a result due independently to Panti [28] and to Kroupa, [16], states can be represented as integrals. Hence, when the elements of an MV-algebra are interpreted as continuous random variables in C(X )[0, 1], states represent their expected values. After the above mentioned seminal papers by Mundici, Panti and Kroupa, many interesting papers on many-valued probability appeared, for instance, with reference to this journal [3,10,11,13,14,18]. De Finetti also extended the interpretation of probability in terms of bets to conditional probability over classical events (represented as elements of a boolean algebra). If the betting-odd for a conditional event φ|ψ is α , then:

• G chooses a real number λ and pays λα to B. • G gets back λ if both φ and ψ are true, λα if ψ is false, and nothing if ψ is true and φ is false. Again, λ may be negative, and paying β < 0 is by definition the same as getting −β , and viceversa. The extension of conditional probability to many-valued events is more complicated, because it has (at least) two interpretations. First of all, the probability of φ|ψ may be interpreted as the probability of φ in a theory having ψ as a new axiom. This approach has been pursued in [26, Theorem 15.1] (see also [25]), where the author shows the existence of a conditional state which satisfies all Rényi axioms of conditional probability, along with other important mathematical properties. An alternative definition of conditional probability on many-valued events in terms of bets has been presented in [22]. According to that definition, betting on φ|ψ is like betting on φ , with the difference that only a part of the bet proportional to v(ψ) is valid. In more detail, the bookmaker’s payoff corresponding to the betting odd α , to the bet λ and to the valuation v, is λv(ψ)(α − v(φ)). According to this interpretation, Rényi’s law Pr(φ|φ) = 1 is no longer valid. For instance, if the truth value of φ is constantly equal to 12 (taking thus ψ = φ = 12 in the previous formula), if the betting odd for φ|φ is 1 and if λ = −1 then B always loses money. In the case of conditional assessments, the absence of sure loss for B does not guarantee the rationality of the assessment. For instance, if there is a valuation v such that v(ψ) = 0, then the assessment φ|ψ  → 0, ¬φ|ψ  → 0 avoids sure loss, although it is certainly not rational. In order to restore the equivalence between coherence and rationality, we first restrict ourselves to complete assessments. Let us say that an assessment is complete if, together with a betting odd on a conditional event φ|ψ , it contains a betting odd for the conditioning event ψ . Hence, a complete assessment has the form

φ1 |ψ1 → α1 , . . . , φn |ψn → αn , ψ1 → β1 , . . . , ψn → βn . When the assessment is complete and the betting odd for each conditioning event is non-zero, de Finetti’s criterion extends to conditional probability. In order to express it, it is convenient to work in MV-algebras with product, also called PMV+ algebras. This is not problematic, because every MV-algebra is a subreduct of a PMV+ -algebra, as we shall prove in Section 3. For instance, in the algebra C(X )[0, 1] we can define a product · by (f · g )(x) = f (x)g (x), thus obtaining a PMV+ -algebra.

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The following result was proved in [22, Theorem 3.9] under the additional assumption that A is a free PMV+ -algebra, but it can be (and it will be) extended to arbitrary PMV+ -algebras: Theorem 1.3. Let  = φ1 |ψ1  → α1 , . . . , φn |ψn  → αn , ψ1  → β1 , . . . , ψn  → βn be a complete assessment on a PMV+ algebra A, where β1 , . . . , βn are non-zero. Then  avoids sure loss iff there is a state s such that for i = 1, . . . , n, s(ψi ) = βi and s(φi · ψi ) = αi · βi . Hence, when the betting odd for the conditioning event is non-zero, the above defined coherence criterion corresponds to Kroupa’s law [17] s(φ|ψ)

=

s(φ

· ψ) . s(ψ)

The situation changes if the betting odd βi for some conditioning event is 0. For instance, in Borel’s example about the probability to choose a point in the Western Hemisphere, given that it belongs to the Equator, any betting odd would avoid sure loss, because the probability to choose a point in the Equator is 0. In order to avoid this inconvenience, we will use non-standard probabilities [27]. Such probabilities are very useful in mathematics. For instance, in [2] the authors are able to find a uniform non-standard distribution over any set. In the present paper, we use hyperstates, the many-valued analogue of non-standard probabilities. A precise definition of hyerstate will be given in Section 3. The idea is that hyperstates are similar to states, but can take values in a non-standard extension [0, 1]∗ of [0, 1]. Another difference between states and hyperstates is that if s is a state, if s∗ is a hyperstate and a is an infinitesimal, 1 then s(a) = 0, while s∗ (a) may be a non-zero infinitesimal. Hence, hyperstates allow us to treat bets in which non-standard betting odds, non-standard truth values and non-standard bets are allowed. An important advantage of hyperstates is that any MV-algebra has a faithful hyperstate, i.e., a hyperstate s∗ such that s∗ (φ) > 0 when φ = 0 (this is not true of usual states; for hyperstates over Boolean algebras, the result can be easily derived from [2]). Moreover, any coherent assessment  : φ1  → α1 , . . . , φn  → αn on an MV-algebra may be extended by a faithful hyperstate modulo an infinitesimal, that is, there is a faithful hyperstate s∗ such that for i = 1, . . . , n, αi − s∗ (φi ) is an infinitesimal. Mundici’s equivalence between coherence and extendability to a state can be extended to non-standard assessments (hypersassessments in the sequel) and to hyperstates. That is, a hyperassessment avoids sure loss if it can be extended to a hyperstate. The presence of many faithful hyperstates allows us to treat conditional probability when the conditioning event has probability 0. Consider a complete assessment of conditional probability  = φ1 |ψ1  → α1 , . . . , φn |ψn  → αn , ψ1  → β1 , . . . , ψn → βn . Let us say that  is stably coherent if there is an assessment  = φ1 |ψ1  → α1 , . . . , φn |ψn  → αn , ψ1  → β1 , . . . , ψn → βn such that α1 , . . . , αn , β1 , . . . , βn belong to a nonstandard extension, [0, 1]∗ , of [0, 1], and in addition: (i)  and  differ by an infinitesimal, that is, for i (ii) for i = 1, . . . , n, βi > 0. (iii)  avoids sure loss.

= 1, . . . , n, αi − αi and βi − βi are infinitesimal;

The idea is that the gambler may force the bookmaker to change his assessment by an infinitesimal in such a way that to any conditioning event a non-zero value is assigned. Now if the assessment is stably coherent, the bookmaker can manage to do that preserving coherence. Then the main result of this paper says: Theorem 1.4. A complete assessment

 = φ1 |ψ1  → α1 , . . . , φn |ψn  → αn , ψ1  → β1 , . . . , ψn  → βn on a PMV+ -algebra is stably coherent iff there is a faithful hyperstate s∗ such that, if for all x ∈ [0, 1]∗ st (x) denotes the standard part of x, that is, unique real number such that x − st (x) is infinitesimal, the following condition hold: (i) For i (ii) For i

= 1, . . . , n, st (s∗ (ψi )) = βi . = 1, . . . , n, αi = st

 ∗ s (φi



· ψi ) . s∗ (ψi )

1 An element a of an MV-algebra is said to be an infinitesimal if for every n, na ≤∼ a, where na = a ⊕ · · · ⊕ a, n times. Moreover, a hyperreal number b is said to be infinitesimal if for every positive standard real number , |b| < .

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In some respects, this result extends a result by Krauss [15] relating non-standard probability to standard conditional probability. However, unlike [16], in this paper we treat conditional probability over MV-events and we relate it to de Finetti’s betting scheme. The paper is organized as follows: Section 2 is devoted to some preliminary notions. In Section 3, we introduce the algebraic background, as well as the notions of hypervaluation and of hyperstate. Section 4 deals with the relationship between hyperassessments and hyperstates. It is shown that a hyperassessment avoids sure loss iff it can be extended to a hyperstate. Section 5 deals with conditional probability. The notion of stable coherence is introduced, and Theorem 1.4, relating stable coherence to faithful hyperstates, is proved. Section 6 deals with a version of Theorem 1.4 which avoids infinitesimals, and in Section 7 we present the conclusions and some open problems.

