Boolean algebras of conditionals, probability and logic
aa r X i v : . [ m a t h . L O ] J un Boolean algebras of conditionals, probability and logic
Tommaso Flaminio , Lluis Godo , Hykel Hosni Artificial Intelligence Research Institute (IIIA) - CSIC, Barcelona, Spain email: { tommaso,godo } @iiia.csic.es Department of Philosophy, University of Milan, Milano, Italy email: [email protected]
Abstract
This paper presents an investigation on the structure of conditional events andon the probability measures which arise naturally in this context. In particular weintroduce a construction which defines a (finite)
Boolean algebra of conditionals fromany (finite) Boolean algebra of events. By doing so we distinguish the propertiesof conditional events which depend on probability and those which are intrinsic tothe logico-algebraic structure of conditionals. Our main result provides a way to re-gard standard two-place conditional probabilities as one-place probability functionson conditional events. We also consider a logical counterpart of our Boolean algebrasof conditionals with links to preferential consequence relations for non-monotonic rea-soning. The overall framework of this paper provides a novel perspective on the richinterplay between logic and probability in the representation of conditional knowledge.
KEYWORDS
Conditional probability; conditional events; Boolean algebras; prefer-ential consequence relations
Conditional expressions are pivotal in representing knowledge and reasoning abilities ofintelligent agents. Conditional reasoning features in a wide range of areas spanning non-monotonic reasoning, causal inference, learning, and more generally reasoning under un-certainty.This paper proposes an algebraic structure for conditional events which serves as alogical basis to analyse the concept of conditional probability – a fundamental tool inArtificial Intelligence.At least since the seminal work of Gaifman [22], who in turn develops the initial ideasof his supervisor Alfred Tarski [31], it has been considered natural to investigate theconditions under which Boolean algebras – i.e. classical logic – played the role of the logicof events for probability. The point is clearly made in [23]:Since events are always described in some language they can be identified withthe sentences that describe them and the probability function can be regardedas an assignment of values to sentences. The extensive accumulated knowledgeconcerning formal languages makes such a project feasible.1e are interested in pursuing the same idea, but taking conditional probability as aprimitive notion and obtain unconditional probability by specialisation. Taking condi-tional probability as primitive has a long tradition which dates back at least to [12] andincludes [32, 44, 45, 51]. The key justification for doing this lies in the methodologicalview that no assessment of probability takes place in a vacuum. On the contrary, eachprobabilistic evaluation must be done in the light of all and only the available evidence.In this sense, any probabilistic assessment of uncertainty is always conditional.The first step in achieving our goal is to clarify how conditional knowledge and in-formation should be represented. To do this we put forward a structure for representingconditional events, taken as the primitive objects of uncertainty quantification. In otherwords we aim to capture the logic/algebra which plays the role of classical logic whenthe focus of probability theory is shifted on conditional probability. In our preliminaryinvestigations [20, 21] on the subject we suggested taking the methodological approach ofasking the following questions:(i) which properties of conditional probabilities depend on properties of the measure and do not depend on the logical properties of conditional events?(ii) which properties do instead depend on the logic – whatever it is – of conditionalevents?Bruno de Finetti was the first not to take the notion of conditional events for grantedand argued that they cannot be described by truth-functional classical logic. He expressedthis by referring to conditional events as trievents [12, 14], with the following motivation.Since, intuitively, conditional events of the form “ a given b ” express some form of hypo-thetical assertion – the assertion of the consequent a based on the supposition that theantecedent b is satisfied – the logical evaluation of a conditional amounts to a two-stepprocedure. We first check the antecedent. If this is not satisfied, the conditional ceasesto mean anything at all. Otherwise we move on to evaluating the consequent and theconditional event takes the same value as the consequent.This interpretation allowed de Finetti to use the classical notion of uncertainty res-olution for conditional events implicitly assumed by Hausdorff and Kolmogorov, exceptfor the fact that de Finetti allowed the evaluation of conditional events to be a partial function. This is illustrated clearly by referring to the betting interpretation of subjectiveprobability, which indeed can be extended to a number of coherence-based measures ofuncertainty [18, 19]. To illustrate this, fix an uncertainty resolving valuation v , or in otherwords a two-valued classical logic valuation. Then de Finetti interprets conditional events“ θ given φ ” as follows:a bet on “ θ given φ ” is won if v ( φ ) = v ( θ ) = 1;lost if v ( φ ) = 1 and v ( θ ) = 0;called-off if v ( φ ) = 0 . This idea has been developed in uncertain reasoning, with links with non monotonicreasoning, in [17, 36, 35, 34]. In the context of probability logic, this approach has beenpursued in detail in [9]. Note that this latter approach is measure-theoretically oriented,and yet the semantics of conditional events is three-valued. The algebra of conditionalevents developed in the present paper, on the contrary, will be a Boolean algebra. Hence,2s we will point out in due time, the three-valued semantics of conditional events is notincompatible with requiring that conditional events form a Boolean algebra. What makesthis possible is that uncertainty-resolving valuations no longer correspond to classical logicvaluations, as in de Finetti’s work. Rather, as it will be clear from our algebraic analysis,they will correspond to finite total orders of valuations of classical logics. This cruciallyallows for the formal representation of the “gaps” in uncertainty resolution which arisewhen the antecedent of a conditional is evaluated to 0, forcing the bet to be called off.Some readers may be familiar with the copious and multifarious literature spanningphilosophical logic, linguistics and psychology which seeks to identify, sometimes proba-bilistically, how “conditionals” depart from Boolean (aka material) implication. Withinthis literature emerged a view according to which conditional probability can be viewed asthe probability of a suitably defined conditional. A detailed comparison with a proposal,due to Van Fraassen, in this spirit will be done in Subsection 8.2. However it may bepointed out immediately that a key contribution of this literature has been the very usefulargument, due to David Lewis [39], according to which the conditioning operator “ | ” can-not be taken, on pain of trivialising probability functions, to be a Boolean connective, andin particular material implication. This clearly reinforces the view, held since de Finetti’searly contributions, that conditional events have their own algebra and logic. A key con-tribution of this paper is to argue that this role can be played by what we term BooleanAlgebras of Conditionals (BAC). Armed with these algebraic structures, we can proceed toinvestigate the relation between conditional probabilities and (plain) probability measureson Boolean Algebras of Conditionals. In particular we construct, for each positive prob-ability measure on a finite Boolean algebra, its canonical extension to a Boolean Algebraof Conditionals which coincides with the conditional probability on the starting algebra.Hence we provide a formal setting in which the probability of conditional events can be re-garded as conditional probability. This contributes to a long-standing question which hasbeen put forward, re-elaborated and discussed by many authors along the years, and whosegeneral form can be roughly stated as follows: conditional probability is the probability ofconditionals [1, 39, 49, 51, 26, 33]. In the late 1960 logic-based Artificial Intelligence started to encompass qualitativeuncertainty. First through the notion of negation-as-failure in logic programming, thenwith the rise of non-monotonic logics, a field which owes substantially to the 1980 doublespecial issue of
Artificial Intelligence edited by D.G. Bobrow. Much of the following decadewas devoted to identifying general patterns in non monotonic reasoning, in the felicitousturn of phrase due to David Makinson [41]. One prominent such pattern emerged fromthe semantical approach put forward by Shoham [48]. According to it, a sentence θ is anon-monotonic consequence of a sentence φ if θ is (classically) satisfied by all preferred ormost normal models of φ . This equipped the syntactic notion of defaults – i.e. conditionalswhich are taken to be defeasibly true – with a natural semantics: defaults are conditionalswhich are “normally” true, where normality is captured by suitably ordering classicalmodels. Ordered models have then been the key to providing remarkable unity [38] tonon-monotonic reasoning, which by the early 2000s encompassed not only a variety of To some extent it can be regarded as a simplified version of Adams’s thesis [1, 2], claiming that the assertibility of a conditional a ⇒ b correlates with the conditional probability P ( b | a ) of the consequent b given the antecedent a . A more concrete statement of this thesis was put forward by Stalnaker by equatingAdam’s notion of assertability with that of probability: P ( a ⇒ b ) = P ( a ∧ b ) /P ( a ), whenever P ( a ) > Stalnaker’s thesis [49]. conditionals . Structure and summary of contributions of the paper
The paper is structured as follows. After this introduction and recalling some basic factsabout Boolean algebras in Section 2, we present in Section 3 the main construction whichallows us to define, starting from any Boolean algebra A of events, a corresponding Booleanalgebra of conditional events C ( A ). These algebras C ( A ), whose elements are objects ofthe form ( a | b ) for a, b ∈ A and their boolean combinations, are finite if the originalalgebras A are so.Section 4 is dedicated to the atomic structure of Boolean algebras of conditionals. Themain result is a full characterization of the atoms of each finite C ( A ) in terms of theatoms of A . This characterization is a fundamental step for the rest of the paper. Furtherelaborating on the atomic structure, Section 5 presents two tree-like representations forthe set of atoms of an algebra of conditionals which will be decisive in establishing themain result of the paper in Section 6.In fact, Section 6 introduces probability measures on Boolean algebras of conditionalsand presents our main result to the effect that every positive probability P on a finiteBoolean algebra A can be canonically extended to a positive probability µ P on C ( A )which agrees with the “conditionalised” version of the former. That is, we prove that, forevery a, b ∈ A with b = ⊥ , µ P (‘( a | b )’) = P ( a ∧ b ) /P ( b ) . As a welcome consequence of our investigation we provide an alternative, finitary,solution to the problem known in the literature as the strong conditional event problem ,introduced and solved in the infinite setting of the Goodman and Nguyen’s ConditionalEvent Algebras of [26].Although our Boolean algebras of conditionals do not allow for an equational descrip-tion, the characterizing properties of these algebras are expressible in an expansion of thelanguage of classical propositional logic and hence they give rise naturally to a simple logicof (non-nested) conditionals, that we name LBC (for
Logic of Boolean Conditionals ). Thisis investigated in Section 7, where we axiomatize the logic and prove soundness and com-pleteness with respect to a class of preferential structures. Moreover, we show that LBCsatisfies the properties of preferential non-monotonic consequence relations, in the sensepioneered by the seminal paper [37] and refined by [38]. Finally, Section 8 draws some4etailed comparisons between our contributions and the research on Measure-free condi-tionals (Subsection 8.1) and with Conditional Event Algebras (Subsection 8.2). Section 9outlines a set of key issues for future work.To facilitate the reading of the paper, most proofs are relegated to an appendix.
The algebraic framework of this paper is that of Boolean algebras and hence its logicalsetting is that of classical propositional logic (CPL). Here, we will briefly recap on someneeded notions and basic results about Boolean algebras and CPL, for a more exhaustiveintroduction about this subject we invite the reader to consult [7, § IV], and [10, 25, 29].Given a countable (finite or infinite) set V of propositional variables , the CPL language L ( V ) (or simply L when V will be clear by the context) is the smallest set containing V and closed under the usual connectives ∧ , ∨ , ¬ , ⊥ , and ⊤ of type (2 , , , , ϕ, ψ , etc (with possible subscript) for formulas. Further,we shall adopt the following abbreviations: ϕ → ψ = ¬ ϕ ∨ ψ , ϕ ↔ ψ = ( ϕ → ψ ) ∧ ( ψ → ϕ ).We shall denote by ⊢ CP L the provability relation of CPL, in particular we will write ⊢ CP L ϕ to denote that ϕ is a theorem.A logical valuation (or simply a valuation ) of L is a map from v : V → { , } , whichuniquely extends to a function, that we denote by the same symbol v , from L to { , } in accordance with the usual Boolean truth functions, i.e. v ( ϕ ∧ ψ ) = min { v ( ϕ ) , v ( ψ ) } , v ( ⊥ ) = 0, v ( ¬ ϕ ) = 1 − v ( ϕ ), etc. We shall denote by Ω the set of all valuations of L . Fora given formula ϕ and a given valuation v ∈ Ω, we will write v | = ϕ whenever v ( ϕ ) = 1.We will broadly adopt, analogously to the above recalled logical frame, the signature( ∧ , ∨ , ¬ , ⊥ , ⊤ ) of type (2 , , , ,
0) for the algebraic language upon which Boolean algebrasare defined. Thus, the same conventions and abbreviations of L can be adopted also inthe algebraic setting. Further, in every Boolean algebra A = ( A, ∧ , ∨ , ¬ , ⊥ , ⊤ ) we shallwrite a ≤ b , whenever a → b = ⊤ . The relation ≤ is indeed the lattice-order in A . Thus, a ≤ b iff a ∧ b = a iff a ∨ b = b .Along this paper, in order to distinguish an algebra from its universe, we will denotethe former by A , B etc, and the latter by A , B etc, respectively.Recall that a map h : A → B between Boolean algebras is a homomorphism if h commutes with the operations of their language, that is, h ( ⊤ A ) = ⊤ B , h ( ¬ A a ) = ¬ B h ( a ), h ( a ∧ A b ) = h ( a ) ∧ B h ( b ) etc, (notice that we adopt subscripts to distinguish the operationsof A from those of B ). Bijective (or 1-1) homomorphisms are called isomorphisms and ifthere is a isomorphism between A and B , they are said to be isomorphic (and we write A ∼ = B ).A congruence of a Boolean algebra A is an equivalence relation ≡ on A which iscompatible with its operations (see [25, § a, a ′ , b, b ′ ∈ A , if a ≡ a ′ and b ≡ b ′ then ¬ a ≡ ¬ a ′ , a ∧ b ≡ a ′ ∧ b ′ and a ∨ b ≡ a ′ ∨ b ′ . The compatibility propertyallows us to equip the set A/ ≡ = { [ a ] | a ∈ A } of equivalence classes with operationsinherited from A , endowing A/ ≡ with a structure of Boolean algebra, written A / ≡ , andwhich is called the quotient of A modulo ≡ . For a later use, we further recall that for all a, a ′ ∈ A such that a ≡ a ′ , the equality [ a ] = [ a ′ ] holds in A / ≡ . Recall that for any subset5 ⊆ A × A , the congruence generated by X is the smallest congruence ≡ X which contains X . The congruence ≡ X always exists [7, § V , the free V -generated algebra Free ( V ) (see [7, § II]) is isomorphic to the Lindenbaumalgebra of CPL over a language whose propositional variables belong to V (see for instance[6]). Since these structures will play a quite important role in the main construction wewill introduce in Section 3, let us briefly recap on them. Given any set V of propositionalvariables, we denote by L ( V ) / ≡ the set of equivalence classes of formulas of the language L ( V ) modulo the congruence relation ≡ of equi-provability , i.e., two formulas ϕ and ψ areequi-provable iff ⊢ CP L ϕ ↔ ψ . The algebra L ( V ) = ( L ( V ) / ≡ , ∧ , ∨ , ¬ , ⊥ , ⊤ ) is a Booleanalgebra called the Lindenbaum algebra of CPL over the language L ( V ). Therefore, a map v : L ( V ) → { , } is a valuation iff it is a homomorphism of L ( V ) into (where denotesthe Boolean algebra of two elements { , } ). Definition 2.1.
