On conditional probabilities and their canonical extensions to Boolean algebras of compound conditionals 1

In this paper we investigate canonical extensions of conditional probabilities to Boolean algebras of conditionals. Before entering into the probabilistic setting, we ﬁrst prove that the lattice order relation of every Boolean algebra of conditionals can be characterized in terms of the well-known order relation given by Goodman and Nguyen. Then, as an interesting methodological tool, we show that canonical extensions behave well with respect to conditional subalgebras. As a consequence, we prove that a canonical extension and its original conditional probability agree on basic conditionals. Moreover, we verify that the probability of conjunctions and disjunctions of conditionals in a recently introduced framework of Boolean algebras of conditionals are in full agreement with the corresponding operations of conditionals as deﬁned in the approach developed by two of the authors to conditionals as three-valued objects, with betting-based semantics, and speciﬁed as suitable random quantities. Finally we discuss relations of our approach with nonmonotonic reasoning based on an entailment relation among conditionals.


Introduction
Conditionals play a key role in different areas of logic, probabilistic reasoning and knowledge representation in AI, and they have been studied from many points of view, see, e.g., [2,3,4,5,6,7,8,9,10,11,12]. In particular, a three-valued calculus of conditional objects has been given in [13], where a simple semantics for the preferential entailment studied in [14,15,16] has been provided. Other approaches to conditional objects in the realm of Boolean algebras have been studied in [17,18]. Further results, from the artificial intelligence perspective, have been given, for instance, in [19,20,21].
In the recent paper [22], an alternative algebraic setting for Boolean conditionals has been put forward. More precisely, given a finite Boolean algebra A " pA,^, _,s, K, Jq of events, the authors build another (much bigger but still finite) Boolean algebra CpAq where basic conditionals, i.e. objects of the form pA|Bq for A P A and B P A 1 " AztKu, can be freely combined with the usual Boolean operations, yielding compound conditional objects, while they are required to satisfy a set of natural properties. Moreover, the set of atoms of CpAq are fully identified and it is shown they are in a one-to-one correspondence with sequences of pairwise different atoms of A of maximal length. Finally, it is also shown that any positive probability P on the set of events from A can be canonically extended to a probability µ P on the algebra of conditionals CpAq in such a way that the probability µ P p"pA|Bq"q of a basic conditional pA|Bq coincides with the conditional probability PpA|Bq " PpA^Bq{PpBq. This is done by suitably defining the probability of each atom of CpAq as a certain product of conditional probabilities.
On the other hand, the recent paper [23] presents results in the setting of conditional random quantities, with values in the unit interval r0, 1s, where the numerical approach to conjunctions and disjunctions of conditional events (see, e.g., [24,25,26]) is extended in general to cover arbitrarily complex compound conditionals. These objects are conditional random quantities obtained by conjunctions, disjunctions, and negations of conditional events and/or compound conditionals.
In this paper we take the more symbolic algebraic approach to conditionals from [22] a step further and bring it closer to the more numerical approach of [23] and [24,25,26]. We do this by first providing new basic results on the algebras CpAq of conditionals themselves, and second by turning operational some of its algebraic and probabilistic definitions. For instance, and in contrast with the above mentioned papers, in [22] precise definitions of conjunction and disjunction of conditionals are not explicitly given. Rather, any compound conditional comes determined by the disjunction of those atoms in CpAq that lie below it. Similarly, the probability of any compound conditional is computed as the sum of the probabilities of the atoms below the conditional. But no operational and systematic procedure to do these computations avoiding a combinatorial explosion is provided in [22]. More precisely, the main novel contributions of the paper are: • We show that the construction of the algebra of conditionals CpAq from a finite algebra of events A is compatible with subalgebras. Also we explore the relationship of Goodman and Nguyen's inclusion relation between basic conditionals with the natural order relation ď in CpAq.
• We extend the definition from [22] of the canonical extension to CpAq of a positive probability on A to the case of starting with a general conditional probability on AˆA 1 , and we show that this extension is compatible with taking restrictions on subalgebras and with Stalnaker's thesis.
• We derive for the canonical extension the formula to compute the probability of a conjunction and a disjunction of conditionals, and check they coincide with the ones proposed in the literature by McGee and Kaufmann, also in accordance with the random quantities approach.
• Finally, we introduce an entailment relation in terms of the lattice order in CpAq and we characterize probabilistically the entailment relation by canonical extensions. Then, we show that a corresponding nonmonotonic consequence relation on the algebra A satisfies the well-known rules of the system P.
The paper is structured as follows. After this introduction and some preliminaries in Section 2, we first examine in Section 3 the relation between the lattice order in a conditional algebra CpAq and the inclusion relation defined by Goodman and Nguyen. In Section 4 we show that the positivity assumption for the probability on A, needed for the canonical extension to the algebra of conditionals CpAq, can be lifted by starting from a conditional probability (in the axiomatic sense) on AˆA 1 . Then in Section 5 we show that, if B is a subalgebra of events of A and P a conditional probability on AˆA 1 , then the restriction of the canonical extension µ P on CpAq to CpBq is, in fact, the canonical extension of the restriction of P on BˆB 1 . This will allow us to prove that µ P is such that µ P p"pA|Bq"q " PpA|Bq and then in Section 6 that the probability of the conjunction coincides with McGee and Kaufmann's expressions obtained within the approach developed by two of the authors to conditionals as three-valued objects, with betting-based semantics, and specified as suitable random quantities. We also obtain the probability of the disjunction and the probability sum rule, in agreement with the approach given in [24]. In Section 7 we introduce an entailment relation in terms of the lattice order in CpAq; then, we characterize probabilistically the entailment relation by canonical extensions. Then, we examine a nonmonotonic consequence relation on the algebra A, which satisfies the well-known rules of the system P. Moreover, we discuss the Rational Monotony and the disjunctive Weak Rational Monotony rules. We also illustrate an example related to the failure of the transitive property. We conclude in Section 8 with some remarks and prospects for future work.