2. Preliminaries We refer to [4] for the basic notions of universal algebras. Algebras will be usually denoted by boldface capital letters, 2 and their domains will be denoted by the corresponding lightface capital letters. Some basic facts about MV-algebras are collected below. For a more complete presentation of the subject, see [6], and for a more advanced treatment, including states and their relationship with coherence, see [26]. Definition 2.1. An MV-algebra [6] is an algebra A

= (A, ⊕, ∼, 0) such that:

(1) (A, ⊕, 0) is a commutative monoid. (2) x⊕ ∼ 0 =∼ 0. (3) ∼∼ x = x. (4) x ⊕ (∼ (∼ y ⊕ x)) = y ⊕ (∼ (∼ x ⊕ y)). The MV operations , →, ↔, , ∨, ∧ and the constant 1 are defined as follows: x  y =∼ (∼ x⊕ ∼ y); x → y =∼ x ⊕ y; x ↔ y = (x → y)  (y → x); x  y = x ∼ y; x ∨ y = x ⊕ (y  x); x ∧ y = x  (x  y); 1 =∼ 0. The variety of MV-algebras is generated by the algebra [0, 1]MV whose domain is the unit interval [0, 1] and whose operations are x ⊕ y = min {x + y, 1} and ∼ x = 1 − x. Definition 2.2. A PMV + -algebra [21] is an algebra A (1) (2) (3) (4)

= (A, ⊕, ∼, ·, 0, 1) such that:

(A, ⊕, ∼, 0) is an MV-algebra, and 1 =∼ 0. (A, ·, 1) is a commutative monoid. The equation x · (y  z ) = (x · y)  (x · z ) holds. The quasi identity (x2 = 0) ⇒ (x = 0) holds. 3

An Ł -algebra (see [12]) is a PMV+ -algebra equipped with a binary operation →π which is the residual of ·, that is, for all x, y, z, one has: x · y ≤ z iff x ≤ y →π z. PMV+ -algebras constitute a quasivariety (not a variety), which is generated by [0, 1]PMV , that is, [0, 1]MV equipped with ordinary product on [0, 1], see [21, Corollary 4.4]. Moreover, [21, Proposition 2.3], a PMV+ -algebra and its MV-reduct have the same congruences, although the quotient of a PMV+ -algebra modulo a congruence need not be a PMV+ -algebra. Unlike PMV+ -algebras, Ł -algebras constitute a variety, and PMV+ -algebras are precisely the subreducts of Ł -algebras, [21, Theorem 4.2]. Combining this result with [21, Lemma 2.11, (ii)], we obtain that every PMV+ -algebra is a subalgebra of a direct product of PMV+ -algebras which are ultrapowers of [0, 1]PMV . Finally, since the first-order theory T of [0, 1]PMV has quantifier-elimination (an easy consequence of quantifier-elimination for real closed fields), the class of its models has the joint embeddability property, that is, any two models of T have a joint embedding in another model of T. It follows that for any set S of ultrapowers of [0, 1]PMV , there is an ultrapower of [0, 1]PMV in which all algebras in S embed. As a consequence, we have: Theorem 2.1. Every PMV+ -algebra embeds into an algebra of the form ([0, 1]∗PMV )I , where I is a suitable index set and [0, 1]∗PMV is a suitable ultrapower of [0, 1]PMV . Definition 2.3. An MV-algebra (resp., a PMV+ -algebra) is said to be finitely presented iff it is isomorphic to the quotient of a finitely generated free MV-algebra (resp., PMV+ -algebra) modulo a finitely generated congruence.

2 3

An exception: we will denote the standard MV-algebra on [0, 1] and the standard PMV+ -algebra on [0, 1] by [0, 1]MV and by [0, 1]PMV , respectively. Algebras satisfying (1), (2) and (3) in the present definition are called PMV-algebras. So, PMV+ -algebras are PMV-algebras satisfying the quasiequation (4).

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Definition 2.4. A filter of an MV-algebra (resp., PMV+ algebra), A, is a subset F of A such that 1 ∈ F and for all a, b, c , d ∈ A, / F and maximal iff it is maximal among if a, b ∈ F then a  b ∈ F and if c ∈ F and c ≤ d, then d ∈ F. A filter is proper if 0 ∈ all proper filters. In both MV and PMV+ algebras, filters correspond to congruences: given a congruence θ , the set Fθ = {x : (x, 1) ∈ θ } is a filter, and given a filter F the set {(x, y) : x ↔ y ∈ F } is a congruence. Moreover the maps θ  → Fθ and F  → θF are mutually inverse isomorphisms between the congruence lattice and the filter lattice of an MV-algebra (resp., PMV+ -algebra). The quotient of A modulo the congruence θF determined by a filter F is denoted by A /F, and the equivalence class of an element a modulo θF is denoted by a/F. Note that, while the quotient of an MV-algebra is always an MV-algebra, the quotient of a PMV+ -algebra need not be a PMV+ algebra, because PMV+ -algebras do not constitute a variety. A filter F is maximal iff A /F is a simple algebra, that is, its only congruences are {(a, a) : a ∈ A} and A × A. We recall [6] that an MV-algebra A is simple iff it can be embedded into into [0, 1]MV ; in this case, the embedding is uniquely determined by A. The same result can be proved for PMV+ -algebras, because there is only one operation on [0, 1]MV which makes it into a PMV+ -algebra, namely, ordinary product. An MV-algebra (resp., PMV+ -algebra) is said to be semisimple iff it is isomorphic to a subdirect product of simple MV-algebras (resp., PMV+ -algebras). Hence, every semisimple MV-algebra (resp., PMV+ -algebra) is a subalgebra of a direct product of copies of [0, 1]MV (resp., [0, 1]PMV ). In the sequel we shall often use the notion of state. Definition 2.5. A state on an MV-algebra A (see [23]) is a unary function s from A into [0, 1] such that s(1)

φ, ψ ∈ A, if φ  ψ = 0, then s(φ ⊕ ψ) = s(φ) + s(ψ).

= 1 and for all

A valuation on an MV-algebra A is a homomorphism from A into [0, 1]MV . + More generally, let [0, 1]MV be an expansion 4 of [0, 1]MV by means of a set of continuous operations on [0, 1], and let A + + be an element of the variety generated by [0, 1]MV . Then a valuation on A is a homomorphism from A into [0, 1]MV . Moreover, a state on A is just a state of its MV-reduct.

The set V (A) of all valuations constitutes a closed subset of [0, 1]A , equipped with the Tychonoff topology. Moreover, every valuation is a state, and a state is a valuation iff it is not a proper convex combination of two different states (such states are called extremal, see [26, page 120]). +

+

Definition 2.6. Let [0, 1]MV be as in Definition 2.5, and let A be an element of the variety generated by [0, 1]MV . A map

: φ1  → α1 , . . . , φn  → αn from a finite subset of A into [0, 1] is called an assessment on A. Let W be a closed set of valuations on A, and let : φ1  → α1 , . . . , φn  → α n be an assessment on A. Then is said to be W -coherent if for every λ1 , . . . , λn ∈ R there is a valuation v ∈ W such that ni=1 λi (αi − v(φi )) ≥ 0. We say that is a convex combination of n + 1 valuations in W if there are valuations v1 , . . . , vn+1 ∈ W and non-negative reals μ1 , . . . , μn+1 such that n +1 j=1

μj = 1 and for i = 1, . . . , n, αi =

n +1 j=1

μj vj (φi ).

When there is no explicit reference to W , it is tacitly assumed that W

= V (A) is the set of all valuations on A.

In [19], the following result is proved: +

+

Theorem 2.2. Let [0, 1]MV be as in Definition 2.5, and let A be an element of the variety generated by [0, 1]MV . Let W be a closed set of valuations on A, and let : φ1  → α1 , . . . , φn  → αn be an assessment on A. The following are equivalent: (1) is W -coherent. (2) is a convex combination of n + 1 valuations in W . Moreover if W = V (A), then (1) and (2) are equivalent to (3) There is a state s on A such that for i = 1, . . . , n, s(φi ) =

αi .

We conclude this section recalling the ultraproduct construction and the Fundamental  Ultraproduct Theorem, see [5, Theorem 4.1.9]. Let {Ai : i ∈ I } be a family of structures of the same language, let A = i∈I Ai denote the product of all ultrafilter structures Ai , and let the superscript A denote the realization of the predicate and function symbols in A. Let U be an  on I (that is, a maximal filter of the boolean algebra consisting of all subsets of I). We define an equivalence θU on i∈I Ai as follows: (a, b) ∈ θU iff {i ∈ I : ai = bi } ∈ U. Let A /U denote the following structure: 4 By the term expansion of an algebra A we mean an algebra B such that the universe of B coincides with the universe of A, the operations of B in the signature of A are as in A, but B has some additional operations not in the signature of A.

