A logical and algebraic treatment of conditional probability

Transcription

1 A logical and algebraic treatment of conditional probability Tommaso Flaminio, Franco Montagna Department of Mathematics and Computer Science Pian dei Mantellini 44, Siena Italy {flaminio, Abstract This paper is devoted to a logical and algebraic treatment of conditional probability. The main ideas are the use of non-standard probabilities and of some kind of standard part function in order to deal with the case where the conditioning event has probability zero, and the use of a many-valued modal logic in order to deal probability of an event ϕ as the truth value of the sentence ϕ is probable, along the lines of Hájek s book [H98] and of [EGH]. To this purpose, we introduce a probabilistic manyvalued logic, called F P (S LΠ), which is sound and complete with respect a class of structures having a non-standard extension [0, 1] of [0, 1] as set of truth values. We also prove that the coherence of an assessment of conditional probabilities is equivalent to the coherence of a suitably defined theory over F P (S LΠ) whose proper axioms reflect the assessment itself. 1 Introduction There are two important mathematical theories of uncertainty: the theory of probability and the theory of fuzzy sets. (Strictly speaking, the theory of fuzzy sets is a mathematical theory of vagueness, but here we are using the word uncertainty in a very broad sense). Although difference, these theories are related each other. Their logic counterparts are probability logics and fuzzy logics. The book [CS] contains an attempt to treat the t-norm s of many-valued logic by means of conditional probability. On the other hand, Hájek s book [H98] contains a treatment of probabilistic logic inside a modal fuzzy logic. The idea is that the probability of an event ϕ may be regarded as the truth s degree of the modal formula ϕ is probable, denoted by P r(ϕ). Of course, the modal logic of probability is not based on classical logic, because intermediate values are needed, but rather on many-valued logic. The most appropriate many-valued logic seems to be Lukasiewicz Logic, because in such logic it is possible to express additivity of the probability operator P r. Moreover, it is useful to have some constants corresponding to rational truth-values. Thus Hájek in [H98] introduces a probabilistic many-valued logic with rational truth values, denoted 1

2 by F P (RP L). Since this logic is crucial in order to understand Hájek s manyvalued approach to probability, we describe it since now. The logic F P (RP L) (Fuzzy Probabilistic Rational Pavelka Logic) has infinitely many propositional variables p 0,..., p n,..., propositional constants q for every rational q [0, 1], the connectives and & of Lukasiewicz Logic, and the modal operator P r. Formulas split into two classes, Boolean formulas and modal formulas, which are inductively defined as follows: (i) The set BF of Boolean formulas is the smallest set containing all propositional variables and closed under the connectives, and. (In Lukasiewicz logic, these connectives are definable by ϕ ψ = ϕ&(ϕ ψ), ϕ ψ = (ϕ ψ) ψ, and ϕ = ϕ 0, but in the definition of Boolean formulas we can use them as primitive symbols). (ii) The set MF of modal formulas is the smallest set containing all formulas of the form P r(ϕ), ϕ is a Boolean formula, as well as all formulas of the form q, q is a rational in [0, 1], and closed under and &. The axioms and rules of F P (RP L) are as follows: (L) All axioms and rules of Lukasiewicz Logic, see [H98], restricted to modal formulas. (C) All axioms of Classical Logic restricted to Boolean formulas. (R) All axioms for rational constants, namely: All axioms of the form r ( p q), (where ϕ ψ stands for (ϕ ψ)&(ψ ϕ)) for all p, q, r Q [0, 1] such that min{1, 1 p+q} = r. All axioms of the form r ( p& q) for all p, q, r Q [0, 1] such that r = max{0, p + q 1}. (Pr) The axiom schemes P r( ϕ ψ) (P r(ϕ) P r(ψ)), P r( ϕ) P r(ϕ) and P r(ϕ ψ) ((P r(ϕ) P r(ϕ ψ)) P r(ϕ)). (N) The Necessitation Rule: ϕ P r(ϕ) for any Boolean formula ϕ. A Boolean evaluation is a map v from the set BF of Boolean formulas into {0, 1} such that v( A) = 1 v(a), v(a B) = min{v(a), v(b)}, v(a B) = max{v(a), v(b)}. The semantics for F P (RP L) is constituted by the class of probabilistic Kripke models. These structures are systems W, µ, e where: W is a non-empty set, whose elements are called nodes, and e is a map from W BF into {0, 1} such that for all w W, e(w, ) is a Boolean evaluation.

3 µ (but possibly not countably additive) probability measure on a Boolean subalgebra of the powerset of W which takes values in [0, 1], such that for every Boolean formula ϕ, the set W ϕ = {w W : e(w, ϕ) = 1} is µ-measurable (i.e., µ(w ϕ ) is defined). Given a probabilistic Kripke model M = W, µ, e and a node w W, the truth value ϕ M,w of a formula ϕ in M at the node w is defined inductively as follows: If ϕ is Boolean, then ϕ M = e(w, ϕ). If ϕ is a propositional constant q, then ϕ M,w = q. P r(ϕ) M,w = µ(w ϕ ). ϕ&ψ M,w = max{0, ϕ M,w + ψ M,w 1}, and ϕ ψ M,w = min{1, 1 ϕ M,w + ψ M,w }. If ϕ is a modal formula, then ϕ M,w is independent of w, so we will omit the subscript w. We say that M is a model of a modal formula ϕ (denoted by M = ϕ) if ϕ M = 1. If Γ is a set of formulas, then we say that M is a model of Γ (denoted by M = Γ) if for all ψ Γ, M = ψ. In [H98] it is proved that F P (RP L) is sound and complete with respect to the class of probabilistic Kripke models. This means the following: as usual, for every logic L, for every set Γ of sentences of L and for every sentence ϕ of L, say that L Γ ϕ iff there is a finite sequence ϕ 1,..., ϕ n of sentences of L such that ϕ n = ϕ, and for i = 1,..., n, either ϕ i is an axiom of L, or ϕ i Γ, or ϕ i can be derived by a rule of L from a finite number of formulas of the form ϕ j with j < i. Also, let Γ = L ϕ L Γ (in a sense which will be specified for every theory we will deal with) is a model of ϕ. Then: Theorem 1.1 Let Γ be a set of modal formulas of F P (RP L), and let ϕ be a modal formula of F P (RP L). Then, F P (RP L) Γ ϕ iff Γ = F P (RP L) ϕ. Thus F P (RP L) is adequate for a treatment of simple probability. Another fundamental contribution to the study of the relationship between probability and many-valued logic is the book chapter [MR], where the Caratheodoryvon Neumann approach to algebraic probability theory is extended to a manyvalued framework. However, this (very deep) investigation is carried on in a purely algebraic setting, and it seems very hard to translate the sophisticated tools involved into a logical language. A crucial concept in probability theory is that of conditional probability. Thus a natural problem related to the connections between probability and manyvalued logic is the investigation of logics which are suitable for the treatment of conditional probability. 3