Preliminaries
In this section we recall basic notions and results from [22] where, for any Boolean algebra of events A " pA,^, _,s, K, Jq, a Boolean algebra of conditionals, denoted CpAq, is built. We will also denote a conjunction A^B simply by AB. Intuitively, a Boolean algebra of conditionals over A allows basic conditionals, i.e. objects of the form pA|Bq for A P A and B P A 1 " AztKu, to be freely combined with the usual Boolean operations up to certain extent.
In mathematical terms, the formal construction of the algebra of conditionals CpAq is done as follows. One first considers the free Boolean algebra FreepA|A 1 q " pFreepA|A 1 q, [, \,s, K, Jq 6 generated by the set A|A 1 " tpA|Bq : A P A, B P A 1 u. Then, one considers the smallest congruence relation " C on FreepA|A 1 q satisfying the following natural properties: Notice that, if A " A 1 and B " B 1 in A, then pA|Bq " pA 1 |B 1 q. Then, in a sense, the partial operation "|" is well-defined. Finally, the algebra CpAq is defined as follows.
Definition 1. For every Boolean algebra A, the Boolean algebra of conditionals of A is the quotient structure CpAq " FreepA|A 1 q{ " C .
Since CpAq is a quotient of FreepA|Aq, elements of CpAq are equivalence classes but, without danger of confusion, one can henceforth identify classes rts " C with one of its representative elements, in particular, by t itself. Conditionals of the form pA|Jq will also be simply denoted as A.
A basic observation is that if A is finite, CpAq is finite as well, and hence atomic. Indeed, if A is a Boolean algebra with n atoms atpAq " tα 1 , . . . , α n u, i.e. |atpAq| " n, it is shown in [22] that the atoms of CpAq are in one-to-one correspondence with sequences α " pα i 1 , . . . , α i n´1 q of n´1 pairwise different atoms of A, each of these sequences giving rise to an atom ω α of CpAq defined as the following conjunction of n´1 basic conditionals: (1) 6 We will continue denoting the top and bottom of FreepAq by J and K respectively without danger of confusion.

3
In what follows the atom in (1) will be also denoted by xα i 1 , . . . , α i n´1 y, or by ω i 1¨¨¨in´1 . It is then clear that the cardinality of the set of atoms of CpAq is |atpCpAqq| " n!. We recall that the lattice order relation ď in CpAq is defined as We also observe that, for every s, t P CpAq, because In [22,Proposition 4.7] it is shown that an atom ω " pα i 1 |Jq [ pα i 2 |s α i 1 q [¨¨¨[ pα i n´1 |s α i 1¨¨¨s α i n´2 q is below a conditional pA|Hq w.r.t. the lattice order ď in CpAq, i.e. ω ď A|H, if and only if, letting ω " xα i 1 , α i 2 , . . . , α i n´1 y, if j is the first index for which α i j ď H, then α i j ď A as well; in other words, the following condition is satisfied: either "α i 1 ď AH", or "α i 1 ď s H and α i 2 ď AH", or . . . , or "α i 1 ď s H and . . . and α i n 2 ď s H and α i n´1 ď AH".
Next we will recall some properties holding in CpAq that will be useful for next sections. For each subvector pi 1 , . . . , i k q of p1, . . . , nq we set that is, the conjunction ω i 1¨¨¨ik , which we also denote by xα i 1 , . . . , α i k y, stands for the initial conjunction of k components of the atom ω i 1¨¨¨in´1 . Indeed, as pα i n |s α i 1¨¨¨s α i n´1 q " pα i n |α i n q " J, for each permutation pi 1 , . . . , i n q of p1, . . . , nq, we obtain the following atom of CpAq: We hence recall that, from [22,Proposition 4.3], for each k, the conjunctions ω i 1¨¨¨ik 's constitute a partition of the algebra CpAq. In particular this implies that Ů pi 1 ,...,i k qPΠ t j 1 ,..., j k u ω i 1¨¨¨ik " J, where Π t j 1 ,..., j k u is the set of all permutations pi 1 , . . . , i k q of the set t j 1 , . . . , j k u. Example 1. An example of an algebra A, with 3 atoms, is obtained by considering the partition tα 1 , α 2 , α 3 u " tEH, s EH, s Hu, where E, H are two uncertain logically independent events. In this case The basic conditionals of CpAq are the elements of the set A|A 1 " tpA|Bq : A P A, B P A 1 u, where A 1 " AztKu. Moreover, the atoms of CpAq are the 3! elements of the form ω i j " α i [ pα j |s α i q, with i ‰ j, that is, atpCpAqq " tω 12 , ω 13 , ω 21 , ω 23 , ω 31 , ω 32 u, where A pictorial representation of the algebra CpAq, with 2 3! " 2 6 " 64 elements, can be found in Figure 1. We observe that The algebra of conditionals CpAq of Example 1, where atpAq " tα 1 , α 2 , α 3 u and atpCpAqq " tω 12 , ω 13 , ω 21 , ω 23 , ω 31 , ω 32 u. The element t is obtained as ω 12 \ ω 31 . The atoms of CpAq, the elements of the original algebra A, and the element t are identified with big dots. Moreover, we have the following properties: Finally any compound conditional t P CpAq is a disjunction of the atoms below t. For instance, let Based on [22,Proposition 4.7], we observe that E|H " ω 12 \ ω 13 \ ω 31 , p s H|EH _ s Hq " ω 31 \ ω 32 \ ω 23 and p s EH| s E _ s Hq " ω 21 \ ω 23 \ ω 12 . Then we have: Proposition 1. Consider two sequences of indices pi 1 , . . . , i k q and p j 1 , . . . , j t q, with k ď t. Then: Moreover, piiiq For every sequence pi 1 , . . . , i k q, it holds that Proof. piq Given two sequences pi 1 , . . . , i k q and p j 1 , . . . , j t q, with k ď t and p j 1 , . . . , j k q " pi 1 , . . . , i k q, it holds that piiq Denote by h the first index in the set ti 1 , . . . , i k u such that i h ‰ j h . Then Therefore, piiiq From piq, ω i 1¨¨¨ik [ ω j 1¨¨¨jt " ω j 1¨¨¨jt , when i r " j r , 1 ď r ď k. In particular for t " n´1 it holds that ω i 1¨¨¨ik [ ω j 1¨¨¨jn´1 " ω j 1¨¨¨jn´1 P atpCpAqq. Then, as J " Ů tp j 1 ,..., j n´1 qPΠ p1,...,n´1q u ω j 1¨¨¨jn´1 , from piiq it follows that ω i 1¨¨¨ik " ω i 1¨¨¨ik [ J " ğ tp j 1 ,..., j n´1 qPΠ p1,...,n´1q : j r "i r , 1ďrďku Let us notice that the construction of the algebra CpAq presented above can be seen as a map that, for every finite Boolean algebra A gives its associated Boolean algebra of conditionals CpAq. For a later use, it is convenient to observe that such construction preserves subalgebras in the sense made clear by the next easy result.
Proposition 2. Let A be a finite Boolean algebra and let B be a subalgebra of A. Then CpBq is a subalgebra of CpAq.
Proof. Since A and B are finite algebras, so are CpAq and CpBq. Moreover, by the way atoms are characterized in every boolean algebra of conditionals, it is clear that |atpCpBqq| ď |atpCpAqq|. Thus, by an easy cardinality argument, it immediately follows that CpBq is isomorphic to a subalgebra of CpAq. More concretely, if β 1 , . . . , β t are the atoms of B, CpBq is the subalgebra of CpAq whose atoms are of the form Notice that each ω B i clearly is an element of CpAq, whence CpBq is indeed the concrete subalgebra of CpAq having the ω B i 's as atoms.
Subalgebras of CpAq of the form CpBq, with B being a subalgebra of A, will be called conditional subalgebras of CpAq.
A particularly useful class of subalgebras of a given algebra A are those generated by partitions of A standing for the different truth conditions of a set F of conditionals of CpAq taken as three-valued objects.
In e.g. [25,26], the elements of such a partition are called the constituents generated by F . The simplest example is the case of a single conditional F 1 " tpA|Hqu, with A and H (uncertain) logically independent events, that generates the partition of A π 1 " tAH, s AH, s Hu, where the event AH makes the conditional pA|Hq true, the event s AH makes the conditional pA|Hq false, while the event s H makes the conditional pA|Hq void. In the case of two conditionals F 2 " tpA|Hq, pB|Kqu, we therefore have in principle 3 2 " 9 different combined truth conditions, leading to the following 9-element partition assuming all these events are different from K (i.e. assuming the uncertain events A, H, B, K logically independent). When one deals with sets of more conditionals, the corresponding partitions can be defined by an easy generalisation of the previous procedure. Now, consider a positive probability P : A Ñ r0, 1s on the algebra of plain events A. It is shown in [22] that P can be extended to a probability µ P : CpAq Ñ r0, 1s on the Boolean algebra of conditionals CpAq, called canonical extension, in such a way that µ P p"pA|Bq"q, the probability of a basic conditional pA|Bq, coincides with the conditional probability of A given B, i.e. µ P p"pA|Bq"q " PpA|Bq " PpA^Bq{PpBq, in accordance with the so-called Stalnaker's thesis, stating that the probability of a conditional is the conditional probability, whenever the antecedent has non-zero probability [27]. In particular, µ P p"pA|Jq"q " PpA|Jq " PpAq for any A P A. Actually, the probability µ P is first defined on the atoms of CpAq as follows: for any atom ω i 1¨¨¨in´1 " α i 1 [ pα i 2 |s α i 1 q [¨¨¨[ pα i n´1 |s α i 1¨¨¨s α i n´2 q, its probability is defined as the following product of conditional probabilities: This is well defined because of the assumption that P is positive. Then µ P is extended to the whole algebra CpAq of conditionals by additivity: for any element t of CpAq, where the lattice order ď in CpAq is as defined in (2). Moreover, it is shown in [22] that for any k, the following factorization holds: We finally notice that, as observed above, since for each k the conjunctions ω i 1¨¨¨ik 's constitute a partition of CpAq, the sum of the probabilities over all of them is 1, that is: µ P pω i j q "¨¨¨" ÿ pi 1 ,...,i n qPΠ t1,...,nu µ P pω i 1¨¨¨in´1 q.