An element a of a Boolean algebra A is said to be an atom of A if a > ⊥ and for any other element b ∈ A such that a ≥ b ≥ ⊥ , either a = b or b = ⊥ .For every algebra A , we shall henceforth denote by at ( A ) the set of its atoms andwe will denote its elements by α, β, γ etc. If at ( A ) = ∅ , A is called atomic , otherwise A is said to be atomless . If A is finite, then it is atomic. In particular, if V is finite, theLindenbaum algebra L ( V ) is finite as well and thus atomic [4]. In fact, if | · | denotesthe cardinality map, if | V | = n then | at ( L ( V )) | = 2 n , and | L ( V ) | = 2 n . The followingproposition collects well-known and needed facts about atoms (see e.g. [7, § I] and [29, § at ( A ) is a partition of an atomic algebra A .Recall that a partition of a Boolean algebra is a collection of pairwise disjoint elementsdifferent from ⊥ whose supremum is ⊤ . Proposition 2.2.
Every finite Boolean algebra is atomic. Further, for every finite Booleanalgebra A the following hold:(i) for every α, β ∈ at ( A ) , α ∧ β = ⊥ ;(ii) for every a ∈ A , a = W α ≤ a α . Thus, in particular, W α ∈ at ( A ) α = ⊤ ;(iii) for each α ∈ at ( A ) , the map h α : A → such that h α ( a ) = 1 if a ≥ α and h α ( a ) = 0 otherwise, is a homomorphism. Furthermore, the map λ : α h α is a1-1 correspondence between at ( A ) and the set of homomorphisms of A in ;(iv) if A = L ( V ) with V finite, the map λ as in (iii) is a 1-1 correspondence between theatoms of L ( V ) and the set Ω of valuations of V .Moreover, a subset X = { x , . . . , x m } ⊆ A coincides with at ( A ) iff the following twoconditions are satisfied:(a) X is a partition of A (i.e. x i ∧ x j = ⊥ if i = j , and W mi =1 x i = ⊤ );(b) every x i ∈ X is such that ⊥ < x i and there is no b ∈ A such that ⊥ < b < x i . Boolean algebras of conditionals
In this section we introduce the notion of Boolean algebras of conditionals and prove somebasic properties. For any Boolean algebra A , the construction we are going to presentbuilds a Boolean algebra of conditionals that we shall denote by C ( A ). In the following,given a Boolean algebra A , we will write A ′ for A \ {⊥} .Intuitively, in a Boolean algebra of conditionals over A we will allow basic conditionals ,i.e. objects of the form ( a | b ) for a ∈ A and b ∈ A ′ , to be freely combined with the usualBoolean operations up to certain extent. Recall from the introduction that our main goal isto distinguish, as far as this is possible, the properties of the uncertainty measure from thealgebraic properties of conditionals. This means that we must pin down properties whichmake sense in the context of conditional reasoning under uncertainty. Those propertiesare summed up in the following four informal requirements, which guide our construction.R1 For every b ∈ A ′ , the conditional ( b | b ) will be the top element of C ( A ), while( ¬ b | b ) will be the bottom;R2 Given b ∈ A ′ , the set of conditionals A | b = { ( a | b ) : a ∈ A } will be the domain of aBoolean subalgebra of C ( A ), and in particular when b = ⊤ , this subalgebra will beisomorphic to A ;R3 In a conditional ( a | b ) we can replace the consequent a by a ∧ b , that is, we requirethe conditionals ( a | b ) and ( a ∧ b | b ) to represent the same element of C ( A );R4 For all a ∈ A and all b, c ∈ A ′ , if a ≤ b ≤ c , then the result of conjunctively combiningthe conditionals ( a | b ) and ( b | c ) must yield the conditional ( a | c ).Whilst conditions R1-R3 do not require delving into particular justifications, it is worthnoting that R4 encodes a sort of restricted chaining of conditionals and it is inspired by thechain rule of conditional probabilities: P ( a | b ) · P ( b | c ) = P ( a | c ) whenever a ≤ b ≤ c .Given these four requirements, the formal construction of the algebra C ( A ) is done inthree steps described next. The first one is to consider the set of objects A | A = { ( a | b ) : a ∈ A, b ∈ A ′ } and thealgebra Free ( A | A ) = ( F ree ( A | A ) , ⊓ , ⊔ , ∼ , ⊥ ∗ , ⊤ ∗ ) . Recall from Section 2 that
Free ( A | A ) is (up to isomorphism) the Boolean algebra whoseelements are equivalence classes (modulo equi-provability) of Boolean terms generated byall pairs ( a | b ) ∈ A | A taken as propositional variables. In other words, in Free ( A | A )two Boolean terms can be identified (i.e. they belong to the same class) only if one term canbe rewritten into the other one by using only the laws of Boolean algebras. For instance( a | b ) ⊓ ( c | b ) and ( c | b ) ⊓ ( a | b ) clearly belong to the same class in Free ( A | A ), but( a ∧ c | b ) does not, a fact that is not in agreement with requirement R2.Therefore, in a second step, in order to accommodate the requirements R1-R4 abovewe need to identify more classes in Free ( A | A ). In particular, we would like ( a ∧ c | b ),( a | b ) ⊓ ( c | b ) and ( c | b ) ⊓ ( a | b ) to represent the same element in the algebra C ( A ). Thus,to enforce this and all the other desired identifications in Free ( A | A ), we consider thecongruence relation on Free ( A | A ) generated by the subset C ⊆ F ree ( A | A ) × F ree ( A | A ) containing the following pairs of terms: Our construction is inspired by the one by Mundici for algebraic tensor products [43]. b | b ) , ⊤ ∗ ), for all b ∈ A ′ ;(C2) (( a | b ) ⊓ ( a | b ) , ( a ∧ a | b )), for all a , a ∈ A , b ∈ A ′ ;(C3) ( ∼ ( a | b ) , ( ¬ a | b )), for all a ∈ A , b ∈ A ′ ;(C4) (( a ∧ b | b ) , ( a | b )), for all a ∈ A , b ∈ A ′ ;(C5) (( a | b ) ⊓ ( b | c ) , ( a | c )), for all a ∈ A , b, c ∈ A ′ such that a ≤ b ≤ c .Note that (C1)-(C5) faithfully account for the requirements R1-R4 where, in particular,(C2) and (C3) account for R2. In particular, observe that, continuing the discussion above,now the elements ( a ∧ c | b ), ( a | b ) ⊓ ( c | b ) and ( c | b ) ⊓ ( a | b ) belong to the same classunder the equivalence ≡ C .Then, we finally propose the following definition. Definition 3.1.
For every Boolean algebra A , we define the Boolean algebra of condi-tionals of A as the quotient structure C ( A ) = Free ( A | A ) / ≡ C . Note that, by construction, if A is finite, so is C ( A ). For the sake of an unambiguousnotation, we will henceforth distinguish the operations of A from those of C ( A ) by adoptingthe following signature: C ( A ) = ( C ( A ) , ⊓ , ⊔ , ∼ , ⊥ C , ⊤ C ) . Remark 3.2 (Notational convention) . Since C ( A ) is a quotient of Free ( A | A ), its genericelement is a class [ t ] ≡ C , for t being a Boolean term, whose members are equivalent to t under ≡ C . For the sake of a clear notation and without danger of confusion, we willhenceforth identify [ t ] ≡ C with one of its representative elements and, in particular, by t itself. Given two elements t , t of C ( A ), we will write t = t meaning that t and t determine the same equivalence class of C ( A ) or, equivalently, that t ≡ C t .It is then clear that, using the above notation convention, the following equalities,which correspond to (C1)–(C5) above, hold in any Boolean algebra of conditionals C ( A ). Proposition 3.3.
Any Boolean algebra of conditionals C ( A ) satisfies the following prop-erties for all a, a ′ ∈ A and b, c ∈ A ′ :(i) ( b | b ) = ⊤ C ;(ii) ( a | b ) ⊓ ( c | b ) = ( a ∧ c | b ) ;(iii) ∼ ( a | b ) = ( ¬ a | b ) ;(iv) ( a ∧ b | b ) = ( a | b ) ;(v) if a ≤ b ≤ c , then ( a | b ) ⊓ ( b | c ) = ( a | c ) . Straightforward consequences of (iv) and (v) above are the following.
Corollary 3.4. (i) ( b → a | b ) = ( a | b ) ; ii) ( a ∧ b | ⊤ ) = ( a | b ) ⊓ ( b | ⊤ ) ;(iii) ( a ∧ b | c ) = ( a | b ∧ c ) ⊓ ( b | c ) . Notice that (iii) above corresponds to the qualitative version of axiom CP3 of [30,Definition 3.2.3].It is convenient to distinguish the elements of C ( A ) in basic and compound conditionals.The former are expressions of the form ( a | b ), while the latter are those terms t whichare (non trivial) Boolean combination of basic conditionals but which are not equivalentmodulo ≡ C (and hence not equal ), to any element of C ( A ) of the form ( a | b ). For instance,if b = b ∈ A ′ there is no general rule, among (C1)–(C5) above, which allows us to identifyin C ( A ) the term ( a | b ) ⊓ ( a | b ) with a basic conditional of the form ( x | y ) whilst,the term ( a | b ) ⊓ ( a | b ) coincides in C ( A ) with the basic conditional ( a ∧ a | b ), asrequired by (C2). Example 3.5.
Let us consider the four elements Boolean algebra A whose domain is {⊤ , a, ¬ a, ⊥} . Then, A | A = { ( ⊤ , ⊤ ), ( ⊤ , a ), ( ⊤ , ¬ a ), ( a, ⊤ ), ( a, a ), ( a, ¬ a ), ( ¬ a, ⊤ ),( ¬ a, a ), ( ¬ a, ¬ a ), ( ⊥ , ⊤ ), ( ⊥ , a ), ( ⊥ , ¬ a ) } has cardinality 12 and Free ( A | A ) is the freeBoolean algebra of 2 elements, i.e. the finite Boolean algebra of 2 atoms. However,in C ( A ) the following equations hold (and the conditionals below are hence identified):1. ⊤ C = ( ⊤ | ⊤ ) = ( a | ⊤ ) ⊔ ( ¬ a | ⊤ ) = ( ⊤ | a ) = ( a | a ) = ( ¬ a | ¬ a );2. ( ⊤ | ⊤ ) ⊓ ( a | ⊤ ) = ( ⊤ ∧ a | ⊤ ) = ( a | ⊤ ) = ∼ ( ¬ a | ⊤ );3. ( ⊤ | ⊤ ) ⊓ ( ¬ a | ⊤ ) = ( ⊤ ∧ ¬ a | ⊤ ) = ( ¬ a | ⊤ ) = ∼ ( a | ⊤ );4. ⊥ C = ∼ ( ⊤ | ⊤ ) = ( ⊥ | ⊤ ) = ( a | ⊤ ) ⊓ ( ¬ a | ⊤ ) = ( a | ⊤ ) ⊓ ∼ ( a | ⊤ ).Thus, it is easy to see that C ( A ) contains only four elements that are not redundantunder ≡ C : ( ⊤ | ⊤ ) , ( a | ⊤ ) , ( ¬ a | ⊤ ) , ( ⊥ | ⊤ ). As we will show in Section 4 (see Theorem4.4) C ( A ) has 2 atoms and it is indeed isomorphic to A .Next, we present some further basic properties of Boolean algebras of conditionalswhich are not immediate from the construction. However, since their proofs are essentiallytrivial, we also omit them. Proposition 3.6.