4 This problem is afforded in [EGH]. The central idea is that product conjunction & Π ( whose truth function is ordinary product in [0, 1]) has a residual Π whose truth function Π behaves like a truncated division, namely: { y x Π y = x if y x 1 otherwise Thus if the truth value of P r(ψ) is non-zero, we can express the conditional probability P r(ϕ ψ) of ϕ given ψ by (the truth value of) P r(ψ) Π P r(ϕ ψ). In [EGH], Esteva, Godo and Hájek introduce a probabilistic logic which is based on LΠ, a logic introduced by Esteva, Godo and Montagna in [EGM] which includes both Lukasiewicz and product conjunctions and their residuals. Roughly speaking, such a probabilistic logic results from LΠ by the adding of a probabilistic modality P r and the axioms and rules of F P (RP L). Following these (very smart) ideas, the authors obtain a logical treatment of conditional probability which works well in all cases, except the one where the conditioning event has probability 0. Indeed, if P r(ψ) has value 0, then the formula P r(ψ) Π P r(ψ ϕ) has always truth-value 1, and this fact does not correctly reflect the mathematical concept of conditional probability. In order to overcome such difficulty, we use a non-stadard approach to probability (cf e.g. [N], [K]). The (not new) idea is the following: we first introduce a non-standard probability P r such that only the impossible event may have probablity 0, but a non-impossible event may have an infinitesimal probability. Then the non-standard conditional probability P r (ϕ ψ) may be expressed as P r (ϕ ψ) P r (ψ), which is the truth value of the formula P r(ψ) Π P r(ψ ϕ). Moreover, we can go back to standard probability (thus allowing for non-impossible events of probability 0) by taking the standard part, that is defining a standard probability P r by P r(ϕ) = St(P r (ϕ)), where St is the standard part function which associates to every non-standard real α [0, 1] the unique standard real β such that the distance α β between α and β is infinitesimal. We illustrate this idea by an example: the unit square [0, 1] has measure 0 in [0, 1] 3. Thus if we choose a point at random in [0, 1] 3 (with uniform distribution), the probability of choosing an element of [0, 1] is 0. Still, it is reasonable to say that the probability of choosing a point in [0, 1 ] given that the point belongs to [0, 1] is 1 4. This can be expressed by means of non-standard probability as follows: think of the unit square as a parallelepiped of dimensions 1, 1 and ε, where ε is a positive infinitesimal. Similarly, [0, 1 ] can be regarded as a parallelepiped with dimensions 1, 1 and ε. Let ϕ denote the event: we choose a point in [0, 1], and let ψ denote the event we choose a point in [0, 1 ]. Then the non-standard probability of ϕ is given by P r (ϕ) = ε, and similarly P r (ψ) = 1 1 ε = ε 4, therefore, taking the standard parts, we get P r(ϕ) = P r(ψ) = 0, and P r(ψ ϕ) = St( ε 4ε ) = St( 1 4 ) = 1 4. The aim of the present paper is to express all these ideas in a logical and algebraic formalism. To this purpose, we introduce a class of algebras which are roughly LΠ 1 -algebras A ( LΠ-algebras added by a constant for 1 ) with a unary 4

5 operation σ which is an idempotent endomorphism of the PMV-reduct of A, i.e. of the reduct of A relative to the Lukasiewicz operations and constants,, 0 and 1, and to product. (σ is supposed to represent the function standard part, but this function is not even first-order axiomatizable, so what we really get is only some approximation of it). The algebras obtained in this way are called S LΠ-algebras. The logic counterpart of such algebras is a logic, S LΠ, which is roughly LΠ 1 with an additional operator S satisfying some axioms which are just the logic translation of the axioms for σ in the definition of S LΠ-algebras. Then we introduce a probabilistic logic F P (S LΠ) whose language is that of F P (RP L) plus product conjunction & Π and product implication Π, and whose axioms and rules are those of S LΠ plus those of F P (RP L). Here, the truth values of P r(ϕ) and P r(ψ) Π P r(ϕ ψ) are supposed to represent the non-standard probability of ϕ and the non-standard conditional probability of ϕ given ψ respectively. The corresponding standard probabilities are represented by S(P r(ϕ)) and by S(P r(ψ) Π P r(ϕ ψ)). The paper is organized as follows: in Section we collect some known results which will be used throughout the paper. In Section 3, we introduce the logic S LΠ and its algebraic semantics, namely, the variety of S LΠ-algebras. We also prove that every S LΠ-algebra is isomorphic to a subdirect product of linearly ordered S LΠ-algebras, and we derive some useful consequences of this result. In Section 4, we prove some kind of non-standard completeness for S LΠ. That is, we prove that S LΠ is complete with respect to interpretations in ultrapowers [0, 1] of the LΠ 1 -algebra [0, 1] LΠ on [0, 1] equipped with an idempotent 1 endomorphism σ of the PMV-reduct of [0, 1]. In Section 5 we introduce the probabilistic variant F P (S LΠ) of S LΠ, and we prove that it is complete with respect to the class of Kripke models in which the probabilistic measure takes values in a non-standard extension [0, 1] of [0, 1 1] LΠ, and S is interpreted as an idempotent endomorphism σ of the PMV- reduct of [0, 1]. Finally, in Section 6 we apply the results of the previous sections to the problem of the coherence of an assessment of conditional probabilities. We prove that probabilistic coherence is equivalent to the logical coherence of a theory over F P (S LΠ) whose proper axioms reflect the probability assessment. Preliminary notions In this section we introduce some algebraic concepts which will be used in order to describe the semantics of the probabilistic logics taken into consideration. Definition.1 (see e.g. [BF00]). A hoop is an algebra H,,, 1 such that H,, 1 is a commutative monoid, and is a binary operation such that the 5