On the relation between the lattice order in CpAq and Goodman-Nguyen's inclusion relation
In [28] Goodman and Nguyen introduced an inclusion relation between conditional objects in the context of measure-free conditionals. Adapted to the setting of conditionals in an algebra of conditionals it amounts to the following definition.
Definition 2. The Goodman-Nguyen inclusion relation between basic conditionals in an algebra CpAq is defined as follows: for any A|H, B|K P CpAq, where ď is the lattice order relation in A.
In this section we explore the relationship of this inclusion relation in CpAq with the natural order relation ď in CpAq, and we provide a full characterisation of ď in terms of Ď, extending partial results in [22]. Theorem 1. Given any conditional events A|H and B|K of an algebra CpAq, it holds that A|H ď B|K ðñ AH " K , or s BK " K , or A|H Ď B|K.
Proof. pùñq If AH " K, or s BK " K, then the statement holds. Assume that AH ‰ K, s BK ‰ K, and by absurd that A|H Ę B|K, that is AH ę BK or s BK ę s AH. If it were AH ę BK, as AH ‰ K there would exist an atom α P atpAq such that α ď AH and α ď s BK_ s K. On the other hand, since s BK ‰ K there would exist an atom β P atpAq such that β ď s BK. Now let ω be an atom of CpAq of the form ω " xα, β, . . .y. Then, it would be ω ď A|H and ω ď s B|K, hence pA|Hq [ p s B|Kq ‰ K because ω ď pA|Hq [ p s B|Kq. This leads to a contradiction because by hypothesis A|H ď B|K, that is, by recalling (3), pA|Hq [ p s B|Kq " K. If it were s BK ę s AH, as s BK ‰ K there would exist an atom α P atpAq such that α ď s BK and α ď AH_ s H. On the other hand, since AH ‰ K, there would exist an atom β P atpAq such that β ď AH. Now let ω be an atom of CpAq of the form ω " xα, β, . . .y. Then, it would be ω ď A|H and ω ď s B|K, which is absurd because, it contradicts the hypothesis A|H ď B|K. pðùq Observe first that if AH " K or s BK " K, then A|H " K or B|K " J, respectively. Thus, in both cases it holds that A|H " pA|Hq [ pB|Kq and hence, by recalling (2), the condition A|H ď B|K is satisfied.
Assume now that AH ‰ K and s BK ‰ K. Moreover, assume that A|H Ď B|K, that is AH ď BK and s BK ď s AH. As AH ď BK and s BK ď s AH, it holds that AH s BK " AH s K " s H s BK " K. Taking into account these last logical relationships, it follows that the partition π 2 of A generated by A|H and B|K is contained in the set tα 1 , . . . , α 6 u, 7 where For the sake of simplicity we first assume all these α i 's are different from K, i.e. that π 2 " tα 1 , . . . , α 6 u, and let B be the subalgebra of A generated by π 2 . In other words we assume atpBq " π 2 . Let us consider any atom ω i 1¨¨¨i5 " pα i 1 |Jq[pα i 2 |s α i 1 q[¨¨¨[pα i 5 |s α i 1¨¨¨s α i 4 q of the conditional algebra CpBq, that is a subalgebra of CpAq. According to Section 2, in order the relation ω i 1¨¨¨i5 ď A|H be satisfied, in the sequence pi 1¨¨¨i5 q the number 1 must appear before the numbers 2, 3, and 4. Then, the sequence must be such that i 1 " 1, or pi 1 , i 2 q " p5, 1q, or pi 1 , i 2 q " p6, 1q, pi 1 , i 2 , i 3 q " p5, 6, 1q, or pi 1 , i 2 , i 3 q " p6, 5, 1q. From Proposition 1, we 7 Note that, in principle, depending on the logical relations among the events A, H, B and K, some of these α i 's might be K.
Then, from (8) and (9) it follows that pA|Hq [ p s B|Kq " K. Therefore that is A|H ď B|K. Finally, notice that in the case where π 2 Ă tα 1 , . . . , α 6 u, by a similar reasoning we would still obtain that A|H ď B|K.
Remark 1. Notice that formula (7) is also valid in terms of a numerical inequality where the conditional events are replaced by their indicators ([29, Equation (15)]). We also observe that in [30,Theorem 6] it has been proved that the condition on the right side of formula (7) is equivalent to the property that, denoting by Π the set of coherent probability assessments px, yq on tA|H, B|Ku, it holds that x ď y, for every px, yq P Π. Therefore, since every coherent probability assessment px, yq can be extended to a conditional probability P (see Remark 2), by Theorem 1 it follows that A|H ď B|K ðñ PpA|Hq ď PpB|Kq, @ P .
The next result directly follows from Theorem 1 and specifies under which conditions the inequality ď and the inclusion relation Ď between two conditional events are equivalent. Corollary 1. Given any conditional events A|H and B|K, with either AH ‰ K and s BK ‰ K, or AH " It is interesting to remark that, regarding the above Corollary 1, when either AH " K and s BK ‰ K, or AH ‰ K and s BK " K, it holds that A|H ď B|K, but it could be that A|H Ę B|K. For instance, if A " s H and s H s BK ‰ K, then K " A|H ď B|K, but s BK ę s AH " H and hence A|H Ę B|K. A slightly different (but still equivalent) characterization of the lattice order relation ď among conditional events can be given as follows.
Theorem 2. Given any conditional events A|H and B|K, it holds that Proof. pùñq. The condition A|H ď B|K amounts to pA|Hq [ p s B|Kq " K. Then, by observing that Then, the atoms are α 1 , . . . , α k`1 , with k ď 7. Moreover, by recalling that H_K " HK_ s HK_H s K, we obtain Then, Notice that Theorem 1 also follows as a corollary from Theorem 2.