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• The universe of A /U is the quotient of A modulo θU . • Let for all a ∈ A, [a]U denote its equivalence class modulo θU . Then for every n-ary function symbol f , and for every [a1 ]U , . . . , [an ]U ∈ A/U, the interpretation, f A/U or f in A /U, is defined by f A/U ([a1 ]U , . . . , [an ]U )

= [f A (a1 , . . . , an )]U

• For every n-ary predicate symbol P and for every [a1 ]U , . . . , [an ]U ∈ A/U, the interpretation, P A/U or P in A /U, is defined by P A/U [a1 ]U , . . . , [an ]U ) iff {i

∈ I : Ai | P (a1i , . . . , ani )} ∈ U .

It is easy to prove that f A/U and P A/U are well-defined, that is, the definitions of f A/U and of P A/U depend on [a1 ]U , . . . , [an ]U but not on a1 , . . . , an . Definition 2.7. The structure A /U just defined is called the ultraproduct of the structures Ai  U and is denoted by i∈I Ai /U.

: i ∈ I with respect to the ultrafilter

The Fundamental Ultraproduct Theorem says: Theorem 2.3. For any formula φ(x1 , . . . , xn ) and for any [a1 ]U , . . . , [an ]U

∈



i∈I

Ai /U, the following are equivalent:



(i) i∈I Ai /U | φ([a1 ]U , . . . , [an ]U ). (ii) The set {i ∈ I : Ai | φ(a1i , . . . , ani )} is in U. When all structures Ai coincide with a fixed structure B, their ultraproduct is called an ultrapower and is denoted by BI /U. Ultrapowers of the real field R, possibly equipped with arbitrary functions or predicates, are usually denoted by R ∗ , and the Ultraproduct Theorem gives us: Transfer Principle. For every first-order sentence φ in the language of R, R

| φ iff R ∗ | φ .

The Transfer Principle will be used sometimes in the paper. However, many constructions will depend on the choice of the ultrafilter, and hence in some cases we will explicitly need the Fundamental Ultraproduct Theorem.

3. Extensions of MV-algebras, hypervaluations and hyperstates In this section we introduce an algebraic apparatus for non-standard probability in many-valued logic. Definition 3.1. Let A be an MV-algebra, and let [0, 1]∗MV be an ultrapower of [0, 1]MV . We say that [0, 1]∗MV is A-amenable if for every x ∈ A \ {0}, there is a homomorphism h from A into [0, 1]∗MV such that h(x) = 0. Similar definition for PMV+ -algebras, with [0, 1]MV and [0, 1]∗MV replaced by [0, 1]PMV and [0, 1]∗PMV , respectively. I  By Di Nola’s theorem [9], any MV-algebra A embeds into [0, 1]∗MV for some index set I and for some ultrapower [0, 1]∗MV of [0, 1]MV , and hence an A-amenable ultrapower of [0, 1]MV always exists. Indeed, if x = 0, then there is an index i ∈ I such that xi = 0, and hence, the projection πi on the ist coordinate is a homomorphism such that πi (x) = 0. By Theorem 2.1, a similar result holds for PMV+ -algebras. Moreover, [0, 1]MV (resp., [0, 1]PMV ) is A-amenable iff A is semisimple. Lemma 3.1. Let A be an MV-algebra, let [0, 1]∗MV be A-amenable, and let H be the set of all homomorphisms from A into [0, 1]∗MV . Then: H  (i) The map : a  → (h(a) : h ∈ H ) is an embedding from A into [0, 1]∗MV .   H (ii) We can make [0, 1]∗MV into a PMV+ -algebra having both A and [0, 1]∗MV as MV-subreducts. Proof. (i) That is a homomorphism is clear. Moreover since [0, 1]∗MV is A-amenable, the kernel, −1 ({0}), of , is {0}, and hence, is one-one. (ii) The product on [0, 1]MV makes [0, 1]MV into a PMV+ -algebra, and hence the ultraproduct construction produces a H  product on [0, 1]∗MV which makes it into a PMV+ -algebra. Hence, the direct product construction makes [0, 1]∗MV into a PMV+ -algebra as well. Finally, the map sending any α ∈ [0, 1]∗MV into the function α ◦ on H which is constantly equal to α H  is an embedding of [0, 1]∗MV into [0, 1]∗MV . 

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Remark 3.1. A similar argument shows that if A is a PMV+ -algebra, if [0, 1]∗PMV is A-amenable and H is the set of all PMV+ H  homomorphisms from A into [0, 1]∗PMV , then both A and [0, 1]∗PMV embed into [0, 1]∗PMV . However, a homomorphism of the MV-reduct of A need not be a homomorphism of A. Corollary 3.1. Any MV-algebra is a subreduct of a PMV+ -algebra. Notation. Let A be an MV-algebra and let [0, 1]∗MV be A-amenable. Then we identify any α from H into [0, 1]∗MV defined by α ◦ (h) = α for every h ∈ H.

∈ [0, 1]∗MV with the function α ◦

H  Moreover, letting, for all a ∈ A, (a) = (h(a) : h ∈ H ), the subalgebra of [0, 1]∗MV generated by (A) and by all ∗ ∗ constants α ∈ [0, 1]MV will be denoted by A , [0, 1]MV . A similar construction, with [0, 1]∗PMV in place of [0, 1]∗MV , can be performed when A is a PMV+ -algebra.   The construction of A , [0, 1]∗MV plays a very important role in this paper. Indeed, we need product in order to represent conditional probability, andhyperreal constants in order to represent non-standard assessments. Both product and hyperreal  constants are present in A , [0, 1]∗MV , and hence the tools we need are available in this algebra. The following lemma is immediate:   Lemma 3.2. [0, 1]∗MV is amenable for A , [0, 1]∗MV . Definition 3.2. Let A be an MV-algebra and let [0, 1]∗MV be A-amenable. A hypervaluation on (A , [0, 1]∗MV ) is a PMV + homomorphism v∗ from (A , [0, 1]∗MV ) into [0, 1]∗MV ) such that v(α) for all α ∈ [0, 1]∗MV .   A hyperstate on (A , [0, 1]∗MV ) is a map s∗ from A , [0, 1]∗MV into [0, 1]∗MV such that: (1) (2) (3) (4)

s∗ (1) = 1; If φ  ψ = 0, then s∗ (φ ⊕ ψ) = s∗ (φ) + s∗ (ψ); For all α ∈ [0, 1]∗MV and for all φ ∈ (A , [0, 1]∗MV ), s∗ (α · φ) = α · s∗ (φ); For all φ ∈ (A , [0, 1]∗MV ) there is a hypervaluation v∗ on (A , [0, 1]∗MV ) such that v∗ (φ)

=α

≤ s∗ (φ).

Remark 3.2. Hyperstates constitute a natural non-standard generalization of the concept of state. Indeed, any MV-algebra without infinitesimals (see footnote 1 for the definition of infinitesimal) can be embedded into an algebra of the form C(X )[0, 1], and since states map all infinitesimals to 0, we can restrict ourselves to states on algebras of this form. Now Panti [28] and Kroupa [16] show that states can be regarder as integrals, and hence, as monotonic, normalized and linear functionals, on C(X )[0, 1] (recall that a functional S is monotonic if f ≤ g implies S (f ) ≤ S (g ), and normalized if S (1) = 1). Linearity, normalization and monotonicity imply:

(+)

min(f )

