Generalized logical operations among conditional events

Noname manuscript No. (will be inserted by the editor) Generalized Logical Operations among Conditional Events ⋆ arXiv:1804.10447v1 [math.PR] 27 Apr 2018 Angelo Gilio · Giuseppe Sanfilippo Received: date / Accepted: date Abstract We generalize, by a progressive procedure, the notions of conjunction and disjunction of two conditional events to the case of n conditional events. In our coherence-based approach, conjunctions and disjunctions are suitable conditional random quantities. We define the notion of negation, by verifying De Morgan’s Laws. We also show that conjunction and disjunction satisfy the associative and commutative properties, and a monotonicity property. Then, we give some results on coherence of prevision assessments for some families of compounded conditionals; in particular we examine the FréchetHoeffding bounds. Moreover, we study the reverse probabilistic inference from the conjunction Cn+1 of n+1 conditional events to the family {Cn , En+1 |Hn+1 }. We consider the relation with the notion of quasi-conjunction and we examine in detail the coherence of the prevision assessments related with the conjunction of three conditional events. Based on conjunction, we also give a characterization of p-consistency and of p-entailment, with applications to several inference rules in probabilistic nonmonotonic reasoning. Finally, we examine some non p-valid inference rules; then, we illustrate by an example two methods which allow to suitably modify non p-valid inference rules in order to get inferences which are p-valid. Keywords Conditional events · Conditional random quantities · Conjunction · Disjunction · Negation · Fréchet-Hoeffding bounds · Coherent prevision ⋆ This paper is a substantially extended version of [30]. A. Gilio Department SBAI, University of Rome “La Sapienza”, Italy E-mail: angelo.gilio@sbai.uniroma1.it Retired G. Sanfilippo Department of Mathematics and Computer Science, University of Palermo, Italy E-mail: giuseppe.sanfilippo@unipa.it Both authors contributed equally to this work. 2 Angelo Gilio, Giuseppe Sanfilippo assessments · Coherent extensions · Quasi conjunction · Probabilistic reasoning · p-entailment · Inference rules. Mathematics Subject Classification (2000) MSC 60A05 · MSC 03B48 · MSC 68T37 1 Introduction The research on combining logic and probability has a long history (see, e.g., [2, 8, 13, 16, 33]). In this paper we use a coherence-based approach to probability, which allows to introduce probability assessments on arbitrary families of conditional events, by properly managing conditioning events of zero probability (see, e.g., [4, 5, 13, 18, 19, 28, 29, 21, 42]). In probability theory and in probability logic a relevant problem, largely discussed by many authors (see, e.g., [3, 14, 15, 32]), is that of suitably defining logical operations among conditional events. In a pioneering paper, written in 1935 by de Finetti ([16]), it was proposed a three-valued logic for conditional events coinciding with that one of Lukasiewicz. A survey of the many contributions by different authors (such as Adams, Belnap, Calabrese, de Finetti, Dubois, van Fraassen, McGee, Goodmann, Lewis, Nguyen, Prade, Schay) to research on three-valued logics and compounds of conditionals has been given in [39]; conditionals have also been extensively studied in [17, 38]. In the literature, the conjunction and disjunction have been usually defined as suitable conditionals; see e.g. [2, 9, 11, 32]. A theory for the compounds of conditionals has been proposed in [38, 34]. A related theory has been developed in the setting of coherence in [25, 26, 29]; in these papers, conjunction and disjunction of two conditional events are not defined as conditional events, but as suitable conditional random quantities, with values in the interval [0, 1]. In the present paper we generalize the notions of conjunction and disjunction of two conditional events to the case of n conditional events; we also give the notion of negation. Then, we examine a monotonicity property for conjunction and disjunction. Moreover, we give some results on coherence of prevision assessments for some families of compounded conditionals; in particular we examine the Fréchet-Hoeffding bounds. Finally, we examine in detail the coherence of prevision assessments related with the conjunction of three conditional events. The paper is organized as described below. In Section 2 we recall some preliminary notions and results which concern coherence, quasi conjunction, conjunction, disjunction, and Fréchet-Hoeffding bounds. In Section 3 we introduce, in a progressive way, the notions of conjunction and disjunction for n conditional events; then, we define the notion of negation and we show that De Morgan’s Laws are satisfied. We define the notion of conjunction (resp., disjunction) for the conjunctions (resp., disjunctions) associated with two families of conditional events, by showing then the validity of commutative and associative properties. In Section 4, after a preliminary result concerning the inequality X|H ≤ Y |K between two conditional random quantities, we show that the conjunction Cn+1 of n + 1 conditional events is a conditional random quantity less than Generalized Logical Operations among Conditional Events 3 or equal to any conjunction Cn of a subfamily of n conditional events. Likewise, we show that the disjunction Dn+1 of n + 1 conditional events is greater than or equal to any disjunction Dn of a subfamily of n conditional events. We also show that Cn and Dn belong to the interval [0, 1]. Moreover, we derive some inequalities from the monotony property. In Section 5, based on a geometrical approach, we characterize by an iterative procedure the set of coherent assessments on the family {Cn , En+1 |Hn+1 , Cn+1 }. In Section 6 we study the (reverse) inference from Cn+1 to {Cn , En+1 |Hn+1 }, by determining the set of coherent extensions (µn , xn+1 ) of any coherent assessment µn+1 , where µn = P(Cn ), xn+1 = P (En+1 |Hn+1 ), and µn+1 = P(Cn+1 ). In Section 7 we show that the prevision of the conjunction Cn satisfies the Fréchet-Hoeffding bounds. Then, by exploiting De Morgan’s Laws, we give the dual result for the disjunction Dn . In Section 8 we examine in detail the conjunction for a family of three conditional events E1 |H1 , E2 |H2 , E3 |H3 . We also consider the relation with the notion of quasi-conjunction studied in [2]; see also [27, 28]. We also determine the set of coherent prevision assessments on the whole family {E1 |H1 , E2 |H2 , E3 |H3 , (E1 |H1 )∧(E2 |H2 ), (E1 |H1 )∧(E3 |H3 ), (E2 |H2 )∧ (E3 |H3 ), (E1 |H1 ) ∧ (E2 |H2 ) ∧ (E3 |H3 )}. Moreover, we consider the particular case where H1 = H2 = H3 = H. In Section 9, by applying our notion of conjunction, we give a characterization of p-consistency and p-entailment and we examine some p-valid inference rules in probabilistic nonmonotonic reasoning. Moreover, based on a suitable notion of iterated conditioning, we briefly describe a characterization of p-entailment in the case of two premises. In Section 10, after examining some non p-valid inference rules, we illustrate two methods which allow to construct p-valid inferences. Finally, in Section 11 we give a summary of results. Notice that for almost all (new) results of this paper the proofs are given in Appendix A. 2 Some Preliminaries In this section we recall some basic notions and results on coherence (see, e.g., [5, 7, 10, 13, 40]). In our approach an event A represents an uncertain fact described by a (non ambiguous) logical proposition; hence A is a two-valued logical entity which can be true, or false. The indicator of A, denoted by the same symbol, is 1, or 0, according to whether A is true, or false. The sure event is denoted by Ω and the impossible event is denoted by ∅. Moreover, we denote by A ∧ B, or simply AB, (resp., A ∨ B) the logical conjunction (resp., logical disjunction). The negation of A is denoted A. Given any events A and B, we simply write A ⊆ B to denote that A logically implies B, that is AB is the impossible event ∅. We recall that n events are logically independent when the number m of constituents, or possible worlds, generated by them is 2n (in general m ≤ 2n ). 4 Angelo Gilio, Giuseppe Sanfilippo 2.1 Conditional events and coherent probability assessments Given two events E, H, with H 6= ∅, the conditional event E|H is defined as a three-valued logical entity which is true, or false, or void, according to whether EH is true, or EH is true, or H is true, respectively. We recall that, agreeing to the betting metaphor, if you assess P (E|H) = p, then, for every real number s, you are willing to pay an amount ps and to receive s, or 0, or ps, according to whether EH is true, or EH is true, or H is true (bet called off), respectively. Then, the random gain associated with the assessment P (E|H) = p is G = sHE + psH − ps = sH(E − p). Given a real function P : K → R, where K is an arbitrary family of conditional events, let us consider a subfamily F = {E1 |H1 , . . . , En |Hn } of K and the vector P = (p1 , . . . , pn ), where pi = P (Ei |Hi ) , i ∈ Jn = {1, . . . , n}. We denote by Hn the disjunction H1 ∨ · V · · ∨ Hn . As Ei Hi ∨ E i Hi ∨ H i = Ω , i ∈ Jn , by expanding the expression i∈Jn (Ei Hi ∨ E i Hi ∨ H i ) we can represent Ω as the disjunction of 3n logical conjunctions, some of which may be impossible. The remaining ones are the constituents generated by F and, of course, are a partition of Ω. We denote by C1 , . . . , Cm the constituents which logically imply Hn and (if Hn 6= Ω) by C0 the remaining constituent Hn = H 1 · · · H n , so that Hn = C 1 ∨ · · · ∨ C m , Ω = H n ∨ Hn = C0 ∨ C1 ∨ · · · ∨ Cm , m + 1 ≤ 3n . In the context P of betting scheme, with the pair (F, P) we associate the random gain G = i∈Jn si Hi (Ei − pi ), where s1 , . . . , sn are n arbitrary real numbers. We P that you receive, P observe that G is the difference between the amount i∈Jn si pi , and repi∈Jn si (Ei Hi + pi H i ), and the amount that you pay, resents the net gain from engaging each transaction Hi (Ei − pi ), the scaling and meaning (buy or sell) of the transaction being specified by the magnitude and the sign of si , respectively. Let gh be the value of G when Ch is true; then G ∈ {g0 , g1 , . . . , gm }. Of course, g0 = 0. We denote by GHn the set of values of G restricted to Hn , that is GHn = {g1 , . . . , gm }. Then, based on the betting scheme of de Finetti, we have Definition 1 The function P defined on K is said to be coherent if and only if, for every integer n, for every finite subfamily F of K and for every real numbers s1 , . . . , sn , one has: min GHn ≤ 0 ≤ max GHn . Notice that the condition min GHn ≤ 0 ≤ max GHn can be written in two equivalent ways: min GHn ≤ 0, or max GHn ≥ 0. As shown by Definition 1, a probability assessment is coherent if and only if, in any finite combination of n bets, it does not happen that the values g1 , . . . , gm are all positive, or all negative (no Dutch Book). 