6 following identities hold: x x = 1, x (y z) = (x y) z and x (x y) = y (y x). A Wajsberg hoop is a hoop satisfying the equation (x y) y = (y x) x. A bounded hoop is a hoop equipped by a constant 0 such that the identity 0 x = 1 holds. A MV-algebra is a structure A,,, 0, 1 such that letting x y = ( x y) and x y = x y, the algebra A,,, 0, 1 is a bounded Wajsberg hoop. In the sequel, in any MV-algebra we put x y = ( x y), x y = x (x y), x y = (x y) y, and x y iff x y = 1. Note that is a distributive lattice order, and and are the corresponding operations of join and meet ([CMO]). A typical MV-algebra, which generates the whole variety of MV-algebras ([CMO]), is [0, 1] MV = [0, 1],,, 0, 1, where [0, 1] is the unit real interval, x y = min{x + y, 1}, and x = 1 x. Let denote the restriction of ordinary product to [0, 1], and let for all x, y [0, 1], { y x Π y = x if y x 1 otherwise Then Π is the residuum of, i.e., one has x y z iff x y Π z. Let [0, 1] LΠ = [0, 1],,, Π, 0, 1, and let [0, 1 1] LΠ = [0, 1],,, Π, 0, 1, 1. In [EGM] it is shown that the elements of the variety generated by [0, 1] LΠ, called LΠ-algebras, can be described as follows: Definition. A LΠ-algebra is an algebra A = A,,,, Π, 0, 1 such that, letting Π x = x Π 0, and δ(x) = Π x, the following conditions hold: A,,, 0, 1 is a MV-algebra. A,, Π, 0, 1 is a bounded hoop. For all x, y, z A one has: x (y z) = (x y) (x z). δ(x y) x Π y x y. δ(x) δ(x y) δ(y) δ(x) δ(x) = 1 δ(x) x δ(δ(x)) = δ(x) δ(1) = 1 δ(x y) = δ(x) δ(y) 6

7 x Π x = 0 A LΠ 1 -algebra is a LΠ-algebra with an additional constant 1 satisfying 1 = 1. We recall the following results ([EGM] and [M01]). Theorem.3 (i) The class of LΠ-algebras ( LΠ 1 -algebras respectively) constitutes a variety which is generated by [0, 1] LΠ ([0, 1 1] LΠ respectively). (ii) Every linearly oredered LΠ ( LΠ 1 respectively) algebra can be embedded into an ultrapower of [0, 1] LΠ ([0, 1 1] LΠ respectively). (iii) The subalgebra of [0, 1 1] LΠ whose domain consists of all rationals in [0, 1] can be embedded into any non-trivial LΠ 1 -algebra. We conclude this section with the axiomatization of the logics LΠ and LΠ 1 whose equivalent algebraic semantics (in the sense of [BlP89]) is constituted by the classes of LΠ-algebras and of LΠ 1 -algebras respectively. Definition.4 The language of the logic LΠ has four binary connectives, &,, & Π and Π, and two constants, 0 and 1. We use the following abbreviations: ϕ ψ (ϕ ψ)&(ψ ϕ), ϕ ϕ 0, Π ϕ ϕ Π 0, ϕ ψ ϕ&(ϕ ψ), ϕ ψ (ϕ ψ) ψ, and (ϕ) Π ϕ. Then the axioms of LΠ are: The axioms of Lukasiewicz Logic for &, and 0 (cf [H98]). The axioms of the logic BL (cf [H98]) for & Π and Π. (ϕ& Π ( Π (ψ γ)) ( ((ϕ& Π ψ) (ϕ&γ))). (ϕ ψ) (ϕ Π ψ). (ϕ Π ψ) (ϕ ψ). (ϕ) ( (ϕ ψ) (ψ)). (ϕ) (ϕ). (ϕ) ϕ. (ϕ) ( (ϕ)). (ϕ ψ) ( (ϕ) (ψ)). (ϕ Π ϕ) The rules of LΠ are Modus Ponens: ϕ ϕ ψ ψ and Necessitation: ϕ (ϕ). The logic LΠ 1 is the extension of LΠ by means of an additional constant 1 and the axiom

8 3 The logic S LΠ and its algebraic models Definition 3.1 The logic S LΠ is obtained from LΠ 1 connective S and of the axioms: by the adding of an unary S(ϕ ψ) S(ϕ) S(ψ) for {&,, & Π }. S( 0). S(ϕ) S(S(ϕ)). (ϕ) S(ϕ). Definition 3. A S LΠ-algebra is a LΠ 1 -algebra A equipped with a unary operator σ such that: σ is an endomorphism of the PMV-reduct (i.e., of the {,,, 0, 1}-reduct) of A. σ is idempotent, i.e., for every x A, σ(σ(x)) = σ(x). For all x A, δ(x) σ(x). Definition 3.3 Let A be any S LΠ-algebra. An evaluation of S LΠ into A is a map e from S LΠ-formulas into A such that e( 0) = 0, and for all formulas ϕ and ψ one has: e(ϕ&ψ) = e(ϕ) e(ψ). e(ϕ ψ) = e(ϕ) e(ψ). e(ϕ& Π ψ) = e(ϕ) e(ψ). e(ϕ Π ψ) = e(ϕ) Π e(ψ). e(s(ϕ)) = σ(e(ϕ)). For every formula ϕ of S LΠ, for every S LΠ-algebra A and for every evaluation e of S LΠ into A, we say that A, e is a model of ϕ (and we write A, e = ϕ) iff e(ϕ) = 1. As usual, Γ = S LΠ ϕ means that for every S LΠ-algebra A and for every evaluation e in A, if A, e = γ for all γ Γ, then A, e = ϕ. Since the axioms for S LΠ-algebras are just the algebraic translations of the axioms for S LΠ, it is clear that the Lindenbaum sentence algebra of S LΠ is (termwise equivalent to) the free S LΠ-algebra on countably many generators, therefore S LΠ is sound and strongly complete with respect to the class of all S LΠ-algebras. In other words, for every set Γ of sentences and for every sentence ϕ, one has: Γ ϕ iff Γ = S LΠ S LΠ ϕ. We are going to prove that S LΠ is strongly complete with respect to the class of all linearly ordered S LΠ-algebras. be its underlying LΠ- Lemma 3.4 Let A be any S LΠ-algebra, and let A algebra. Then A and A have the same congruences. 8