= S(min(f )) ≤ S(f ) ≤ S(max(f )) ≤ max(f ). 5

In the standard case the above formula is equivalent to property (4). In the non-standard case, the scenario changes. First of all, we do not ignore infinitesimals. Hence, we cannot restrict ourselves to MV-algebras of the form C(X )[0, 1], and it is more convenient to consider MV-algebras of functions taking values in [0, 1]∗ . Moreover, in the definition of hyperstate, we require linearity over R ∗ , and not only over R. Finally, in the hyperreal case property (4) is weaker than property (+), because min(f ) and max(f ) need not exist in R ∗ . Summing-up, the definition of hyperstate involves property (4), a weak version of (+), and the fact that a hyperstate corresponds to a monotonic and normalized functional which is linear over R ∗ . Definition 3.3. A hyperstate (resp., a hypervaluation) on A is a map on A which can be extended to a hyperstate (resp., to a  hypervaluation) on A , [0, 1]∗MV for some A-amenable ultrapower [0, 1]∗MV of [0, 1]MV . A hyperstate s∗ is said to be faithful if for all φ ∈ (A , [0, 1]∗MV ), if s∗ (φ) = 0, then φ = 0. Similar definitions for PMV+ -algebras, with [0, 1]∗PMV in place of [0, 1]∗MV . Remark 3.3. A hypervaluation v∗ on A has a unique extension to a hypervaluation v∗ on (A , [0, 1]∗MV ). Indeed, every element of (A , [0, 1]∗MV ) has the form t (φ1 , . . . , φm , γ1 , . . . , γk ) for some PMV+ -term t, for some φ1 , . . . , φm ∈ A and for some γ1 , . . . , γk ∈ [0, 1]∗MV . Then the desired extension v must satisfy v (t (φ1 , . . . , φm , γ1 , . . . , γk ) = t (v(φ1 ), . . . , v(φm ), γ1 , . . . , γk ), where the term to the left is interpreted in (A , [0, 1]∗MV ), and the term to the right is interpreted in [0, 1]∗MV . v +v A hypervaluation is a hyperstate, but not viceversa. For instance, if v1 , v2 are different hypervaluations, then 1 2 2 ∗ is a hyperstate but not a hypervaluation. Moreover when [0, 1]MV = [0, 1]MV , conditions (3) and (4) in Definition 3.2 5

min(f ) and max(f ) exist because f is a continuous function on a compact space.

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become redundant, because in this case (1) and (2) constitute the definition of state, and (3) and (4) hold when hyperstates, hypervaluations and hyperreals are replaced by states, valuations and reals, respectively. This is no longer true when [0, 1]∗MV is a proper extension of [0, 1]MV : let for every x ∈ [0, 1]∗MV , st (x) denote the standard    H part of x. Let H be the set of all endomorphisms of [0, 1]∗MV . Then [0, 1]∗MV , [0, 1]∗MV is a subalgebra of [0, 1]∗MV , and   since st ∈ H, the projection, πst , into the factor corresponding to st, is a PMV+ -homomorphism from [0, 1]∗MV , [0, 1]∗MV ∗ ∗ into [0, 1]MV . It is clear that such projection satisfies (1) and (2), but it does not satisfy any of (3) or (4). Indeed, if ∈ [0, 1]MV is a positive infinitesimal and if we identify with the function on H which is constantly equal to , then 0 = πst ( ) = πst ( · 1) = · πst (1). Moreover, for every hypervaluation v∗ , v∗ ( ) = > πst ( ). Finally, we observe that condition (4) also gives us a hypervaluation w∗ such that w∗ (x) ≥ s∗ (x): let, by (4), w∗ be such that w∗ (∼ x) ≤ s∗ (∼ x). Then s∗ (x) = 1 − s∗ (∼ x) ≤ 1 − w∗ (∼ x) = w∗ (x). Examples . (1) Let A = [0, 1]ω MV . Then [0, 1]MV is A-amenable, and hence, every state on A is a hyperstate. However, there is no faithful state s such that its restriction to the boolean elements of A is a uniform distribution, that is, all atoms in the boolean algebra of all idempotent elements of A have the same probability. To the contrary, along the lines of [2], it is possible to find a faithful hyperstate whose restriction to the boolean elements is uniform. (2) The algebra B faithful hyperstate.

= [0, 1]ℵMV1 has no faithful state, while we will see in the next section that every MV-algebra has a

The next observation will be repeatedly used in the sequel.  I Let A be an MV-algebra, let [0, 1]∗MV be A-amenable and [0, 1]◦MV = [0, 1]∗MV /U (where I is an index set and U is an ultrafilter on I) be an ultrapower of [0, 1]∗MV . Let for i ∈ I, vi (resp., si ) be a hypervaluation (resp., a hyperstate) on ◦ ◦ ◦

(A , [0, 1]∗MV ). We define  two maps  v and s , on A , [0, 1]MV , as follows. ◦  , where: (i) t is a term in the language of Every element of A , [0, 1]◦MV can be written as t φ1 , . . . , φk , α1◦ , . . . , αm PMV+ -algebras; (ii) φ1 , . . . , φk are elements of A; (iii) for j = 1, . . . , m, αj◦ = (αj,i : i ∈ I )/U is an element of [0, 1]◦MV and, for i ∈ I, αj,i ∈ [0, 1]∗MV . Then define: ◦ ) = (v (t (φ , . . . , φ , α , . . . , α ) : i ∈ I )/U , (◦) v◦ (t (φ1 , . . . , φk , α1◦ , . . . , αm i 1 k 1,i n,i ◦ ◦ ◦ ) = (s (t (φ , . . . , φ , α , . . . , α ) : i ∈ I )/U . s (t (φ1 , . . . , φk , α1 , . . . , αm i 1 k 1,i n,i

  Lemma 3.3. v◦ and s◦ are well defined. Moreover, v◦ is a hypervaluation on A , [0, 1]◦MV , and s◦ is a hyperstate on 

A , [0, 1]◦MV . Proof. The result is a direct consequence of the Ultraproduct Theorem. As regards to property (4) in Definition 3.2, with ◦ reference to formulas (◦), let for  every i ∈ I, ti be an abbreviation for t (φ1 , . . . , φk , α1,i , . . . , αn,i ), and t be an abbreviation  for t φ1 , . . . , φn , α1◦ , . . . , αn◦ . Then for every i ∈ I there is a hypervaluation vi such that vi (ti ) ≤ si (ti ). Hence, by the Ultraproduct Theorem, the hypervaluation v◦ and the hyperstate s◦ defined by the formulas (◦) satisfy the condition v◦ (t ◦ ) ≤ s◦ (t ◦ ).  When all hyperstates si coincide with a fixed hyperstate s∗ and all hypervaluations vi coincide with a fixed hypervaluation v∗ , we obtain: of[0, 1]∗MV and [0, 1]∗MV is A-amenable, then every hyperstate on (A , [0, 1]∗MV ) can Corollary 3.2. If [0, 1]◦MV is an ultraproduct  ◦ be extended to a hyperstate on A , [0, 1]MV , and every hypervaluation on (A , [0, 1]∗MV ) can be extended to a hypervaluation   ◦ on A , [0, 1]MV . 4. Hyperassessments and coherence In this section, we extend Theorem 2.2 to hyperstates and to non-standard assessments. To this purpose, we introduce a new notion of coherence for betting games, in which betting odds and truth values can take hyperreal values, and hyperreal bets are allowed. The results will be given for MV-algebras, but we wish to point-out that they can be easily extended, by very similar proofs, to PMV+ -algebras. Notation. In the remainder of this paper, unless explicitly stated otherwise, A will denote an arbitrary MV-algebra and

[0, 1]∗MV an A-amenable ultrapower of [0, 1]MV .

Definition 4.1. A hyperassessment on (A , [0, 1]∗MV ) is a map : φ1  → α1 , . . . , φn  → αn from a finite subset {φ1 , . . . , φn } of (A , [0, 1]∗MV ) into [0, 1]∗MV . that the hyperassessment is W -coherent if for all Let W be a set of hypervaluations on (A , [0, 1]∗MV ). We say λ1 , . . . , λn ∈ [0, 1]∗MV , there is a hypervaluation v∗ ∈ W such that ni=1 λi (αi − v∗ (φi )) ≥ 0.

F. Montagna et al. / International Journal of Approximate Reasoning 54 (2013) 573–589

581

We say that is [0, 1]∗MV -coherent if is W -coherent, where W is the set of all hypervaluations on (A , [0, 1]∗MV ). We say that is coherent if it is [0, 1]◦MV -coherent for some ultrapower, [0, 1]◦MV , of [0, 1]MV , in which [0, 1]∗MV embeds. We say that is a convex combination of n + 1 hypervaluations if there are an ultrapower [0, 1]◦MV of [0, 1]MV in which    [0, 1]∗MV embeds, hypervaluations v1◦ , . . . , vn◦+1 on A , [0, 1]◦MV and μ1 , . . . , μn+1 ∈ [0, 1]◦MV , such that ni=+11 μi = 1 n+1 and for i = 1, . . . , n, αi = j=1 μi vj◦ (φi ).   Remark 4.1. Note that assessments with domain included in A are special cases of assessments on A , [0, 1]∗MV . Moreover the definitions of coherence and of convex combination of hypervaluations are independent of the choice of [0, 1]∗MV , because any ultraproduct [0, 1]◦MV witnessing the above definitions can be extended to one in which [0, 1]∗MV embeds. We are ready to state the main result of this section: Theorem 4.1. Let

: φ1 → α1 , . . . , φn → αn be a hyperassessment on (A , [0, 1]∗MV ). The following are equivalent:

on (A , [0, 1]∗MV ). (1) is a convex combination of n + 1 hypervaluations  (2) There is a hyperstate s∗ on A , [0, 1]∗MV such that for j = 1, . . . , n, s∗ (φj ) (3) is coherent.