The following conditions hold in every Boolean algebra of conditionals C ( A ) :(i) for all a, c ∈ A , ( a | ⊤ ) = ( c | ⊤ ) iff a = c ;(ii) for all b ∈ A ′ , ( ¬ b | b ) = ⊥ C ;(iii) for all a, c ∈ A , and b ∈ A ′ , ( a | b ) ⊔ ( c | b ) = ( a ∨ c | b ) ; For every fixed b ∈ A ′ , we can now consider the set A | b = { ( a | b ) | a ∈ A } of allconditionals having b as antecedent. The following is an immediate consequence of (i-iii)of Proposition 3.3 and Proposition 3.6 (iii) above. Corollary 3.7.
For every algebra C ( A ) and for every b ∈ A ′ the structure A | b = ( A | b, ⊓ , ⊔ , ¬ , ⊥ C , ⊤ C ) is a Boolean subalgebra of C ( A ) . In particular, the algebra A | ⊤ isisomorphic to A .
9s in any Boolean algebra, the lattice order relation in C ( A ), denoted by ≤ , is definedas follows: for every t , t ∈ C ( A ), t ≤ t iff t ⊓ t = t iff t ⊔ t = t . The following propositions collect some general properties related to the lattice order ≤ defined above. Nevertheless, some further and stronger properties on the ≤ -relationbetween basic conditionals will be provided at the end of Section 4, once the atomicstructure of the algebras of conditionals C ( A ) will be characterised in that section. Proposition 3.8.
In every algebra C ( A ) the following properties hold for every a, c ∈ A and b ∈ A ′ :(i) ( a | b ) ≥ ( b | b ) iff a ≥ b ;(ii) if a ≤ c , then ( a | b ) ≤ ( c | b ) ; in particular a ≤ c iff ( a | ⊤ ) ≤ ( c | ⊤ ) ;(iii) if a ≤ b ≤ d , then ( a | b ) ≥ ( a | d ) ; in particular ( a | b ) ≥ ( a | a ∨ b ) ;(iv) if ( a | b ) = ( c | b ) , then a ∧ b = c ∧ b ;(v) ( a ∧ b | ⊤ ) ≤ ( a | b ) ≤ ( b → a | ⊤ ) ;(vi) if a ∧ d = ⊥ and ⊥ < a ≤ b , then ( a | ⊤ ) ⊓ ( d | b ) = ⊥ C ;(vii) ( b | ⊤ ) ⊓ ( a | b ) ≤ ( a | ⊤ ) ;Proof. See Appendix.Some properties in the proposition above have a clear logical reading. For instance,(v) tells us that in a Boolean algebra of conditionals, a basic conditional ( a | b ) is a weakerconstruct than the conjunction a ∧ b but stronger than the material implication b → a ,in accordance to previous considerations in the literature, see e.g. [17]. As a consequence,this suggests that a conditional ( a | b ) can be evaluated to true when both b and a areso (i.e. when a ∧ b is true), while ( a | b ) can be evaluated as false when a is false and b is true (i.e. when falsifying b → a ). Furthermore, (vii) can be read as a form of modusponens with respect to conditional expressions: from b and ( a | b ) it follows a . We referthis discussion on logical issues of conditionals to Section 7 in which we will introduce andstudy a logic of conditionals and where we will propose a formal definition of truth forthem.We now end this section presenting a few further properties of Boolean algebras ofconditionals regarding the disjunction in the antecedents. Proposition 3.9.
In every algebra C ( A ) the following properties hold for all a, a ′ ∈ A and b, b ′ ∈ A ′ :(i) ( a | b ) ⊓ ( a | b ′ ) ≤ ( a | b ∨ b ′ ) ; in particular, ( a | b ) ⊓ ( a | ¬ b ) ≤ ( a | ⊤ ) ;(ii) if a ≤ b ∧ b ′ , then ( a | b ) ⊓ ( a | b ′ ) = ( a | b ∨ b ′ ) ;(iii) ( a | b ) ≤ ( b → a | b ∨ b ′ ) ;(iv) ( a | b ) ⊓ ( a ′ | b ′ ) ≤ (( b → a ) ∧ ( b ′ → a ′ ) | b ∨ b ′ ) . roof. See Appendix.Observe that the logical reading of property (i) above is the well-known
OR-rule ,typical of nonmonotonic reasoning (see [2, 17]). This fact, although not being particularlysurprising, will be further strengthened in Section 7 where we will show that, indeed,Boolean algebras of conditionals provide a sort of algebraic semantics for a nonmonotoniclogic related to System P. Further, (iv) shows that the algebraic conjunction of two basicconditionals is stronger than the operation of quasi-conjunction introduced in the settingof measure-free conditionals (see [2] and [17, Lemma 2]) and recalled in Subsection 8.1.Also notice that, the point (iv) above, in the special case in which a ′ = b , b ′ = c and a ≤ b ≤ c actually gives ( a | b ) ⊓ ( b | c ) = ( a | c ) = (( b → a ) ∧ ( c → b ) | b ∨ c ). Therefore,the requirement R4 is in agreement with the definition of quasi-conjunction. As we already noticed, if A is finite, C ( A ) is finite as well and hence atomic. This sectionis devoted to investigate the atomic structure of finite Boolean algebras of conditionals. Inparticular, in Subsection 4.1 we provide a characterization of the atoms of C ( A ) in termsof the atoms of A . That characterization will be employed in Subsections 4.2 and 4.3 togive, respectively, a full description of the atoms which stand below a basic conditional( a | b ) and to prove results concerning equalities and inequalities among conditionals whichimprove those of Section 3.In this section and in rest of the paper, we will only deal with finite Boolean algebras. C ( A ) Let us recall the notation introduced in Section 2: for every Boolean algebra A , we denoteby at ( A ) the set of its atoms, that will be denoted by lower-case greek letters, α, β, γ etc. Proposition 4.1.
In a conditional algebra C ( A ) , the following hold:(i) each element t of C ( A ) is of the form t = d i ( F j ( a i j | b i j )) ;(ii) each basic conditional is of the form ( a | b ) = F α ≤ a ( α | b ) ;(iii) in particular, every element of C ( A ) is a ⊓ - ⊔ combination of basic conditionals inthe form ( α | W X ) where α ∈ at ( A ) and X ⊆ at ( A ) .Proof. (i). It readily follows by recalling that (1) every element t of C ( A ) is, by construc-tion, a Boolean combination of basic conditionals, (2) it can be expressed in conjunctivenormal form, and (3) the negation of a basic conditional ( a | b ) is the basic conditional( ¬ a | b ).(ii). The claim directly follows from Proposition 3.6 (iii) taking into account that a = W α ≤ a α (recall Proposition 2.2 (ii)).(iii). It is a direct consequence of (i) and (ii).Now, let A be a Boolean algebra with n atoms, i.e. | at ( A ) | = n . For each i ≤ n −
1, letus define
Seq i ( A ) to be the set of sequences h α , α , . . . , α i i of i pairwise different elements11f at ( A ). Thus, for every α = h α , α , . . . , α i i ∈ Seq i ( A ), let us consider the compoundconditional of C ( A ) defined in the following way: ω α = ( α | ⊤ ) ⊓ ( α | ¬ α ) ⊓ . . . ⊓ ( α i | ¬ α ∧ . . . ∧ ¬ α i − ) . (1)Intuitively, such a conjunction of conditonals encodes a sort of chained ‘defeasible’conditional statements about a set of mutually disjoint events: in principle α holds, butif α turns out to be false then in principle α holds, but if besides α turns out to be falseas well, then in principle α holds, and so on . . .These conjunctions of basic conditionals will play an important role in describing theatomic structure of C ( A ) and enjoy suitable properties. To begin with, let us consider setsof those compound conditionals of a given length: for each 1 ≤ i ≤ n −
1, let
P art i ( C ( A )) = { ω α | α ∈ Seq i ( A ) } . Example 4.2.
Let A be the Boolean algebra with 4 atoms, at ( A ) = { α , . . . , α } . For i = 1, the set P art ( C ( A )) is easily built by considering all sequences of length 1 of atomsof A , Seq ( C ( A )) = {h α i , h α i , h α i , h α i} , and hence: P art ( C ( A )) = { ω h α i , . . . , ω h α i } = { ( α | ⊤ ) , . . . , ( α | ⊤ ) } . For i = 2, we have to consider sequences of atoms of length 2, i.e. Seq ( C ( A )) = {h α , α i , h α , α , i , h α , α i , h α , α i , . . . } , and then the corresponding set P art ( C ( A )) is composed by 12 Boolean terms like ω h α ,α i = ( α | ⊤ ) ⊓ ( α | ¬ α ); ω h α ,α i = ( α | ⊤ ) ⊓ ( α | ¬ α ); ω h α ,α i = ( α | ⊤ ) ⊓ ( α | ¬ α ); ω h α ,α i = ( α | ⊤ ) ⊓ ( α | ¬ α );. . .Finally, for i = 3, consider the set sequences of atoms of length 3, Seq ( C ( A )) = {h α , α , α i , h α , α , α i , h α , α , α i , . . . } . Therefore, the set
P art ( C ( A )) contains 24 Boolean terms: ω h α ,α ,α i = ( α | ⊤ ) ⊓ ( α | ¬ α ) ⊓ ( α | ¬ α ∧ ¬ α ); ω h α ,α ,α i = ( α | ⊤ ) ⊓ ( α | ¬ α ) ⊓ ( α | ¬ α ∧ ¬ α ); ω h α ,α ,α i = ( α | ⊤ ) ⊓ ( α | ¬ α ) ⊓ ( α | ¬ α ∧ ¬ α );. . .The following result shows that, for each i , P art i ( C ( A )) is a partition of C ( A )(recallSection 2), and the higher the index i , the more refined the partition is. Proposition 4.3.
P art i ( C ( A )) is a partition of C ( A ) .Proof. See Appendix. 12s we already saw in Example 4.2, if at ( A ) = { α , . . . , α n } , then P art ( C ( A )) = { ( α | ⊤ ) , . . . , ( α n | ⊤ ) } and it gives the coarsest partition among those that we denotedby P art i ( C ( A )).In the following, if | at ( A ) | = n , for simplicity, we will write Seq ( A ) and P art ( C ( A ))instead of Seq n − ( A ) and P art n − ( C ( A )) respectively. Note that in this case, for every α = h α , α , . . . , α n − i ∈ Seq ( A ), the compound conditional ω α defined as in (1), can beequivalently written as ω α = ( α | ⊤ ) ⊓ ( α | α ∨ · · · ∨ α n ) ⊓ ( α | α ∨ . . . ∨ α n ) ⊓ . . . ⊓ ( α n − | α n − ∨ α n ) . Next theorem shows that these conditionals are in fact the atoms of C ( A ). Theorem 4.4.
Let A be a Boolean algebra such that | at ( A ) | = n . Then, at ( C ( A )) = { ω α | α ∈ Seq ( A ) } = P art ( C ( A )) . As a consequence, | at ( C ( A )) | = n ! and |C ( A ) | = 2 n ! .Proof. To show that
P art ( C ( A )) coincides with at ( C ( A )), by Proposition 2.2, we have toprove the following two conditions:(a) P art ( C ( A )) is a partition of C ( A ) . (b) Any ω α ∈ P art ( C ( A )) is such that ⊥ C < ω α and there is no t ∈ C ( A ) such that ⊥ C < t < ω α . It is clear that (a) is just the case i = n − ω α ∈ P art ( C ( A )) is different from ⊥ C . Indeed, it follows from (a) and a symmetryargument on the elements of P art ( C ( A )). Thus, it is enough to show that, if t is anyelement of C ( A ) which is not ⊥ C , then either t ⊓ ω α = ω α , or t ⊓ ω α = ⊥ C . We show thisclaim by cases on the form of t :(i) Assume t is basic conditional of the form t = ( γ | b ) with γ ∈ at ( A ). Let α = h α , α , . . . , α n − i . Since γ ∈ at ( A ), then γ = α i for some 1 ≤ i ≤ n . Then we havetwo cases: either b = α i ∨ · · · ∨ α n , and in that case ω α ⊓ ( γ | b ) = ω α , or otherwise b is of the form b = α i ∨ α k ∨ c , for some k < i and some c ∈ A . If the latter is thecase, we have ( γ | b ) ⊓ ( α k | α k ∨ . . . ∨ α n ) = ( α i | α i ∨ α k ∨ c ) ⊓ ( α k | α k ∨ . . . ∨ α n ) ≤ ( α i | α i ∨ α k ) ⊓ ( α k | α k ∨ α i ) = ⊥ C , and hence ( γ | b ) ⊓ ω α = ⊥ C as well.(ii) Assume t is a basic conditional t = ( a | b ). By (ii) of Proposition 4.1, we can express( a | b ) = F α ∈ at ( A ): α ≤ a ( α | b ). Hence, ω α ⊓ ( a | b ) = F γ ω α ⊓ ( γ | b ), but by (i), foreach γ , ω α ⊓ ( γ | b ) is either ω α or ⊥ C . So this is also the case for ω α ⊓ t .(iii) Finally, assume t is an arbitrary element of C ( A ). By Proposition 4.1 above, t is a ⊓ - ⊔ combination of basic conditionals, i.e. it can be displayed as t = d i ( F j ( a i j | b i j )).Then we have ω α ⊓ t = ω α ⊓ l i G j ( a i j | b i j ) = l i G j ω α ⊓ ( a i j | b i j ) . By (ii), each ω α ⊓ ( a i j | b i j ) is either equal to ⊥ C or to ω α , and hence so is ω α ⊓ t .13 α α ⊥ ¬ α ¬ α ¬ α ⊤ Figure 1:
The Boolean algebra with 3 atoms and 8 elements.