Canonical extension of a conditional probability
In the definition of the canonical extension µ P on CpAq in [22] that we recalled in Section 2, a crucial assumption is that P is positive, i.e. that Ppαq ą 0 for every α P atpAq, otherwise µ P pωq will be undefined for some ω P atpCpAqq (it would be of the form 0{0). A way to overcome this limitation is, instead of starting with a positive (unconditional) probability on A, to directly start with a conditional probability on AˆA 1 in the axiomatic sense, that is to say, with a binary map P : AˆA 1 Ñ r0, 1s, where A 1 " AztKu, such that (CP1) For all B P A 1 , Pp¨|Bq : A Ñ r0, 1s is a finitely additive probability on A; (CP2) For all A P A and B P A 1 , PpA|Bq " PpA^B|Bq; As usual, we will also denote PpA|Jq simply by PpAq, for every A P A.
Remark 2. As has already been mentioned above, differently from the approach in [22], we do not assume here the positivity of the (conditional) probability P. Then, the function P may be such that PpA|Bq " 0 and/or PpBq " 0 for some A P A and B P A 1 . Moreover, we recall that, requesting P : AˆA 1 Ñ r0, 1s to satisfy the above three postulates, assures that P is a coherent conditional probability assessment in the sense of de Finetti to all the conditional objects pA|Bq, with A, B P A and B ‰ K. In fact, a conditional probability assessment on an arbitrary family of (basic) conditional events PpA 1 |B 1 q " x 1 , . . . , PpA n |B n q " x n , is coherent iff it can be extended to a full conditional probability (in the above sense) on AˆA 1 (see, e.g., [31]). In this paper, instead of starting from a (coherent) probability assessment on an arbitrary family of conditional events, we directly start with a full conditional probability P defined on AˆA 1 .
Then, given a conditional probability P : AˆA 1 Ñ r0, 1s, we can proceed as in the previous section to define a (unconditional) probability µ P in CpAq.
One can check that µ P so defined is a probability distribution on atpCpAqq.
Iterating this procedure for sets of n´1 atoms of A we finally get: Then, we can extend µ P to a probability on the whole algebra CpAq in the usual way by additivity, as in the previous case: for any t P CpAq, µ P ptq " ř ωďt µ P pωq. We will keep referring to µ P as the canonical extension of P.
The question of whether µ P actually extends P, in the sense that, for any basic conditional pA|Bq P CpAq, it holds µ P p"pA|Bq"q " PpA|Bq is deferred to Theorem 5 in the next section. From now on, we will simply use the notation µ P pA|Bq instead of µ P p"pA|Bq"q without danger of confusion.

The canonical extension and subalgebras, Stalnaker's thesis, coherence and convexity
In this section we first prove two basic properties of the canonical extensions of conditional probabilities on an algebra of events A to the algebra of conditionals CpAq, namely their compatibility with taking subalgebras, and based on this, that on basic conditionals they agree with the initial conditional probability. Then we show that t0, 1u-valued probabilities on CpAq are in fact always canonical extensions, and as a consequence it follows that the set of canonical extensions on CpAq is not a convex set of probabilities.