= αj .

Proof. (1) ⇒ (2). It is readily seen that any convex combination of hypervaluations is a hyperstate, and hence (1) ⇒ (2) holds. way of (2) ⇒ (3). Let s∗ be a hyperstate on (A , [0, 1]∗MV ) such that for i = 1, . . . , n, s∗ (φi ) = αi . Suppose, by , 1]∗MV ), ni=1 λi contradiction, that there are λ1 , . . . , λn ∈ [0, 1]∗MV such that for every hypervaluation v∗ on (A , [0 (αi − v∗ (φi )) < 0. After multiplying all the λi by a suitable positive constant in R ∗ , we may suppose that ni=1 |λi | ≤ 1. Now for every hypervaluation v∗ into [0, 1]∗MV , 1 2

<

1 2

−

n

i=1

λi (αi − v∗ (φi )) 2

We construct an element φ v∗ (φ)

=

1 2

−

n

i=1

≤ 1.

∈ (A , [0, 1]∗MV ) such that for every hypervaluation v∗ on (A , [0, 1]∗MV ),

λi (αi − v∗ (φi )) 2

.

To this purpose, define a sequence ψ0 , ψ1 , . . . , ψn by induction as follows 6 : ψ0 = 12 ; for i < n, set ⎧ ⎧ ⎨ λ (α ⎨ λ (φ i+1 i+1  φi+1 ) if λi+1 ≥ 0, i+1 i+1  αi+1 ) if λi+1 ≥ 0, θi = χi = ⎩ |λ |(φ ⎩ |λi+1 |(αi+1  φi+1 ) otherwise. i+1 i+1  αi+1 ) otherwise,

χ θ Now define: ψi+1 = ψi ⊕ 2i  2i . It is readily seen that φ = ψn meets our requirements. Moreover, since for i On the other hand, for every hypervaluation w∗ on (A , [0, 1]∗MV ), we have w∗ (φ)

=

1 2

−

n

i=1

λi (αi − w∗ (φi )) 2

>

1 2

= 1, . . . , n, s∗ (φi ) = αi , we have s∗ (φ) = 12 .

,

contradicting condition (4) in Definition 3.2. The proof of (3) ⇒ (1) is more involved, and requires several lemmas.  To begin with, we prove the claim in the special case where A = F(X ) is the free MV-algebra on a set X of variables. Note that in this case any ultrapower of [0, 1]MV is A-amenable. We adopt the following notation: Notation. A capital letter with subscript k will denote the sequence of lowercase letters indexed with subscripts 1, . . . , k. Thus, for instance, Xk will denote x1 , . . . , xk , m will denote ω1 , . . . , ωm , etc. Moreover, if Q is either ∃ or ∀, then QXk will denote Qx1 . . . Qxk . Finally, we write Xm ∈ S for x1 ∈ S , . . . , xn ∈ S. We adopt a similar notation for other letters, and hence, e.g., QZn and Zn ∈ S are defined analogously. 6

Intuitively, ψi represents

1 2

−

i

j =1

λj (αj −v∗ (φj )) 2

.

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By Xm

= x1 , . . . , xm we will denote a string with length m of (distinct) variables in X. Recall that any element of

(F(X ), [0, 1]∗MV ) can be represented as t (Xm , k ) for some PMV+ -term t, for some string Xm of variables in X and for some string k of constants in [0, 1]∗MV . Lemma 4.1. Given any PMV+ -term t (Xm , Yk ) in the variables Xm , Yk , there is a formula Valt (Xm , Yk , z ) in the language of ordered fields such that, for all m , k , γ in [0, 1]MV (resp., in [0, 1]∗MV ), if v is a valuation (resp., a hypervaluation) on (F(X ), [0, 1]∗MV ) such that v(Xm ) = m (i.e., v(xi ) = ωi , for i = 1, . . . , m), then v(t (Xm , k )) = γ iff the formula Valt (m , k , γ ) is true in R (resp., in R ∗ ). Proof. In the sequel, in order to distinguish classical connectives from MV-operations, we denote classical conjunction by

, classical disjunction by , classical negation by ¬ and classical implication by ⇒. Moreover, iterated conjunction and   disjunctions will be denoted by and , respectively.   Let T = {t1 , . . . , tk } be the set of all subterms of t, and let S = s1, . . . , sh be the set of all its atomic subterms. For every s ∈ T, let us introduce a new variable vs (distinct variables for different terms). Define: O(x, y, z ) := ((x + y ≤ 1)  (z = x + y))  ((x + y > 1)  z = 1), = 1 − x and P (x, y, z) := z = x · y, N (x, z ) := z 

[0,1]MV := {(0 ≤ vt )  (vt ≤ 1) : t ∈ T },

⊕ := {O(vu1 , vu2 , vu3 ) : u1 , u2 , u3 ∈ T , u3 = u1 ⊕ u2 },

∼ :=  {N (vu1 , vu2 ) : u1 , u2 ∈ T , u2 =∼ u1 },

π :=  {P (vu1 , vu2 , vu3 ) : u1 , u2 , u3 ∈ T , u3 = u1 · u2 },

S := {vs = s : s ∈ S }. Then the formula Valt (Xm , Yk , z )

:= ∀vt1 . . . ∀vtn (( [0,1]MV  ⊕  ∼  π  S ) ⇒ vt = z)

settles the claim.  Lemma 4.2. Let t1 , . . . , tn be PMV+ -terms in the variables Xm , Yk . Then: (1) There is a formula CoerTn (Yk , Zn ) in the language of ordered fields such that for all k , n in [0, 1]MV (resp., in [0, 1]∗MV ), the following condition holds: CoerTn (k , n ) is true in R (resp., in R ∗ ) iff the assessment (resp., the hyperassessment)

 : t1 (Xm , k )  → σ1 , . . . , tn (Xm , k )  → σn on (F(X ), [0, 1]∗MV ) is coherent. (2) There is a formula ConvTn (Yk , Zn ) such that for all k , n in [0, 1]MV (resp., in [0, 1]∗MV ), the following condition holds: ConvTn (k , n ) is true in R (resp., in R ∗ ) iff the assessment (resp., the hyperassessment)  in (1) is a convex combination of n + 1 valuations (resp., hypervaluations) on (F(X ), [0, 1]∗MV ), with coefficients in R (resp., in R ∗ ). Proof. (1) After multiplying λ1 , . . . , λn by a positive constant, we may assume without loss of generality that the coefficients λi in the definition of coherence satisfy the inequalities −1 ≤ λi ≤ 1. Now for (1), let VALTn (Xm , Yk , Un ) Norm(n )

:=

n  i=1

Pos(n , Zn , Un )

:=

n  i=1

Valti (Xm , Yk , ui ),

|λi | ≤ 1,

:=

n  i=1

λi (zi − ui ) ≥ 0.

Then the desired formula CoerTn (Yk , Zn ) is

∀n ∃Xm ∀Un ((Norm(n )  VALTn (Xm , Yk , Un )) ⇒ Pos(n , Zn , Un )). As regards to (2), let: Un,n+1

:= Un1 , . . . , Unn+1 , Xm,n+1 := Xm1 , . . . , Xmn+1 ,

∃Xm,n+1 := ∃Xm1 . . . ∃Xmn+1 , ∃Un,n+1 := ∃Un1 . . . ∃Unn+1 .

F. Montagna et al. / International Journal of Approximate Reasoning 54 (2013) 573–589

Moreover let for i

583

= 1, . . . , n + 1,

i , Yk , Uni ) VALTn (Xm

:=

n  j=1

i Valtj (Xm , Yk , uij ),

VALT+n (Xm,n+1 , Yk , Un,n+1 ) ⎛ C (n+1 , Zn , Un,n+1 )

:=

n +1

:=

i=1

i VALTn (Xm , Yk , Uni ),

n +1

⎝

r =1

(0 ≤

⎞

σr )⎠ 

n +1 r =1

⎛

σr =

1⎝

n n +1  i=1 r =1

⎞

σr uri

= zi ⎠ .