9 Proof. In [EGM] it is shown that the congruence lattice of any LΠ-algebra is isomorphic to the lattice of δ-filters (ordered by inclusion), where a δ-filter is a subset F of the algebra such that 1 F, if x F and x y F, then y F, and if x F, then δ(x) F. The isomorphism is defined, for every congruence θ by F θ = {x : (x, 1) θ}, and its inverse is defined for every δ-filter F, by θ F = {(x, y) : x y F and y x F }. Hence in order to prove the claim it is sufficient to prove that for every δ-filter of A, θ F is a congruence of A. We already know that θ F is a congruence of A, therefore we only need to prove that if (x, y) θ F, then (σ(x), σ(y)) θ F. Now if (x, y) θ F, then x y F, therefore δ(x y) F. Since δ(x y) σ(x y), we obtain that σ(x y) F, therefore σ(x) σ(y) F. Similarly we can prove that σ(y) σ(x) F, therefore (σ(x), σ(y)) θ F, and the claim is proved. Theorem 3.5 Every S LΠ-algebra is isomorphic to a subdirect product of a family of linearly ordered S LΠ-algebras. Proof. By the Birkhoff subdirect representation theorem (cf [MMT87]), every S LΠ-algebra is isomorphic to a subdirect product of a family of subdirectly irreducible S LΠ-algebras. Now being subdirectly irreducible only depends on the congruence lattice, therefore by Lemma 3.4, a S LΠ-algebra is subdirectly irreducible iff its underlying LΠ-algebra is subdirectly irreducible. Since any subdirectly irreducible LΠ-algebra is linearly ordered, the claim follows. Corollary 3.6 Let A be any S LΠ-algebra. Then: (i) A satisfies the equation ( Π (σ(x)) Π (σ(y))) (σ(x Π y) ((σ(x) Π σ(y))) = 1, which (if A is linearly ordered) expresses that whenever at least one of σ(x), σ(y) is non-zero, then σ commutes with product implication. (ii) Let A be the LΠ 1 -reduct of A. Then the set F ix A(σ) = {x A : σ(x) = x} is the domain of a LΠ 1 -subalgebra of A (in fact, a S LΠ subalgebra, as σ is the identity on F ix A (σ)). Hence in particular F ix A (σ) contains (an isomorphic copy of) all rationals in [0, 1]. (iii) For all a A, σ( a σ(a) ) = 0, where x y stands for (x y) (y x). Proof. (i). It is sufficient to prove that the claim holds in any linearly ordered S LΠ-algebra A. Let x, y A. If σ(x) = σ(y) = 0, then Π (σ(x)) Π (σ(y)) = 1, and the claim follows. Otherwise, we distinguish the following cases: (a) If x y, then since σ is order-preserving, σ(x Π y) = σ(x) Π σ(y) = 1, and once again the claim follows. (b) Suppose y < x. Let z = x Π y. Then z x = y, therefore σ(z) σ(x) = σ(y), since σ preserves. It cannot be σ(x) = 0, because otherwise σ(y) = 0, and we have excluded this possibility. Then σ(z) σ(x) = σ(y), and for u > σ(z), u σ(x) > σ(y) (because product is strictly increasing if the factors are nonzero). This says that σ(z) = σ(x Π y) is the residual of σ(x) and σ(y), i.e., σ(x Π y) = σ(x) Π σ(y), as desired. 9

10 (ii) Since σ is compatible with and with all Lukasiewicz operators, F ix A (σ) is closed under those operations. Moreover, σ( 1 ) = σ( 1 ) = σ( 1 ), which implies that σ( 1 ) = 1, and 1 F ix A(σ). So it remains to prove that it is closed under Π. Let x, y F ix A (σ). If σ(x) = σ(y) = 0, then x Π y = σ(x) Π σ(y) = 1 F ix A (σ). Otherwise, by (i), σ(x Π y) = σ(x) Π σ(y) = x Π y, and x Π y F ix A (σ). (iii) One has: σ( x σ(x) ) = σ(x) σ(σ(x)) = σ(x) σ(x) = 0. Another consequence of Theorem 3.5 is the strong completeness theorem for S LΠ with respect to the class of linearly ordered S LΠ-algebras. Theorem 3.7 Let Γ be any set of S LΠ-sentences, and let ϕ be any S LΠsentence. If Γ S LΠ ϕ, then there are a linearly ordered S LΠ-algebra B and an evaluation v on B such that v(ϕ) 1 and v(ψ) = 1 for all ψ Γ. Proof. If Γ S LΠ ϕ, then by the algebraic strong completeness of S LΠ, there are an S LΠ-algebra A and an evaluation e in A such that e(ϕ) 1 and e(ψ) = 1 for all ψ Γ. By Theorem 3.5, we can decompose A as a subdirect product of a family of linearly ordered S LΠ-algebras A i : i I. Since for every i I the projection π i on A i is a homomorphism, it preserves 1, therefore π i (e(ψ)) = 1 for all ψ Γ. Moreover, since e(ϕ) 1, there is an i 0 I such that π i0 (e(ϕ)) 1. Now let B = A i0, and let v = π i0 e be the composition of π i0 and e. Then v is an evaluation in the linearly ordered S LΠ-algebra B = A i0 such that v(ϕ) 1 and v(ψ) = 1 for all ψ Γ. 4 Non-standard completeness of S LΠ A general task of many-valued logic is proving standard completeness, i.e., proving the completeness of a many-valued logic with respect to algebras with truth values in the standard interval [0, 1]. For S LΠ, this is not possible: the formula S(ϕ) ϕ is valid in any S LΠ-algebra whose LΠ 1 -reduct is [0, 1] LΠ, because 1 the only endomorphism of the PMV-reduct of [0, 1 1] LΠ is the identity, but this formula is not provable in S LΠ (as a counterexample, take a non-trivial ultrapower [0, 1] of [0, 1 1] LΠ, and interpret σ as the standard part function which associates to every element α [0, 1] the unique real at infinitesimal distance from α). This is the reason why, instead of the standard one, we prove a form of non-standard completeness, i.e, the (strong) completeness of S LΠ with respect to the class of ultrapowers [0, 1] of [0, 1] LΠ with σ interpreted as a suitable idempotent endomorphism of [0, 1],. To begin with, we recall the following definition and the following result from [M00]. Definition 4.1 Let F be any linearly ordered field. By Π(F) we denote the algebra [0, 1],,,, Π, 0, 1, 1, where [0, 1] = {x F : 0 x 1}, x y = 10