The canonical extension and subalgebras
Given the canonical extension µ P to CpAq of a conditional probability P on CpAq, and given a subalgebra B of A, in this subsection we first examine the restriction of µ P to the conditional subalgebra CpBq of CpAq, and we show that such restriction coincides with the canonical extension of the restriction of P to B. Then we use this result to show that µ P is such that, for every basic conditional pA|Bq P CpAq, µ P pA|Bq " PpA|Bq. This in fact can be regarded as a slight generalisation of the Stalnaker thesis mentioned in Section 2 as in this case the antecedent need not have strictly positive probability.
To start with, let A be a finite Boolean algebra with atpAq " tα 1 , α 2 , . . . , α n u, and let B be a subalgebra of A. If β 1 , . . . , β k are the atoms of B, it means that the set of atoms of A can be partitioned in non-empty subsets A 1 , . . . , A k such that for all j " 1, . . . , k, β j " Ž αPA j α. We will first consider the following particular case of a subalgebra B of A: take an index i ă n, and let B the subalgebra of A generated by α 1 , . . . , α i´1 , α i _α i`1 , α i`2 , . . . , α n . In other words, for j " 1, . . . , n´1, let Then B is the subalgebra of A generated by β 1 , . . . , β n´1 and atpBq " tβ 1 , . . . , β n´1 u. Now let us consider P : AˆA 1 Ñ r0, 1s a conditional probability and µ P : CpAq Ñ r0, 1s its canonical extension to CpAq. Further, let P 1 : BˆB 1 Ñ r0, 1s be the restriction of P to BˆB 1 , and let µ P 1 : CpBq Ñ r0, 1s be its canonical extension to CpBq. The question of interest is whether µ P 1 is in fact the restriction of µ P to CpBq. Next theorem shows this is actually the case. Indeed, given any permutation p j 1 , . . . , j n´1 q of p1, . . . , n´1q, we let and we recall that, by definition of the canonical extension of µ P 1 , In the next result we show that µ P pω 1 j 1¨¨¨jn´2 q " µ P 1 pω 1 j 1¨¨¨jn´2 q. But first we state a preliminary remark that will be useful in the proof.
Remark 3. For any events A, B, and C in an algebra A with A ď B ď C and B ‰ K, and any conditional probability P : AˆA 1 Ñ r0, 1s, it holds that Indeed, when A ď B ď C, from (CP3) one has PpA|Cq " PpA|BqPpB|Cq. Then one also has: PpA|Bq " PpA|BqPp s B|Cq`PpA|BqPpB|Cq " Pp s B|CqPpA|Bq`PpA|Cq.
Theorem 3. For each atom ω 1 j 1¨¨¨jn´2 P atpCpBqq, the following holds: Proof. Due to its length, and to easy the reading of the paper, the proof has been moved to Appendix A.
As an illustration of Theorem 3, we examine the case of a simple example for n " 4.
Thus pβ 3 |Jq [ pβ 1 | s β 3 q " ω 31 \ ω 341 \ ω 41 \ ω 431 and hence Notice that, by suitably reordering the subscripts, the result of Theorem 3, still holds for the case where β i " α i _α t , with t ą i`1. More in general, for each conditional subalgebra CpBq of CpAq, by a suitable iterated application of Theorem 3 it can be verified that (17) is satisfied. This yields the following result.
Theorem 4. Let A be a finite Boolean algebra. For any subalgebra B of A, and any conditional probability P : AˆA 1 Ñ r0, 1s, let P 1 be its restriction on BˆB 1 . Then, (i) for every atom ω 1 P atpCpBqq it holds that µ P pω 1 q " µ P 1 pω 1 q, and hence, (ii) for each C P CpBq it also holds that µ P pCq " µ P 1 pCq.
To exemplify the iterative procedure to prove (i) in the above theorem, let us consider the following simple example.
Theorem 4 shows that the restriction of the canonical extension µ P on the conditional algebra CpAq to the conditional subalgebra CpBq coincides with the canonical extension µ P 1 on the conditional subalgebra CpBq, see the commutative diagram in Figure 2.
This result enables a local approach in order to study properties of basic and compound conditionals, as A, P : AˆA 1 Ñ r0, 1s BĎA Restriction

Canonical extension
Cp¨q, µ p¨q + 3 CpBq, µ P 1 : CpBq Ñ r0, 1s Figure 2: Compatibility of the canonical extension construction with respect to taking subalgebras. Notice that in the above diagram, µ P 1 " µ P ae CpBq . That is to say, the canonical extension µ P 1 of the conditional probability P 1 obtained by restricting P to BˆB 1 is the restriction of the canonical extension µ P to the subalgebra CpBq of CpAq.
done in the next section. Actually, the above Theorem 4 is very powerful, because when dealing with a set of conditional events F " tpA i |H i qu iPI from CpAq and their probabilities, one needs not resort to probabilities defined on the whole algebra CpAq (and thus specified on all the atoms of CpAq) but only on a relevant subalgebra of CpAq. Indeed, it is enough to consider the conditional subalgebra CpBq where B is the subalgebra of A generated by a suitable partition determined by the family F of conditional events along the lines studied in [26, Sec. 2.1].
In the next result we extend a main result of [22] in the following sense. In [22,Theorem 6.13] the authors show that if P is a positive probability on A, then its canonical extension on CpAq is such that µ P pA|Hq " PpAHq{PpHq. That is, for positive probabilities, conditional probability can be understood as a simple probability over conditionals, a result that it is related to the well-known Stalnaker's thesis. Thanks to Theorem 4, here we show that this still holds if we start with a conditional probability on AˆA 1 .
Theorem 5. Let P be a conditional probability on AˆA 1 and µ P its canonical extension to CpAq. Then, for every basic conditional pA|Hq P CpAq, it holds that µ P pA|Hq " PpA|Hq.
A direct consequence of the previous result is that, given two conditional probabilities P and P 1 , with P ‰ P 1 , it follows that µ P ‰ µ P 1 . Indeed, from Theorem 5, if µ P " µ P 1 , then P " P 1 because PpA|Hq " µ P pA|Hq " µ P 1 pA|Hq " P 1 pA|Hq , @ A|H P CpAq .
Remark 4. In light of Theorem 5, the result recalled in Remark 1 can be further extended so to involve also the canonical extensions of conditional probabilities. Indeed, we can now characterize the order ď between conditional events in the following way: for every conditional events A|H and B|K, it holds that A|H ď B|K ðñ µ P pA|Hq ď µ P pB|Kq @P.
Remark 5. Given three events A, B, C, with A ď B ď C, by (CP3) it holds that PpA|Cq " PpA|BqPpB|Cq. Moreover, by recalling pC5q, we observe that
As we can see, (20) shows that the "independence" between A|B and B|C, when A ď B ď C, still holds between A|B and s B|C. In particular, given any events E and H, by applying (20) with A " EH, B " H,and C " J, as s H|J " s H, we obtain Formula (21) will be generalized in Theorem 6, where s H is replaced by any K such that HK " K.