Then the desired formula ConvTn (Yk , Zn ) is

∃Xm,n+1 ∃Un,n+1 ∃n+1 (C (n+1 , Zn , Un,n+1 )  VALT+n (Xm,n+1 , Yk , Un,n+1 )).



Now the equivalence, for an assessment : φ1  → α1 , . . . , φn  → αn , between coherence and being a convex combination of n + 1 valuations holds in R (Theorem 2.2). Since such an equivalence can be expressed by a first-order formula, (Lemma 4.2) by the Transfer Principle it also holds in R ∗ . It follows: Lemma 4.3. For a hyperassessment on (F(X ), [0, 1]∗MV ), conditions (1), (2) and (3) in Theorem 4.1 are equivalent. Our task is to extend Lemma 4.3 to (A , [0, 1]∗MV ), where A is an arbitrary MV-algebra. We start with the following remark. Remark 4.2. Let A be an arbitrary MV-algebra, and let [0, 1]∗MV be A-amenable. Then there is a free MV-algebra F(X ) such that A is an epimorphic image, via an epimorphism h, of F(X ). Note that h induces an epimorphism h from (F(X ), [0, 1]∗MV ) onto

(A , [0, 1]∗MV ), defined, for every Xm ∈ X, for every PMV+ -term t (Xm , Yk ) and for every k ∈ [0, 1]∗MV , by h (t (Xm , k )) = t (h(Xm ), k ). Hence, every hypervaluation v on (A , [0, 1]∗MV ) determines the hypervaluation v on (F(X ), [0, 1]∗MV ), defined by v = v ◦ h where ◦ denotes composition. Clearly, (h )−1 ({1}) ⊆ (v )−1 ({1}). Conversely, let v be a hypervaluation on (F(X ), [0, 1]∗MV ), and suppose that h is a homomorphism from

(F(X ), [0, 1]∗MV ) into (A , [0, 1]∗MV ) such that (h )−1 ({1}) ⊆ (v )−1 ({1}). Then v factorizes through h , that is, there is a unique hypervaluation v on A such that v = v ◦ h . This is an immediate consequence of the homomorphism theorems. Hence, hypervaluations v on (A , [0, 1]∗MV ) are in bijection with hypervaluations v on (F(X ), [0, 1]∗MV ) satisfying  (h )−1 ({1}) ⊆ (v )−1 ({1}). It follows: Lemma 4.4. With the same assumptions and the same notation as in Remark 4.2, let W be the set of all hypervaluations v on (F(X ), [0, 1]∗MV ) such that (h )−1 ({1}) ⊆ (v )−1 ({1}). Let : φ1  → α1 , . . . , φn  → αn be a hyperassessment on

(A , [0, 1]∗MV ), and let, for i = 1, . . . , n, ti ∈ (F(X ), [0, 1]∗MV ) be such that h (ti ) = φi . Define an assessment  on (F(X ), [0, 1]∗MV ) by

 : t1  → α1 , . . . , tn  → αn . Then: (1) is coherent iff  is W-coherent. (2) is a convex combination of n+1 hypervaluations on (A , [0, 1]∗MV ) iff  is a convex combination of n+1 hypervaluations in W . Lemma 4.5. Let Tn = t1 , . . . , tn , Wh = w1 , . . . , wh be PMV+ terms in the variables Xm , Yk . Let W be the set of all hypervaluations v on (F(X ), [0, 1]∗MV ) such that v (w1 ) = · · · = v (wh ) = 1. Then there are formulas CoerT∗n ,Wh (Yk , Zn ) and ConvT∗n ,Wh (Yk , Zn ) in the language of ordered fields such that for all k , n ∈ [0, 1]MV (resp., [0, 1]∗MV ), the following conditions hold: (1) CoerT∗n ,Wh (k , n ) is true in R (resp., in R ∗ ) iff the assessment (resp., the hyperassessment)

: t1 (Xm , k )  → σ1 , . . . , tn (Xm , k )  → σn is W -coherent.

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F. Montagna et al. / International Journal of Approximate Reasoning 54 (2013) 573–589

(2) ConvT∗n ,Wh (k , n ) is true in R (resp., in R ∗ ) iff is a convex combination of n + 1 valuations (resp., hypervaluations) in W , with coefficients in R (resp., in R ∗ ). Proof. The proof is similar to the proof of Lemma 4.2, the only difference being that we have to add the condition that w1 , . . . , wh are evaluated at 1. More formally, with reference to Lemma 4.2 (1), let: SWh (Xm , Yk )

:=

h  i=1

Valwi (Xm , Yk , 1),

and G(n , Xm , Yk , Un )

:= Norm(n )  VALTn (Xm , Yk , Un )).

Then the desired formula CoerT∗n ,Wh (Yk , Zn ) is

∀n ∃Xm (SWh (Xm , Yk )  ∀Un (G(n , Xm , Yk , Un ) ⇒ Pos(n , Zn , Un ))). In order to prove (2), let ∗ (Xm,n+1 , Yk ) SW h

:=

n +1 i=1



i SWh Xm , Yk ,

and let H (Xm,n+1 , Yk , Un,n+1 , n+1 , Zn ) denote the formula VALT+n (Xm,n+1 , Yk , Un,n+1 )  C (n+1 , Zn , Un,n+1 ). Then the desired formula ConvT∗n ,Wh (Yk , Zn ) is ∗ ∃Xm,n+1 ∃Un,n+1 ∃n+1 (SW (Xm,n+1 , Yk )  H (Xm,n+1 , Yk , Un,n+1 , n+1 , Zn )). h



Corollary 4.1. If A is a quotient of a free MV-algebra modulo a finitely generated congruence, then (1), (2) and (3) of Theorem 4.1 are equivalent. Proof. By Lemmas 4.5 and 4.4, both [0, 1]∗MV -coherence and being a convex combination of n + 1 hypervaluations are first-order properties which are equivalent in R, and hence, by the Transfer Principle, they are equivalent in R ∗ . This settles the claim.  We are ready to conclude the proof of Theorem 4.1. ∗ Lemma 4.6. Let A, [0, 1]∗MV and  be as in Theorem 4.1. If is [0, 1]MV -coherent, then it is a convex combination of n hypervaluations on A , [0, 1]◦MV for some ultrapower, [0, 1]◦MV , of [0, 1]∗MV .

+1

Proof. Let h be a homomorphism from F(X ) onto A, and let h be its extension to (F(X ), [0, 1]∗MV ) defined as in Remark 4.2. Let H = (h )−1 {1}, and let k ∈ [0, 1]∗MV and, for i = 1, . . . , n, ti (Xm , k ) ∈ (F(X ), [0, 1]∗MV ) be such that, h (ti (Xm , k )) = φi . Let W be the set of all hypervaluations v such that v (H ) = {1}, and consider the hyperassessment:

 : t1 (Xm , k )  → α1 , . . . , tn (Xm , k )  → αn . By Lemma 4.4, it suffices to prove that if  is W -coherent, then it is a convex combination of n + 1 hypervaluations in W . Hence, suppose that  is W -coherent. Let S be any finite subset of H, and let WS be the set of all hypervaluations w such that S ⊆ (w )−1 ({1}). Clearly, W ⊆ WS , and hence  is WS -coherent. By Corollary 4.1, there are non-negative hyperreals   μS1 , . . . , μSn+1 and hypervaluations v1S , . . . , vnS +1 in WS such that nj=+11 μSj = 1 and, for i = 1, . . . , n, αi = nj=+11 μSj vjS (ti ). Now let Fin(H ) be the family of all finite subsets of H, let for S ∈ Fin(H ), S = {Z ∈ Fin(H ) : S ⊆ Z }, and  = {S : S ∈ Fin(H )}. Then  has the finite intersection property, and hence it can be extended to an ultrafilter U. Consider the ultrapower ([0, 1]∗MV )Fin(H ) /U, and let us denote it by [0, 1]◦MV . We define μ◦1 , . . . , μ◦n+1 ∈ [0, 1]◦MV by μ◦i = (μSi : S ∈ Fin(H ))/U. Moreover for j = 1, . . . , n + 1, we define vj◦ from the hypervaluations vjS

: S ∈ Fin(H ) by means of the formulas (◦) introduced just before Lemma 3.3.  = 1, . . . , n, αi = nj=+11 μ◦j vj◦ (ti ).