11 min{x + y, 1}, x = 1 x, is the restriction of product in F to [0, 1], 1 is the unique solution in F of the equation x + x = 1, and Π is defined by: { x x Π y = 1 y if x > y 1 otherwise Theorem 4. ([M00]) (i) For every ordered field F, Π(F) is a linearly ordered LΠ 1 -algebra. (ii) For every linearly ordered LΠ 1 -algebra A, there is an (unique up to isomorphism) ordered field F such that A = Π(F). Definition 4.3 Let F be an ordered field. Then F fin denotes the ordered subring of F constituted by all x F such that x n for some n N (where x stands for max{x, x}). A convex ideal of a commutative ordered ring G is a ring ideal J of G such that if a J and b a, then b J. Given a convex ideal J of a commutative ordered ring G, J denotes the set {x G : a J, n N : x n a }. The next lemma is easy to demonstrate, and its proof is left to the reader: Lemma 4.4 If G is a commutative ordered ring and J is a convex ideal of G, then J is in turn a convex ideal of G, and the quotient ring G/ J is an ordered integral domain. Lemma 4.5 Let A = A,,,, Π, σ, 0, 1, 1 be a linearly ordered S LΠ-algebra, let A be its LΠ 1 -reduct, let F be an ordered field such that A = Π(F), and let R be the real closure of F. Then there is an operator σ on Π(R) such that: σ extends σ. σ makes Π(R) a S LΠ-algebra, and A is a subalgebra of it. Proof. First note that σ can be (uniquely) extended to an idempotent orderpreserving endomorphism σ + of the ordered ring F fin as follows: every element a F fin can be uniquely represented as a = z + b, where z Z, and 0 b < 1 (hence b A). Then it is sufficient to define σ + (z + b) = z + σ(b). Also, note that F fin is an ordered integral domain, and that F is its fraction field. Now let F ix(f fin ) = {x F fin : σ + (x) = x}, and let J = {x F fin : σ + (x) = 0}. It is readily seen that F ix(f fin ) is (the domain of) an ordered subring of F fin, and that J is a convex ideal of F fin. Claim A. F ix(f fin ) is isomorphic to F fin /J under the natural homomorphism ρ: x ρ x/j. Proof of Claim A. That ρ is a homomorphism is clear. We prove that ρ is oneone. If x/j = y/j, then x y J, therefore σ + (x y) = 0, σ + (x) = σ + (y), and finally x = y, as x, y F ix(f fin ). We prove that ρ is onto. For all x F fin, we have σ + ( x σ + (x) ) = σ + (x) σ + (σ + (x)) = 0, therefore x σ + (x) J, 11

12 x/j = σ + (x)/j, ρ(x) = ρ(σ + (x)), and σ + (x) F ix(f fin ). Hence ρ is onto, and the proof of Claim A is complete. Note that since the restriction of ρ to F ix(f fin ) is one-one, we may identify modulo isomorphism F ix(f fin ), F ix(f fin )/J and F fin /J. Now let I denote the ideal of R fin generated by J, and let J + = I. Claim B. J + F fin = J. Hence F fin /J and F fin /J + are isomorphic under the isomorphism γ: x/j γ x/j +. Proof of Claim B. Let a J + F fin. Then for some b J and for some n N, a n b. It follows that σ + (a n ) = σ + (a) n σ + (b) = 0, as b J. Since a F fin and F fin is an integral domain, σ + (a) = 0, and a J. For the last part of Claim B, we only need to verify that γ is one-one. Now if x, y F fin and x/j + = y/j +, then x y F fin J + = J, hence x/j = y/j. This concludes the proof of Claim B. We continue the proof of Lemma 4.5. By claims A and B above we may identify modulo isomorphism F fin /J + with F ix(f fin ) and with F ix(f fin )/J +. Now consider any α R fin. Then α is algebraic over F, as R is an algebraic extension of F. Let P (X) = n i=0 a ix i be the minimal polynomial of α over F. After multiplying all coefficients a i of P (X) by a suitable constant, we may assume without loss of generality that all the a i are in F fin. Next consider F(α), the algebraic extension of F by α. Let P s (X) be the polynomial obtained from P (X) by replacing each coefficient a i of P (X) by σ + (a i ). Then recalling that α R fin, we can find a natural number k such that P s (α) = P s (α) P (α) k n i=0 a i σ + (a i ). Since for every i, a i σ + (a i ) J, P s (α) J +. Hence P s (α)/j + = 0. Since all coefficients of P s are in F ix(f fin ), and since we may identify F ix(f fin ) with F ix(f fin )/J +, we can conclude that α/j + is algebraic over F ix(f fin )/J + = F ix(f fin ). By the arbitrariness of α, we can conclude that (the fraction field of) R fin /J + is isomorphic to a linearly ordered algebraic extension of (the fraction field of) F ix(f fin ). Hence it is isomorphic to a linearly ordered subfield K of the real closure R of F. Thus we may assume up to isomorphism that R fin /J + R. Hence the natural homomorphism τ defined for all x R fin by τ(x) = x/j + is an endomorphism of R fin, and after identifying (by Claims A and B) F ix(f fin ) with F fin /J +, we have that τ extends σ +. In order to conclude the proof, let σ be the restriction of τ to Π(R). Then σ is an endomorphism of the PMV-reduct of Π(R) which extends σ, and Lemma 4.5 is proved. Theorem 4.6 (Non-standard completeness of S LΠ). Let Γ be a set of sentences and ϕ be a sentence such that Γ S LΠ ϕ. Then there are an ultrapower [0, 1] of [0, 1 1] LΠ, a S LΠ algebra [0, 1] σ whose LΠ 1 -reduct is [0, 1] and an evaluation e in [0, 1] σ such that e(ψ) = 1 for all ψ Γ, and e(ϕ) 1. 1