On the canonical extensions of t0, 1u-valued conditional probabilities
It is very well-known that homomorphisms from a Boolean algebra A into the two-element Boolean algebra 2 " t0, 1u are in fact the t0, 1u-valued probabilities on the algebra A. We now show that the homomorphisms from CpAq into to 2 are the canonical extensions of t0, 1u-valued conditional probabilities on AˆA 1 . Specifically, we prove that for each ω P atpCpAqq the map µ : CpAq Ñ t0, 1u, such that µpsq " 1, if ω ď s, and µpsq " 0, otherwise, is a canonical extension of a suitable conditional probability on AˆA 1 . Lemma 1. For any atom ω P atpCpAqq, there is a conditional probability P ω : AˆA 1 Ñ r0, 1s whose canonical extension µ P ω is such that µ P ω pωq " 1 and µ P ω pω 1 q " 0 for any atom ω 1 ‰ ω.
Proof. Assume ω " xα 1 , . . . , α n y. For any event B P A, let at ď pBq be the set of atoms of A below B, and let minpBq " minti P t1, ..., nu|α i P at ď pBqu. Then define P ω : AˆA 1 Ñ r0, 1s as follows: Notice that, from the definition, it directly follows that P ω pA|Bq " 1 if ω ď pA|Bq in CpAq, and P ω pA|Bq " 0 otherwise. Moreover P ω is indeed a t0, 1u-valued mapping. It is not difficult to check that, so defined, P ω is a conditional probability: (CP1) We have to show that for all B P A 1 , P ω p¨|Bq : A Ñ r0, 1s is a finitely additive probability on A. Indeed, it is clear from the definition that P ω pJ|Bq " 1 and P ω pK|Bq " 0. As for the additivity, assume A^C " K. Then P ω pA _ C|Bq " 1 iff ω ď pA _ C|Bq " pA|Bq \ pC|Bq iff ω ď pA|Bq or ω ď pC|Bq but not both, that is, iff P ω pA|Bq " 1 and P ω pC|Bq " 0, or P ω pC|Bq " 1 and P ω pA|Bq " 0. Therefore, P ω pA _ C|Bq " 1 iff P ω pA|Bq`P ω pC|Bq " 1.
(CP2) For all A P A and B P A 1 , we have to show that P ω pA|Bq " P ω pA^B|Bq. As observed above, P ω pA|Bq " 1 iff ω ď pA|Bq iff ω ď pA^B|Bq iff P ω pA^B|Bq " 1.
(CP3) Finally, we show that for all A P A, B, C P A 1 , if A ď B ď C, then P ω pA|Cq " P ω pA|Bq¨P ω pB|Cq.
In the last part of the proof of the above lemma we have proved that, for any atom ω " xα 1 , . . . , α n y P atpCpAqq, µ P ω is such that P ω pα 1 |Jq " P ω pα 2 |α 2 _..._α n q " . . . " P ω pα n´1 |α n´1 _α n q " 1, and according to Remark 2, this means that the probability assessment P " p1, . . . , 1q on the family F " tpα 1 |Jq, pα 2 |s α 1 q, . . . , pα n´1 |s α 1¨¨¨s α n´2 qu (as a restriction of P ω ) is coherent. In the rest of this subsection we show that this is in accordance with what can be obtained by applying the Algorithm 1 in [32,25], in order to check coherence of P, by considering a suitable sequence of linear systems and by verifying the solvability of each linear system.
By the algorithm, we have I 0 " H; then, the procedure ends by declaring P coherent.
We observe that the conditional probability P on AˆA 1 , extension of the assessment P on F , is such that µ P pωq " 1 and µ P pω 1 q " 0 for every atom ω 1 ‰ ω. Then, since µ P " µ P ω and, as we have observed above, the canonical extensions are unique, it holds that P " P ω , that is P ω is the unique extension of the assessment P on F as a (full) conditional probability to AˆA 1 . Notice that, for each r " 1, . . . , n´1, the probability assessment π r is a restriction of P ω on the family tα r |s α 1¨¨¨s α r´1 , . . . , α n |s α 1¨¨¨s α r´1 u, that is on the family tα r |α r _¨¨¨_ α n , . . . , α n |α r _¨¨¨_ α n u.
We remark that, in order to verify the coherence of the assessment P on F , an equivalent procedure is given in [31] which exploits a suitable class of (unconditional) probability assessements tP r u agreeing with P. For each conditional event pα h |α h _¨¨¨_ α n q in F , its probability is represented as a ratio by using an element of the class. In our case the (unique) agreeing class is tP 0 , P 1 , . . . , P n´1 u, where, for each h, the assessment P h´1 is defined as P h´1 pα h q " 1 , P h´1 pα j q " 0 , j ą h.

Indeed, we have
We recall that in [31], given any event E P A, with E ‰ K, the zero-layer of E with respect to an agreeing class tP r u is the first index k such that P k pEq ą 0, denoted opEq " k. In our case, for each atom α h the first index k such that P k pα h q ą 0 is k " h´1 and hence opα h q " h´1. Thus, the events α 1 , α 2 , . . . , α n belong to different zero-layers, because opα 1 q " 0 , opα 2 q " 1 , . . . , opα n´1 q " n´2 , opα n q " n´1 .
Finally, in [31] the zero-layer of a conditional event E|H, with respect to an agreeing class tP r u, is defined as opE|Hq " opEHq´opHq. Then, by observing that for every h it holds that opα h q " opα h _¨¨¨_ α n q, with respect to tP 0 , P 1 , . . . , P n´1 u we obtain opα h |α h _¨¨¨_ α n q " 0, @ pα h |α h _¨¨¨_ α n q P F .

On the non convexity of the set of canonical extensions
Based on Lemma 1 we can verify that the set of canonical extensions µ P on CpAq is not convex. Indeed, as we have seen, for every ω P atpCpAqq, the probability µ P ω as in Lemma 1 is a homomorphism of CpAq to the two element Boolean algebra t0, 1u. In other words, the set of all probability measures µ P on CpAq that are canonical extensions of some conditional probability P contains all the homomorphisms of CpAq to t0, 1u. Thus, if the set of canonical extensions were convex, this set would coincide with the set of all probability measures of CpAq and this is known not to be the case, since there are probabilities on algebras CpAq that are not conditional probabilities and hence are not canonical extensions, see for instance [22,Example 6.3]. Next we provide another example. Example 4. Let us consider an algebra A with atoms tα 1 , α 2 , . . . , α 5 u, and let A " α 1 _α 2 , H " α 1 _¨¨¨_α 4 and B " α 1 _α 3 _α 5 . Notice that ABH " α 1 , A s BH " α 2 , s ABH " α 3 , s A s BH " α 4 , s H " α 5 . Then, consider in CpAq the two atoms ω 1 " ω 1234 and ω 2 " ω 2314 . We also consider the conditional events: A|H " pα 1 _ α 2 |α 1 _¨¨¨_ α 4 q, s B|H " pα 2 _ α 4 |α 1 _¨¨¨_ α 4 q, A|BH " pα 1 |α 1 _ α 3 q.