Therefore, we have proved that
P art ( C ( A )) = at ( C ( A )). Example 4.5.
Let A be the Boolean algebra of 3 atoms α , α , α , and 8 elements, seeFigure 1. Theorem 4.4 tells us that the atoms of the conditional algebra C ( A ) are asfollows: at ( C ( A )) = { ( α i | ⊤ ) ⊓ ( α j | ¬ α i ) : i, j = 1 , , i = j } . Therefore, C ( A ) has six atoms and 2 = 64 elements (see Figure 2). In particular, theatoms are: ω = ( α | ⊤ ) ⊓ ( α | ¬ α ), ω = ( α | ⊤ ) ⊓ ( α | ¬ α ), ω = ( α | ⊤ ) ⊓ ( α | ¬ α ), ω = ( α | ⊤ ) ⊓ ( α | ¬ α ), ω = ( α | ⊤ ) ⊓ ( α | ¬ α ), ω = ( α | ⊤ ) ⊓ ( α | ¬ α ).Notice that ω ⊔ ω = ( α | ⊤ ). Indeed, (( α | ⊤ ) ⊓ ( α | ¬ α )) ⊔ (( α | ⊤ ) ⊓ ( α |¬ α )) = ( α | ⊤ ) ⊓ ( α ∨ α | ¬ α ) = ( α | ⊤ ) ⊓ ( ¬ α | ¬ α ) = ( α | ⊤ ) ⊓ ⊤ C = ( α | ⊤ ).Analogously, we can also derive ω ⊔ ω = ( α | ⊤ ) and ω ⊔ ω = ( α | ⊤ ).Now, let us consider the conditional t = ( α | ¬ α ). Obviously, t = F { ω i : ω i ≤ t } andthanks to Proposition 3.8, we can check that t = ω ⊔ ω ⊔ ω . As a matter of fact, since ω ⊔ ω = ( α | ⊤ ), we have ω ⊔ ω ⊔ ω = ( α | ⊤ ) ⊔ (( α | ⊤ ) ⊓ ( α | ¬ α )) = (( α | ⊤ ) ⊔ ( α | ⊤ )) ⊓ (( α | ⊤ ) ⊔ ( α | ¬ α )) . Now, ( α | ⊤ ) ⊔ ( α | ⊤ ) = ( α ∨ α | ⊤ ) = ( ¬ α → α | ⊤ ), while Proposition 3.8 (iii)implies that ( α | ⊤ ) ⊔ ( α | ¬ α ) = ( α | ¬ α ) because α ≤ ¬ α ≤ ⊤ . Finally, byProposition 3.8 (v) ( α | ¬ α ) ≤ ( ¬ α → α | ⊤ ) and hence ω ⊔ ω ⊔ ω = ( α | ¬ α ) = t . 14 C ω ω ω ω ω ω t ( α | ⊤ ) ( α | ⊤ ) ( α | ⊤ ) ⊤ C Figure 2:
The algebra of conditionals C ( A ) of Example 4.5, where at ( A ) = { α , α , α } and at ( C ( A )) = { ω , ω , ω , ω , ω , ω } . The element t = ( α | ¬ α ) (squared node) is obtained as ω ⊔ ω ⊔ ω . The atoms of C ( A ) are marked by grey dots, while the elements a of the originalalgebra A , regarded as conditionals ( a | ⊤ ), correspond to the bigger black dots. Thanks to Theorem 4.4 and without danger of confusion, for every sequence α ∈ Seq ( A ), we will henceforth denote by ω α the atom of C ( A ) which is uniquely associatedto the sequence α .A simple argument about the cardinality of Boolean algebras shows that not everyfinite Boolean algebra is isomorphic to a finite Boolean algebra of conditionals. In fact,from Theorem 4.4, there is no Boolean algebras of conditionals of cardinality different from2 n ! for every n whence, for instance, there is no Boolean algebra of conditionals with 2 elements. Therefore, the class of all finite Boolean algebras of conditionals forms a propersubset of the class of all finite Boolean algebras. The following corollary shows that thecardinality of a finite Boolean algebra is enough to describe it as an isomorphic copy ofsome C ( A ). Corollary 4.6.
Let B be a Boolean algebra of cardinality n ! . Then there exists a Booleanalgebra A of cardinality n such that B ∼ = C ( A ) .Proof. Let β , . . . , β n ! be the atoms of B and let P = { P , . . . , P n } be a partition of at ( B )in subsets of ( n − P i = { β i , . . . , β i ( n − } , let a i = W ( n − j =1 β i j . Since P is a partition, if a i = a j , then a i ∧ a j = ⊥ B and W ni =1 a i = ⊤ B . Thus, the subalgebra A P of B generated by a , . . . , a n is such that at ( A P ) = { a , . . . , a n } . Notice, that, if P and P ′ are two different partitions of at ( B ) in subsets of cardinality ( n − A P ∼ = A P ′ .Thus, C ( A P ) ∼ = B since | at ( C ( A P )) | = | at ( B ) | .15 .2 Characterising the atoms below a basic conditional Now, we will be concerned with the description of the set of atoms of C ( A ) below (accordingto the lattice ordering in C ( A )) a given basic conditional ( a | b ). In general, for everyBoolean algebra B and for every b ∈ B , we will henceforth write at ≤ ( b ) to denote thesubset of at ( B ) below b . Thus, in particular, for every ( a | b ) ∈ C ( A ), at ≤ ( a | b ) = { ω α ∈ at ( C ( A )) | ω α ≤ ( a | b ) } . Proposition 4.7.
Let A be a Boolean algebra, let α = h α , α , . . . , α n − i ∈ Seq ( A ) andlet ( a | b ) ∈ C ( A ) be such that a ≤ b . Then, the following conditions are equivalent:(i) ω α ∈ at ≤ ( a | b ) ;(ii) there is an index i ≤ n − such that α j ≤ ¬ b for all j < i and α i ≤ a ;(iii) α i ≤ a for the smallest index i ≤ n − such that α i ≤ b .Proof. Let us start by showing that (ii) implies (i). Since ω α ≤ ( a | b ) = W β ≤ a ( β | b ) iff ω α ≤ ( β | b ) for some β ≤ a , it is sufficient to prove the claim for the case in which a = β .In such a case, if α i = β and ¬ α ∧ . . . ∧ ¬ α i − ≥ b then, since a ≤ b and α i = β = a , onehas b ≥ α i whence ¬ α ∧ . . . ∧ ¬ α i − ≥ b ≥ α i . Therefore, by Proposition 3.8 (iii),( α i | ¬ α ∧ . . . ∧ ¬ α i − ) ≤ ( α i | b ) = ( β | b ) , and hence ω α = ( α | ⊤ ) ⊓ . . . ⊓ ( α i | ¬ α ∧ . . . ∧ ¬ α i − ) ⊓ · · · ≤ ( β | b ) as well.To prove the other direction, (i) implies (ii), we consider two cases:(a) There is i ≤ n − α i = β . If i = 1, since ⊤ ≥ b , then the claim is fulfilled.Thus, assume i > ω α ⊓ ( β | b ) = ω α , and let us prove ¬ α ∧ . . . ∧ ¬ α i − ≥ b . Indeed,we have the following subcases: • If α ≤ b , we would have ( α | ⊤ ) ⊓ ( β | b ) ≤ ( α | b ) ⊓ ( α i | b ) = ⊥ C , and hence ω α ⊓ ( β | b ) = ⊥ C , contradiction. Therefore α ≤ ¬ b . • If α ≤ b , since α ≤ ¬ b , we would have ( α | ¬ α ) ⊓ ( β | b ) ≤ ( α | b ) ⊓ ( β | b ) = ⊥ C ,and hence ω α ⊓ ( β | b ) = ⊥ C , contradiction. Therefore, α ≤ ¬ b .. . . • If α i − ≤ b , since α ≤ ¬ b , α ≤ ¬ b , . . . , α i − ≤ ¬ b , we would have ( α i − |¬ α ∧ . . . ∧ α i − ) ⊓ ( β | b ) ≤ ( α | b ) ⊓ ( β | b ) = ⊥ C , and hence ω α ⊓ ( β | b ) = ⊥ ,contradiction. Therefore, α i − ≤ ¬ b .As a consequence, α ∨ . . . ∨ α i − ≤ ¬ b or, equivalently, ¬ α ∧ . . . ∧ ¬ α i − ≥ b .(b) β = α n , where α n is the remaining atom not appearing in α . In this case, one canshow that ω α ⊓ ( β | b ) = ⊥ C , and hence ω α ( β | b ). Indeed, if β = α n < b , it means that b ≥ α i ∨ α n , with i ≤ n −
1. Then, the expression ( α i | ¬ α ∧ . . . ¬ α i − ) = ( α i | α i ∨ . . . ∨ α n )appears as a conjunct in the atom ω α . Thus, ω α ⊓ ( β | b ) ≤ ( α i | α i ∨ . . . ∨ α n ) ⊓ ( β | b ) ≤ ( α i | α i ∨ α n ) ⊓ ( β | α i ∨ α n ) = ⊥ C .Finally, notice that (iii) is just an equivalent rewriting of (ii) observing that if α i ≤ a then α i ≤ b as well, since we are assuming a ≤ b .16s a consequence of the above characterisation, one can compute the number of atomsbelow a given conditional. Corollary 4.8.
Let A be a Boolean algebra with | at ( A ) | = n . For every basic conditional ( a | b ) ∈ C ( A ) with a ≤ b , | at ≤ ( a | b ) | = n ! · | at ≤ ( a ) || at ≤ ( b ) | .Proof. See Appendix.
The results proved in the previous subsections allow us to determine when two basicconditionals are equal. Further, in this subsection, we will investigate properties regardingthe order among conditionals which improve those of Section 3. Let us start with twopreliminary lemmas.
Lemma 4.9.
Let a, b, c ∈ A be such that ⊥ < a < b and a < c . Then ( a | b ) ≥ ( a | c ) iff b ≤ c .Proof. One direction is easy. If b ≤ c then b ∨ c = c , and by Proposition 3.9 (ii), we have( a | b ) ⊓ ( a | c ) = ( a | b ∨ c ) = ( a | c ), that is, ( a | b ) ≥ ( a | c ).As for the other, let α be an atom of A such that α ≤ a . If b (cid:2) c , there is an atom β such that β ≤ b but β c (and thus β a ). Then let ω β ∈ at ( C ( A )) be of the form ω β = ( β | ⊤ ) ⊓ ( α | ¬ β ) ⊓ . . . . Then it follows that ω β ≤ ( a | c ), since β (cid:2) c and both α ≤ c and α ≤ a , but ω β ( a | b ), since β ≤ b but β (cid:2) a . Hence ( a | b ) (cid:3) ( a | c ).As a direct consequence of this lemma, we have the following property that strengthensProposition 3.6 (i). Corollary 4.10.
Let a, b, c ∈ A be such that ⊥ < a < b and a < c . Then ( a | b ) = ( a | c ) iff b = c . Lemma 4.11.
Let a, b, c, d ∈ A be such that ⊥ < a < b and ⊥ < c < d . Then ( a | b ) =( c | d ) iff a = c and b = d .Proof. One direction is trivial. As for the other, assume a = c , and hence there is an atom α of A such that α ≤ a but α c . Further, since c < d there exists an atom γ of A suchthat γ ≤ d but γ (cid:2) c . Consider any atom ω α of C ( A ) of the form ω α = ( α | ⊤ ) ⊓ ( γ |¬ α ) ⊓ . . . . Then, using Proposition 4.7, one can check that ω α ≤ ( a | b ) since α ≤ a , but ω α ( c | d ) since α (cid:2) c and both γ ≤ d and γ (cid:2) c . Therefore, there are atoms below( a | b ) that are not below ( c | d ), hence ( a | b ) = ( c | d ), contradiction. Hence, it must be a = c .Finally, we apply Corollary 4.10 to get b = d as well.As a consequence of the above two lemmas, we can provide necessary and sufficientconditions for the equality between two basic conditionals. Theorem 4.12.