2.2 Coherent conditional prevision assessments Given a prevision function P defined on an arbitrary family K of finite conditional random quantities, consider a finite subfamily F = Generalized Logical Operations among Conditional Events 5 {X1 |H1 , . . . , Xn |Hn } ⊆ K and the vector M = (µ1 , . . . , µn ), where µi = P(Xi |Hi ) is the assessed prevision for the conditional random quantity Xi |Hi , i ∈ Jn . With the pair (F, M) we associate the random gain G = P i∈Jn si Hi (Xi − µi ); moreover, we denote by GHn the set of values of G restricted to Hn = H1 ∨ · · · ∨ Hn . Then, by the betting scheme, we have Definition 2 The function P defined on K is coherent if and only if, ∀n ≥ 1, ∀ F ⊆ K, ∀ s1 , . . . , sn ∈ R, it holds that: min GHn ≤ 0 ≤ max GHn . Given a family F = {X1 |H1 , . . . , Xn |Hn }, for each i ∈ Jn we denote by {xi1 , . . . , xiri } the set of possible values for the restriction of Xi to Hi ; then, for each i ∈ Jn and j = 1, . . . , ri , we set Aij = (Xi = xij ). Of course, for each i ∈ Jn , the family {H iW , Aij Hi , j = 1, . . . , ri } is a partition of the sure ri event Ω, with Aij Hi = Aij , j=1 Aij = Hi . Then, the constituents generated by the family F areV(the elements of the partition of Ω) obtained by expanding the expression i∈Jn (Ai1 ∨ · · · ∨ Airi ∨ H i ). We set C0 = H 1 · · · H n (it may be C0 = ∅); moreover, we denote by C1 , . . . , Cm the V Wm constituents contained in Hn . Hence i∈Jn (Ai1 ∨ · · · ∨ Airi ∨ H i ) = h=0 Ch . With each Ch , h ∈ Jm , we associate a vector Qh = (qh1 , . . . , qhn ), where qhi = xij if Ch ⊆ Aij , j = 1, . . . , ri , while qhi = µi if Ch ⊆ H i ; with C0 it is associated Q0 = M = (µ1 , . . . , µn ). Denoting by I the convex hull of Q1 , . . . , Qm , the condition M ∈ I amounts P P to the existence of a vector (λ1 , . . . , λm ) such that: λ Q = M , h h h∈Jm h∈Jm λh = 1 , λh ≥ 0 , ∀ h; in other words, M ∈ I is equivalent to the solvability of the system (Σ), associated with (F, M), P P (Σ) (1) h∈Jm λh = 1 ; λh ≥ 0 , h ∈ Jm . h∈Jm λh qhi = µi , i ∈ Jn ; Given the assessment M = (µ1 , . . . , µn ) on F = {X1 |H1 , . . . , Xn |Hn }, let S be the set of solutions Λ = (λ1 , . . . , λm ) of system (Σ) defined in (1). Then, the following characterization theorem for coherent assessments on finite families of conditional events can be proved ([6]) Theorem 1 [Characterization of coherence]. Given a family of n conditional random quantities F = {X1 |H1 , . . . , Xn |Hn }, with finite sets of possible values, and a vector M = (µ1 , . . . , µn ), the conditional prevision assessment P(X1 |H1 ) = µ1 , . . . , P(Xn |Hn ) = µn is coherent if and only if, for every subset J ⊆ Jn , defining FJ = {Xi |Hi , i ∈ J}, MJ = (µi , i ∈ J), the system (ΣJ ) associated with the pair (FJ , MJ ) is solvable. We point out that the solvability of system (Σ) (i.e., the condition M ∈ I) is a necessary (but not sufficient) condition for coherence of M on F. Moreover, assuming the system (Σ) solvable, that is S 6= ∅, we define: P I0 = {i : maxΛ∈S h:Ch ⊆Hi λh = 0}, F0 = {Xi |Hi , i ∈ I0 }, M0 = (µi , i ∈ I0 ) . (2) Then, the following theorem can be proved ([6, Theorem 3]) Theorem 2 [Operative characterization of coherence] A conditional prevision assessment M = (µ1 , . . . , µn ) on the family F = {X1 |H1 , . . . , Xn |Hn } is coherent if and only if the following conditions are satisfied: 6 Angelo Gilio, Giuseppe Sanfilippo (i) the system (Σ) defined in (1) is solvable; (ii) if I0 6= ∅, then M0 is coherent. By Theorem 2, the following algorithm checks in a finite number of steps the coherence of the prevision assessment M on F. Algorithm 1 Let be given the pair (F, M). 1. Construct the system (Σ) defined in (1) and check its solvability; 2. If the system (Σ) is not solvable then M is not coherent and the procedure stops, otherwise compute the set I0 ; 3. If I0 = ∅ then M is coherent and the procedure stops, otherwise set (F, M) = (F0 , M0 ) and repeat steps 1-3. By following the approach given in [12, 20, 25, 26, 29, 35] a conditional random quantity X|H can be seen as the random quantity XH + µH, where µ = P(X|H). In particular, in numerical terms, A|H is the random quantity AH + xH, where x = P (A|H). Then, when H ⊆ A, coherence requires that P (A|H) = 1 and hence A|H = H + H = 1. Notice that, as H|H = 0 and XH|H = X|H, it holds that: (XH + µH)|H = XH|H + µH|H = X|H, where µ = P(X|H). Moreover, the negation of A|H is defined as A|H = 1 − A|H = A|H. Coherence can be characterized in terms of proper scoring rules ([7, 24]), which can be related to the notion of entropy in information theory ([36, 37]). We recall a result (see [29, Theorem 4]) which shows that, given two conditional random quantities X|H, Y |K, if X|H = Y |K when H ∨ K is true, then X|H = Y |K also when H ∨ K is false, so that X|H = Y |K. Theorem 3 Given any events H 6= ∅, K 6= ∅, and any r.q.’s X, Y , let Π be the set of the coherent prevision assessments P(X|H) = µ, P(Y |K) = ν. (i) Assume that, for every (µ, ν) ∈ Π, X|H = Y |K when H ∨ K is true; then µ = ν for every (µ, ν) ∈ Π. (ii) For every (µ, ν) ∈ Π, X|H = Y |K when H ∨ K is true if and only if X|H = Y |K. 2.3 Quasi conjunction, conjunction, and disjunction of two conditional events. The notion of quasi conjunction plays an important role in nonmonotonic reasoning. In particular for two conditional events A|H, B|K the quasi conjunction QC(A|H, B|K) is the conditional event (H ∨ A) ∧ (K ∨ B) | (H ∨ K). Note that: QC(A|H, B|K) is true, when a conditional event is true and the other one is not false; QC(A|H, B|K) is false, when a conditional event is false; QC(A|H, B|K) is void, when H ∨ K is false. In other words, the quasi conjunction is the conjunction of the two material conditionals H ∨ A, K ∨ B given the disjunction of the conditioning events H, K. In numerical terms one has (3) QC(A|H, B|K) = min {H ∨ A, K ∨ B} | (H ∨ K) and, if we replace the material conditionals H ∨ A, K ∨ B by the conditional events A|H, B|K, from formula (3) we obtain the definition below ([26]). Generalized Logical Operations among Conditional Events 7 Definition 3 Given any pair of conditional events A|H and B|K, with P (A|H) = x, P (B|K) = y, we define their conjunction as the conditional random quantity (A|H) ∧ (B|K) = Z | (H ∨ K), where Z = min {A|H, B|K}. Then, defining z = P[(A|H) ∧ (B|K)], we have  1, if AHBK is true,      0, if AH ∨ BK is true, (A|H) ∧ (B|K) = x, if HBK is true,    y, if AHK is true,   z, if HK is true. (4) Remark 1 We recall that A|H = AH + xH, where x = P (A|H). Then, by Definition 3, it holds that (A|H) ∧ (A|H) = (A|H)|H = (AH + xH)|H = AH|H = A|H. From (4), the conjunction (A|H) ∧ (B|K) is the following random quantity (A|H) ∧ (B|K) = 1 · AHBK + x · HBK + y · AHK + z · HK . (5) For the quasi conjunction it holds that QC(A|H, B|K) = AHBK + HBK + AHK + ν · HK, (6) where ν = P (QC(A|H, B|K)). We recall that, if P (A|H) = P (B|K) = 1, then ν = 1 (see, e.g., [28, Section 3]). We also recall a result which shows that Fréchet-Hoeffding bounds still hold for the conjunction of conditional events ([29, Theorem 7]). Theorem 4 Given any coherent assessment (x, y) on {A|H, B|K}, with A, H, B, K logically independent, H 6= ∅, K 6= ∅, the extension z = P[(A|H) ∧ (B|K)] is coherent if and only if the following Fréchet-Hoeffding bounds are satisfied: max{x + y − 1, 0} = z ′ ≤ z ≤ z ′′ = min{x, y}. Remark 2 We observe that, if x = y = 1, then coherence requires that z = ν = 1 and then by (5) and (6) it follows that (A|H) ∧ (B|K) = QC(A|H, B|K). We recall now the notion of disjunction of two conditional events. Definition 4 Given any pair of conditional events A|H and B|K, with P (A|H) = x, P (B|K) = y, we define their disjunction as (A|H) ∨ (B|K) = W | (H ∨ K), where W = max {A|H, B|K}. Then, defining w = P[(A|H) ∨ (B|K)], we  1, if      0, if (A|H) ∨ (B|K) = x, if   y, if    w, if have AH ∨ BK is true, AHBK is true, HBK is true, AHK is true, HK is true. (7) 8 Angelo Gilio, Giuseppe Sanfilippo Remark 3 We recall that A|H = AH + xH, where x = P (A|H). Then, by Definition 4, it holds that (A|H) ∨ (A|H) = (A|H)|H = (AH + xH)|H = AH|H = A|H. From (7), the disjunction (A|H) ∨ (B|K) is the following random quantity (A|H) ∨ (B|K) = 1 · AH ∨ BK + x · HBK + y · AHK + w · HK. (8) 3 Conjunction, Disjunction, and Negation We now define the conjunction and the disjunction of n conditional events in a progressive way by specifying the possible values of the corresponding conditional random quantities. Given a family of n conditional events F = {E1 |H1 , . . . , En |Hn }, we denote by C0 , C1 , . . . , Cm , with m + 1 ≤ 3n , the constituents associated with F, where C0 = H 1 H 2 · · · H n . With each Ch , h = 0, 1, . . . , m, we associate a tripartition (Sh′ , Sh′′ , Sh′′′ ) of the set {1, . . . , n}, such that, for each i ∈ {1, . . . , n} it holds that: i ∈ Sh′ , or i ∈ Sh′′ , or i ∈ Sh′′′ , according to whether Ch ⊆ Ei Hi , or Ch ⊆ E i Hi , or Ch ⊆ H i . In other words, for each h = 0, 1, . . . , m, we have Sh′ = {i : Ch ⊆ Ei Hi }, Sh′′ = {i : Ch ⊆ E i Hi }, Sh′′′ = {i : Ch ⊆ H i } . (9) Definition 5 (Conjunction of n conditionals) Let be given a family of n conditional events F = {E1 |H1 , . . . , En |Hn }. For V each non-empty subset S of {1, . . . , n}, let xS be a prevision assessment on i∈S (Ei |Hi ). Then, the conjunction C(F) = (E1 |H1 ) ∧ · · · ∧ (En |Hn ) is defined as  if Sh′ = {1, . . . , n},  1, Pm if Sh′′ 6= ∅, Zn |(H1 ∨ · · · ∨ Hn ) = h=0 zh Ch , where zh = 0,  ′′′ xSh , if Sh′′ = ∅ and Sh′′′ 6= ∅ . (10) Remark 4 As shown by (10), the conjunction (E1 |H1 ) ∧ · · · ∧ (En |Hn ) assumes one of the following possible values: 1, when every conditional event is true; 0, when at least one conditional event is false; xS , when the conditional event Ei |Hi is void, for every i ∈ S, and is true for every i ∈ / S. In the case S = {i}, we simply set xS = xi . Notice that the notion of conjunction given in (10) has been already proposed, with positive probabilities for the conditioning events, in [38]. But, our approach is developed in the setting of coherence, where conditional probabilities and conditional previsions are primitive notions. Moreover, coherence allows to properly manage zero probabilities for conditioning events. Generalized Logical Operations among Conditional Events 9 Remark 5 We observe that to introduce the random quantity defined by formula (10) we need to specify in a coherent way the set of prevision assessments {xS : S ⊆ {1, 2, . . . , n}}. In particular, when the conditioning events H1 , . . . , Hn are all false, i.e. C0 is true, the associated tripartition is (S0′ , S0′′ , S0′′′ ) = (∅, ∅, {1, 2, . . . , n}) and the value of the conjunction C(F) is its prevision xS0′′′ = P[C(F)]. Moreover, we observe that the set of the constituents {C0 , . . . , Cm } associated with F is invariant with respect to any permutation of the conditional events in F. Then, the operation of conjunction introduced by Definition 5 is invariant with respect to any permutation of the conditional events in F. Definition 6 Given two finite families of conditional events F ′ and F ′′ , based on Definition 5, we set C(F ′ ) ∧ C(F ′′ ) = C(F ′ ∪ F ′′ ). Proposition 1 The operation of conjunction is associative and commutative. Proof Concerning the commutative property, let be given two finite families of conditional events F ′ and F ′′ . As F ′′ ∪ F ′ = F ′ ∪ F ′′ , it holds that C(F ′′ ) ∧ C(F ′ ) = C(F ′′ ∪F ′ ) = C(F ′ ∪F ′′ ) = C(F ′ )∧C(F ′′ ). Concerning the associative property, let be given three finite families of conditional events F ′ , F ′′ and F ′′′ . We have [C(F ′ ) ∧ C(F ′′ )] ∧ C(F ′′′ ) = C(F ′ ∪ F ′′ ) ∧ C(F ′′′ ) = C(F ′ ∪ F ′′ ∪ F ′′′ ) = = C(F ′ ) ∧ C(F ′′ ∪ F ′′′ ) = C(F ′ ) ∧ [C(F ′′ ) ∧ C(F ′′′ )] = C(F ′ ) ∧ C(F ′′ ) ∧ C(F ′′′ ). ⊔ ⊓ Definition 7 (Disjunction of n conditionals) Let be given a family of n conditional events F = {E1 |H1 , . . . , En |Hn }. Morever, for W each non-empty subset S of {1, . . . , n}, let yS be a prevision assessment on i∈S (Ei |Hi ). Then, the disjunction D(F) = (E1 |H1 ) ∨ · · · ∨ (En |Hn ) is defined as the following conditional random quantity  if Sh′ 6= ∅,  1, Pm if Sh′′ = {1, 2, . . . , n}, Wn |(H1 ∨ · · · ∨ Hn ) = h=0 wh Ch , where wh = 0,  ′′′ ySh , if Sh′ = ∅ and Sh′′′ 6= ∅ . (11) W n We recall that S0′′′ = {1, 2, . . . , n}; thus yS0′′′ = P[ i=1 (Ei |Hi )] = P[D(F)]. As shown by (11), the disjunction D(F) assumes one of the following possible values: 1, when at least one conditional event is true; 0, when every conditional event is false; yS , when the conditional event Ei |Hi is void, for every i ∈ S, and is false for every i ∈ / S. Definition 8 Given two finite families of conditional events F ′ and F ′′ , based on Definition 7, we set D(F ′ ) ∨ D(F ′′ ) = D(F ′ ∪ F ′′ ). Proposition 2 The operation of disjunction is associative and commutative. Proof The proof is analogous to that of Proposition 1. ⊔ ⊓ 10 Angelo Gilio, Giuseppe Sanfilippo We give below the notion of negation for the conjunction and the disjunction of a family of conditional events. Definition 9 Given a family of conditional events F, the negations for the conjunction C(F) and the disjunction D(F) are defined as C(F) = 1 − C(F) and D(F) = 1 − D(F), respectively. Given a family of n conditional events F = {E1 |H1 , . . . , En |Hn }, we denote by F the family {E 1 |H1 , . . . , E n |Hn }. Of course F = F. In the next result we show that De Morgan’s Laws are satisfied. Theorem 5 Given a family of n conditional events F = {E1 |H1 , . . . , En |Hn }, it holds that: (i) D(F) = C(F), that is D(F) = C(F); (ii) C(F) = D(F), that is C(F) = D(F). Proof See Appendix A. 4 Monotonicity property For any given n conditional events E1 |H1 , . . . , En |Hn , we set Cn = V Wn n (E |H ) and D = (E |H i i ). Moreover, for every non empty subset S i i n i=1 i=1 of {1, 2, . . . , n} we set _ ^ (Ei |Hi ) . (Ei |Hi ), DS = CS = i∈S i∈S In this section, among other results, we will show the monotonicity property of conjunction and disjunction, that is Cn+1 ≤ Cn and Dn+1 ≥ Dn , for every n ≥ 1. We first prove a preliminary result, which in particular shows that, given two conditional random quantities X|H, Y |K, if X|H ≤ Y |K when H ∨ K is true, then X|H ≤ Y |K also when H ∨ K is false, so that X|H ≤ Y |K. This result generalizes Theorem 3, as the symbol = is replaced by ≤, and it will be used in Theorem 7. Theorem 6 Given any events H 6= ∅, K 6= ∅, and any r.q.’s X, Y , let Π be the set of the coherent prevision assessments P(X|H) = µ, P(Y |K) = ν. (i) Assume that, for every (µ, ν) ∈ Π, X|H ≤ Y |K when H ∨ K is true; then µ ≤ ν for every (µ, ν) ∈ Π. (ii) For every (µ, ν) ∈ Π, X|H ≤ Y |K when H ∨ K is true if and only if X|H ≤ Y |K. Proof See Appendix A. The next two results illustrate the monotonicity property of conjunction and disjunction. Generalized Logical Operations among Conditional Events 11 Theorem 7 Given n + 1 arbitrary conditional events E1 |H1 , . . . , En+1 |Hn+1 , with n ≥ 1, for the conjunctions Cn and Cn+1 it holds that Cn+1 ≤ Cn . Proof See Appendix A. Theorem 8 Given n + 1 arbitrary conditional events E1 |H1 , . . . , En+1 |Hn+1 , with n ≥ 1, for the disjunctions Dn and Dn+1 it holds that Dn+1 ≥ Dn . Proof Defining Fn = {E1 |H1 , . . . , En |Hn } and Fn+1 = Fn ∪ {En+1 |Hn+1 }, by Theorems 5 and 7 it holds that Dn+1 = D(Fn+1 ) = C(F n+1 ) = 1 − C(F n+1 ) ≥ 1 − C(F n ) = C(F n ) = Dn . ⊔ ⊓ The next result shows that the conjunction and the disjunction are random quantities with values in the interval [0, 1]. Theorem 9 Given n arbitrary conditional events E1 |H1 , . . . , En |Hn , it holds that: (i) Cn ∈ [0, 1]; (ii) Dn ∈ [0, 1]. Proof See Appendix A. Remark 6 From Theorem 7, it holds that Cn ≤ Cn−1 ≤ . . . ≤ C1 ; in particular P(Cn ) ≤ P(Ck ), k = 1, 2, . . . , n − 1. More generally, for every non empty subset S of {1, . . . , n}, it holds that P(Cn ) ≤ P(CS ). In particular, P(Cn ) ≤ P (En |Hn ). Then, P(Ck ) ≤ min{P(Ck−1 ), P (Ek |Hk )}, k = 2, 3, . . . , n, and by iterating it follows P(Cn ) ≤ min{P (E1 |H1 ), . . . , P (En |Hn )}. (12) Likewise, by Theorem 8, P(Dk ) ≥ max{P(Dk−1 ), P (Ek |Hk )}, k = 2, 3, . . . , n, and by iterating it follows P(Dn ) ≥ max{P (E1 |H1 ), . . . , P (En |Hn )}. (13) 5 Coherent assessments on {Cn , En+1 |Hn+1 , Cn+1 } Given any n + 1 arbitrary conditional events E1 |H1 , . . . , En+1 |Hn+1 , let us consider the conjunctions Cn = (E1 |H1 ) ∧ · · · ∧ (En |Hn ) and Cn+1 = (E1 |H1 ) ∧ · · · ∧ (En+1 |Hn+1 ). We set P(Cn ) = µn , P(Cn+1 ) = µn+1 and P (En+1 |Hn+1 ) = xn+1 . Remark 7 Let us consider the points Q1 = (1, 1, 1), Q2 = (1, 0, 0), Q3 = (0, 1, 0), Q4 = (0, 0, 0). We observe that the equations of the three planes containing the points Q1 , Q2 , Q3 , or Q1 , Q2 , Q4 , or Q1 , Q3 , Q4 , are z = x + y − 1, or z = x, or z = y, respectively. It can be shown that a point (x, y, z) belongs to the convex hull I of Q1 , Q2 , Q3 , Q4 if and only if (x, y) ∈ [0, 1]2 , max{x + y − 1, 0} ≤ z ≤ min{x, y} . (14) The convex hull I, which is a tetrahedron with vertices Q1 , Q2 , Q3 , Q4 , is depicted in Figure 1. 12 Angelo Gilio, Giuseppe Sanfilippo We observe that the lower and upper bounds in (14) are the Fréchet-Hoeffding bounds, which characterize the next result. 1 Q1 0.9 0.8 0.7 z 0.6 P 00 0.5 0 0.4 0.3 0.2 Q3 P0 0.1 0 0 0 Q4 1 0.2 0.8 0.4 0.6 0.6 0.4 0.8 x 0.2 1 Q2 0 y Fig. 1 Convex hull of the points Q1 , Q2 , Q3 , Q4 . P ′ = (x, y, z ′ ), P ′′ = (x, y, z ′′ ), where (x, y) ∈ [0, 1]2 , z ′ = max{x + y − 1, 0}, z ′′ = min{x, y}. In the figure the numerical values are: x = 0.6, y = 0.5, z ′ = 0.1, and z ′′ = 0.5. Theorem 10 Assume that the events E1 , . . . , En+1 , H1 , . . . , Hn+1 are logically independent. Let I be the convex hull of the points Q1 = (1, 1, 1), Q2 = (1, 0, 0), Q3 = (0, 1, 0), Q4 = (0, 0, 0). Then, the assessment M = (µn , xn+1 , µn+1 ) on the family {Cn , En+1 |Hn+1 , Cn+1 } is coherent if and only if M ∈ I, that is if and only if (µn , xn+1 ) ∈ [0, 1]2 , µ′n+1 ≤ µn+1 ≤ µ′′n+1 , where µ′n+1 = max{µn + xn+1 − 1, 0} and µ′′n+1 = min{µn , xn+1 }. Proof See Appendix A. Remark 8 We observe that the representation of each coherent assessment M = (µn , xn+1 , µn+1 ) as a linear convex combination λ1 Q1 + λ2 Q2 + λ3 Q3 + P4 λ4 Q4 (where h=1 λh = 1, λh ≥ 0, h = 1, 2, 3, 4 ) is unique, with  λ1 = µn+1 = P(Cn+1 ) ≥ 0,    λ2 = µn − µn+1 = P(Cn ) − P(Cn+1 ) ≥ 0, λ3 = xn+1 − µn+1 = P (En+1 |Hn+1 ) − P(Cn+1 ) ≥ 0,    λ4 = 1 − µn − xn+1 + µn+1 = 1 − P(Cn ) − P (En+1 |Hn+1 ) + P(Cn+1 ) ≥ 0 . In particular, concerning the extreme cases µn+1 = µ′n+1 , or µn+1 = µ′′n+1 , we can examine four cases: 1) µ′n+1 = µn + xn+1 − 1 > 0; 2) µ′n+1 = 0; 3) µ′′n+1 = µn and 4) µ′′n+1 = xn+1 . Generalized Logical Operations among Conditional Events 13 In the case 1 the point M = (µn , xn+1 , µn+1 ) is a linear convex combination λ1 Q1 + λ2 Q2 + λ3 Q3 + λ4 Q4 , with λ1 = µ′n+1 = µn + xn+1 − 1, λ2 = 1 − xn+1 , λ3 = 1 − µn , λ4 = 0. In the case 2 it holds that λ1 = µ′n+1 = 0, λ2 = µn , λ3 = xn+1 , λ4 = 1 − µn + xn+1 . In the case 3 it holds that λ1 = µ′′n+1 = µn , λ2 = 0 , λ3 = xn+1 − µn , λ4 = 1 − xn+1 . In the case 4 it holds that λ1 = µ′′n+1 = xn+1 , λ2 = µn − xn+1 , λ3 = 0 , λ4 = 1 − µn . 6 Probabilistic Inference from Cn+1 to {Cn , En+1 |Hn+1 } In this section, given any coherent prevision assessment µn+1 on Cn+1 , we find the set of coherent extensions (µn , xn+1 ) on {Cn , En+1 |Hn+1 }. As we will see, it is enough to illustrate the case n = 1, by finding the set of coherent extensions (x, y) on {E1 |H1 , E2 |H2 } of any assessment z = P[(E1 |H1 ) ∧ (E2 |H2 )] ∈ [0, 1]. Theorem 11 Given any prevision assessment z on (E1 |H1 ) ∧ (E2 |H2 ), with z ∈ [0, 1], with E1 , H1 , E2 , H2 logically independent, with H1 6= ∅ and H2 6= ∅, the extension x = P (E1 |H1 ), y = P (E2 |H2 ) is coherent if and only if (x, y) belongs to the set Tz = {(x, y) : x ∈ [z, 1], y ∈ [z, 1 + z − x]}. Proof We recall that, by logical independence of E1 , H1 , E2 , H2 , the assessment (x, y) is coherent for every (x, y) ∈ [0, 1]2 . From Theorem 4, the set Π of all coherent assessment (x, y, z) on {E1 |H1 , E2 |H2 , (E1 |H1 ) ∧ (E2 |H2 )} is Π = {(x, y, z) : (x, y) ∈ [0, 1]2 , max{x + y − 1, 0} ≤ z ≤ min{x, y}}. We note that Π = {(x, y, z) : z ∈ [0, 1], x ∈ [z, 1], y ∈ [z, z + 1 − x]} = = {(x, y, z) : z ∈ [0, 1], (x, y) ∈ Tz }. Then, (x, y) is a coherent extension of z if and only if (x, y) ∈ Tz . ⊔ ⊓ Remark 9 We observe that, given any S z ∈ [0, 1] and defining Πz = {(x, y, z) : (x, y) ∈ Tz }, it holds that Π = z∈[0,1] Πz (see Figure 2). The set Π is the tetrahedron depicted in Figure 1. Hence, contrarily to the general case, for the family {E1 |H1 , E2 |H2 , (E1 |H1 ) ∧ (E2 |H2 )} the set of coherent prevision assessments Π is convex. Indeed, Π is also the (convex) set of coherent probability assessment (x, y, z) on the family of unconditional events {E1 , E2 , E1 E2 }. We recall that, assuming H1 ∧ H2 = ∅, the set of coherent prevision assessments (x, y, z) on {E1 |H1 , E2 |H2 , (E1 |H1 ) ∧ (E2 |H2 )} is the surface {(x, y, z) : (x, y) ∈ [0, 1]2 , z = xy}, which is a strict non-convex subset of Π (see [26, Section 5]). Theorem 12 Given any prevision assessment µn+1 = P(Cn+1 ) ∈ [0, 1], with µn+1 ∈ [0, 1], the extension µn = P(Cn ), xn+1 = P (En+1 |Hn+1 ) is coherent if and only if (µn , xn+1 ) ∈ {(µn , xn+1 ) : µn ∈ [µn+1 , 1], xn+1 ∈ [µn+1 , 1 + µn+1 − µn ]}. 