Probability of the conjunction and the disjunction of conditionals
In this section we start by showing that the probability of the conjunction K [ pA|Hq, when HK " K, is the product of the probabilities of the conjuncts. Then, we represent the conjunction of two conditional events as a suitable disjunction. Finally, based on the canonical extension, we obtain the probability for the conjunction and the disjunction of two conditional events A|H and B|K, which are related with analogous results given in the setting of coherence in [24,25,26,29,33]. In the next result we generalize formula (21). Theorem 6. Given an algebra A and any events A, H, K P A, with H ‰ K and HK " K, given a conditional probability P on AˆA 1 and its canonical extension µ P to CpAq, it holds Proof. As HK " K, it holds that H s K " H, H_ s K " s K, and s HK " K; then We consider the partition tβ 1 , . . . , β 4 u, where and the associated subalgebra B; moreover we consider the atoms ω 1 Let P 1 be the restriction of P to BˆB 1 and µ P 1 its canonical extension to CpBq. As H s K " H it holds that PpAH| s Kq " PpA|H s KqPpH| s Kq " PpA|HqPpH| s Kq. Then, from Theorem 5 and from (13) we obtain µ P rK [ pA|Hqs " µ P 1 rK [ pA|Hqs " µ P 1 pω 1 21 q`µ P 1 pω 1 241 q " " Ppβ 2 qPpβ 1 | s β 2 q`Ppβ 2 qPpβ 4 | s β 2 qPpβ 1 | s β 2 s β 4 q " PpKqPpAH| s Kq`PpKqPp s H| s KqPpA|Hq " " PpKqrPpAH| s Kq`Pp s H| s KqPpA|Hqs " PpKqrPpA|HqPpH| s Kq`PpA|HqPp s H| s Kqs " PpKqPpA|Hq .
Using the above result, we can now provide a suitable representation of the conjunction of two conditionals based on which we will show in Theorem 8 how to compute the probability of such a compound conditional object. In the next result we finally obtain the probability for the conjunction of two conditionals A|H [ B|K in terms of conditional probabilities of events related to the partition obtained from the family tA|H, B|Ku.
Theorem 8. Given an algebra A and a conditional probability P on AˆA 1 , let µ P be the canonical extension to CpAq. For any conditional events A|H, B|K P CpAq it holds that Proof. From (24) HBK|H_Kqs " PpA|HqPp s HBK|H_Kq (27) and We first assume that the uncertain events A, H, B, K logically independent and we consider the subalgebra B generated by the partition tβ 1 , . . . , β 9 u, where Notice that, by logical independence, β j ‰ K, for each j " 1, . . . , 9. Moreover, we consider the compound conditionals ω 1 i 1¨¨¨ik 's of CpBq, 1 ď k ď 8 . Let P 1 be the restriction of P to BˆB 1 and µ P 1 its canonical extension to CpBq. We recall that from Theorem 4, µ P pCq " µ P 1 pCq, for every C P CpBq. By exploiting the distributivity property, we decompose the conjunction pA|Hq [ p s HBK|H_Kq as

Algebraic and probabilistic entailment with conditionals and nonmonotonic reasoning
In this section we first consider an entailment relation |ù among conditionals of a conditional algebra CpAq defined in terms of the lattice order in CpAq. We show that this algebraic definition can be probabilistically characterised by means of canonical extensions as a generalisation of Adams' p-entailment. Then we show that these two equivalent entailments induce a nonmonotonic consequence relation on the algebra of plain events A satisfying the well-known rules of the system P and we discuss the Rational Monotony rule. The underlying algebraic nature of |ù allows us to provide simple algebraic proofs based on results given in the paper, like the decomposition property of the conjunction of Theorem 7.
Let us say that an atom ω P CpAq satisfies a compound t when ω ď t. We also say that t is satisfiable if t ‰ K, i.e. if there exists ω P CpAq such that ω ď t. Then, it is clear that for every atom ω and every compound t, either ω satisfies t or ω satisfies s t. In particular, if t " pA|Hq, then either ω satisfies pA|Hq, or falsifies it (i.e. it satisfies p s A|Hq). This reflects the fact that, by construction, conditionals are Boolean objects in conditional algebras CpAq. But this is compatible with the 3-valued nature of basic conditionals when viewed from the original algebra of (plain) events A. Indeed, for every atom α of A, exactly one of the following three conditions holds: either α ď AH, or α ď s AH, or α ď s H, that respectively correspond to the cases in which either α satisfies pA|Hq, or α falsifies pA|Hq, or α makes pA|Hq undefined or void.