By Lemma 3.3, each vj◦ is a hypervaluation. Moreover, by the Ultraproduct Theorem, for i

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= 1, . . . , n + 1, vj◦ is a W -hypervaluation. Indeed, for each t ∈ H, {t } ∈ , and hence, {t } ∈ U. Moreover if S ∈ {t } , then for j = 1, . . . , n + 1, vjS (t ) = 1, and hence, by the Ultraproduct Theorem, vj◦ (t ) = 1. Summing-up, we have proved that  is a convex combination of n + 1 hypervaluations in W , as desired. At this point, the implication (3) ⇒ (1) of Theorem 4.1 follows from Lemma 4.6. This concludes the proof of Theorem 4.1.  Finally, for j

We conclude this section with a proof that every coherent hyperassessment may be extended, up to infinitesimals, to a faithful hyperstate. Theorem 4.2. Every MV-algebra A has a faithful hyperstate. Moreover, for every coherent hyperassessment  : φ1  → α1 , . . . , φn → αn on (A , [0, 1]∗MV ), there is a faithful hyperstate s∗ on A such that for i = 1, . . . , n, s∗ (φi ) − αi is infinitesimal. Proof. Let by Theorem 4.1, s be a hyperstate on (A , [0, 1]∗MV ) such that for i = 1, . . . , n, s(φi ) = αi . Let B = A \ {0}. Let be a positive infinitesimal in [0, 1]∗MV . For every finite subset S = {ψ1 , . . . , ψk } of B, take hypervaluations v1 , . . . , vk on [0, 1]∗MV such that, for i = 1, . . . , k, vi (ψi ) > 0, and set, for all φ ∈ (A , [0, 1]∗MV ) sS (φ)

= (1 − )s∗ (φ) +

k  vi (φ) i=1

k

.

Clearly, sS is a hyperstate. Now let Fin(B) be the set of all finite subsets of B. Moreover for S ∈ Fin(B), let S = {T ∈ Fin(B) : S ⊆ T } and let  = {S : S ∈ Fin(B)}. Then  has the finite intersection property, and hence it may be extended to an ultrafilter, U, say.  Fin(B) /U. Let m be an abbreviation for φ1 , . . . , φm . Then by Lemma 3.3, the map s◦ defined by Let [0, 1]◦MV = [0, 1]∗MV s◦ (t ( m , α1◦ , . . . , αh◦ ))

= (sS (t ( m , α1S , . . . , αhS )) : S ∈ Fin(B))/U

is a hyperstate. Moreover for every ψ ∈ B, if S ∈ Fin(B) and ψ ∈ S, then sS (ψ) > 0. Hence, the set of all S ∈ Fin(B) such that sS (ψ) > 0 contains {ψ} , and hence it is in U. By the Ultraproduct Theorem, s◦ (ψ) > 0 for every ψ ∈ B. Hence, s◦ is faithful. Finally, for every (standard) natural number n, for every φ ∈ A and for every S again by the Ultraproduct Theorem, s◦ (φ) − s(φ) is an infinitesimal. 

∈ Fin(B), |sS (φ) − s(φ)| < 1n , and

5. Conditional probability In this section, we introduce a variant of the usual notion of coherence, namely, stable coherence, which is more adequate for the treatment of conditional probability, especially when the conditioning event has probability 0. The main result will be a characterization of stable coherence as mentioned in Section 1 (Theorem 1.4). We start from a generalization of Theorem 1.3 (which was proved in [22] for free PMV+ -algebras), to arbitrary PMV+ algebras. Recall that given a complete assessment

 : φ1 |ψ1  → α1 , . . . , φn |ψn  → αn , ψ1  → β1 , . . . , ψn  → βn on a PMV+ -algebra B, the payoff function for the bookmaker corresponding to the valuation v, when the gambler bets λi on φi |ψi and μi on ψi (i = 1, . . . , n), is n  i=1

λi v(ψi )(αi − v(φi )) +

n  i=1

μi (βi − v(ψi )).

The assessment  is said to be coherent if there is no system of bets leading to a sure loss for the bookmaker, that is, if there is no system of bets such that the payoff is (strictly) negative independently of the valuation v. We are going to prove: Lemma 5.1. Let

 : φ1 |ψ1  → α1 , . . . , φn |ψn  → αn , ψ1  → β1 , . . . , ψn  → βn be an assessment on a PMV+ -algebra B. Suppose that β1 , . . . , βn are strictly positive. Let

 : ψ1  → β1 , . . . , ψn  → βn , φ1 · ψ1  → α1 β1 , . . . , φn · ψn  → αn βn . Then  is coherent iff  is coherent. Proof. Suppose that  is coherent. Let λ1 , . . . , λn , μ1 , . . . , μn be given. We want a valuation (resp., a hypervaluation) v∗ such that

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F. Montagna et al. / International Journal of Approximate Reasoning 54 (2013) 573–589 n  i=1

λi (αi βi − v∗ (φi )v∗ (ψi )) +

n  i=1

μi (βi − v∗ (ψi )) ≥ 0.

This is equivalent to n  i=1

λi (αi v∗ (ψi ) − v∗ (φi )v∗ (ψi )) +

n  i=1

(λi αi + μi )(βi − v∗ (ψi )) ≥ 0.

The existence of such a valuation (resp., hypervaluation) follows from the coherence of . Conversely, suppose that  is coherent, and consider λ1 , . . . , λn , μ1 , . . . , μn . We want a valuation (resp., a hypervaluation) v◦ such that n  i=1

λi v◦ (ψi )(αi − v◦ (φi )) +

n  i=1

μi (βi − v◦ (ψi )) ≥ 0.

The last condition is equivalent to n  i=1

λi (αi βi − v◦ (φi )v◦ (ψi )) +

n  i=1

(μi − λi αi )(βi − v◦ (ψi )) ≥ 0,

and the existence of such a valuation (resp., hypervaluation) v◦ follows from the coherence of  . Now if  is a complete assessment as in Theorem 1.3, then by Lemma 5.1, the coherence of  is equivalent to the coherence of  , and since  only involves unconditional events, the claim follows from Theorem 2.2.  to hyperassessments on PMV+ -algebras of the form Lemma 5.1 can be extended, essentially with the same proof, ∗ ∗

A , [0, 1]MV (where, as usual, A is an MV-algebra and [0, 1]MV is A-amenable) and together with Theorem 4.1, it gives:   Corollary 5.1. Let  and  be hyperassessments defined on A , [0, 1]∗MV according to Lemma 5.1. The following are equivalent: (1) (2) (3) (4)

 is a coherent.  is coherent.

There exists a hyperstate on (A , [0, 1]∗MV ) which extends  .  is a convex combination of 2n + 1 hypervaluations.

By Theorem 1.3, coherence is a good rationality criterion for complete conditional assessments, as far as the betting odds for conditioning events are non-zero. For complete (real-valued) assessments in which the betting odd of some conditioning event is 0 we propose a new concept of coherence. Definition 5.1. A complete assessment

 : φ1 |ψ1  → α1 , . . . , φn |ψn  → αn , ψ1  → β1 , . . . , ψn  → βn on an MV-algebra A (where all coherent hyperassessment

ψi are non-zero and each βi is allowed to be 0), is said to be stably coherent iff there is a

+ : φ1 |ψ1  → α1 , . . . , φn |ψn  → αn , ψ1  → β1 , . . . , ψn  → βn   on A , [0, 1]∗MV such that, for i

= 1, . . . , n, βi > 0, and αi − αi , as well as βi − βi , are infinitesimal.