14 Angelo Gilio, Giuseppe Sanfilippo 1 0.9 (1 1 1) 0.8 0.7 0.6 z & 0.5 0.4 0.3 &z 0.2 0.1 (0 1 0) 0 0 (0 0 0) 0.5 x 1 (1 0 0) 0 0.4 0.2 0.6 0.8 1 y Fig. 2 Set Π of all S coherent assessments (x, y, z) on {E1 |H1 , E2 |H2 , (E1 |H1 ) ∧ (E2 |H2 )}. Notice that Π = z∈[0,1] Πz , where for each given z ∈ [0, 1] the set Πz is the triangle {(x, y, z) : (x, y) ∈ Tz }, with Tz = {(x, y) : x ∈ [z, 1], y ∈ [z, 1 + z − x]}. Proof From Theorem 10, the set Π of all coherent assessment (µn , xn+1 , µn+1 ) on {Cn , En+1 |Hn+1 , Cn+1 } is Π = {(µn , xn+1 , µn+1 ) : (µn , xn+1 ) ∈ [0, 1]2 , max{µn + xn+1 − 1, 0} ≤ µn+1 ≤ min{µn , xn+1 }}. Moreover, as observed in the proof of Theorem 11, the set Π coincides with the set {(µn , xn+1 , µn+1 ) : µn+1 ∈ [0, 1], µn ∈ [µn+1 , 1], xn+1 ∈ [µn+1 , 1 + µn+1 − µn ]}. Then, (µn , xn+1 ) is a coherent extension of µn+1 if and only if (µn , xn+1 ) belongs to the set {(µn , xn+1 ) : µn ∈ [µn+1 , 1], xn+1 ∈ [µn+1 , 1 + µn+1 − µn ]}. ⊔ ⊓ 7 Fréchet-Hoeffding Bounds In the next result we show that the prevision of the conjunction Cn = E1 |H1 ∧ · · · ∧ En |Hn satisfies the Fréchet-Hoeffding bounds. Theorem 13 Let be given n conditional events E1 |H1 , E2 |H2 , . . . , En |Hn , with xi = P (Ei |Hi ), i = 1, 2, . . . , n, and with P(Cn ) = µn . Then max{x1 + · · · + xn − (n − 1), 0} ≤ µn ≤ min{x1 , . . . , xn } . (15) Proof From Theorem 10, it holds that µn ≥ µn−1 + xn − 1 ≥ µn−2 + xn−1 + xn − 2 ≥ · · · ≥ x1 + · · · + xn − (n − 1). Then, by inequality (12) and by Theorem 9 it holds that the inequalities in (15) are satisfied. ⊔ ⊓ Generalized Logical Operations among Conditional Events 15 Likewise, the following result holds for the prevision ηn of the disjunction Dn = E1 |H1 ∨ E2 |H2 ∨ · · · ∨ En |Hn . Theorem 14 Let be given n conditional events E1 |H1 , E2 |H2 , . . . , En |Hn , with xi = P (Ei |Hi ), i = 1, 2, . . . , n, and with P(Dn ) = ηn . Then max{x1 , . . . , xn } ≤ ηn ≤ min{x1 + · · · + xn , 1} . Proof By Definition 9, Theorems 5 {E 1 |H1 , E 2 |H2 , . . . , E n |Hn } it holds that and 13, defining (16) Fn = Vn P(Dn ) = 1 − P(Dn ) = 1 − P(C(F n )) = 1 − P( i=1 (Ei |Hi )) ≤ ≤ 1 − [(1 − x1 ) + · · · + (1 − xn ) − (n − 1)] = x1 + · · · + xn . Then, by (13) and by Theorem 9, the inequalities in (16) are satisfied. ⊔ ⊓ 8 Conjunction of Three Conditional Events Given a family of three conditional events F = {E1 |H1 , E2 |H2 , E3 |H3 }, we set P (Ei |Hi ) = xi , i = 1, 2, 3, P[(Ei |Hi ) ∧ (Ej |Hj )] = xij = xji , i 6= j, and x123 = P[(E1 |H1 ) ∧ (E2 |H2 ) ∧ (E3 |H3 )]. Then, by Definition 5, the conjunction of E1 |H1 , E2 |H2 , E3 |H3 is the conditional random quantity  1,     0,     x1 ,      x2 , C(F ) = (E1 |H1 ) ∧ (E2 |H2 ) ∧ (E3 |H3 ) = x3 ,    x12 ,      x13 ,      x23 , x123 , if if if if if if if if if E1 H1 E2 H2 E3 H3 is true E 1 H1 ∨ E 2 H2 ∨ E 3 H3 is true, H 1 E2 H2 E3 H3 is true, H 2 E1 H1 E3 H3 is true, H 3 E1 H1 E2 H2 is true, H 1 H 2 E3 H3 is true, H 1 H 3 E2 H2 is true, H 2 H 3 E1 H1 is true, H 1 H 2 H 3 is true. (17) Remark 10 Notice that in the betting scheme x123 is the quantity to be paid in order to receive C(F). Assuming that the assessment (x1 , x2 , x3 , x12 , x13 , x23 ) on {E1 |H1 , E2 |H2 , E3 |H3 , (E1 |H1 ) ∧ (E2 |H2 ), (E1 |H1 ) ∧ (E3 |H3 ), (E2 |H2 ) ∧ (E3 |H3 )} is coherent, we are interested in finding the values x123 which are a coherent extension of (x1 , x2 , x3 , x12 , x13 , x23 ). Of course, as xi ∈ [0, 1], i = 1, 2, 3, and xij ∈ [0, 1], i 6= j, a necessary condition for coherence is x123 ∈ [0, 1]. From Remark 5 and Proposition 1 the conjunction C(F) is invariant with respect to any given permutation (i1 , i2 , i3 ) of (1, 2, 3); that is C(F) = (Ei1 |Hi1 ) ∧ (Ei2 |Hi2 )) ∧ (Ei3 |Hi3 ), for any permutation (i1 , i2 , i3 ) of (1, 2, 3). 16 Angelo Gilio, Giuseppe Sanfilippo 8.1 Study of Coherence Notice that in general, if we assess the values xS = P(CS ) for some S ⊂ {1, 2 . . . , n}, then the study of coherence may be very complex. In this section we study coherence in the case n = 3 when we assess the prevision xS = P(CS ) for every S ⊆ {1, 2, 3}. In the next result we determine the set of coherent assessments M = (x1 , x2 , x3 , x12 , x13 , x23 , x123 ) on the family F = {E1 |H1 , E2 |H2 , E3 |H3 , (E1 |H1 ) ∧ (E2 |H2 ), (E1 |H1 ) ∧ (E3 |H3 ), (E2 |H2 ) ∧ (E3 |H3 ), (E1 |H1 ) ∧ (E2 |H2 ) ∧ (E3 |H3 )} = {CS : ∅ = 6 S ⊆ {1, 2, 3}}. Theorem 15 Assume that the events E1 , E2 , E3 , H1 , H2 , H3 are logically independent, with H1 6= ∅, H2 6= ∅, H3 6= ∅. Then, the set Π of all coherent assessments M = (x1 , x2 , x3 , x12 , x13 , x23 , x123 ) on F = {E1 |H1 , E2 |H2 , E3 |H3 , (E1 |H1 ) ∧ (E2 |H2 ), (E1 |H1 ) ∧ (E3 |H3 ), (E2 |H2 ) ∧ (E3 |H3 ), (E1 |H1 ) ∧ (E2 |H2 ) ∧ (E3 |H3 )} is the set of points (x1 , x2 , x3 , x12 , x13 , x23 , x123 ) which satisfy the following conditions  (x1 , x2 , x3 ) ∈ [0, 1]3 ,     max{x1 + x2 − 1, x13 + x23 − x3 , 0} ≤ x12 ≤ min{x1 , x2 },      max{x1 + x3 − 1, x12 + x23 − x2 , 0} ≤ x13 ≤ min{x1 , x3 }, max{x2 + x3 − 1, x12 + x13 − x1 , 0} ≤ x23 ≤ min{x2 , x3 }, (18)   1 − x − x − x + x + x + x ≥ 0,  1 2 3 12 13 23    x123 ≥ max{0, x12 + x13 − x1 , x12 + x23 − x2 , x13 + x23 − x3 },    x123 ≤ min{x12 , x13 , x23 , 1 − x1 − x2 − x3 + x12 + x13 + x23 }. Proof See Appendix A. We observe that, from (18) it follows that (x1 , x2 , x3 , x12 , x13 , x23 ) amounts to the inequality the coherence of min{x12 , x13 , x23 , 1 − x1 − x2 − x3 + x12 + x13 + x23 } ≤ ≤ max{0, x12 + x13 − x1 , x12 + x23 − x2 , x13 + x23 − x3 } . Then, by Theorem 15 it follows Corollary 1 For any coherent assessment (x1 , x2 , x3 , x12 , x13 , x23 ) on {E1 |H1 , E2 |H2 , E3 |H3 , (E1 |H1 )∧(E2 |H2 ), (E1 |H1 )∧(E3 |H3 ), (E2 |H2 )∧(E3 |H3 )} the extension x123 on (E1 |H1 ) ∧ (E2 |H2 ) ∧ (E3 |H3 ) is coherent if and only if x123 ∈ [x′123 , x′′123 ], where x′123 = max{0, x12 + x13 − x1 , x12 + x23 − x2 , x13 + x23 − x3 }, x′′123 = min{x12 , x13 , x23 , 1 − x1 − x2 − x3 + x12 + x13 + x23 }. (19) Proof As shown in (18), (see also (46) in the Appendix A), the coherence of (x1 , x2 , x3 , x12 , x13 , x23 , x123 ) amounts to the condition min{x12 , x13 , x23 , 1 − x1 − x2 − x3 + x12 + x13 + x23 } ≤ x123 ≤ ≤ max{0, x12 + x13 − x1 , x12 + x23 − x2 , x13 + x23 − x3 } . Generalized Logical Operations among Conditional Events 17 Then, in particular, the extension x123 on (E1 |H1 ) ∧ (E2 |H2 ) ∧ (E3 |H3 ) is coherent if and only if x123 ∈ [x′123 , x′′123 ], where x′123 = max{0, x12 + x13 − x1 , x12 + x23 − x2 , x13 + x23 − x3 }, x′′123 = min{x12 , x13 , x23 , 1 − x1 − x2 − x3 + x12 + x13 + x23 }. ⊔ ⊓ 8.2 The Case H1 = H2 = H3 We recall that in case of logical dependencies, the set of all coherent assessments may be smaller than that one associated with the case of logical independence. However, in this section we show that the results of Theorem 15 and Corollary 1 still hold when the conditioning events H1 , H2 , and H3 coincide. Theorem 16 Let be given any logically independent events E1 , E2 , E3 , H, with H 6= ∅. Then, the set Π of all coherent assessments M = (x1 , x2 , x3 , x12 , x13 , x23 , x123 ) on F = {E1 |H, E2 |H, E3 |H, (E1 |H) ∧ (E2 |H), (E1 |H) ∧ (E3 |H), (E2 |H) ∧ (E3 |H), (E1 |H) ∧ (E2 |H) ∧ (E3 |H)} is the set of points (x1 , x2 , x3 , x12 , x13 , x23 , x123 ) which satisfy the conditions in formula (18). Proof See Appendix A. Corollary 2 For any coherent assessment (x1 , x2 , x3 , x12 , x13 , x23 ) on {E1 |H, E2 |H, E3 |H, (E1 E2 )|H, (E1 E3 )|H, (E2 E3 )|H} the extension x123 on (E1 E2 E3 )|H is coherent if and only if x123 ∈ [x′123 , x′′123 ], where x′123 and x′′123 are defined in (19). Proof The proof is the same as for Corollary 1. ⊔ ⊓ Of course, the results of Theorem 16 and Corollary 2 still hold in the unconditional case where H = Ω. Remark 11 As shown in this section, a consistent management of conjunctions (and/or disjunctions) defined on a given family of conditional events F essentially requires an (iterative) coherence checking and propagation of probability and prevision assessments on compounded conditionals, for each subfamily of F. Then, an analysis of complexity in our context would be of the same kind of the exhaustive complexity analysis given in [5] for probabilistic reasoning under coherence. 18 Angelo Gilio, Giuseppe Sanfilippo 9 Characterization of p-consistency and p-entailment with applications to nonmonotonic reasoning In this section we apply our notion of conjunction to characterize the notions of p-consistency and p-entailment. Then, we examine some inference rules related with probabilistic nonmonotonic reasoning. We also briefly describe a characterization of p-entailment by a notion of iterated conditioning, in the case of two premises. We recall below the notions of p-consistency and pentailment of Adams ([2]) as formulated for conditional events in the setting of coherence (see, e.g., [5, 23, 27]). Definition 10 Let F = {Ei |Hi , i = 1, . . . , n} be a family of n conditional events. Then, F is p-consistent if and only if the probability assessment (p1 , p2 , . . . , pn ) = (1, 1, . . . , 1) on F is coherent. Definition 11 A p-consistent family F = {Ei |Hi , i = 1, . . . , n} p-entails a conditional event En+1 |Hn+1 if and only if for any coherent probability assessment (p1 , . . . , pn , pn+1 ) on F ∪ {En+1 |Hn+1 } it holds that: if p1 = · · · = pn = 1, then pn+1 = 1. We recall below the notion of logical implication ([31]) between two conditional events. Definition 12 Given two conditional events A|H and B|K we say that A|H logically implies B|K, which we denote by A|H ⊆ B|K, if and only if AH true implies BK true and BK true implies AH true; that is: AH ⊆ BK and BK ⊆ AH. We observe that, by coherence, it holds that (see, e.g., [28, Theorem 7]). A|H ⊆ B|K =⇒ P (A|H) ≤ P (B|K). (20) We also recall the notion of quasi conjunction for a general family of n conditional events. Definition 13 Given a family F = {Ei |Hi , i = 1, . . . , n} of n conditional events, the quasi conjunction QC(F) of the conditional events in F is defined as the following conditional event QC(F) = n ^ i=1 (H i ∨ Ei Hi )|( n _ Hi ). i=1 Remark 12 We observe that, by Definition 13, based on (9) the quasi conjunction can be represented as   1, if Sh′ 6= ∅ and Sh′′ = ∅, Pm QC(F) = h=0 νh Ch , where νh = 0, if Sh′′ 6= ∅ (21)  ν, if Sh′′′ = {1, . . . , n} , Generalized Logical Operations among Conditional Events 19 where ν = P (QC(F)). Therefore, by (10), (21), and by also recalling Theorem 6, it holds that zh ≤ νh , h = 0, 1, . . . , m; thus C(F) ≤ QC(F). (22) In particular, if F is p-consistent and P (Ei |Hi ) = 1, i = 1, . . . , n, then from (15) it holds that xS = P(C(FS )) = 1 for every S ⊆ {1, 2, . . . , n}, where FS = {Ei |Hi ∈ F : i ∈ S}; then zh = νh , h = 0, 1, . . . , m, and C(F) = QC(F). 9.1 Characterization of p-consistency and p-entailment We illustrate below a characterization of p-consistency of a family F in terms of the coherence of the prevision assessment P[C(F)] = 1. Theorem 17 A family of n conditional events F = {E1 |H1 , . . . , En |Hn } is p-consistent if and only if the prevision assessment P[C(F)] = 1 is coherent. Proof (⇒) By Definition 10, as F is p-consistent, the probability assessment (x1 , x2 , . . . , xn ) = (1, 1, . . . , 1) on F is coherent. Then, by (15) the extension P[C(F)] = 1 is unique and of course coherent. (⇐) By (15) it holds that P[C(F)] ≤ min{x1 , . . . , xn } and hence P[C(F)] = 1 implies (x1 , x2 , . . . , xn ) = (1, 1, . . . , 1) on F. Moreover, the coherence of P[C(F)] = 1 requires that the (unique) extension (1, 1, . . . , 1) on F be coherent. Thus, F is p-consistent. We observe that, in the case where H1 = . . . = Hn = H, the assessment P (E1 |H) = . . . P (En |H) = 1 is coherent (that is, F is p-consistent) if and only if P [(E1 · · · En )|H] = 1 is coherent. The next theorem gives a characterization of p-entailment in terms of a result which involves suitable conjunctions associated with the premise set and the conclusion of the given inference rule. Theorem 18 Let be given a p-consistent family of n conditional events F = {E1 |H1 , . . . , En |Hn } and a further conditional event En+1 |Hn+1 . Then, the following assertions are equivalent: (i) F p-entails En+1 |Hn+1 ; (ii) the conjunction Cn+1 = (E1 |H1 ) ∧ · · · ∧ (En |Hn ) ∧ (En+1 |Hn+1 ) coincides with the conjunction Cn = (E1 |H1 ) ∧ · · · ∧ (En |Hn ); (iii) the inequality Cn ≤ (En+1 |Hn+1 ) is satisfied. Proof See Appendix A. As a first simple application of Theorem 18 we observe that, given two conditional events A|H, with AH 6= ∅, and B|K, the p-entailment of B|K from A|H amounts to the