For any pair of conditionals ( a | b ) , ( c | d ) ∈ C ( A ) , we have ( a | b ) = ( c | d ) iff either ( a | b ) = ( c | d ) = ⊤ , or ( a | b ) = ( c | d ) = ⊥ , or a ∧ b = c ∧ d and b = d . Thanks to the characterisation Theorem 4.12, one can easily compute the number of(distinct) basic conditionals in a given algebra of conditionals C ( A ). Indeed, to computethe number of basic conditionals different from ⊤ and ⊥ amounts to counting all pairs( M, N ) with ∅ 6 = M ⊂ N ⊆ at ( A ). 17 orollary 4.13. Let A be a Boolean algebra with n atoms, i.e. with | at ( A ) | = n . Thenthe number of basic conditionals in C ( A ) is bc ( n ) = 2 + P nr =2 (cid:0) nr (cid:1) · (2 r − .Proof. The counting of pairs (
M, N ) with ∅ 6 = M ⊂ N ⊆ at ( A ) results from the followingobservations:(1) For each N with r elements, there are 2 r − M of N with lessthan r elements (note that necessarily | N | ≥ ∅ 6 = M ⊂ N );(2) Hence, there are (cid:0) nr (cid:1) · (2 r −
2) such pairs (
M, N ) such that | N | = r ;(3) Thus, the total number of such pairs will be n X r =2 (cid:18) nr (cid:19) · (2 r − ⊥ and ⊤ conditionals) to the above quantity.For example, in the algebra C ( A ) of Fig. 2, built from the algebra A with 3 atoms( n = 3), we have bc (3) = 14 basic conditionals, out of 64 elements in total. Hence itcontains 50 proper compound conditionals.Thanks to the results about the atomic structure of the algebras of conditionas C ( A ),we can now also provide similar, but partial, characterisation results of when an inequalitybetween two basic conditionals holds, stronger than those shown e.g. in Proposition 3.8. Lemma 4.14.
Let a, b, c, d ∈ A be such that ⊥ < a < b , ⊥ < c < d . Then:(i) if a ≤ c and b ≥ d then ( a | b ) ≤ ( c | d ) ;(ii) if ( a | b ) ≤ ( c | d ) then a ≤ c ;(iii) if c ≤ b and ( a | b ) ≤ ( c | d ) then b ≥ d .Proof. (i) Assume a ≤ c and b ≥ d . In such a case we have ⊥ < c ∧ b < b . Indeed,if it were c ∧ b = b , it would mean b ≤ c , and then we would have a < b ≤ c < d ,hence b < d that is in contradiction with the hypothesis b ≥ d . Therefore we havethe following chain of inequalities:( a | b ) ≤ ( c | b ) = ( c ∧ b | b ) ≤ ( c ∧ b | d ) ≤ ( c | d ) . Observe that the first and third inequalities are clear from (iii) of Proposition 3.8,while the second one follows from Lemma 4.9 due to the fact that c ∧ b < b and c ∧ d < d .(ii) Assume a (cid:2) c , and hence assume there is an atom α of A such that α ≤ a but α c . Further, since c < d there exists an atom γ of A such that γ ≤ d but γ (cid:2) c .Consider any atom ω α of C ( A ) of the form ω α = ( α | ⊤ ) ⊓ ( γ | ¬ α ) ⊓ . . . . Then,using Proposition 4.7, one can check that ω α ≤ ( a | b ) since α ≤ a , but ω α ( c | d )since α (cid:2) c and both γ ≤ d and γ (cid:2) c . Therefore, ( a | b ) (cid:2) ( c | d ).18iii) Assume b (cid:3) d . In this case, we can further assume a ≤ c , otherwise the previousitem can be applied. Let α be an atom of A such that α ≤ a , and hence α ≤ b and α ≤ c ≤ d as well.If b (cid:3) d , there is an atom β such that β ≤ d but β b (and thus β a and β c as well since a < b and c ≤ b ). Then let ω β ∈ at ( C ( A )) be of the form ω β = ( β | ⊤ ) ⊓ ( α | ¬ β ) ⊓ . . . . Then it follows that ω β ≤ ( a | b ), since β (cid:2) b and both α ≤ a and α ≤ b , but ω β ( c | d ), since β ≤ d but β (cid:2) c . Therefore, ( a | b ) (cid:2) ( c | d ).Note that the property (i) in the above lemma is stronger than both (ii) and (iii) ofProposition 3.8.From these properties, we can express the following characterisation result, althoughwith an additional assumption (the one in (iii) above) that restricts the scope of its appli-cation. It remains as an open problem whether this extra condition could be eventuallyremoved. Corollary 4.15.
For any pair of conditionals ( a | b ) , ( c | d ) ∈ C ( A ) such that c ∧ d ≤ b ,we have ( a | b ) ≤ ( c | d ) iff either ( c | d ) = ⊤ , ( a | b ) = ⊥ , or a ∧ b ≤ c ∧ d and b ≥ d . at ( C ( A )) In this section we provide two representations of the set at ( C ( A )) of atoms of a conditionalBoolean algebra as trees that will turn very helpful in the proof of the main result of nextSection 6.2. We advise readers who are not interested in the full detail of the proof toskim through this section to get acquainted with the notation, which will be useful lateron. Let A be a Boolean algebra with n atoms, say at ( A ) = { α , . . . , α n } , and let us inductivelydefine the following rooted trees T ( m ) of increasing depth, for m = 0 , , . . . , n :(0) T (0) consists of only one node: ( ⊤ | ⊤ ).(1) T (1) is obtained from T (0) by attaching the following n child nodes to ( ⊤ | ⊤ ):( α | ⊤ ) , . . . , ( α n | ⊤ ).( k ) For k = 1 , . . . , n − T ( k + 1) is obtained by spanning T ( k ) in the following way.For each leaf ( α i | b ) of T ( k ), let Atup ( α i , b ) be the set of atoms that appear in theconsequents of conditionals along the path from ( α i | b ) to the root, including α i itself. Then attach ( α i | b ) with the n − k child nodes of the form ( α j | b ∧ ¬ α i ), foreach α j ∈ at ( A ) \ Atup ( α i , b ).( n −
1) Put T = T ( n − Figure 5.1 clarifies the above construction in the case of an algebra A with four atoms.The following hold: In the above construction, if we would proceed to the stage n , the leaves of T ( n ) would be of the form( α j | V α i = α j ¬ α i ) = ( α j | α j ) = ( ⊤ | ⊤ ). For this reason, and in order not to trivialize the construction,we stop the definition at level n − T , but the root, are of the form ( α i | b ) for α i ∈ at ( A ) and b ∈ A ;Fact 2: at each level of the tree, all conditionals which are children of the same node share thesame antecedent. For instance the leafs of T ( k + 1) are all of the form ( α j | b ∧ ¬ α i ),for each α j ∈ at ( A ) \ Atup ( α i , b ) (see the construction above).Fact 3: there is a bijective correspondence between the atoms of C ( A ) and the paths fromthe root to the leaves in T . α | ⊤ α | ¬ α α | ¬ α α | ⊤ α | ⊤ α | ⊤⊤ | ⊤ α | ¬ α ∧ ¬ α Figure 3:
The tree T = T (3) whose 24 paths describe the atoms of C ( A ) with | at ( A ) | = 4. Inparticular, notice that the left-most path (dashed in the figure) provides a description for the atom ω α = ( α | ⊤ ) ⊓ ( α | ¬ α ) ⊓ ( α | ¬ α ∧ ¬ α ) where α = h α , α , α i . Let A be such that at ( A ) = { α , . . . , α n } . Consider a basic conditional of the form ( α | b ),with α ∈ at ( A ). Without loss of generality, we will henceforth assume α = α . In whatfollows, we provide another description of at ≤ ( α | b ), alternative to the one given inProposition 4.7. To this end, let us start introducing the following notation: for every j = 1 , . . . , n , let S j be the subset of atoms of C ( A ) below ( α | b ) whose first conjunct is( α j | ⊤ ), that is, equivalently, S j = { ω γ | γ = h γ , . . . , γ n − i ∈ Seq ( A ) , ω α ≤ ( α | b ) and γ = α j } . (2)Obviously at ≤ ( α | b ) = S nj =1 S j . Moreover, by construction, the sets S j satisfy thefollowing properties. Lemma 5.1.
For every basic conditional of the form ( α | b ) , the family { S j } j =1 ,...,n is apartition of at ≤ ( α | b ) . Further, the following properties hold:(i) S = { ω γ | h γ , . . . , γ n − i ∈ Seq ( A ) , γ = α } .(ii) For every j ≥ , if α j ≤ b , then S j = ∅ .Proof. We already noticed that at ≤ ( α | b ) = S nj =1 S j and moreover, it is immediate tocheck that, for j = j , S j ∩ S j = ∅ . Thus, it is left to prove (i) and (ii).(i). It is clear that any atom with first coordinate α is below ( α | b ) since ( α | ⊤ ) ⊓ ( β |¬ α ) ⊓ . . . ≤ ( α | ⊤ ) ≤ ( α | b ).(ii). It follows from Proposition 4.7. 20or every pair of sequences of symbols σ , σ , we will write σ ≪ σ to denote that σ is an initial segment of σ . Definition 5.2.
Let A be a Boolean algebra with n atoms and let 1 ≤ i ≤ n −
1. Then,for any sequence α = h α , . . . , α i i ∈ Seq i ( A ), we define: J α , . . . , α i K = { ω γ ∈ at ( C ( A )) | h α , . . . , α i i ≪ γ } . In other words, J α , . . . , α i K denotes the subset of atoms of C ( A ) of the form ω γ , with γ = h γ , . . . , γ n − i where γ = α , . . . , γ i = α i . Proposition 5.3.
Let A be a Boolean algebra with atoms α , . . . , α n , and let b ∈ A besuch that ¬ b = β ∨ . . . ∨ β k with β k = α n . For every t = 2 , . . . , k − , let Π t denote theset of all permutations π : { , . . . , t } → { , . . . , t } . Then, S n = J α n , α K ∪ k − [ i =1 J α n , β i , α K ∪ k − [ t =2 [ π ∈ Π t J α n , β π (1) , . . . , β π ( t ) , α K . Proof.
Immediate consequence of the definition of J α n , α K , J α n , β i , α K and J α n , β π (1) , . . . , β π ( t ) , α K .Obviously, renaming the indexes of the atoms of A , the above proposition provides adescription for all S j ’s. However, we preferred to show the case of S n since that will bethe case we shall need in Subsection 6.2.Let us start defining a tree whose nodes are sets of atoms in S n as in Proposition 5.3above. Further, keeping the same notation as in the statement of Proposition 5.3 above,we denote by β , . . . , β k the elements of at ≤ ( ¬ b ) and β k = α n . Definition 5.4.
With the above premises, we define the tree B in the following inductiveway.0. The root of B is J α n , α K .1. The child nodes of J α n , α K are J α n , β , α K , . . . , J α n , β k − , α K .. . .t. The child nodes of J α n , β π (1) , . . . , β π ( t ) , α K are J α n , β π (1) , . . . , β π ( t ) , β l , α K ,. . . , J α n , β π (1) , . . . , β π ( t ) , β l m , α K , where { β l , . . . , β l m } = { β , . . . , β k − }\{ β π (1) , . . . , β π ( t ) } .Given any node J α n , β π (1) , . . . , β π ( t ) , α K of B , we will henceforth denote by B J α n , β π (1) , . . . , β π ( t ) , α K the subtree of B generated by J α n , β π (1) , . . . , β π ( t ) , α K .We clarify the above construction with the following example. Example 5.5.
In order not to trivialize the example, let us consider the algebra C ( A )with at ( A ) = { α , . . . , α } . Let b = α ∨ α , whence ¬ b = α ∨ α ∨ α , and S = { ω γ ∈ at ( C ( α )) | γ = α , ω γ ≤ ( α | b ) } . Then, according to the above definition, we first needto consider the following subsets of S : J α , α K = { ω γ | h , i ≪ γ } = { ω γ | γ ∈ {h , , , i , h , , , i , h , , , i , h , , , i , h , , , i , h , , , i}} ;21 α , α , α K = { ω γ | h , , i ≪ γ } = { ω γ | γ ∈ {h , , , i , h , , , i}} ; J α , α , α K = { ω γ | h , , i ≪ γ } = { ω γ | γ ∈ {h , , , i , h , , , i}} ; J α , α , α , α K = { ω γ | γ = h , , , i} ; J α , α , α , α K = { ω γ | γ = h , , , i} .It is now clear that S = J α , α K ∪ J α , α , α K ∪ J α , α , α K ∪ J α , α , α , α K ∪ J α , α , α , α K .The resulting tree B is hence depicted as in Figure 5.2. J α , α KJ α , α , α K J α , α , α KJ α , α , α , α K J α , α , α , α K Figure 4:
The tree B for S when A has 5 atoms and b = α ∨ α . With the desired Boolean algebraic structure for conditionals in place, we are now in aposition to tackle the main issue of this paper which is the relation between conditionalprobability functions on a Boolean algebra A and simple (i.e. “unconditional”) probabil-ities on the conditional algebra C ( A ). In the following subsections we shall address twoquestions that go in the direction of clarifying when, and under which conditions, a prob-ability function on a conditional algebra can be regarded as a conditional probability. Inparticular, in Subsection 6.1 we will determine under which conditions a simple measureon C ( A ) satisfies the axioms of a conditional probability function on A , while in Subsec-tion 6.2 we will prove our main result, namely, a canonical way to define a simple measureon C ( A ) which agrees on each basic conditional with a given conditional probability on A . C ( A ) We assume the reader to be familiar with the usual and well-known notion of (simpleor unconditional) finitely-additive probability . Let us recall the notion of a conditionalprobability map which we take, with inessential variations, from [30, Definition 3.2.3]. Seealso [44] and [12] where conditional probability was firstly considered as a primitive notion.