A stably coherent complete assessment is coherent. Indeed, if there is a system of (real-valued) bets causing a sure loss to the bookmaker, then the same system of bets causes a sure loss when the assessment is modified by an infinitesimal. As a consequence of Theorem 5.1 below, which provides for a characterization of stable coherence, the converse does not hold. Theorem 5.1. Let  : φ1 |ψ1 following are equivalent:

→ α1 , . . . , φn |ψn → αn , ψ1 → β1 , . . . , ψn → βn be a complete assessment on A. The

(1)  is stably coherent.   (2) There is a faithful hyperstate s∗ on A , [0, 1]∗MV such that for i   ∗ s (φi · ψi ) ∗ st (s (ψi )) = βi and st = αi . s∗ (ψi )

= 1, . . . , n,

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Proof. (1) ⇒ (2). Condition (1) is equivalent to the existence of β1 , . . . , βn , α1 , . . . , αn such that (a) for i = 1, . . . , n, βi > 0, (b) αi − αi and βi − βi are infinitesimal, and (c) the assessment + : ψ1 → β1 , . . . , ψn → βn , φ1 |ψ1 → α1 , . . . , φn |ψn → αn avoids sure loss. By Lemma 5.1, condition (c) is equivalent to the coherence of the assessment

 : ψ1  → β1 , . . . , ψn  → βn , φ1 · ψ1  → α1 β1 , . . . φn · ψn  → αn βn . By Theorem 4.1, this implies the existence of a hyperstate s∗ on (A , [0, 1]∗MV ) which extends  . Then   ∗    αβ s (φi · ψi ) ∗  st (s (ψi )) = st (βi ) = βi and st = st i  i = αi . ∗ s (ψi ) βi If s∗ is faithful, the claim  is proved.  Otherwise, let H = c ∈ (A , [0, 1]∗MV ) : c > 0 and s∗ (c ) = 0 . For each finite set S = {c1 , . . . , cn } ⊆ H, take a positive infinitesimal such that, for i = 1, . . . , n, < (βi )2 , and if αi > 0, then < (αi βi )2 . Take hypervaluations v1∗ , . . . , vn∗ such that for i = 1, . . . , n, vi∗ (ci ) > 0, and define sS∗ (x)

= (1 − )s∗ (x) +

v1∗ (x) + · · · + vn∗ (x) n

Then sS∗ is a hyperstate. Moreover, |s∗ (ψi ) − sS∗ (ψi )| s∗ (φ

i

· ψi ) − < i +

β

s∗ (φ S

i

S

It follows that   st sS∗ (ψi )

· ψi ) < i)

s∗ (ψ

= βi and st

s∗ (φ

i

· . < and

· ψi ) + . i −

β



 ∗ sS (φi

· ψi ) = st (αi ) = αi (i = 1, . . . , n). ∗ sS (ψi )

Now let Fin(H ), S and  be as in the proof of Theorem 4.2, and let U be an ultrafilter on Fin(H ) such that  ⊆ U. Let, ∗ (a) = s∗ (a) : C ∈ Fin(H ) /U. Then along the lines of the proof of Theorem 4.2, we can prove for all a ∈ (A , [0, 1]∗MV ), sU C ∗ is a faithful hyperstate (taking values in [0, 1]∗ Fin(H ) /U). Moreover for every S ∈ Fin(H ), for all m ∈ N \ {0} and that sU MV for every i = 1, . . . , n,     s∗ (φ · ψ )   ∗ 1 1 i  s (ψi ) − βi  < , and  S i < . − α i U   s∗ (ψi ) m m S Hence,  ∗  st sU (ψi )

= βi , and st



 ∗ sU (φi

· ψi ) = αi . ∗ (ψ ) sU i

(2) ⇒ (1). Let s∗ is a faithful hyperstate on (A , [0, 1]∗MV ) satisfying (2). Then letting, for i

βi = s∗ (ψi ) and αi =

= 1, . . . , n,

s∗ (φi

· ψi ) , βi

the hyperassessment

 : ψi  → βi , φi · ψi  → αi βi , i = 1 . . . , n is coherent, and by Lemma 5.1, the hyperassessment

 : φi |ψi  → αi , ψi  → βi , i = 1, . . . , n is also coherent. Moreover for i This settles the claim. 

= 1, . . . , n, βi > 0 and both αi − αi and βi − βi are infinitesimal.

Example . Let φ and ψ be such that v(ψ)

= 0 for some valuation v. Then the assessment

: φ|ψ  → 1, ¬φ|ψ  → 1, ψ  → 0 is coherent, because for any system of bets, if v(ψ) = 0, then the bookmaker’s payoff is 0. However, is not stably coherent, otherwise there would be a hyperstate s∗ and infinitesimals α, β and γ such that s∗ (ψ) = α > 0, s∗ (φ · ψ) = α(1 − β) and s∗ ((¬φ) · ψ) = α(1 − γ ). But then the axioms of hyperstates imply

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F. Montagna et al. / International Journal of Approximate Reasoning 54 (2013) 573–589

α = s∗ (ψ) = s∗ (φ · ψ) + s∗ ((¬φ) · ψ) = α(1 − β) + α(1 − γ ), which is clearly impossible.

6. Avoiding non-standard reals In this section we propose an alternative coherence condition for complete assessments of conditional probability, which does not refer to non-standard reals and, at least in the case of finitely presented MV-algebras is equivalent to stable coherence. For conditional probability over boolean events, a coherence condition not involving infinitesimals, and using layers, was presented in [7]. Theorem 6.1. Let : φ1 |ψ1  → α1 , . . . , φn |ψn  → αn , ψ1  → β1 , . . . , ψn  → βn be a complete (standard) assessment on a finitely presented MV-algebra A. Let, for every positive real (resp., hyperreal) ε , (ε) denote the following condition: There is a coherent assessment (resp. hyperassessment)

ε : φ1 |ψ1  → α1ε , . . . , φn |ψn  → αnε , ψ1  → β1ε , . . . , ψn  → βnε on (A , [0, 1]∗MV ) such that, for i The following are equivalent:

= 1, . . . , n, βiε > 0, |αi − αiε | < ε and |βi − βiε | < ε .

(1) is stably coherent. (2) every positive standard real ε , (ε) holds. + Proof. We start   from the following observation. First of all, can be regarded as hyperassessment on the PMV -algebra ∗

A , [0, 1]MV , and by Lemma 5.1, the coherence of a complete assessment of conditional events is equivalent to the coherence of an assessment on unconditional events.   By Lemma 4.5, the coherence of an unconditional hyperassessment on A , [0, 1]∗MV when A is finitely presented, may be expressed by a first-order formula. Hence, there is a first-order formula  (x) in the language of ordered fields such that for every positive standard real ε ,  (ε) holds in R iff (ε) holds. By the Transfer Principle, (ε) holds in R iff it holds in R ∗ . (1) ⇒ (2). Suppose that is stably coherent. Then there are hyperreals α1 , . . . , αn , β1 , . . . , βn such that the hyper







: φ1 |ψ1 → α1 , . . . , φn |ψn → αn , ψ1 → β1 , . . . , ψn → βn is coherent, and, for i = 1, . . . , n, βi > 0,  αi − αi and βi − βi are infinitesimal. Hence, for any standard positive real number ε , |αi − αi | < ε and |βi − βi | < ε , and ∗ (ε) holds in R . It follows that for every ε > 0, (ε) holds in R. (2) ⇒ (1). If for every standard positive real ε , (ε) holds, then ∀x( (x)) holds in R, and hence it holds in R ∗ . It follows that for every infinitesimal ε > 0 there are hyperreals α1ε , . . . , αnε , β1ε , . . . , βnε such that the hyperassessment

ε : φ1 |ψ1  → α1ε , . . . , φn |ψn  → αnε , ψ1  → β1ε , . . . , ψn  → βnε is coherent, and, for i = 1, . . . , n, βiε > 0, |αi − αiε | < ε  and |βi − βi | < ε . This immediately implies that is stably coherent.  assessment 

7. Conclusions and future work Motivated by the treatment of conditional probability over many-valued events when the conditioning event has probability zero, we have proposed a non-standard approah to MV-probability and we have proved an analogue of de Finetti’s equivalence between coherence and extendability to a probability measure. For conditional probability, however, we proposed an alternative to de Finetti’s coherence criterion, namely stable coherence, and we proved that it corresponds to Kroupa’s law s∗ (φ|ψ)

=

s∗ (φ·ψ) s∗ (ψ)

for some faithful hyperstate s∗ . Finally, in the case of finitely presented MV-algebras we

found a criterion which is equivalent to stable coherence and only refers to standard real numbers. This paper leaves some problems open. The first problem is very general: there are plenty of applications of non-standard probability to mathematics, see [27,2], and one of them has been presented in this paper. But in our opinion it is not completely clear whether non-standard probabilities exist, so to speak, in the real world. For instance, is there an event with infinitesimal probability which is relevant to common people? An interesting reference to this problem, unfortunately, only in Italian, is [20, Chapter I]. We also mention two technical problems:

• Investigate the computational complexity of stable coherence. • Extend the investigation to imprecise conditional probability, both in the case where the conditioning event is manyvalued and in the case where it has lower probability zero.

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Acnowledgements The authors wish to thank the anonymous referees for their suggestions and remarks, which contributed to improve the present paper. The first author acknowledges support of the FP7-PEOPLE-2009-IRSES Project MaToMuVi. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

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