condition (ii), i.e., A|H ∧ B|K = A|H, or equivalently condition (iii), i.e., A|H ≤ B|K. In particular, (ii) and (iii) are both satisfied when A|H ⊆ B|K. 20 Angelo Gilio, Giuseppe Sanfilippo 9.2 Applications to some p-valid inference rules We recall that an inference from a p-consistent family F to E|H is p-valid if and only if F p-entails E|H. We will examine some p-valid inference rules by verifying that conditions (ii) and (iii) in Theorem 18 are satisfied. In particular we consider the following inference rules of System P: And, Cut, CM, and Or. In what follows, if not specified otherwise, the basic events are assumed to be logically independent. And rule: The family {B|A, C|A} p-entails BC|A. It holds that (B|A) ∧ (C|A) = BC|A = (B|A) ∧ (C|A) ∧ (BC|A) and (B|A) ∧ (C|A) = BC|A ≤ BC|A; that is, conditions (ii) and (iii) are satisfied. Cut rule: The family {C|AB, B|A} p-entails C|A. By (5), as ABAB = AABC = ∅, it holds that (C|AB) ∧ (B|A) = ABC + zA, where z = P[(C|AB) ∧ (B|A)]. Moreover, BC|A = ABC + xA, where x = P (BC|A). As (C|AB) ∧ (B|A) and BC|A coincide conditionally on A being true, by Theorem 3, it follows that (C|AB) ∧ (B|A) = BC|A. Then, condition (ii) is satisfied, that is (C|AB) ∧ (B|A) ∧ (C|A) = (BC|A) ∧ (C|A) = BC|A = (C|AB) ∧ (B|A). Moreover, C|AB ∧ B|A = BC|A ≤ C|A, that is condition (iii) is satisfied too. Remark 13 As shown in the analysis of Cut rule, it holds that C|AB ∧ B|A = BC|A. Then, the family {C|AB, B|A} p-entails BC|A. This p-valid rule is called CCT (Conjunctive Cumulative Transitivity); see, e.g., [43]. CM rule: The family {C|A, B|A} p-entails C|AB. It holds that (C|A) ∧ (B|A) = BC|A. Moreover, (C|A) ∧ (B|A) ∧ (C|AB) = (BC|A) ∧ (C|AB). By (5), it holds that (BC|A) ∧ (C|AB) = ABC + zA, where z = P[(BC|A) ∧ (C|AB)]. Moreover, BC|A = ABC + xA, where x = P (BC|A). As (BC|A) ∧ (C|AB) and BC|A coincide conditionally on A being true, by Theorem 3 it follows that (BC|A) ∧ (C|AB) = BC|A; so that (C|A) ∧ (B|A) ∧ (C|AB) = BC|A = (C|A) ∧ (B|A), so that condition (ii) is satisfied. Moreover, based on Definition 12, it holds that (C|A) ∧ (B|A) = BC|A ⊆ C|AB, then (C|A) ∧ (B|A) ≤ C|AB, so that condition (iii) is satisfied too. Generalized Logical Operations among Conditional Events 21 Or rule: The family {C|A, C|B} p-entails C|(A ∨ B). We set P (C|A) = x, P (C|B) = y, and P((C|A) ∧ (C|B)) = z; then, by observing that the family {ABC, ABC, ABC, (A ∨ B)C, AB} is a partition of the sure event, we obtain  1,      0, (C|A) ∧ (C|B) = x,   y,    z, if if if if if ABC is true, (A ∨ B)C is true, ABC is true, ABC is true, AB is true. (23) Moreover, by defining P[(C|A) ∧ (C|B) ∧ (C|(A ∨ B))] = t, we obtain  1,      0, (C|A) ∧ (C|B) ∧ (C|(A ∨ B)) = x,   y,    t, if if if if if ABC is true, (A ∨ B)C is true, ABC is true, ABC is true, AB is true. As we can see, (C|A) ∧ (C|B) ∧ (C|(A ∨ B)) and (C|A) ∧ (C|B) coincide when A ∨ B is true; then, by Theorem 3 it holds that t = z, so that (C|A) ∧ (C|B) ∧ (C|(A ∨ B)) = (C|A) ∧ (C|B), that is condition (ii) is satisfied. Moreover, defining P (C|(A ∨ B)) = w, we have  1, if ABC is true,      0, if (A ∨ B)C is true, (24) C|(A ∨ B) = 1, if ABC is true,   is true, 1, if ABC    w, if AB is true. Based on (23) and (24), it holds that (C|A) ∧ (C|B) ≤ C|(A ∨ B) conditionally on A ∨ B being true. Then, from Theorem 6 it holds that P((C|A) ∧ (C|B)) = t ≤ w = P (C|(A ∨ B)); thus (C|A) ∧ (C|B) ≤ C|(A ∨ B), that is condition (iii) is satisfied. An inference rule related to Or rule [1, Rule 5, p. 189]. In this inference rule the premise set is {C|(A∨B), C|A} and the conclusion is C|B. We first observe that the premise set F = {C|(A ∨ B), C|A} is p-consistent because the assessment P (C|(A ∨ B)) = P (C|A) = 1 is coherent. Indeed, by applying Algorithm 1 to the pair (F, M) = ({C|(A ∨ B), C|A}, (1, 1)), it holds that the starting system (Σ) is solvable, with F0 = {C|A}. Then, by repeating the steps of the algorithm, the assessment P (C|A) = 1 is coherent. Thus, the assessment (1, 1) on F is coherent and hence F is p-consistent. We also note that, defining P (C|(A ∨ B)) = x, P (C|A) = y, and P((C|(A ∨ B)) ∧ (C|A)) = z, the coherence of (x, y) = (1, 1) from (15) amounts to coherence of z = 1, which by 22 Angelo Gilio, Giuseppe Sanfilippo Theorem 17 is another characterization for the p-consistency of F. Concerning p-entailment, we observe that    0,   0,      0, (C|A ∨ B) ∧ (C|A) = 0,   y,      0,   z, if if if if if if if ABC is true, ABC is true,  ABC is true,  0, if A ∨ ABC is true, ABC is true, = y, if ABC is true, (25)  ABC is true, z, if AB is true. ABC is true, AB is true, Moreover, by defining P[(C|(A ∨ B)) ∧ (C|A) ∧ (C|B)] = t, we obtain   0, if A ∨ ABC is true, (C|(A ∨ B)) ∧ (C|A) ∧ (C|B) = y, if ABC is true,  t, if AB is true. (26) As we can see from (25) and (26), the two quantities (C|(A∨B))∧(C|A)∧(C|B) and (C|(A ∨ B)) ∧ (C|A) coincide when A ∨ B is true; then, by Theorem 3 it holds that t = z, so that (C|(A ∨ B)) ∧ (C|A) ∧ (C|B) = (C|(A ∨ B)) ∧ (C|A), that is condition (ii) is satisfied. Moreover, defining P (C|B) = w, we have  1,     0,       1, if BC is true,  w, C|B = 0, if BC is true, = w,    w, if B is true.  1,     0,   w, if if if if if if if ABC is true, ABC is true, ABC is true, ABC is true, ABC is true, ABC is true, AB is true, (27) Based on (25) and (27), it holds that (C|(A ∨ B)) ∧ (C|A) ≤ C|B conditionally on A ∨ B being true. Then, from Theorem 6 it holds that P((C|(A ∨ B)) ∧ (C|A)) = t ≤ w = P (C|B); thus (C|(A ∨ B)) ∧ (C|A) ≤ C|B, that is condition (iii) is satisfied. Thus, this inference rule is p-valid. Notice that the p-validity of the rule could be also derived by using the lower and upper bounds given for Or rule in [18]. Indeed, using Or rule, when P (C|A) = 0 and P (C|B) = y it holds that z = P (C|A ∨ B) ∈ [0, y], so that P (C|(A ∨ B)) ≤ P (C|B). Then, P (C|(A ∨ B)) = 1 and P (C|A) = 1 implies P (C|B) = 1, that is {C|(A ∨ B), C|A} p-entails C|B. Generalized Logical Operations among Conditional Events 23 Generalized Or rule: In this p-valid rule, studied in [19] (see also [28]), the p-consistent premise set is {C|A1 , C|A2 , . . . , C|An } and the conclusion is C|(A Vnsubset S ⊂ {1, 2, . . . , n}, we define V 1 ∨ A2 ∨ · · · ∨ An ). For each nonempty P[ i∈S (C|Ai )] = xS ; moreover, we set P[ i=1 (C|Ai )] = z. Then,  1, if A1 A2 · · · An C is true,    0, if (A ∨ A ∨ · · · ∨ A )C is true, 2V n V1 (28) (C|A1 ) ∧ · · · ∧ (C|An ) = A C is true, A x , if  j i S j ∈S / i∈S   z, if A1 A2 · · · An is true. Moreover, by defining P[(C|A1 ) ∧ · · · ∧ (C|An ) ∧ (C|(A1 ∨ A2 ∨ · · · ∨ An ))] = t, we obtain  1, if A1 A2 · · · An C is true,    0, if (A ∨ A ∨ · · · ∨ A )C is true, 2V n V1 (C|A1 )∧· · ·∧(C|An )∧(C|(A1 ∨A2 ∨· · ·∨An )) =  xS , if i∈S Ai j ∈S / Aj C is true,   t, if A1 A2 · · · An is true. (29) As we can see from (28) and (29), (C|A1 )∧· · ·∧(C|An )∧(C|(A1 ∨A2 ∨· · ·∨An )) and (C|A1 )∧· · ·∧(C|An ) coincide when A1 ∨· · ·∨An is true; then, by Theorem 3 it holds that t = z, so that (C|A1 ) ∧ · · · ∧ (C|An ) ∧ (C|(A1 ∨ A2 ∨ · · · ∨ An )) = (C|A1 ) ∧ · · · ∧ (C|An ), that is condition (ii) is satisfied. Moreover,    1, if A1 A2 · · · An C is true,  0, if (A V 1 ∨ A2V∨ · · · ∨ An )C is true, C|(A1 ∨ A2 ∨ · · · ∨ An ) = 1, if  j ∈S / Aj C is true, i∈S Ai   w, if A1 A2 · · · An is true, (30) where w = P (C|(A1 ∨ A2 ∨ · · · ∨ An )). Based on (28) and (30), it holds that (C|A1 ) ∧ · · · ∧ (C|An ) ≤ C|(A1 ∨ A2 ∨ · · · ∨ An ) conditionally on A1 ∨ · · · ∨ An being true. Then, from Theorem 6 it holds that t ≤ w; thus (C|A1 )∧· · ·∧(C|An ) ≤ C|(A1 ∨A2 ∨· · ·∨An ), that is condition (iii) is satisfied. 9.3 Iterated conditioning and p-entailment We now briefly describe a characterization of p-entailment of a conditional event E3 |H3 from a p-consistent family {E1 |H1 , E2 |H2 }, which exploits a suitable notion of iterated conditioning. Definition 14 Let be given n + 1 conditional events E1 |H1 , . . . , En+1 |Hn+1 , with (E1 |H1 ) ∧ · · · ∧ (En |Hn ) 6= 0. We denote by (En+1 |Hn+1 )|((E1 |H1 ) ∧ · · · ∧ (En |Hn )) = (En+1 |Hn+1 )|Cn the random quantity (E1 |H1 ) ∧ · · · ∧ (En+1 |Hn+1 ) + µ(1 − (E1 |H1 ) ∧ · · · ∧ (En |Hn )) = = Cn+1 + µ(1 − Cn ), where µ = P[(En+1 |Hn+1 )|Cn ]. 24 Angelo Gilio, Giuseppe Sanfilippo We observe that, based on the betting metaphor, the quantity µ is the amount to be paid in order to receive the amount Cn+1 + µ(1 − Cn ). Definition 14 generalizes the notion of iterated conditional (E2 |H2 )|(E1 |H1 ) given in previous papers (see, e.g., [25, 26, 29]). We also observe that, defining P(Cn ) = zn and P(Cn+1 ) = zn+1 , by the linearity of prevision it holds that µ = zn+1 +µ(1−zn ); then, zn+1 = µzn , that is P(Cn+1 ) = P[(En+1 |Hn+1 )|Cn ]P(Cn ), which is the compound prevision theorem. By applying Definition 14 with n = 2, given a p-consistent family {E1 |H1 , E2 |H2 } and a further event E3 |H3 , it can be proved that ([22]) {E1 |H1 , E2 |H2 } p-entails E3 |H3 ⇐⇒ (E3 |H3 )|((E1 |H1 ) ∧ (E2 |H2 )) = 1 , that is: {E1 |H1 , E2 |H2 } p-entails E3 |H3 if and only if the iterated conditional (E3 |H3 )|((E1 |H1 ) ∧ (E2 |H2 )) is constant and equal to 1. 10 From non p-valid to p-valid inference rules In this section we first examine some non p-valid inference rules, by showing that conditions (ii) and (iii) of Theorem 18 are not satisfied. Then, we illustrate by an example two different methods which allow to get p-valid inference rules starting by non p-valid ones. 10.1 Some non p-valid inference rules We start by showing that Transitivity is not p-valid. Transitivity. In this rule the p-consistent premise set is {C|B, B|A} and the conclusion is C|A. The rule is not p-valid ([21]), indeed we can show that (C|B) ∧ (B|A) ∧ (C|A) 6= (C|B) ∧ (B|A) and (C|B) ∧ (B|A)  C|A. Defining P (B|A) = x, P (BC|A) = y, P (C|A) = t, P[(C|B) ∧ (B|A) ∧ (C|A)] = µ, P[(C|B) ∧ (B|A)] = z, we have  1,     0,      0, (C|B) ∧ (B|A) ∧ (C|A) = (C|B) ∧ (BC|A) = 0,    y,     0,   µ, if if if if if if if ABC is true, ABC is true, ABC is true, ABC is true, ABC is true, ABC is true, AB is true, (31) Generalized Logical Operations among Conditional Events 25 and  1,     0,      0, (C|B) ∧ (B|A) = 0,   x,      0,   z, if if if if if if if ABC is true, ABC is true, ABC is true, ABC is true, ABC is true, ABC is true, AB is true. (32) Then, as (in general) x 6= y, it holds that (C|B) ∧ (B|A) ∧ (C|A) 6= (C|B) ∧ (B|A), so that condition (ii) is not satisfied. Moreover,    1,   0,      0, C|A = 0,   t,      t,   t, if if if if if if if ABC is true, ABC is true, ABC is true, ABC is true, ABC is true, ABC is true, AB is true. (33) Then, by observing that (in general) x  t it follows that (C|B)∧(B|A)  C|A, so that condition (iii) is not satisfied. Therefore, Transitivity rule is not pvalid. Denial of the antecedent. We consider the rule where the premise set is {A, C|A} and the conclusion is C. The premise set {A, C|A} is p-consistent because, by applying the Algorithm 1, the assessment P (A) = P (C|A) = 1 is coherent. We verify that A ∧ (C|A) ∧ C 6= A ∧ (C|A) and that A ∧ (C|A)  C, that is the Denial of the antecedent is not p-valid. We set P (C|A) = y, then and   0, if A is true, A ∧ (C|A) ∧ C = 0, if AC is true,  y, if AC is true,   0, if A is true, A ∧ (C|A) = y, if AC is true,  y, if AC is true. Assuming y > 0, when AC is true it holds that A ∧ (C|A) ∧ C = 0 < y = A ∧ (C|A), A ∧ (C|A) = y > 0 = C, thus: A ∧ (C|A) ∧ C 6= A ∧ (C|A) and A ∧ (C|A)  C, that is conditions (ii) and (iii) are not satisfied. 