13 Proof. By Theorem 3.7, if Γ S LΠ ϕ, then there are a linearly ordered S LΠalgebra A and an evaluation e on A such that e (ψ) = 1 for all ψ Γ, and e (ϕ) 1. By Lemma 4.5, we may embed A into an algebra B whose LΠ 1 - reduct is of the form Π(R) for a suitable real closed field R. Since the theory of real closed fields is complete, R is elementarily equivalent to the real field R. Hence R and R have isomorphic ultrapowers, therefore Π(R) and Π(R) have isomorphic ultrapowers. It follows that there is an elementary embedding of the LΠ 1 -reduct of B into an ultrapower [0, 1] of [0, 1] LΠ. Moreover we can define σ on [0, 1] as follows: let B I /U, where I is the index set and U is an ultrafilter on I, be an ultrapower of B isomorphic to [0, 1] under the isomorphism ξ say. Then we may define σ (α) as follows: let ξ 1 (α) = (α i : i I)/U, and let σ B denote the realization of σ in B. Then we define σ (α) = ξ((σ B (α i ) : i I)/U). It is readily seen that this definition is correctly given, i.e., it does not depend on the choice of (α i : i I) in the equivalence class (α i : i I)/U. Now let [0, 1] σ denote the resulting algebra. It is clear that with this definition ξ preserves σ, hence it is an isomorphism of S LΠ-algebras. It follows that A embeds into [0, 1] σ. Let Ψ be such an embedding. Then [0, 1] σ and the evaluation e defined for every formula γ by e(γ) = Ψ(e (γ)) meet our requirements. This concludes the proof. 5 A probabilistic logic over S LΠ In this section we introduce a probabilistic logic which allows us to treat conditional probability in a logical setting. This logic will be used in the next section for a logic treatment of coherent assessments of conditional probability. Definition 5.1 The logic F P (S LΠ) is defined as follows: (a) The symbols of F P (S LΠ) are those of S LΠ plus the modal operator P r. (b) The formulas of F P (S LΠ) split into two classes, the class BF Π of Boolean formulas, defined exactly as in the case of F P (RP L), and the class of modal formulas, which is the minimal class MF Π such that: 0 MF Π and 1 MF Π. For every Boolean formula ϕ, P r(ϕ) MF Π. If ϕ, ψ MF Π and {&,, & Π, Π }, then ϕ ψ MF Π. If ϕ MF Π, then S(ϕ) MF Π. (c) The axioms and rules of F P (S LΠ) are those of S LΠ (restricted to modal formulas), plus the axiom schemes (C) and (Pr) and the rule (N) as in the definition of F P (RP L). Note that in F P (S LΠ) we have variable-free formulas representing rationals in [0, 1], and modulo this representation, all axioms of the form (R) are derivable in F P (S LΠ). Definition 5. A Kripke model for F P (S LΠ) is a system M = [0, 1], W, µ, e, where: 13

14 equipped with an idempotent endomor- [0, 1] is an ultrapower of [0, 1 1] LΠ phism σ of its P MV -reduct. W is a non-empty set, and e is a map from W BF Π such that for every w W, λϕ.e(w, ϕ) is a Boolean evaluation. µ is an additive probability measure on a Boolean algebra of subsets of W which takes values in [0, 1] such that for every Boolean formula ϕ the set W ϕ = {w W : e(w, ϕ) = 1} is µ-measurable. For every formula ψ and every node w W, we can define ψ M,w in analogy with the case of Kripke models for F P (RP L) and with the following additional clauses: ψ& Π γ M,w = ψ M,w γ M,w. ψ Π γ M,w = ψ M,w Π γ M,w. S(ψ) M,w = σ( ψ M,w ). Once again, if ψ is a modal formula, then ψ M,w is independent of w, therefore we will omit the subscript w. Moreover, we write M = ψ to mean that ψ M = 1. If Γ MF Π and ψ MF Π, we write Γ = F P (S LΠ) ψ to mean that for every probabilistic Kripke model M for F P (S LΠ) if for all γ Γ, M = γ, then M = ψ. Theorem 5.3 F P (S LΠ) is sound and strongly complete with respect to the class of all Kripke models of F P (S LΠ). In other words, if Γ MF Π and ψ MF Π, then Γ = ψ iff Γ F P (S LΠ) F P (S LΠ) ψ. Proof. The right-to-left direction is straightforward. For the other direction, the argument is analogous to the argument for F P (RP L) in [H98]. We sketch the parts where the proofs are quite parallel, and we add more details in the parts where the two proofs diverge. Let for every modal formula χ, χ be obtained from χ by replacing every occurrence of a subformula of the form P r(ϕ) by a new variable p ϕ (the same variable for different occurrences of the same formula P r(ϕ), and different variables for different ϕ). Let for every set X MF Π, X = {χ : χ X}. Then let T be the set of all axioms of F P (S LΠ) of the form (Pr), and let Γ be T Γ plus all formulas of the form p ϕ, ϕ a Boolean tautology. Then, along the lines of [H98], it is easy to prove that for every modal formula ψ, F P (S LΠ) Γ ψ iff S LΠ Γ ψ. Hence if Γ ψ, then Γ ψ, there are a S LΠ-algebra A and an evaluation v in A such that for all γ Γ, v(γ ) = 1, and v(ψ ) 1. By the non-standard completeness of S LΠ, we may assume that A is an ultrapower of [0, 1 1] LΠ equipped with an idempotent endomorphism σ of its P MV -reduct. Now let W be the set of all evaluations of Boolean formulas of F P (S LΠ). Define, for every Boolean formula ϕ and for every w W, e(w, ϕ) = w(ϕ). As regards to µ, its domain is the family of all subsets of W of the form W ϕ, where ϕ 14