Definition 6.1.
For a Boolean algebra A , a two-place function CP : A × A ′ → [0 , conditional probability if it satisfies the following conditions, where we write, as usual,( x | y ) instead of ( x, y ):(CP1) for all b ∈ A ′ , CP ( b | b ) = 1; 22CP2) if a , a ∈ A , a ∧ a = 0 and b ∈ A ′ , CP ( a ∨ a | b ) = CP ( a | b ) + CP ( a | b );(CP3) if a ∈ A and b ∈ A ′ , CP ( a | b ) = CP ( a ∧ b | b );(CP4) if a ∈ A and b, c ∈ A ′ with a ≤ b ≤ c , then CP ( a | c ) = CP ( a | b ) · CP ( b | c ).In what follows, for the sake of a simpler notation, if µ : C ( A ) → [0 ,
1] is a (uncondi-tional) probability we will write µ ( a | b ) instead of µ (( a | b )). Remark 6.2.
Take any probability µ on C ( A ) and fix an element b ∈ A ′ . Then, A | b isa Boolean subalgebra of C ( A ) from Corollary 3.7, and the restriction µ b of µ to A | b is aprobability measure. Notice that µ b also satisfies axiom (CP3).Therefore, a simple probability on C ( A ), once restricted to basic conditionals, alwayssatisfies properties (CP1), (CP2) and (CP3) above. However, axiom (CP4), also knownas chain rule , does not always hold as we show in the next example. Example 6.3.
In the conditional algebra C ( A ) of Example 4.5, consider the elements( α | ⊤ ), ( α | α ∨ α ) and ( α ∨ α | ⊤ ). Clearly, in the Boolean algebra A , we have α ≤ α ∨ α ≤ ⊤ and hence ( α | ⊤ ) = ( α | α ∨ α ) ⊓ ( α ∨ α | ⊤ ) by Proposition 3.3(v). As usual, for every element t ∈ C ( A ), let us write at ≤ ( t ) to denote the set of atomsof C ( A ) below t . Then, in particular,1. at ≤ ( α | ⊤ ) = { ω , ω } ,2. at ≤ ( α ∨ α | ⊤ ) = at ≤ ( α | ⊤ ) ⊔ at ≤ ( α | ⊤ ) = { ω , ω , ω , ω } ,3. at ≤ ( α | α ∨ α ) = { ω , ω , ω } .Notice that at ≤ ( α | ⊤ ) ∩ at ≤ ( α ∨ α | ⊤ ) = at ≤ ( α | ⊤ ) = ∅ and in particular at ≤ ( α ∨ α | ⊤ ) \ at ≤ ( α | ⊤ ) = at ≤ ( α | ⊤ ) = ∅ . Let p : at ( C ( A )) → [0 ,
1] be the probability distribution defined by the following stipula-tion: p ( x ) = (cid:26) x ∈ { ω , ω } , / x ∈ { ω , ω , ω , ω } . Thus, let µ p : C ( A ) → [0 ,
1] be the probability measure on C ( A ) induced by p : for all t ∈ C ( A ), µ p ( t ) = P ω ∈ at ≤ ( t ) p ( ω ). In particular we have: µ p ( α | ⊤ ) = p ( ω ) + p ( ω ) = 0,while µ p ( α ∨ α | ⊤ ) = p ( ω ) + p ( ω ) + p ( ω ) + p ( ω ) = 1 / µ p ( α | α ∨ α ) = p ( ω ) + p ( ω ) + p ( ω ) = 1 /
4. Hence0 = µ p ( α | ⊤ ) = µ p ( α | α ∨ α ) · µ p ( α ∨ α | ⊤ ) = 1 / µ p does not satisfy (CP4) and is not a conditional probability.One might wonder if the failure of (CP4) is a consequence of the fact that the distri-bution p assigns 0 to some atoms, and hence the measure µ p is not positive. This is notthe case. Indeed, consider the following distribution parametrized by ǫ : p ǫ ( x ) = (cid:26) ǫ if x ∈ { ω , ω } , / − ǫ/ x ∈ { ω , ω , ω , ω } . The equation µ p ǫ ( α | ⊤ ) = µ p ǫ ( α | α ∨ α ) · µ p ǫ ( α ∨ α | ⊤ ) has solution only for ǫ = 1 / ǫ = 1 /
6. In particular, for every 1 / < ǫ < /
2, the probability µ p ǫ is positive butdoes not satisfy (CP4). 23he above example can be easily generalised to algebras C ( A ) with A having more than3 atoms. In other words, discarding trivial cases, only a proper subclass of probabilitieson a conditional algebra C ( A ) gives rise to conditional probabilities on A , and obviously,these are those satisfying the condition µ (( a | b ) ⊓ ( b | c )) = µ ( a | b ) · µ ( b | c ) , for all a, b, c ∈ A ′ s.t. a ≤ b ≤ c, which is equivalent to the chain rule in condition (CP4) above as in C ( A ), ( a | c ) = ( a | b ) ⊓ ( b | c ) under the hypothesis that a ≤ b ≤ c . Note that it amounts in turn to requirethat, in C ( A ), any pair of conditional events ( a | b ) and ( b | c ), with a ≤ b ≤ c , tobe stochastically independent with respect to µ . This motivates our terminology in thefollowing definition. Definition 6.4 (Separable probabilities) . A probability µ : C ( A ) → [0 ,
1] is said to be separable if µ satisfies the chain rule (CP4).The following are two significant examples of separable probabilities on C ( A ). Example 6.5. (i) First notice that a map P from a finite Boolean algebra A to the Boolean algebra ofclassical truth-values = { , } is a probability iff it is a homomorphism and hencea truth-valuation. Now, every probability P of C ( A ) into is separable accordingto the previous definition. Indeed, P : C ( A ) → { , } satisfies (CP4) since, due tocondition (C5) in Proposition 3.3, if a ≤ b ≤ c then P ( a | c ) = P (( a | b ) ⊓ ( b | c )) = min( P ( a | b ) , P ( b | c )) = P ( a | b ) · P ( b | c ) . Therefore, every { , } -valued probability on C ( A ) is a conditional probability on A .(ii) For every finite Boolean algebra A , the probability measure µ u : C ( A ) → [0 , C ( A ), i.e. the one defined by µ u ( ω ) = 1 / | at ( C ( A )) | for every ω ∈ at ( C ( A )), is separable. Indeed, by Corollary4.8, one has µ u ( a | b ) = | at ≤ ( a ) | / | at ≤ ( b ) | if a ≤ b .We finish this section with an observation about convexity. It is well-known that everyprobability on a finite Boolean algebra B is a convex combination of homomorphisms of B into = { , } . So in particular this is the case for C ( A ). As shown in the exampleabove, all these homomorphisms are separable. Therefore, since not every probability on C ( A ) with | at ( A ) | ≥ Corollary 6.6.
Let A be a Boolean algebra such that | at ( A ) | ≥ . Then, the set ofseparable probabilities on C ( A ) is not convex. C ( A ) In this subsection we finally address the fundamental question that has motivated ourinvestigation, namely: 24Given a (positive) probability on an algebra of events A , P : A → [0 , C ( A ), µ P : C ( A ) → [0 , µ P ( a | b ) = P ( a ∧ b ) P ( b ) (3)for any basic conditional ( a | b ) ∈ C ( A ).”This is, in a slightly different setting, what Goodman and Nguyen call in [26] the strong conditional event problem . They solve it positively by defining conditional eventsas countable unions of special cylinders in the infinite algebra A ∞ whose set of atoms is at ( A ∞ ) = { ( α , α , . . . ) : α i ∈ at ( A ) , i = 1 , , . . . } = ( at ( A )) N i.e. infinite sequences of atoms of A , and by defining a probability ˆ P on A ∞ as the productprobability measure with identical marginals P on each factor space.In this section we show we can also solve the problem in a finitary setting by defininga suitable probability µ P : C ( A ) → [0 ,
1] on the finite
Boolean algebra C ( A ).To this end, given a positive probability on A , we start defining the following map on at ( C ( A )). Definition 6.7.
Let P : A → [0 ,
1] be a positive probability. Then we define the map µ P : at ( C ( A )) → [0 ,
1] by the following stipulation: for every α = h α , . . . , α n − i ∈ Seq ( A ), µ P ( ω α ) = P ( α ) · P ( α | ¬ α ) · . . . · P ( α n − | ¬ α ∧ . . . ∧ ¬ α n − )or, equivalently, µ P ( ω α ) = P ( α ) · P ( α | α ∨ . . . ∨ α n − ∨ α n ) · . . . · P ( α n − | α n − ∨ α n ) . Now, we show that the map µ P is indeed a probability distribution. Lemma 6.8.
The map µ P is a probability distribution on at ( C ( A )) , that is, X α ∈ Seq ( A ) µ P ( ω α ) = 1 . Proof.
See Appendix.Having a distribution µ P on the atoms of C ( A ), we can define the correspondingprobability measure (that we shall keep denoting by µ P ) on C ( A ) in the obvious way. Definition 6.9.
Given a positive probability P on A , the probability measure on C ( A )induced by the distribution µ P on at ( C ( A )), i.e. for t ∈ C ( A ), µ P ( t ) = X ω ∈ at ( C ( A )) ,ω ≤ t µ P ( ω ) (4)will be called the canonical extension of P to C ( A ).25he next three lemmas provide a necessary technical preparation for the main resultof this section, namely Theorem 6.13. For them, we invite the reader to recall the maindefinitions and constructions of Subsection 4.2 and, in particular, Definitions 5.2 and5.4. With an abuse of notation, for every subset X of at ( C ( A )) we write µ P ( X ) for µ P ( W ω ∈ X ω ) = P ω ∈ X µ P ( ω ). Lemma 6.10.
Let α i , . . . , α i t ∈ at ( A ) . Then(i) µ P ( J α i , . . . , α i t K ) = P ( α i ) · P ( α i ) P ( ¬ α i ) · . . . · P ( α it ) P ( ¬ α i ∧¬ α i ∧ ... ∧¬ α it − ) . (ii) µ P ( J α i , . . . , α i t K ) = µ P ( J α i , . . . , α i j − , α i j +1 , . . . , α i t K ) · P ( α ij ) P ( ¬ α i ∧¬ α i ∧ ... ∧¬ α it − ) . Proof.
See Appendix.In the next two lemmas, we will present results which allow us to compute the measure µ P of a basic conditional ( α | b ), for α ∈ at ( A ). In particular, adopting the same con-ventions used in Subsection 5.2 we will henceforth fix, without loss of generality, α = α .Also, we invite the reader to remind the definition of J α , . . . , α i K as the set of atoms of C ( A ) whose initial segment is h α , . . . , α i i (Definition 5.2) and the definition of the tree B (Definition 5.4) together with the definition of subtree of B generated by a J α , . . . , α i K . Lemma 6.11.
Let ( α | b ) be a basic conditional such that ¬ b = β ∨ . . . ∨ β k and β k = α n .Then the following holds: for all t ∈ { , . . . , k − } , µ P ( B J α n , β , . . . , β t , α K ) = µ P ( J α n , β , . . . , β t − , α K ) · P ( β t ) P ( b ) , where B J α n , β , . . . , β t , α K is understood as the set of all the atoms belonging to its nodes.Proof. See Appendix.For the next one, recall the definition of the sets S j as we did in Subsection 5.2 and,in particular, Lemma 5.1. Lemma 6.12.
Let ( α | b ) be a basic conditional and let b ≥ α . Then,(i) µ P ( S ) = P ( α ) ;(ii) for any ≤ j ≤ n , µ P ( S j ) = P ( α ) · P ( α j ) P ( b ) .Proof. See Appendix.We can now prove the following.
Theorem 6.13.