26 Angelo Gilio, Giuseppe Sanfilippo Affirmation of the consequent. We consider the rule where the (p-consistent) premise set is {C, C|A} and the conclusion is A. We verify that C ∧(C|A)∧A 6= C ∧ (C|A) and C ∧ (C|A)  A, that is the Affirmation of the consequent rule is not p-valid. We set P (C|A) = y, then  1, if AC is true, C ∧ (C|A) ∧ A = AC = 0, if AC is true, and   1, if AC is true, C ∧ (C|A) = 0, if C is true,  y, if AC is true. (34) Assuming y > 0, when AC is true it holds that C ∧ (C|A) ∧ A = 0 < y = C ∧ (C|A), C ∧ (C|A) = y > 0 = A, thus: C ∧ (C|A) ∧ A 6= C ∧ (C|A) and C ∧ (C|A)  A, that is conditions (ii) and (iii) are not satisfied. Remark 14 We now will make a comparison between the two objects C ∧(C|A) and C|(A ∨ C), by showing they do not coincide. Defining P (C|(A ∨ C)) = t, it holds that   1, if AC is true, (35) C|(A ∨ C) = 0, if C is true,  t, if AC is true. It could seem, from (34) and (35), that y and t should be equal and then C ∧(C|A) and C|(A∨C) should coincide. However, in this case the conditioning event for C ∧ (C|A) is Ω ∨ A = Ω, so that the disjunction of the conditioning events is Ω ∨ (A ∨ C) = Ω; the two objects C ∧ (C|A) and C|(A ∨ C) do not coincide conditionally on Ω; then C ∧ (C|A) and C|(A ∨ C) do not coincide (condition (i) of Theorem 3 is not satisfied). We also observe that, defining P(C ∧(C|A)) = µ, (in general) µ does not belong to the set {1, 0, y} of possible values of C ∧ (C|A), because µ is a linear convex combination of the values {1, 0, y}. As a further aspect, we verify below that t ≤ µ ≤ y. The constituents generated by {A, C} are: AC, AC, AC, AC; then, the associated values for the random vector (C|(A ∨ C), C ∧ (C|A), C|A) are (1, 1, 1) , (0, 0, 0) , (t, y, y) , (0, 0, y). (36) Based on Theorem 6, we observe that: • C|(A ∨ C) ≤ C|A conditionally on A ∨ C ∨ A = A ∨ C, hence P (C|(A ∨ C)) = t ≤ y = P (C|A); • C ∧ (C|A) ≤ C|A conditionally on Ω ∨ A = Ω, hence P(C ∧ (C|A)) = µ ≤ y = P (C|A); • C|(A∨C) ≤ C ∧(C|A) conditionally on A∨C ∨Ω = Ω, hence P (C|(A∨C)) = t ≤ µ = P(C ∧ (C|A)). In other words: t ≤ µ ≤ y. We observe that these inequalities also follow because coherence requires that the prevision point (t, µ, y) must be a linear convex combination of points in (36). Generalized Logical Operations among Conditional Events 27 On combining evidence: An example from Boole. We now examine an example studied in [8, p. 632] (see also [33, Theorem 5.45]), where p-entailment does not hold. Indeed, it can be proved that the extension w = P (C|AB) of any (coherent) assessment (x, y) on {C|A, C|B} is coherent for every w ∈ [0, 1]. Using conditions (ii) and (iii) of Theorem 18, we show that the p-consistent family {C|A, C|B} does not p-entail C|AB. We set P (C|A) = x, P (C|B) = y, and P((C|A) ∧ (C|B)) = z; then,  1, if ABC is true,      0, if ABC is true, 1, if ABC is true,         is true, 0, if ABC  0, if (A ∨ B)C is true,  (C|A) ∧ (C|B) = 0, if ABC is true, = x, if ABC is true, (37)     is true, y, if ABC   x, if ABC is true,       y, if ABC is true, z, if AB is true.   z, if AB is true. Moreover, by defining P[(C|A) ∧ (C|AB)] = u, P[(C|B) ∧ (C|AB)] = v and P[(C|A) ∧ (C|B) ∧ (C|AB)] = t, we obtain    1, if ABC is true,    0, if (A ∨ B)C is true, (C|A) ∧ (C|B) ∧ (C|AB) = u, if ABC is true,   v, if ABC is true,    t, if AB is true. As in general x 6= u and y 6= v, then (C|A)∧(C|B)∧(C|AB) and (C|A)∧(C|B) do not coincide, so that condition (ii) is not satisfied. Moreover,  1, if ABC is true,     0, if ABC is true,       1, if ABC is true,  w, if ABC is true, (38) C|AB = w, if ABC is true, = 0, if ABC is true,    ABC is true, AB is true. w, if w, if      w, if ABC is true,   w, if AB is true, Based on (37) and (38), we can see that (C|A) ∧ (C|B)  C|(AB), so that condition (iii) is not satisfied. Thus, the inference from {C|A, C|B} to C|AB is not p-valid. 10.2 Two methods for constructing p-valid inference rules We now illustrate by an example two different methods by means of which, starting by a non p-valid inference rule, we get p-valid inference rules: a) to add a suitable premise; b) to add a suitable logical constraint. The further premise, or logical constraint, (must preserve p-consistency and) is determined by analyzing the possible values of conjunctions. 28 Angelo Gilio, Giuseppe Sanfilippo Weak Transitivity. In our example we start by the (non p-valid) Transitivity rule where the premise set is {C|B, B|A} and the conclusion is C|A. Method a). We add the premise A|(A ∨ B), so that the premise set is {C|B, B|A, A|(A ∨ B)}, while the conclusion is still C|A. The premise set {C|B, B|A, A|(A ∨ B)} is p-consistent; indeed as ABC 6= ∅, by evaluating P (ABC) = 1 we get P (C|B) = P (B|A) = P (A|(A ∨ B)) = 1. We show that (C|B) ∧ (B|A) ∧ (A|(A ∨ B)) ∧ (C|A) = (C|B) ∧ (B|A) ∧ (A|(A ∨ B)) and (C|B) ∧ (B|A) ∧ (A|(A ∨ B)) ≤ C|A. Defining P[(C|B) ∧ (B|A) ∧ (A|(A ∨ B)) ∧ (C|A)] = µ, we have  1, if ABC is true,     0, if ABC is true,      0, if ABC is true, (C|B) ∧ (B|A) ∧ (A|(A ∨ B)) ∧ (C|A) = 0, if ABC is true,   0, if ABC is true,      0, if ABC is true,   µ, if AB is true. Moreover, defining P[(C|B) ∧ (B|A) ∧ (A|(A ∨ B))] = z, we have    1, if ABC is true,     0, if ABC is true,    0, if ABC is true, (C|B) ∧ (B|A) ∧ (A|(A ∨ B)) = 0, if ABC is true,   0, if ABC is true,      0, if ABC is true,   z, if AB is true. Conditionally on A ∨ B being true it holds that (C|B) ∧ (B|A) ∧ (A|(A ∨ B)) ∧ (C|A) = (C|B) ∧ (B|A) ∧ (A|(A ∨ B)) = ABC|(A ∨ B). Then, by Theorem 3 we have (C|B) ∧ (B|A) ∧ (A|(A ∨ B)) ∧ (C|A) = (C|B) ∧ (B|A) ∧ (A|(A ∨ B)) = ABC|(A ∨ B), so that condition (ii) is satisfied. Finally, as ABC|(A ∨ B) ⊆ C|A, it holds that (C|B)∧(B|A)∧(A|(A∨B)) = ABC|(A∨B) ≤ C|A, so that condition (iii) is satisfied. Therefore this Weak Transitivity rule is p-valid. We observe that another p-valid version of Weak Transitivity would be obtained by adding the premise A|B instead of A|(A ∨ B). Method b). We add the logical constraint ABC = ∅, that is BC ⊆ A. The p-consistency of the premise set {C|B, B|A} is preserved because, as before ABC 6= ∅ and by evaluating P (ABC) = 1 we get P (C|B) = P (B|A) = 1. Based on (31), (32) , (33) it holds that  1, if ABC is true,     0, if ABC is true,    0, if ABC is true, (C|B) ∧ (B|A) ∧ (C|A) = (C|B) ∧ (BC|A) = 0, if ABC is true,      0, if ABC is true,   µ, if AB is true, Generalized Logical Operations among Conditional Events and  1,     0,    0, (C|B) ∧ (B|A) = 0,      0,   z, if if if if if if 29 ABC is true, ABC is true, ABC is true, ABC is true, ABC is true, AB is true. As we can see (C|B) ∧ (B|A) ∧ (C|A) = (C|B) ∧ (B|A) conditionally on A ∨ B being true. Then, by Theorem 3 condition (ii) is satisfied. Moreover,  1, if ABC is true,     0, if ABC is true,    0, if ABC is true, C|A = 0, if ABC is true,      t, if ABC is true,   t, if AB is true. Then, (C|B) ∧ (B|A) ≤ C|A conditionally on A ∨ B being true. Thus, by Theorem 6 condition (iii) is satisfied too. Therefore, under the logical constraint ABC = ∅, the family {C|B, B|A} p-entails C|A, which is another p-valid version of Weak Transitivity. We observe that in [21, Theorem 5] it has been shown that another p-valid version of Weak Transitivity is obtained by adding the probabilistic constraint P (A|(A ∨ B)) > 0, that is P (C|B) = 1, P (B|A) = 1, P (A|(A ∨ B)) > 0 =⇒ P (C|A) = 1. 11 Conclusions We generalized the notions of conjunction and disjunction of two conditional events to the case of n conditional events. We introduced the notion of negation and we showed that De Morgans Laws still hold. We also verified that the associative and commutative properties are satisfied. We studied the monotonicity property, by proving that Cn+1 ≤ Cn and Dn+1 ≥ Dn for every n. We computed the set of all coherent assessments on the family {Cn , En+1 |Hn+1 , Cn+1 }, by showing that Fréchet-Hoeffding bounds still hold in this case; then, we examined the (reverse) probabilistic inference from Cn+1 to the family {Cn , En+1 |Hn+1 }. Moreover, given a family F = {E1 |H1 , E2 |H2 , E3 |H3 } of three conditional events, with E1 , E2 , E3 , H1 , H2 , H3 logically independent, we determined the set Π of all coherent prevision assessments for the set of conjunctions {CS : ∅ = 6 S ⊆ {1, 2, 3}}. In particular, we verified that the set Π is the same in the case where H1 = H2 = H3 and we also considered the relation between conjunction and quasi-conjunction. By using conjunction we also characterized p-consistency and p-entailment; then, we examined several examples of p-valid inference rules. We briefly described a characterization of p-entailment, in the case of two premises, by using a suitable notion of iterated 30 Angelo Gilio, Giuseppe Sanfilippo conditioning. Then, after examining some non p-valid inference rules, we illustrated by an example two methods for constructing p-valid inference rules. In particular, we applied these methods to Transitivity by obtaining p-valid versions of the rule (Weak Transitivity). Future work could concern the extension of the results of this paper to more complex cases, with possible applications to the psychology of cognitive reasoning under uncertainty. This work should lead, for instance, to further developments of the results given in [41, 42]. Acknowledgements We thank the anonymous referees for their useful criticisms and suggestions. A Appendix Proof of Theorem 5. We observe that (ii) follows by (i), by replacing F by F ; indeed, by (i) it holds that D(F ) = C(F ) = C(F ). Then, it is enough to proof the assertion (i). We will prove the assertion by induction. Step 1: n = 1, F = {E1 |H1 }. We have D(F ) = E1 |H1 = 1 − E1 |H1 = E 1 |H1 = C(F ). Thus the assertion holds when n = 1. Step 2: n = 2, F = {E1 |H1 , E2 |H2 }. We set P (E1 |H1 ) = x, P (E2 |H2 ) = y, P[(E1 |H1 ) ∨ (E2 |H2 )] = w, P[(E 1 |H1 ) ∧ (E 2 |H2 )] = t. We observe that the family {E1 H1 ∨ E2 H2 , E 1 HE 2 H2 , H 1 E 2 H2 , E 1 H1 H 2 , H 1 H 2 } is a partition of the sure event Ω. Moreover, by Definitions 3 and 4 we have  0, if E1 H1 ∨ E2 H2 is true,    1,  if E 1 H1 E 2 H2 is true,  D(F ) = 1 − (E1 |H1 ) ∨ (E2 |H2 ) = 1 − x, if H 1 E 2 H2 is true, (39)    1 − y, if E 1 H1 H 2 is true,   1 − w, if H 1 H 2 is true. and  0,    1,   C(F ) = (E 1 |H1 ) ∧ (E 2 |H2 ) = 1 − x,    1 − y,   t, if if if if if E1 H1 ∨ E2 H2 is true, E 1 HE 2 H2 is true, H 1 E 2 H2 is true, E 1 H1 H 2 is true, H 1 H 2 is true. (40) We observe that D(F ) and C(F ) coincide when H1 ∨ H2 is true. Thus, by Theorem 3, P(D(F )) = P(C(F )) and hence 1 − w = t. Therefore D(F ) still coincides with C(F ) when H1 ∨ H2 is false, so that D(F ) = C(F ). Step 3: F = {E1 |H1 , E2 |H2 , . . . , En |Hn }. (Inductive Hypothesis) Let us assume that for any (strict) subset S ⊂ {1, . . . , n}, by defining FS = {Ei |Hi , i ∈ S}, it holds that D(FS ) = C(FS ). Now we will prove that D(FS ) = C(FS ) when S = {1, . . . , n}, in which case FS = F . By Definition 7 we have  if Sh′ 6= ∅,  0, Pm if Sh′′ = {1, 2, . . . , n}, (41) D(F ) = h=0 wh Ch , where wh = 1,  1 − y ′′′ , if S ′ = ∅ and S ′′′ 6= ∅. Sh h h Generalized Logical Operations among Conditional