15 ranges over all Boolean formulas. Moreover, for every Boolean formula ϕ we define µ(w ϕ ) = v(p ϕ ). Then M = [0, 1], W, µ, e is a probabilistic Kripke model for F P (S LΠ). Moreover, by induction on ψ we can easily prove that for every modal formula ψ one has: ψ M = v(ψ ). Hence for all γ Γ we have γ M = 1, and ψ M 1. This completes the proof. 6 Applications to the coherence problem An important problem in probability theory is the coherence of a probabilistic assessment. We recall that an assessment P (ϕ i ) = α i : i = 1,..., n is coherent iff there is a probability measure θ on the space of events such that for i = 1,..., n one has: θ(ϕ i ) = P (ϕ i ). Similarly, an assessment of conditional probabilities P (ϕ i γ i ) = α i : i = 1,..., n is coherent if there is a conditional probability ρ (i.e., a function from the set of pairs (ψ, ψ ) of events with ψ ) into [0, 1] which satisfies all axioms of conditional probability as given e.g. in [CS]) such that for i = 1,..., n one has: ρ(ϕ i γ i ) = P (ϕ i γ i ). An immediate consequence of the completeness of F P (RP L) is a logical characterization of coherence of an assessment of simple probabilities when the probabilistic values are rationals. Theorem 6.1 Let χ : P (ϕ i ) = α i : i = 1,..., n be an assessment of simple probabilities, where the α i are rationals in [0, 1], and let T χ = F P (RP L) {P r(ϕ i ) α i : i = 1,..., n}. Then χ is is a coherent assessment iff T χ is a coherent logic (i.e.,if T χ 0). Proof. By the strong completeness of F P (RP L), T χ is coherent iff there is a probabilistic Kripke model M such that for every axiom ψ of T χ one has ψ M = 1. Hence, in order to prove the claim it is sufficient to prove that this is the case iff χ is coherent. Suppose that χ is coherent. Let B be the Boolean algebra of all events modulo probable equivalence, and let us identify, by abuse of language, any event with its equivalence class. Then there is a probability measure θ on B that for i = 1,..., n, θ(ϕ i ) = α i. We obtain a probabilistic Kripke model letting W be the set of all Boolean evaluations, and letting for every Boolean formula ϕ and for every w W, e(w, ϕ) = w(ϕ) and µ(w ϕ ) = θ(ϕ). Then since θ(ϕ i ) = α i, we have P r(ϕ i ) M = α i. Therefore, the proper axioms of T χ, P r(ϕ i ) α i : i = 1,..., n, are evaluated to 1, and M is a model of T χ. Conversely, if M = W, µ, e is a model of T χ, then letting for every Boolean formula ϕ, θ(ϕ) = µ(w ϕ ), we obtain a probability measure θ which extends χ, therefore χ is coherent. We now turn our attention to the coherence problem for conditional probability. We will use a characterization of coherence which is contained in [CS] and one contained in [K]. Let κ = P (ϕ j γ j ) = α j : j = 1,..., n be an assessment of conditional probabilities, let B be the Boolean algebra generated by {ϕ i, γ i : 15

16 i = 1,..., n}, (here we identify any event with its equivalence class modulo provable equivalence), let and Ω denote the minimum and the maximum of B respectively, and let A denote the set of atoms of B. Then, rephrasing a characterization of coherence given in [CS], see also [C], [C3] and [CS], we obtain: Theorem 6. κ is coherent iff there are a finite sequence τ = A 0 A 1... A h of non-empty subsets of A, and numbers x i a : i = 0,..., h, a A i, such that the following set C τ of conditions is satisfied: (i) 0 x i a 1, and x i a = 0 iff i < h and a A i+1. (ii) For i = 0,..., h, a A i x i a = 1. (iii) Let for every atom a, l(a) = max{m h : a A m }, and let for every Moreover: γ B, l(γ) = min{l(a) : a A; a γ}. Then, α j = Pa A l(γj ) :a γ j ϕ j xl(γ j ) a Pa A l(γj ) :a γ j xl(γ j ) a Theorem 6.3 (See [K]). Let κ, B, Ω,, etc. be as above. The following are equivalent: (i) κ is coherent. (ii) There is a non-standard probability measure on B (i.e., a map µ from B into a non-standard extension [0, 1] of [0, 1] such that µ (Ω) = 1 and if ϕ, ψ are mutually uncompatible, then µ (ϕ ψ) = µ (ϕ) + µ (ψ)) such that: If γ B, then µ (γ) 0. For j = 1,..., n one has St standard part function. ( µ (ϕ j γ j) µ (γ j) ) = α j, where St denotes the As a consequence of Theorems 6. and 6.3 and of the non-standard completeness of F P (S LΠ), we obtain: Theorem 6.4 With reference to Theorems 6. and 6.3, if in addition κ takes values α i Q, then, κ is coherent iff the theory T κ constituted by F P (S LΠ) plus all axioms of the form π π P r(ϕ) with ϕ B \ { } plus the axioms (S(P r(γ j ) Π P r(ϕ j γ j ))) α j : j = 1,..., n is coherent. Proof.. Suppose that κ is coherent. Let µ be a non-standard probability measure whose existence is granted by Theorem 6.3, and let us extend it to a non-standard probability measure defined on all formulas. (For instance, we may let for every propositional variable p not in B, µ (p) = 1, and then we may extend µ to all formulas in the obvious way). Then we obtain a probabilistic Kripke model of T κ as follows: let A be [0, 1] with the non-standard extensions 16