For every positive probability measure P on A , the canonical extension µ P on C ( A ) is such that, for every basic conditional ( a | b ) , µ P ( a | b ) = P ( a ∧ b ) P ( b ) . roof. Let P be given as in the hypothesis, and let µ P be defined on C ( A ) as in (4). Bydefinition and Lemma 6.8, µ P is a positive probability function. Thus, it is left to provethat, for every conditional ( a | b ), µ P ( a | b ) = P ( a ∧ b ) /P ( b ).To this end, recall that for each basic conditional we have ( a | b ) = ( W α i ≤ a α i | b ) = F α i ≤ a ( α i | b ). Thus, since µ P is additive, it is sufficient to prove the claim for thoseconditionals of the form ( α | b ) where α ∈ at ( A ). Without loss of generality we willassume α = α .Notice first that, if α b , then ( α | b ) = ⊥ C , whence the claim is trivial. Thus, weshall henceforth assume that α ≤ b . Now, from Lemma 5.1, { S j } j =1 ,...,n is a partition of at ≤ ( α | b ). Thus, µ P ( α | b ) = n X j =1 µ P ( S j ) = µ P ( S ) + n X j =2 µ P ( S j ) . (5)From Lemma 5.1 (ii), µ P ( S j ) = 0 for all j such that α j ≤ b , therefore, n X j =2 µ P ( S j ) = X j : α j ≤¬ b µ P ( S j ) . By Lemma 6.12 (ii) above, we also have: X j : α j ≤¬ b µ P ( S j ) = X j : α j ≤¬ b P ( α ) · P ( α j ) P ( b ) = P ( α ) P ( b ) · X j : α j ≤¬ b P ( α j ) = P ( α ) P ( b ) · P ( ¬ b ) . Finally, using this and Lemma 6.12 (i), from (5) we get: µ P ( α | b ) = P ( α ) + P ( α ) P ( b ) · (1 − P ( b ))= P ( α ) + P ( α ) P ( b ) − P ( α )= P ( α | b ) . An immediate consequence of the above theorem is that, for every positive probability P on A and for every α = h α , . . . , α n − i ∈ Seq ( A ), one has µ P ( ω α ) = µ P (( α | ⊤ ) ⊓ ( α | ¬ α ) ⊓ . . . ⊓ ( α n − | ¬ α ∧ ¬ α ∧ . . . ∧ ¬ α n − ))= P ( α | ⊤ ) · P ( α | ¬ α ) · . . . · P ( α n − | ¬ α ∧ ¬ α ∧ . . . ∧ ¬ α n − )= µ P ( α | ⊤ ) · µ P ( α | ¬ α ) · . . . · µ P ( α n − | ¬ α ∧ ¬ α ∧ . . . ∧ ¬ α n − ) . (6)That is, the basic conditionals which conjunctively define an atom of C ( A ) are jointlyindependent with respect to the probability µ P .Furthermore, the following result provides a characterization for the positive and sep-arable probabilities on C ( A ). Corollary 6.14.
For every finite Boolean algebra A and for every positive probability µ on C ( A ) , the following are equivalent:(i) µ is separable; ii) there is a positive probability P µ on A such that for every basic conditional ( a | b ) , µ ( a | b ) = P µ ( a ∧ b ) P µ ( b ) = µ P µ ( a | b ) . Proof. ( i ) ⇒ ( ii ). Let P µ be the restriction of µ to the Boolean subalgebra A | ⊤ of C ( A ).Thus, as A | ⊤ is isomorphic to A by Corollary 3.7, P µ is a positive probability of A (Remark 6.2 plus the trivial observation that the positivity of µ induces the positivity of P µ ). Furthermore, for every basic conditional ( a | b ) of C ( A ), from Theorem 6.13 and thedefinition of P µ , µ P µ ( a | b ) = P µ ( a ∧ b ) P µ ( b ) = µ ( a ∧ b | ⊤ ) µ ( b | ⊤ ) . In order to conclude the proof notice that, since µ is separable, µ ( a | b ) · µ ( b | ⊤ ) = µ ( a ∧ b | b ) · µ ( b | ⊤ ) = µ ( a ∧ b | ⊤ ). Therefore, µ P µ ( a | b ) = µ ( a ∧ b | ⊤ ) µ ( b | ⊤ ) = µ ( a | b ).(2) ⇒ (1). For each basic conditional ( a | b ) ∈ C ( A ), µ ( a | b ) = µ P ( a | b ), and since µ P isseparable, for all a ≤ b ≤ c ∈ A , one has µ ( a | c ) = µ P ( a | c ) = µ P ( a | b ) · µ P ( b | c ) = µ ( a | b ) · µ ( b | c ) which settles the claim.Corollary 6.14 implies that, if µ is separable, the values on basic conditionals ( a | b )only depend on the values on basic conditionals of the form ( α | ⊤ ), with α an atom of A . Indeed, if µ is separable, then µ ( a | b ) = µ ( a ∧ b | ⊤ ) µ ( b | ⊤ ) . Therefore, any other measure µ ′ that coincides with µ on the basic conditionals of theform ( α | ⊤ ), will coincide as well on each basic conditional.However, although positive separable measures on C ( A ) are characterized throughCorollary 6.14 above as those measures which coincide with µ P on basic conditionals, themeasures of the form µ P are not the unique positive and separable probability functionson C ( A ). Indeed, let µ P be the canonical extension to C ( A ) of a positive probability P : A → [0 , at ( A ) = α , . . . , α n (with n ≥
3) and fix two atoms ω , ω ∈ at ≤ ( α | ⊤ ). Assume, without loss of generality, that µ P ( ω ) ≤ µ P ( ω ) and take any0 < ε < min( µ P ( ω ) , − µ P ( ω ))2 so that µ P ( ω ) − ε > µ P ( ω ) + ε <
1. Now, for every ω ∈ at ( C ( A )), define µ ′ ( ω ) = µ P ( ω ) if ω = ω , ω = ω ,µ P ( ω ) − ε if ω = ω ,µ P ( ω ) + ε if ω = ω . Clearly, P ω ∈ at ( C ( A )) µ ′ ( ω ) = 1 and µ ′ ( ω ) > ω ∈ at ( C ( A )). Thus, µ ′ extends to apositive probability, that we still denote by µ ′ , on the whole C ( A ). Further, by definition, µ ′ ( α | ⊤ ) = µ P ( α | ⊤ ), whence for all α i ∈ at ( A ), µ ′ ( α i | ⊤ ) = µ P ( α i | ⊤ ) and therefore,by the above considerations, µ ′ ( a | b ) = µ P ( a | b ) for every basic conditional ( a | b ) ∈ C ( A ) . (7)28oreover, the above property and the separability of µ P imply the separability of µ ′ .Indeed, for every a ≤ b ≤ c ∈ A ′ , µ ′ (( a | b ) ⊓ ( b | c )) = µ ′ ( a | c ) = µ P ( a | c ) = µ P ( a | b ) · µ P ( b | c ) = µ ′ ( a | b ) · µ ′ ( b | c ) . However, µ ′ is not expressible as the canonical extension µ P ′ for some positive P ′ on A .For otherwise, assuming that ω = ( α | ⊤ ) ⊓ ( α | ¬ α ) ⊓ . . . ⊓ ( α n − | V n − i =1 ¬ α i ), from(6) and (7), one would have µ ′ ( ω ) = µ ′ ( α | ⊤ ) · . . . · µ ′ ( α n − | n − ^ i =1 ¬ α i ) = µ P ( α | ⊤ ) · . . . · µ P ( α n − | n − ^ i =1 ¬ α i ) = µ P ( ω )while µ ′ ( ω ) = µ P ( ω ) − ε and ε > C ( A ), we can identify three relevant subclasses:1. Σ = { µ : C ( A ) → [0 , | µ is a positive probability function }
2. Γ = { µ : C ( A ) → [0 , | µ is positive and separable }
3. Π = { µ : C ( A ) → [0 , | µ = µ P for some positive probability P on A } Obviously Π ⊆ Γ ⊆ Σ. Furthermore, for | at ( A ) | ≥ In this section we undertake a natural step towards defining a logic to reason with con-ditionals whose semantics is in accordance with the notion of the Boolean algebra ofconditionals as described above. This logic, that we call the Logic of Boolean Condition-als (LBC) is axiomatized in the following subsection, while its semantics is introducedin Subsection 7.2 where we also prove a completeness theorem. In Subsection 7.3, weinvestigate the relation between LBC and non-monotonic reasoning.
Let us start by considering the classical propositional logic language L , built from a fi-nite set of propositional variables p , p , . . . p m . Let ⊢ CP L denote derivability in ClassicalPropositional Logic. Based on L , we define the language CL of conditionals by the followingstipulations:- Basic (or atomic) conditional formulas, expressions of the form ( ϕ | ψ ) where ϕ, ψ ∈ L and such that CP L ¬ ψ , are in CL .- Further, if Φ , Ψ ∈ CL , then ¬ Φ , Φ ∧ Ψ ∈ CL . Other connectives like ∨ , → and ↔ are defined as usual. We use the same symbols for connectives in L and in CL without danger of confusion. L to be part of CL but, as a matter of fact, any proposition ϕ will be always identifiable with the conditional( ϕ | ⊤ ). Next an axiomatic system for a logic of Boolean conditionals over the language CL is presented. Definition 7.1.
The
Logic of Boolean conditionals (LBC for short) has the followingaxioms and rules:(CPL) For any tautology of CPL, the formula resulting from a uniform replacement of thevariables by basic conditionals.(A1) ( ψ | ψ )(A2) ¬ ( ϕ | ψ ) ↔ ( ¬ ϕ | ψ )(A3) ( ϕ | ψ ) ∧ ( δ | ψ ) ↔ ( ϕ ∧ δ | ψ )(A4) ( ϕ | ψ ) ↔ ( ϕ ∧ ψ | ψ )(A5) ( ϕ | ψ ) ↔ ( ϕ | χ ) ∧ ( χ | ψ ), if ⊢ CP L ϕ → χ and ⊢ CP L χ → ψ (R1) from ⊢ CP L ϕ → ψ derive ( ϕ | χ ) → ( ψ | χ )(R2) from ⊢ CP L χ ↔ ψ derive ( ϕ | χ ) ↔ ( ϕ | ψ )(MP) Modus Ponens: from Φ and Φ → Ψ derive ΨThe notion of proof in LBC, ⊢ LBC , is defined as usual from the above axioms and rules.As we pointed out at the beginning of this section, in the language CL , basic condi-tional formulas ( ϕ | ψ ) can be regarded, in logical terms, as new (complex) propositionalvariables where the compound conditional formulas are built upon using the classical CPLconnectives ∧ and ¬ . Therefore, deductions in LBC employ axioms (A1)-(A5) and rules(R1), (R2) to reason about the internal structure of basic conditional formulas, and theaxioms and rules of CPL (captured by the axiom schema (CPL) and the Modus Ponensrule) for compound conditional formulas. This means that, if we denote by AX the set ofall instantiations of the axioms (A1)-(A5) once closed by rules (R1) and (R2), deductionsin LBC can in fact be translated to deductions from AX (taken as a theory) only using therules and axioms of CPL. This is formally expressed in the first item of following lemma.As a consequence of this translation, among other facts, it follows that the logic LBCsatisfies the deduction theorem, also stated in the same lemma. Lemma 7.2.