Events 31 We continue to use the subsets Sh′ , Sh′′ , Sh′′′ as defined in formula (9) also with the family V F ; moreover we set tS = P[ i∈S (E i |Hi )] = P[C(F S )]. Based on Definition 5, we have C(F ) =  if Sh′ 6= ∅,  0, 1, if Sh′′ = {1, 2, . . . , n}, h=0 zh Ch , where zh =  tS ′′′ , if Sh′ = ∅ and Sh′′′ 6= ∅. Pm Then, D(F ) − C(F ) = (42) h Pm h=0 (wh − zh )Ch , where  if Sh′ 6= ∅,  0, if Sh′′ = {1, 2, . . . , n}, wh − zh = 0,  1 − y ′′′ − t ′′′ , if S ′ = ∅ and S ′′′ 6= ∅. S S h h h (43) h By the inductive hypothesis, it holds that 1 − yS ′′′ = P[D(FS ′′′ )] = P[C(F S ′′′ )] = tS ′′′ for h h h h P h = 1, . . . , m, because Sh′′′ ⊂ {1, 2, . . . , n}. Then, D(F ) − C(F ) = m h=0 (wh − zh )Ch , where  0, h = 1, . . . , m, wh − zh = 1 − y ′′′ − t ′′′ , h = 0. (44) S0 S0 By recalling that S0′′′ = {1, 2, . . . , n}, D(F ) and C(F ) coincide when H1 ∨ H2 ∨ · · · ∨ Hn is true. Thus, by Theorem 3, P[D(F )] = P[C(F ))], that is 1 − yS ′′′ = tS ′′′ . Therefore D(F ) 0 0 still coincides with C(F ) when H1 ∨ H2 ∨ · · · ∨ Hn is false, so that D(F ) = C(F ). ⊓ ⊔ Proof of Theorem 6. (i) Assume that, for every (µ, ν) ∈ Π, the values of X|H and Y |K associated with the constituent Ch are such that X|H ≤ Y |K, for each Ch contained in H ∨ K; then for each given coherent assessment (µ, ν), by choosing s1 = 1, s2 = −1 in the random gain, we have G = H(X − µ) − K(Y − ν) = (X|H − µ) − (Y |K − ν) = (X|H − Y |K) + (ν − µ) . Then, by the hypothesis, GH∨K ≤ (ν − µ) and by coherence 0 = P(GH∨K ) ≤ ν − µ . Then µ ≤ ν, ∀(µ, ν) ∈ Π. (ii) By hypothesis, it holds that (XH + µH c )(H ∨ K) ≤ (Y K + νK c )(H ∨ K); moreover, from condition (i), µ ≤ ν for every (µ, ν) ∈ Π; then X|H = XH + µH c = (XH + µH c )(H ∨ K) + (XH + µH c )H c K c = (XH + µH c )(H ∨ K) + µH c K c ≤ (Y K + νK c )(H ∨ K) + νH c K c = (Y K + νK c )(H ∨ K) + (Y K + νK c )H c K c = Y K + νK c = Y |K . Vice versa, X|H ≤ Y |K trivially implies X|H ≤ Y |K when H ∨ K is true. ⊓ ⊔ Proof of Theorem 7. We distinguish three cases: (a) the value of Cn is 0, with some EiV|Hi false, i ≤ n; (b) the value of Cn is 1, with Ei |Hi true, i = 1, . . . , n; (c) the value of Cn is P[ i∈S (Ei |Hi )] = P(CS ) = xS , for some subset S ⊆ {1, 2, . . . , n}. Case (a). It holds that Cn+1 = 0 = Cn . Case (b). The value of Cn+1 is 1, or 0, or xn+1 , according to whether En+1 |Hn+1 is true, or false, or void; thus Cn+1 ≤ Cn . Case (c). We distinguish three cases: (i) En+1 |Hn+1 is true; (ii) En+1 |Hn+1 is false; (iii) En+1 |Hn+1 is void. In the case (i) the value of Cn+1 is xS , thus Cn+1 = Cn . In the case (ii) the value V of Cn+1 is 0, thus Cn+1 ≤ Cn . In the case (iii) the value of Cn+1 is xS∪{n+1} = P[ i∈S∪{n+1} (Ei |Hi )]; then, in order to prove that Cn+1 ≤ Cn , we need to prove that xS∪{n+1} ≤ xS . We proceed by induction on the cardinality of S, denoted by s. Let be s = 1, with CS = Ei |Hi , for some i ∈ {1, . . . , n}. We note that xS = P(Ei |Hi ) = xi , xS∪{n+1} = P((Ei |Hi ) ∧ (En+1 |Hn+1 )) = x{i,n+1} and by Theorem 4 it holds that xS∪{n+1} = x{i,n+1} ≤ xi = xS . Now, let be s ≥ 2 and xS∪{n+1} ≤ xS 32 Angelo Gilio, Giuseppe Sanfilippo for every s < n, so that, based on Definition 5, Cn+1 ≤ Cn when S is a strict subset of {1, 2, . . . , n}. If S = {1, 2, . . . , n}, as Ei |Hi is void for all i = 1, . . . , n + 1, it holds that Cn = P(Cn ) = x{1,...,n} and Cn+1 = x{1,...,n+1} = P(Cn+1 ) and, in order to prove that Cn+1 ≤ Cn , it remains to prove that P(Cn+1 ) ≤ P(Cn ). By applying Theorem 6, with X|H = Cn+1 = Zn+1 |(H1 ∨ · · · ∨ Hn+1 ) and Y |K = Cn = Zn |(H1 ∨ · · · ∨ Hn ), as Cn+1 ≤ Cn when H1 ∨ · · · ∨ Hn+1 is true (i.e., s < n ), it follows that P(Cn+1 ) ≤ P(Cn ); therefore Cn+1 ≤ Cn . ⊓ ⊔ Proof of Theorem 9. Case (i). We proceed by induction. The property is satisfied for n = 1; indeed, if C1 = E1 |H1 ∈ {1, 0, x1 }, where x1 = P(E1 |H1 ) ∈ [0, 1], then C1 ∈ [0, 1]. Let us assume that the property holds for k < n, that is Ck ∈ [0, 1], for every k < n. Based on Definition 5 we distinguish V three cases: (a) the value of Cn is 0; (b) the value of Cn is 1; (c) the value of Cn is P[ i∈S (Ei |Hi )] = xS , for some subset S ⊆ {1, 2, . . . , n}. In the cases (a) and (b), Cn ∈ [0, 1]. In the case (c), if S = {i1 , . . . , ik } ⊂ {1, 2, . . . , n}, then Cn ∈ [0, 1], because V V xS = P( kj=1 (Eij |Hij )) is a possible value of Ck = kj=1 (Eij |Hij ), with k < n. Finally, if S = {1, 2, . . . , n} (that is the conditioning events H1 , . . . , Hn are all false), then Cn = P(Cn ) and P(Cn ) ∈ [0, 1] because the values of Cn restricted to H1 ∨ · · · ∨ Hn all belong to [0, 1]. Therefore Cn ∈ [0, 1]. By a similar reasoning, based on Definition 7 we can prove that Dn ∈ [0, 1]. ⊓ ⊔ Proof of Theorem 10. Let C0 , . . . , Cm , with m = 3n − 1 be the constituents associated with Fn+1 = {E1 |H1 , . . . , En+1 |Hn+1 }, where C0 = H 1 · · · H n+1 . With each Ch , h = 1, . . . , m, we associate the point Qh = (qh1 , qh2 , qh3 ), which represents the value of the random vector (Cn , En+1 |Hn+1 , Cn+1 ) when Ch is true, where qh1 is the value of Cn , qh2 is the value of En+1 |Hn+1 , and qh3 is the value of Cn+1 . With C0 it is associated the point Q0 = (µn , xn+1 , µn+1 ) = M. We observe that the set of points {Qh , h = 1, . . . , m} contains in particular the points Q1 = (1, 1, 1) , Q2 = (1, 0, 0) , Q3 = (0, 1, 0) , Q4 = (0, 0, 0) , which are respectively associated with the following constituents or logical disjunction of constituents E1 H1 · · · En Hn En+1 Hn+1 , E1 H1 · · · En Hn E n+1 Hn+1 , (E 1 H1 ∨ · · · ∨ E n Hn ) ∧ En+1 Hn+1 , (E 1 H1 ∨ · · · ∨ E n Hn ) ∧ E n+1 Hn+1 . Based on Remark 7, we need to prove that the set of coherent assessments Π on {Cn , En+1 |Hn+1 , Cn+1 } coincides with the convex hull I of Q1 , Q2 , Q3 , Q4 . We recall that coherence of (µn , xn+1 , µn+1 ) implies coherence of all the sub-assessments on the associated subfamilies of {Cn , En+1 |Hn+1 , Cn+1 }. The coherence of the single assessments µn on Cn , or xn+1 on En+1 |Hn+1 , or µn+1 on Cn+1 , simply amounts to conditions µn ∈ [0, 1] , xn+1 ∈ [0, 1] , µn+1 ∈ [0, 1] , respectively. Then, by the hypothesis of logical independence, the sub-assessment (µn , xn+1 ) is coherent, for every (µn , xn+1 ) ∈ [0, 1]2 . By Remark 6, the coherence of the subassessments (µn , µn+1 ) and (xn+1 , µn+1 ) amounts to the conditions 0 ≤ µn+1 ≤ µn ≤ 1 and 0 ≤ µn+1 ≤ xn+1 ≤ 1. Finally, assuming that the above conditions are satisfied, to prove coherence of (µn , xn+1 , µn+1 ), by Theorem 1, it is enough to show that the point (µn , xn+1 , µn+1 ) belongs to the convex hull of the points Q1 , . . . , Qm . Moreover, in order M belongs to the convex hull of Q1 , . . . , Qm the following system (Σ) must solvable M= m X h=1 λh Qh , m X λh = 1, λh ≥ 0, ∀h. (45) h=1 We show that the convex hull of the points Q1 , . . . , Qm coincides with the convex hull I of the points Q1 , Q2 , Q3 , Q4 , described in Remark 7, because all the other points Q5 , . . . , Qm , are linear convex combinations of Q1 , Q2 , Q3 , Q4 , that is Qh ∈ I for each h = 5, . . . , m. Generalized Logical Operations among Conditional Events 33 We examine the following different cases which depend on the logical value of En+1 |Hn+1 : a) En+1 |Hn+1 is true; b) En+1 |Hn+1 is false; c) En+1 |Hn+1 is void. a) In this case Qh = (qh1 , 1, qh1 ) = qh1 (1, 1, 1) + (1 − qh1 )(0, 1, 0) = qh1 Q1 + (1 − qh1 )Q3 . b) In this case Qh = (qh1 , 0, 0) = qh1 (1, 0, 0) + (1 − qh1 )(0, 0, 0) = qh1 Q2 + (1 − qh1 )Q4 . c) Vn In this case Qh = (qh1 , xn+1 , qh3 ) and we distinguish Wn the following subcases: (i) i=1 Ei Hi true, so that Qh = (1, xn+1 , xn+1 ); (ii) i=1 E i Hi true, so that Qh = (0, xn+1 , 0); (iii) Ei |Hi void, for every i ∈ S and Ei |Hi true for every i ∈ {1, 2, . . . , n} \ S,. for some ∅ 6= S ⊂ {1, 2, . . . , n}, so that Qh = (xS , xn+1 , xS∪{n+1} ). In subcase (i) it holds that Qh = (1, xn+1 , xn+1 ) = xn+1 (1, 1, 1) + (1 − xn+1 )(1, 0, 0) = xn+1 Q1 + (1 − xn+1 )Q2 . In subcase (ii) it holds that Qh = (0, xn+1 , 0) = xn+1 (0, 1, 0) + (1 − xn+1 )(0, 0, 0) = xn+1 Q3 + (1 − xn+1 )Q4 . In subcase (iii), it can be verified by a finite iterative procedure that the point Qh = (xS , xn+1 , xVS∪{n+1} ) ∈ I. We examine the different cases on the cardinality s of S. We recall that i∈S (Ei |Hi ) is denoted by CS . Step 1. s = 1. Without loss of generality we assume S = {1}, so that Qh = (xS , xn+1 , xS∪{n+1} ) = (x1 , xn+1 , x{1,n+1} ), where x1 = P (E1 |H1 ), x{1,n+1} = P[(E1 |H1 )∧(En+1 |Hn+1 )]. By Theorem 4 it holds that max{xS +xn+1 −1, 0} ≤ xS∪{n+1} ≤ min{xS , xn+1 }, with (xS , xn+1 ) ∈ [0, 1]2 . In other words, Qh = (xS , xn+1 , xS∪{n+1} ) ∈ I. The reasoning is the same for S = {i}, i = 2, . . . , n. Step 2. s = 2. Without loss of generality we assume S = {1, 2}, so that xS = P[(E1 |H1 ) ∧ (E2 |H2 )], xS∪{n+1} = P[CS∪{n+1} ] = P[(E1 |H1 ) ∧ (E2 |H2 ) ∧ (En+1 |Hn+1 )]. ∗ , the constituents associated with {E |H , i ∈ S ∪ {n + We denote by C0∗ , C1∗ , . . . , Cm ∗ i i V ∗ ∗ ∗ 1}}, where C0 = i∈S∪{n+1} H i . Moreover, with Ch , h = 0, 1, . . . , m , we asso∗ , q ∗ , q ∗ ) which represents the value of the random vector ciate the point Q∗h = (qh1 h2 h3 {CS , En+1 |Hn+1 , CS∪{n+1} } when Ch∗ is true. We observe that Q∗0 = (xS , xn+1 , xS∪{n+1} ) and that Q1 , Q2 , Q3 , Q4 still belongs to the set of points {Q∗h , h = 1, . . . , m∗ }. In order that the assessment (xS , xn+1 , xS∪{n+1} ) on {CS , En+1 |Hn+1 , CS∪{n+1} } be coherent, the point Q∗0 = (xS , xn+1 , xS∪{n+1} ) must belong to the convex hull of points Q∗1 , Q∗2 , . . . , Q∗m . We show that for each point Q∗h 6= Qi , i = 1, 2, 3, 4, it holds that Q∗h ∈ I. By repeating the previous reasoning we only need to analyze the subcase (iii) of case c). We have to show that, for every nonempty subset S ′ ⊂ S, the point Q∗h = (xS ′ , xn+1 , xS ′ ∪{n+1} ) belongs to the convex hull I of Q1 , . . . , Q4 . As S = {1, 2}, it holds that S ′ = {1}, or S ′ = {2}, so that Q∗h = (xS ′ , xn+1 , xS ′ ∪{n+1} ) = (x1 , xn+1 , x{1,n+1} ), or Q∗h = (x2 , xn+1 , x{2,n+1} ). By Step 1, in both cases Q∗h ∈ I. Thus Qh = (xS , xn+1 , xS∪{n+1} ) ∈ I. In other words, max{xS + xn+1 − 1, 0} ≤ xS∪{n+1} ≤ min{xS , xn+1 }, with (xS , xn+1 ) ∈ [0, 1]2 . The reasoning is the same for every S = {i, j} ⊂ {1, 2, . . . , n}. ............................................................................................................... Step k + 1. s = k + 1, 2 < k + 1 < n. By induction, assume that (xS ′ , xn+1 , xS ′ ∪{n+1} ) ∈ I for every S ′ = {i1 , i2 , . . . , ik } ⊂ {1, 2, . . . , n}. Then, by the previous reasoning, it follows that Qh = (xS , xn+1 , xS∪{n+1} ) ∈ I for every S = {i1 , i2 , . . . , ik+1 }. In other words, max{xS + xn+1 − 1, 0} ≤ xS∪{n+1} ≤ min{xS , xn+1 }, with (xS , xn+1 ) ∈ [0, 1]2 , for every S = {i1 , i2 , . . . , ik+1 }. Thus, by this iterative procedure, also in the subcase (iii) of case c) it holds that Qh ∈ I. Then, Qh ∈ I, h = 5, . . . , m. Finally, the condition (45) is equivalent to M ∈ I, so that the assessment M is coherent if and only if (µn , xn+1 ) ∈ [0, 1]2 , max{µn + xn+1 − 1, 0} ≤ µn+1 ≤ min{µn , xn+1 }. ⊓ ⊔ 34 Angelo Gilio, Giuseppe Sanfilippo Proof of Theorem 15. The computation of the set Π is based on Section 2.2. The constituents Ch ’s and the points Qh ’s associated with (F , M) are illustrated