17 of the operations of LΠ 1 -algebras, and with σ interpreted as the standard part function St. Let W be the set of all Boolean evaluations, and let e be defined for every w W and for every Boolean formula ϕ, by e(w, ϕ) = w(ϕ). Finally let for every Boolean formula ϕ, ρ(w ϕ ) = µ (ϕ). Then A, W, e, ρ is a probabilistic Kripke model for F P (S LΠ). Moreover, since for ϕ B \ { }, µ (ϕ) 0, all axioms of the form ( π π P r(ϕ) ) with ϕ B \ {0} are satisfied. Similarly, since µ for j = 1,..., n, St (ϕ j γ j) = α j and since σ is interpreted as St, all axioms µ (γ j) of the form (S(P r(γ j ) Π P r(ϕ j γ j ))) α j are also satisfied.. Suppose that T κ is coherent. Then there is a probabilistic Kripke model M = [0, 1], W, e, µ for F P (S LΠ) such that every axiom of T κ has truthvalue 1 in M. We define a sequence τ: A = A 0 A 1... A h of non-empty subsets of A and sequences x i a : i = 0,..., h, a A i of elements of [0, 1] by induction as follows: A 0 = A, and x 0 a = σ ( ) P r(a) M. Assume that we have defined A 0,... A r and for i = 0,..., r and for a A i, the numbers x i a [0, 1]. If for all a A r one has x r a 0, then the construction stops, h = r, and τ is the sequence A 0,..., A r. Otherwise, let A r+1 = {a A r : x r a = 0}, let Φ r+1 be the disjunction ) of all atoms in A r+1, and let for a A r+1, x r+1 a = σ( P r(a) M P r(φ r+1) M = σ( P r(φ r+1 ) Π P r(a) M ). Clearly A 0 A 1 A..., and since A 0 is finite, the construction stops after finitely many steps. Let A h be the last set constructed. For i h and for a A i one has 0 x i a 1, and by construction, x i a = 0 iff i < h and a A i+1. Finally, by induction on i, we can see that for i h one has a A i x i a = 1. Now let for a A, l(a) = max{i : a A i }, and let for every ϕ B \ { }, l(ϕ) = min{l(a) : a A; a ϕ}. Claim C. For j = 1,..., n we have: Pa A l(γj ) ;a ϕ j γ j xl(γ j ) a Pa A l(γj ) ;a γ j xl(γ j ) a = α j. Proof of Claim C. Since M is a model of T κ, for j = 1,..., n, we have: ( P r(ϕj γ j ) ) M σ( P r(γ j ) Π P r(ϕ j γ j ) M ) = σ = α j. P r(γ j ) M Moreover, P r(ϕ j γ j ) M = a A;a ϕ j γ j P r(a) M, and P r(γ j ) M = P r(a) M. It follows: a A;a γ j α j = σ P r(a) M ( a A;a ϕ j γ j P r(φ l(γj )) M P r(a) M a A;a γ j P r(φ l(γj )) M Now from the definition of l(γ j ) it follows that there is an a A l(γj) such that ) σ( P r(a) M P r(φ l(γj ) M = x l(γj) a 0: Therefore by Corollary 3.6 (i) we obtain: α j = ) ( ) a A;a ϕ j γ j σ P r(a) M P r(φ l(γj )) M ( P r(a) M a A;a γ j σ P r(φ l(γj )) M ).. 17

18 ) Now if a γ j and l(a) > l(γ j ), then σ( P r(a) M P r(φ l(γj )) M = 0. Furthermore, if ) a A l(γj), then σ( P r(a) M P r(φ l(γj )) M = x l(γj) a. It follows: a A l(γj );a ϕ j γ j x l(γj) a a A l(γj );a γ j x l(γj) a = α j, as desired. This completes the proof of Claim C. We conclude the proof of Theorem 6.4. So far, we have shown that if T κ is coherent, then there are a sequence τ: A = A 0 A 1... A h and numbers x i a : i h; a A i satisfying the hypotheses of Theorem 6. with the only exception that the x i a are in [0, 1] and not necessarily in [0, 1]. However, we can reason as follows: the existence of a sequence τ and of numbers x i a as shown above can be expressed as a disjunction over all sequences τ: A = A 0 A 1... A h (finitely many!) of the existential formulas saying that the corresponding sets of conditions C τ have a solution. Hence this fact can by means of a first-order formula Ψ in the language of LΠ 1 -algebras. Since the LΠ 1 -reduct of [0, 1] and [0, 1 1] LΠ are elementarily equivalent, Ψ, being true in [0, 1], is true in [0, 1 1] LΠ, and by Theorem 6., κ is coherent. 7 Conclusions We have shown that conditional probability can be treated by logical means, even in the case where the conditioning event has probability 0. To reach this goal, we have introduced a many-valued logic S LΠ which reflects the properties of a non-standard extension [0, 1] of the real interval [0, 1], equipped with an idempotent endomorphism σ, which is supposed to describe the standard part function on [0, 1]. Then we have added a probabilistic modality to this logic, in the style of [H98], thus obtaining the fuzzy probabilistic logic F P (S LΠ). Even though it is impossible to obtain logical propositional axioms which force σ to be the standard part function, our logic F P (S LΠ) captures enough properties of such function, in the sense that in F P (S LΠ) we are able to treat conditional probability by means of non-standard simple probability, and to find a logical equivalent of coherence of an assessment of conditional probabilities. As far as we can see, our results do not provide for good algorithms for testing coherence of conditional probability assessments. However, we believe that this paper has a theoretical interest, in that it closely relates logical and probabilistic concepts like probabilistic coherence and logical coherence. References [BKW77] A. Bigard, K. Keimel and S. Wolfenstein, Groupes at anneaux reticulés, Lecture Notes in Mathematics, 608, Springer Verlag, Berlin

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