Definition 4. Given any set of (compound) conditionals
We define a consequence relation between sets of conditionals and conditionals from a conditional algebra CpAq by means of the order relation ď.
Definition 5. Let F Y ttu Ď CpAq be a set of (compound) conditionals, with F consistent. Then we say that t is a consequence of F , written F |ù t, whenever [ts|s P F u ď t.
Note that, by definition, F |ù t holds iff every atom of CpAq which is below every compound conditional of F is also below t. Note that if F is not consistent, then F |ù t trivially holds.
The consequence relation |ù among (compound) conditionals can also be characterized in probabilistic terms.
Proposition 4. For any set of (compound) conditionals F Y ttu Ď CpAq, with F consistent, it holds that F |ù t iff, for all conditional probability P, µ P p[tr|r P F uq ď µ P ptq.
Proof. By letting s " [tr|r P F u, it amounts to prove that s ď t iff µ P psq ď µ P ptq for any conditional probability P. From consistency of F it holds that s ą K. The left-to-right direction is direct. For the converse direction, suppose K ă s ę t. Then there is an atom ω P CpAq such that ω ď s but ω ę t. By Lemma 1, µ P ω is such that µ P ω psq " 1 while µ P ω ptq " 0.
As an immediate consequence of this proposition and its proof, in the following corollary we have a stronger version of the previous result, that in fact shows that |ù coincides with a generalised form of Adams' p-entailment for (basic) conditionals [2]. In the following we will say that a conditional probability P is a p-model of a (compound) conditional s when µ P psq " 1, and we will say that a set F of (compound) conditionals p-entails another (compound) conditional t, written F $ p t, when every conditional probability P that is a p-model of all conditionals s P F is a p-model of t as well.
Definition 6. For any set of (compound) conditionals F Y ttu Ď CpAq, with F consistent, we say that F p-entails t, written F $ p t, when every p-model of F is a p-model of t as well.
Lemma 2. For any set of consistent (compound) conditionals F Ď CpAq, it holds that F $ p [ts|s P F u. (43) Proof. Let P be a p-model of F , that is µ P psq " 1 for all s P F . If P were not a p-model of [ts|s P F , that is µ P p[ts|s P F q ă 1, then there would exist an atom ω of CpAq such that ω ę [ts|s P F u and µ P pωq ą 0. Moreover, there would exist s P F such that ω ę s and hence µ P psq ă 1 contradicting the assumption.
Proposition 5. For any set of (compound) conditionals F Y ttu Ď CpAq, it holds that F |ù t iff F $ p t.
Proof. pùñq. If F |ù t, then by Proposition 4, µ P p[ts|s P F q ď µ P ptq for every P. Then, from (43) for every p-model P of F it holds that µ P p[ts|s P F q " 1 and hence µ P ptq " 1. Thus F $ p t. pðùq. Assume that F $ p t. If t were not a consequence of F , then there would exist an atom ω of CpAq such that ω ď [ts|s P F u and ω ę t. Then, from Lemma 1, it would be µ P ω p[ts|s P F q " 1 and µ P ω ptq " 0. Moreover, as µ P ω p[ts|s P F q ď µ P ω psq for all s P F , it would follow that µ P ω psq " 1 for all s P F , with µ P ω pt q " 0. Thus, P ω would be a p-model of F , but not a p-model of t, which contradicts the assumption.
Now we turn to some properties of the entailment |ù related to core properties of the well-known System P for nonmonotonic inference relations.
(vi) pA|Hq |ù p s B|Hq \ pA|BHq, since, obviously pA|Hq ď p s B|Hq \ pA|BHq iff pA|Hq [ pB|Hq ď pA|BHq. Now, each set of (compound) conditionals defines a nonmonotonic consequence relation on the algebra of events A. Definition 7. Let F be a consistent set of (compound) conditionals. Then we define the consequence relation |" F Ď 2 AˆA on events from A as follows: Equivalently, by Corollary 5, tB 1 , . . . , B n u |" F A if F $ p pA|B 1 ...B n q, that is, if PpA|B 1 , .., B n q " 1 for every conditional probability P model of F . Hence |" F is in fact the p-entailment relative to F . Theorem 12. The consequence relation |" F satisfies the core properties of System P: Moreover it satisfies the following additional property related to disjunction: • Orm: if H_K |" F A and H |" F s A then K |" F A Proof. Reflexivity trivially holds, and the rest of properties directly follow from properties (i)-(vi) in Prop. 6.
As a consequence, |" F is a preferential consequence relation in the sense of [15]. We recall that a probabilistic analysis of System P inference rules has been given, in the setting of coherence, in [6]; moreover, the p-validity of such rules has been verified in [25].
It has been shown elsewhere, see e.g. [22], that the Rational Monotony rule for entailments |« similar to |" F is not valid. In fact RM or dRM is not valid in general for |" F , unless the set F consists of only one atom of CpAq, since in such a case H | F s B is equivalent to H |" F B and then RM becomes just CM. The following is a counter-example of the validity of the rule already in the case that F is a disjunction of two atoms of CpAq.
Proposition 7. For any events A, B, H P A with BH ‰ K and for any conditional probability P on AˆA 1 , the following rule • dWRM: if PpA|Hq " 1, then either Pp s B|Hq " 1 or PpA|BHq " 1 holds.
Of course F $ p A|H. Moreover, as P ω 1 is a p-model of F but not of s B|H, then F & p s B|H. Similarly, as P ω 2 is a p-model of F but not of A|BH, then F & p A|BH. Thus, F $ p A|H, F & p s B|H, and F & p A|BH, that is RM rule is not valid.
Therefore, tC|B, B|Au is consistent and we have ω 52 ď pC|Bq, ω 52 ď pB|Aq, but ω 52 ę pC|Aq. In other words, F |ù pC|Bq, F |ù pB|Aq and F |ù pC|Aq, or equivalently Thus, we have shown a counter-example of the validity of the transitivity rule for the consequence relation |" F . Moreover, we observe that tC|B, B|A, pA|A_Bqu |ù C|A or equivalently that tC|B, B|Au |ù pC|Aq \ p s A|A_Bq.
Then, we obtain a weaker version of transitivity (see [25]): if B |" F C, A |" F B and A_B |" F B, then A |" F C.

Conclusions and future work
In this paper we have advanced the study of conditionals in the setting of the Boolean algebras of conditionals as proposed in [22]. More precisely, after a first analysis on the lattice order of our algebras and the known Goodman and Nguyen order relation, we have considered the canonical extension µ P of a conditional probability P on AˆA 1 to the Boolean algebra of conditionals CpAq. Our first main result establishes that for every basic conditional pA|Hq, PpA|Hq " µ P pA|Hq and hence the conditional probability P coincides with the restriction of µ P to basic conditionals.
In turn we get an operational computation of the probability of a conjunction and a disjunction of conditionals, in agreement with previous approaches in the literature, in particular with the one developed by Gilio and Sanfilippo by formalising conditionals as random quantities [24].
Finally, we have discussed relations of our approach with nonmonotonic reasoning. First we have introduced a (monotonic) entailment relation among conditionals defined in terms of the lattice order of CpAq and then we have examined a nonmonotonic consequence relation on the algebra A, which satisfies the well-known rules of the system P. Moreover, we have discussed the Rational Monotony and the disjunctive Weak Rational Monotony rules.
As for future work, an aspect to be deepened concerns the notion of iterated conditional, say pB|Kq|pA|Hq, and its probability in the realm of Boolean algebras of conditionals. Indeed, if we define µ P ppB|Kq|pA|Hqq " de f µ P pA|Hq[pB|Kqq µ P pA|Hq , then, under the hypothesis PpA|Hq ą 0, it holds that which is the prevision of the iterated conditional pB|Kq|pA|Hq obtained in the setting of coherence in [35,Section 6] (see also [39]). Under the further assumption PpH_Kq ą 0, formula (49) coincides with the result given in [8,Thm. 3]. For some applications of iterated conditionals see e.g. [40,41].
By iterating the previous reasoning, for every k " 2, . . . , n´i´2 it holds that µ P pW 1¨¨¨i i`2¨¨¨i`k , i`k`1¨¨¨n´1 q " µ P pω 1¨¨¨i i`2¨¨¨i`k qPpα i`k`1 |α i`k`1 _¨¨¨_α n q¨¨¨Ppα n´1 |α n´1 _α n q.