in Table 1. We recall that Table 1 Constituents Ch ’s and corresponding points Qh ’s associated with (F , M), where M = (x1 , x2 , x3 , x12 , x13 , x23 , x123 ) is a prevision assessment on F = {E1 |H1 , E2 |H2 , E3 |H3 , (E1 |H1 )∧(E2 |H2 ), (E1 |H1 )∧(E3 |H3 ), (E2 |H2 )∧(E3 |H3 ), (E1 |H1 )∧ (E2 |H2 ) ∧ (E3 |H3 )}. C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20 C21 C22 C23 C24 C25 C26 C0 Ch E1 H 1 E2 H2 E3 H3 E1 H 1 E2 H2 E 3 H 3 E1 H 1 E2 H2 H 3 E1 H 1 E 2 H2 E3 H 3 E1 H 1 E 2 H2 E 3 H 3 E1 H 1 E 2 H2 H 3 E1 H 1 H 2 E3 H 3 E1 H 1 H 2 E 3 H 3 E1 H 1 H 2 H 3 E 1 H 1 E2 H2 E3 H 3 E 1 H1 E2 H2 E 3 H3 E 1 H1 E2 H2 H 3 E 1 H1 E 2 H2 E3 H3 E 1 H1 E 2 H2 E 3 H3 E 1 H1 E 2 H2 H 3 E 1 H 1 H 2 E3 H 3 E 1 H1 H 2 E 3 H3 E 1 H1 H 2 H 3 H 1 E2 H 2 E3 H 3 H 1 E2 H2 E 3 H3 H 1 E2 H2 H 3 H 1 E 2 H 2 E3 H 3 H 1 E 2 H2 E 3 H3 H 1 E 2 H2 H 3 H 1 H 2 E3 H3 H 1 H 2 E 3 H3 H1H2H3 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 x1 x1 x1 x1 x1 x1 x1 x1 x1 1 1 1 0 0 0 x2 x2 x2 1 1 1 0 0 0 x2 x2 x2 1 1 1 0 0 0 x2 x2 x2 1 0 x3 1 0 x3 1 0 x3 1 0 x3 1 0 x3 1 0 x3 1 0 x3 1 0 x3 1 0 x3 Qh 1 1 1 0 0 0 x2 x2 x2 0 0 0 0 0 0 0 0 0 x1 x1 x1 0 0 0 x12 x12 x12 1 0 x3 1 0 x3 1 0 x3 0 0 0 0 0 0 0 0 0 x1 0 x13 x1 0 x13 x1 0 x13 1 0 x3 0 0 0 x2 0 x23 1 0 x3 0 0 0 x2 0 x23 1 0 x3 0 0 0 x2 0 x23 1 0 x3 0 0 0 x2 0 x23 0 0 0 0 0 0 0 0 0 x1 0 x13 0 0 0 x12 0 x123 Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20 Q21 Q22 Q23 Q24 Q25 Q26 Q0 Qh = (qh1 , . . . , qh7 ) represents the value associated with Ch of the random vector (E1 |H1 , E2 |H2 , E3 |H3 , (E1 |H1 ) ∧ (E2 |H2 ), (E1 |H1 ) ∧ (E3 |H3 ), (E2 |H2 ) ∧ (E3 |H3 ), (E1 |H1 ) ∧ (E2 |H2 ) ∧ (E3 |H3 )), h = 1, . . . , 26. With C0 = H1 H2 H3 it is associated Q0 = M. Denoting by I the convex hull generated by Q1 , Q2 , . . . , Q26 , the coherence of the prevision assessment M on F requires that the condition M ∈ I be satisfied; this amounts to the solvability of the following system (Σ) M= P26 h=1 λh Qh , P26 h=1 λh = 1, λh ≥ 0, h = 1, . . . , 26 . Generalized Logical Operations among Conditional Events 35 We observe that Q3 = x3 Q1 + (1 − x3 )Q2 , Q6 = x3 Q4 + (1 − x3 )Q5 , Q7 = x2 Q1 + (1 − x2 )Q4 , Q8 = x2 Q2 + (1 − x2 )Q5 , Q9 = x23 Q1 + (x2 − x23 )Q2 + (x3 − x23 )Q4 + (x23 − x2 − x3 + 1)Q5 , Q12 = x3 Q10 + (1 − x3 )Q11 , Q15 = x3 Q13 + (1 − x3 )Q14 , Q16 = x2 Q10 + (1 − x2 )Q13 , Q17 = x2 Q11 + (1 − x2 )Q14 , Q18 = x23 Q10 + (x2 − x23 )Q11 + (x3 − x23 )Q13 + (x23 − x2 − x3 + 1)Q14 , Q19 = x1 Q1 + (1 − x1 )Q10 , Q20 = x1 Q2 + (1 − x1 )Q11 , Q21 = x13 Q1 + (x1 − x13 )Q2 + (x3 − x13 )Q10 + (x13 − x1 − x3 + 1)Q11 , Q22 = x1 Q4 + (1 − x1 )Q13 , Q23 = x1 Q5 + (1 − x1 )Q14 , Q24 = x13 Q4 + (x1 − x13 )Q5 + (x3 − x13 )Q13 + (x13 − x1 − x3 + 1)Q14 , Q25 = x12 Q1 + (x1 − x12 )Q4 + (x2 − x12 )Q10 + (x12 − x1 − x2 + 1)Q13 , Q26 = x12 Q2 + (x1 − x12 )Q5 + (x2 − x12 )Q11 + (x12 − x1 − x2 + 1)Q14 . Thus, I coincides with the convex hull of the points Q1 , Q2 , Q4 , Q5 , Q10 , Q11 , Q13 , Q14 . For the sake of simplicity, we set: Q′1 = Q1 , Q′2 = Q2 , Q′3 = Q4 , Q′4 = Q5 , Q′5 = Q10 , Q′6 = Q11 , Q′7 = Q13 , Q′8 = Q14 . Then, the condition M ∈ I amounts to the solvability of the following system P P8 ′ ′ (Σ ′ ) M = 8h=1 λ′h Q′h , h=1 λh = 1, λh ≥ 0, h = 1, . . . , 8 that is  ′  λ1 + λ′2 + λ′3 + λ′4 = x1 , λ′1 + λ′2 + λ′5 + λ′6 = x2 , λ′1 + λ′3 + λ′5 + λ′7 = x3 , ′ λ′ + λ′ = x12 , λ′1 + λ′3 = x13 , λ′1 + λ′5 = x23 , λ′1 = x123 , (Σ ) P  18 2 ′ ′ h=1 λh = 1, λh ≥ 0, h = 1, 2, . . . , 8. System (Σ ′ ) can be written as  ′  λ1 = x123 , λ′2 = x12 − x123 , λ′3 = x13 − x123 , λ′4 = x1 − x12 − x13 + x123 , ′ (Σ ) λ′5 = x23 − x123 , λ′6 = x2 − x12 − x23 + x123 , λ′7 = x3 − x13 − x23 + x123 ,  ′ λ8 = 1 − x1 − x2 − x3 + x12 + x13 + x23 − x123 , λ′h ≥ 0, h = 1, 2, . . . , 8. As it can be verified, by non-negativity of λ′1 , . . . , λ′8 it follows that (Σ ′ ) is solvable (with a unique solution) if and only if  x123 ≥ max{0, x12 + x13 − x1 , x12 + x23 − x2 , x13 + x23 − x3 }, (46) x123 ≤ min{x12 , x13 , x23 , 1 − x1 − x2 − x3 + x12 + x13 + x23 }, or, in a more explicit way, if and only if the following conditions are satisfied  (x1 , x2 , x3 ) ∈ [0, 1]3 ,    max{x  1 + x2 − 1, x13 + x23 − x3 , 0} ≤ x12 ≤ min{x1 , x2 },     max{x1 + x3 − 1, x12 + x23 − x2 , 0} ≤ x13 ≤ min{x1 , x3 }, max{x2 + x3 − 1, x12 + x13 − x1 , 0} ≤ x23 ≤ min{x2 , x3 },   1 − x1 − x2 − x3 + x12 + x13 + x23 ≥ 0,      x123 ≥ max{0, x12 + x13 − x1 , x12 + x23 − x2 , x13 + x23 − x3 },  x123 ≤ min{x12 , x13 , x23 , 1 − x1 − x2 − x3 + x12 + x13 + x23 }. (47) Notice that the conditions in (47) coincide with that ones in (18). Moreover, assuming (Σ ′ ) solvable, with the solution (λ′1 , . . . , λ′8 ), we associate the vector (λ1 , λ2 , . . . , λ26 ), with λ1 = λ′1 , λ2 = λ′2 , λ4 = λ′3 , λ5 = λ′4 , λ10 = λ′5 , λ11 = λ′6 , λ13 = / {1, 2, 4, 5, 10, 11, 13, 14}, whichWis a solution of (Σ). Moreλ′7 , λ14 = λ′8 , λh = 0, h ∈ over, defining 2, 4, 5, 10, 11, 13, 14}, it holds that Ph∈J Ch = H1 ∧ H2 ∧ H3 . P J = {1,P Therefore, h:Ch ⊆Hi λh = 1, i = 1, 2, 3, h∈J λh = h:ChP ⊆H1 H2 H3 λh = 1 and hence P h:Ch ⊆Hi ∨Hj λh = 1, i 6= j, h:Ch ⊆H1 ∨H2 ∨H3 λh = 1; thus, by (2), I0 = ∅. Then, by Theorem 2, the solvability of (Σ) is also sufficient for the coherence of M. Finally, Π is the set of conditional prevision assessments (x1 , x2 , x3 , x12 , x13 , x23 , x123 ) which satisfy the conditions in (18). ⊓ ⊔ 36 Angelo Gilio, Giuseppe Sanfilippo Proof of Theorem 16. Notice that, (Ei |H)∧(Ej |H) = (Ei Ej )|H, for every {i, j} ⊂ {1, 2, 3}, and (E1 |H)∧(E2 |H)∧ (E3 |H) = (E1 E2 E3 )|H. Then F = {E1 |H, E2 |H, E3 |H, (E1 E2 )|H, (E1 E3 )|H, (E2 E3 )|H, (E1 E2 E3 )|H}. The computation of the set Π is based on Section 2.2. The constituents Ch ’s and the points Qh ’s associated with (F , M) are illustrated in Table 2. We re- Table 2 Constituents Ch ’s and corresponding points Qh ’s associated with (F , M), where M = (x1 , x2 , x3 , x12 , x13 , x23 , x123 ) is a prevision assessment on F = {E1 |H, E2 |H, E3 |H, (E1 E2 )|H, (E1 E3 )|H, (E2 E3 )|H, (E1 E2 E3 )|H}. C1 C2 C3 C4 C5 C6 C7 C8 C0 Ch E1 E2 E3 H E1 E2 E 3 H E1 E 2 E3 H E1 E 2 E 3 H E 1 E2 E3 H E 1 E2 E 3 H E 1 E 2 E3 H E1E2E3H H 1 1 1 1 0 0 0 0 x1 1 1 0 0 1 1 0 0 x2 Qh 1 1 0 0 0 0 0 0 x12 1 0 1 0 1 0 1 0 x3 1 0 1 0 0 0 0 0 x13 1 0 0 0 1 0 0 0 x23 1 0 0 0 0 0 0 0 x123 Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q0 call that Qh = (qh1 , . . . , qh7 ) represents the value associated with Ch of the random vector (E1 |H, E2 |H, E3 |H, (E1 E2 )|H, (E1 E3 )|H, (E2 E3 )|H, (E1 E2 E3 )|H), h = 1, . . . , 8. With C0 = H it is associated Q0 = M. Denoting by I the convex hull generated by Q1 , Q2 , . . . , Q8 , as all the conditioning events coincide with H the assessment M on F is coherent if and only if M ∈ I; that is, if and only if the following system is solvable P P8 (48) M = 8h=1 λh Qh , h=1 λh = 1, λh ≥ 0, h = 1, . . . , 8. The points Q1 , Q2 , . . . , Q8 coincide with the points Q′1 , Q′2 , . . . , Q′8 in the proof of Theorem 15, respectively. Then, system (48) coincides with system (Σ)′ in the proof of Theorem 15. Therefore, it is solvable if and only if the conditions in (18) are satisfied. In other words, the set Π of all coherent assessments M on F coincides with the set of points (x1 , x2 , x3 , x12 , x13 , x23 , x123 ) which satisfy the conditions in (18). ⊓ ⊔ Proof of Theorem 18. In order to prove the theorem it is enough to prove the following implications: a) (i) ⇒ (ii); b) (ii) ⇒ (iii); c) (iii) ⇒ (i). a) (i) ⇒ (ii). We recall that F p-entails En+1 |Hn+1 if and only if either Hn+1 ⊆ En+1 , or there exists a nonempty FΓ ⊆ F , where Γ ⊆ {1, . . . , n}, such that QC(FΓ ) implies En+1 |Hn+1 (see, e.g. [27, Theorem 6]). Let us first consider the case where Hn+1 ⊆ En+1 . In this case P (En+1 |Hn+1 ) = 1 and En+1 |Hn+1 = Hn+1 + H n+1 = 1. We have Cn+1 = Cn ∧ (En+1 |Hn+1 ), with En+1 |Hn+1 = 1. We distinguish two cases: (α) Hn+1 is true; (β) Hn+1 is false. In case (α), by Definition 5 and Remark 4, as En+1 |Hn+1 is true it follows that the values of Cn+1 and of Cn coincide. In case (β), let C0 , . . . , Cm be the constituents ′ associated associated with F , where C0 = H 1 · · · H n . Then, the constituents C0′ , . . . , Cm ′ = C H with F ∪ {En+1 |Hn+1 } and contained in H n+1 are C0′ = C0 H n+1 , . . . , Cm m n+1 . For each constituent Ch′ , h = 1, . . . , m, by formula (10) the corresponding value of Cn is ′ the value of C zh ∈ {1, 0, xS ′′′ }. We denote by zh n+1 associated with zh and we recall that h ′ = 1; if z = 0, then Ch′ ⊆ H n+1 , h = 0, 1, . . . , m. For each index h, if zh = 1, then zh h ′ = 0; if z = x ′′′ , then z ′ = x ′′′ zh . We set P (E |H n+1 n+1 ) = xn+1 ; in our case h Sh Sh ∪{n+1} h xn+1 = 1. Moreover, by Theorem 10 max{xS ′′′ + xn+1 − 1, 0} ≤ xS ′′′ ∪{n+1} ≤ min{xS ′′′ , xn+1 }; h h h therefore xS ′′′ ∪{n+1} = xS ′′′ . Then, the values of Cn+1 and of Cn coincide for every Ch′ . h h Thus, Cn+1 = Cn when Hn+1 ⊆ En+1 . Generalized Logical Operations among Conditional Events 37 We consider now the case where there exists FΓ ⊆ F , FΓ 6= ∅, such that QC(FΓ ) ⊆ En+1 |Hn+1 . First of all we prove that C(FΓ ∪ {En+1 |Hn+1 }) = C(FΓ ). For the sake of simplicity, we set C(FΓ ) = CΓ and C(FΓ ∪ {En+1 |Hn+1 }) = CΓ ∪{n+1} . If the value of CΓ is 1 (because all the conditional events in FΓ are true), then QC(FΓ ) is true and hence En+1 |Hn+1 is also true; thus CΓ ∪{n+1} = 1, so that CΓ ∪{n+1} = CΓ . If the value of CΓ is 0 (because some conditional event in FΓ is false), then CΓ ∪{n+1} is 0 too, so that CΓ ∪{n+1} = CΓ . If CΓ is xS for some nonempty subset S ⊂ Γ (that is, all the conditional events in FS are void and the other ones in FΓ \S are true), then QC(FΓ ) is true and and hence En+1 |Hn+1 is also true; thus CΓ ∪{n+1} = xS , so that CΓ ∪{n+1} = CΓ . If CΓ is xΓ because all the conditional events in FΓ are void, then QC(FΓ ) is void and for En+1 |Hn+1 there are two cases: 1) En+1 |Hn+1 true; 2) En+1 |Hn+1 void. In case 1), by also recalling Remark 4, it holds that CΓ ∪{n+1} = xΓ so that CΓ ∪{n+1} = CΓ . In case 2) it holds that CΓ ∪{n+1} = xΓ ∪{n+1} , where xΓ ∪{n+1} = P(CΓ ∪{n+1} ). Now, we observe that the random quantities CΓ and CΓ ∪{n+1} coincide conditionally on W i∈Γ ∪{n+1} Hi being true; then by Theorem 3 it holds that P(CΓ ) = P(CΓ ∪{n+1} ), that is xΓ = xΓ ∪{n+1} ; thus CΓ ∪{n+1} = CΓ . Finally, denoting by Γ c the set {1, . . . , n} \ Γ , by the associative property of conjunction we obtain Cn+1 = Cn ∧ En+1 |Hn+1 = CΓ c ∧ CΓ ∧ En+1 |Hn+1 = CΓ c ∧ CΓ = Cn . b) (ii) ⇒ (iii). By monotonicity property of conjunction it holds that Cn+1 ≤ En+1 |Hn+1 . Then, by assuming Cn = Cn+1 , it follows Cn ≤ En+1 |Hn+1 . c) (iii) ⇒ (i). Let us assume that Cn ≤ En+1 |Hn+1 , so that P(Cn ) ≤ P (En+1 |Hn+1 ). Moreover, by assuming that P (Ei |Hi ) = 1, i = 1, . . . , n, from (15) it follows P(Cn ) = 1 and hence P (En+1 |Hn+1 ) = 1, that is F p-entails En+1 |Hn+1 . ⊓ ⊔ References 1. E. W. Adams. The logic of conditionals. Inquiry, 8:166–197, 1965. 2. E. W. Adams. The logic of conditionals. Reidel, Dordrecht, 1975. 3. S. Benferhat, D. Dubois, and H. Prade. Nonmonotonic reasoning, conditional objects and possibility theory. Artificial Intelligence, 92:259–276, 1997. 4. V. Biazzo and A. Gilio. 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