For any set of CL -formulas Γ ∪ { Φ , Ψ } , the following properties hold, where AX is as above: • Γ ⊢ LBC Φ iff Γ ∪ AX ⊢ CP L Φ . Actually, Axiom (A4) is not independent since it follows from (A1) and (A3), but we leave it for thesake of coherence with the algebraic part. In the same vein, we could replace the equivalence connective ↔ by the implication connective → in Axioms (A2) and (A3), since the other direction can be deduced byclassical reasoning and using axiom (A4) and the rule (R1) respectively. Deduction theorem: Γ ∪ { Ψ } ⊢ LBC Φ iff Γ ⊢ LBC Ψ → Φ . The above axiomatic system is clearly inspired by the key properties of Boolean al-gebras of conditionals we discussed in Section 3. Indeed, as expected, we can prove thatthe Lindenbaum algebra of provably equivalent formulas of LBC can be seen, in fact,as a Boolean algebra of conditionals. In slightly more detail, for any Φ , Ψ ∈ CL , let uswrite Φ ≡ Ψ if ⊢ LBC Φ ↔ Ψ. Notice that rules (R1) and (R2) ensure that ≡ preservespropositional logical equivalence in the sense that if ⊢ CP L ϕ ↔ ϕ ′ and ⊢ CP L ψ ↔ ψ ′ ,then ( ϕ | ψ ) ≡ ( ϕ ′ | ψ ′ ). Moreover, ≡ is compatible, in the sense of Section 2, withBoolean operations. Thus, the quotient set CL / ≡ endowed with the following operationsis a Boolean algebra: ⊤ ≡ = [( ⊤ | ⊤ )] ≡ and ⊥ ≡ = [( ⊥ | ⊤ )] ≡ ;[Φ] ≡ ⊓ [Ψ] ≡ = [Φ ∧ Ψ] ≡ ;[Φ] ≡ ⊔ [Ψ] ≡ = [Φ ∨ Ψ] ≡ ; ∼ [Φ] ≡ = [ ¬ Φ] ≡ .We claim that CL = ( CL / ≡ , ⊓ , ⊔ , ∼ , ⊤ ≡ , ⊥ ≡ ) is isomorphic to C ( L ), the Boolean algebraof conditionals of the Lindenbaum algebra L of CPL over the language L . Note that, bydefinition, ⊢ LBC
Φ iff [Φ] ≡ = ⊤ ≡ . Theorem 7.3. CL ∼ = C ( L ) . The proof is not difficult but the details are a bit tedious and can be found in theAppendix. However, it is convenient to point out that the isomorphism ι between CL and C ( L ) acts on basic conditionals in the expected way: for each pair of formulas ϕ, ψ suchthat CP L ¬ ψ , the element [( ϕ | ψ )] ≡ of CL is mapped by ι to ([ ϕ ] | [ ψ ]) ∈ C ( L ). As foroperations it is enough to ask that ι commutes with ⊓ and ∼ . Then, if for instance wehave two basic conditionals of CL of the form [( ϕ | ψ )] ≡ and [( δ | ψ )] ≡ , by definition of ≡ , ι ([( ϕ | ψ )] ≡ ⊓ [( δ | ψ )] ≡ ) = ι ([( ϕ ∧ δ | ψ )] ≡ ) = ([ ϕ ∧ ψ ] | [ δ ]) and, in turn by the property ofthe conditional algebra C ( L ), ([ ϕ ∧ δ ] | [ ψ ]) = ([ ϕ ] | [ ψ ]) ∧ ([ δ ] | [ ψ ]) = ι ([( ϕ | ψ )] ≡ ) ∧ ι ([( δ | ψ )] ≡ ). The intuitive idea in defining a semantics for the logic LBC is that, as in classical logic(recall Proposition 2.2 (iv)) the evaluations of CL -formulas should be in 1-1 correspon-dence with the atoms of the algebra C ( L ), which is, by Theorem 7.3, isomorphic to theLindenbaum algebra CL of LBC. As we have seen in Section 4, atoms of C ( L ) can bedescribed by sequences of atoms of L .In the following let Ω be the set of (classical) evaluations v : L → { , } for thepropositional language L . Recall from Section 2 that if L is built from m propositionalvariables, then | Ω | = 2 m . Moreover, we can identify every evaluation v ∈ Ω with itscorresponding minterm and hence with an atom of Lindenbaum algebra L . Therefore, itfollows from Section 4 that the atoms of C ( L ) are of the form( v | ⊤ ) ⊓ ( v | ¬ v ) ⊓ . . . ⊓ ( v m − | ¬ v ∧ . . . ∧ ¬ v m − ) , h v , . . . , v m − i ∈ Seq ( L ). The idea is then to define CL -interpretations as sequences e = h v , . . . , v m i of pair-wise distinct 2 m L -interpretations of Ω and to stipulate that sucha CL -interpretation e makes true a conditional ( ϕ | ψ ) when the atom in C ( L ) determinedby e ω e = ( v | ⊤ ) ⊓ ( v | ¬ v ) ⊓ . . . ⊓ ( v m − | ¬ v ∧ . . . ∧ ¬ v m − ) , (8)is below ( ϕ | ψ ), the latter thought of as an element of C ( L ) as well. This condition has aneasier expression according to (iii) of Proposition 4.7, that leads to the following naturaldefinition. In the following we will assume | Ω | = m and we fix n = 2 m to simplify thereading. Definition 7.4. A CL -interpretation is a sequence e = h v , v , . . . , v n i of n pairwise dis-tinct v , . . . , v n ∈ Ω. We denote also by e the corresponding evaluation of CL -formulas,i.e. the mapping e : CL → { , } defined as follows:- for basic CL -formulas: e ( ϕ | ψ ) = 1 if v i | = ϕ for the lowest index i such that v i | = ψ ,and e ( ϕ | ψ ) = 0 otherwise.- for compound CL -formulas: e is extended using Boolean truth-functions.The corresponding notion of consequence is as expected: for any set of CL -formulas Γ ∪{ Φ } ,Γ | = LBC
Φ if for every CL -interpretation e such that e (Ψ) = 1 for all Ψ ∈ Γ, it also holds e (Φ) = 1.By Proposition 2.2, given any homomorphism h : C ( L ) → { , } , let ω h be its cor-responding atom, i.e., λ ( ω h ) = h in the terminology of Proposition 2.2 (iii). We furtherdenote by Λ( h ) the sequence h v , . . . , v n − , v n i such that h v , . . . , v n − i univocally deter-mines ω h via (8) and v n is the only evaluation left in Ω \ { v , . . . , v n − } . Lemma 7.5.
For every homomorphism h : C ( L ) → { , } , Λ( h ) is a CL -interpretation.Further, for each CL -formula Φ , h (Φ) = 1 iff Λ( h )(Φ) = 1 .Proof. If h : C ( L ) → { , } is a homomorphism, Λ( h ) is a CL -interpretation by definition.Now we prove, by structural induction on the complexity of Φ, that Λ( h )(Φ) = 1 iff h (Φ) =1. The interesting case is when Φ is a basic conditional ( ϕ | ψ ) such that ( ϕ | ψ )
6≡ ⊤ . Inthat case, h (Φ) = 1 iff ω h ≤ ( ϕ | ψ ) iff (by Proposition 4.7) there exists an index i ≤ n − v i ( ϕ ) = v i ( ψ ) = 1 and v j ( ψ ) = 0 for all j < i , and thus, iff Λ( h )(Φ) = 1.Now the soundness and completeness of LBC easily follow from the above. Theorem 7.6 (Soundness and completeness) . LBC is sound and complete w.r.t. CL -evaluations, i.e. ⊢ LBC = | = LBC .Proof.
Soundness is easy. As for completeness, assume that Γ LBC
Φ. Thus, in particularΦ is false in C ( L ), meaning that there exists a homomorphism h : C ( L ) → { , } such that h ( γ ) = 1 for all γ ∈ Γ, and h (Φ) = 0. Thus, by Lemma 7.5, Λ( h ) is a CL -interpretationsuch that Λ( h )( γ ) = 1 for every γ ∈ Γ and Λ( h )(Φ) = 0, i.e. Γ = LBC
Φ .
Remark 7.7.
One could also define an alternative semantics for LBC in terms of aclass of Kripke models with order relations, more in the style of conditional logics (see[30]). This Kripke-style semantics can also be shown to be adequate for LBC (and hence32quivalent to the previous more algebraic semantics), although it is not fully exploited asthe language of LBC does not contain nested applications of the conditioning operatornor pure propositional formulas. Indeed, one can define
LBC-Kripke models as structures M = ( W, {≻ w } w ∈ W , e ), where (i) W is a non-empty set of worlds, (ii) for each world w ∈ W , ≻ w is a (dual) well-order such that w ≻ w w ′ for every w = w ′ ∈ W , and (iii) e : W × V → { , } is a valuation function for propositional variables, that naturally extendsto any Boolean combination of variables. For every w ∈ W and B ⊆ W , let best w ( B )denote the greatest element in B according to ≻ w . Moreover, for any propositional (non-conditional) ϕ , we will use J ϕ K to denote the set of worlds in M evaluating ϕ to true, i.e. J ϕ K = { w ∈ W | e ( w, ϕ ) = 1 } .Truth in a pointed model is defined by induction as follows. For each w ∈ W we define: • ( M, w ) | = K ( ϕ | ψ ) if either J ψ K = ∅ or best w ( J ψ K ) ∈ J ϕ K . Equivalently, in case J ψ K = ∅ , then ( M, w ) | = K ( ϕ | ψ ) if ( M, best w ( J ψ K )) | = K ϕ . • Satisfaction for Boolean combinations of conditionals is defined in the usual way.It is not difficult to see that LBC is also sound w.r.t. to the class of LBC-Kripke models.As for completeness, assume as usual that Γ LBC
Φ, with Γ a finite set, and let us provethat Γ = K Φ. From Lemma 7.2, the condition Γ LBC
Φ is equivalent to LBC Φ ′ , whereΦ ′ = V Ψ ∈ Γ Ψ → Φ. This means that [Φ ′ ] ≡ = [ ⊤ ] ≡ in the (finite) Lindenbaum algebra CL .By Theorem 7.3, for practical purposes we can identify CL with the Boolean algebra ofconditionals C ( L ). Therefore, there is an atom ω ∈ C ( L ) such that ω [Φ ′ ] ≡ . As we haveseen in Section 4, atoms of C ( L ) can be described by sequences of pair-wise different atoms ω i of L . But each atom ω i of L can in turn be identified with (the minterm associatedto) a Boolean evaluation v i of the propositional variables V . Therefore if we let Ω bethe set of evaluations of the propositional variables V , we can assume ω = h v , . . . , v n − i ,with n = 2 | V | and where v , . . . , v n − ∈ Ω. Finally, we consider an LBC-Kripke model M ∗ = (Ω , {≻ ∗ v } v ∈ Ω , e ∗ ) where: • ≻ ∗ v ⊂ Ω × Ω is such that v i ≻ ∗ v v i +1 for i = 1 , . . . , n − and for all i = 1, ≻ ∗ v i is anarbitrary (dual) well-order on Ω; • e ∗ : Ω × V → { , } is such that e ∗ ( v, p ) = v ( p ) for every p ∈ V .Then, it is clear that the condition ω [Φ ′ ] ≡ guarantees that ( M ∗ , v ) = K Φ ′ .An immediate consequence of Theorem 7.6 and the fact that there are only finitelymany CL -evaluations is the decidability of the calculus LBC. Furthermore, a basic con-ditional ( ϕ | ψ ) is satisfiable in LBC iff there exists a classical valuation v such that v ( ϕ ∧ ψ ) = 1. Indeed, if a CL -interpretation e = h v , v , . . . , v n i is such that e ( ϕ | ψ ) = 1,there is i such that v i ( ϕ ∧ ψ ) = 1. Conversely, if v ( ϕ ∧ ψ ) = 1, then any CL -interpretation e = h v, v , . . . , v n i , by definition, is a model of the conditional ( ϕ | ψ ). Thus, the satisfi-ability of a basic conditional ( ϕ | ψ ) reduces to the classical satisfiability. However, thisdirect reduction does not generally apply to the cases in which the conditional formula isa (non-trivial) Boolean combination of basic conditionals. A dual well-order in W is a total order ≻ such that every subset B ⊆ W has a greatest element. Where v n = Ω \ { v , . . . , v n − } . ϕ | ⊤ ) are equivalent to ϕ , it is alsoeasy to see that the classical satisfiability is a subproblem of the satisfiability of LBC whichshows the latter to be NP-hard.Determining a possible NP-containment for the LBC-satisfiability is out of the scopeof the present paper and it will be addressed in our future work. Conditionals possess an implicit non-monotonic behaviour. Given a conditional ( ϕ | ψ ),it does not follow in general that we can freely strengthen its antecedent, i.e. in general,( ϕ | ψ ) LBC ( ϕ | ψ ∧ χ ). For instance, ϕ, ψ, χ can be such that ϕ ∧ ψ = ⊥ while ϕ ∧ ψ ∧ χ | = ⊥ . Actually, and not very surprisingly, the logic ⊢ LBC satisfies the analogues ofthe KLM-properties which characterize the well-known system P of preferential entailment[38, 42].
Lemma 7.8. ⊢ LBC satisfies the following properties:Reflexivity: ⊢ LBC ( ϕ | ϕ ) Left logical equivalence: if | = CP L ϕ ↔ ψ then ( χ | ϕ ) ⊢ LBC ( χ | ψ ) Right weakening: if | = CP L ϕ → ψ then ( ϕ | χ ) ⊢ LBC ( ψ | χ ) Cut: ( ϕ | ψ ) ∧ ( χ | ϕ ∧ ψ ) ⊢ LBC ( χ | ψ ) OR: ( ϕ | ψ ) ∧ ( ϕ | χ ) ⊢ LBC ( ϕ | ψ ∨ χ ) AND: ( ϕ | ψ ) ∧ ( δ | ψ ) ⊢ LBC ( ϕ ∧ δ | ψ ) Cautious Monotony: ( ϕ | ψ ) ∧ ( χ | ψ ) ⊢ LBC ( χ | ϕ ∧ ψ ) Proof. Reflexivity , Left Logical Equivalence , Right Weakening and
AND correspond to(A1), (R2), (R1), and (A3) of LBC, respectively. The other cases are proved as follows.
Cut : by (A4), ( χ | ϕ ∧ ψ ) ∧ ( ϕ | ψ ) is equivalent to ( χ ∧ ϕ ∧ ψ | ϕ ∧ ψ ) ∧ ( ϕ ∧ ψ | ψ ), andby (A5), it is equivalent to ( χ ∧ ϕ ∧ ψ | ψ ), and by (R1) this clearly implies ( χ | ψ ). Cautious Monotony : by (A3), ( ϕ | ψ ) ∧ ( χ | ψ ) is equivalent to ( ϕ ∧ χ | ψ ), which by (A4)is in turn equivalent ( ϕ ∧ χ ∧ ψ | ψ ), and by (A5) implies ( ϕ ∧ χ ∧ ψ | ϕ ∧ ψ ), which by(A3) is equivalent to ( χ | ϕ ∧ ψ ). OR : ( ϕ | ψ ) ∧ ( ϕ | χ ) is equivalent to [( ϕ | ψ ) ∧ ( ϕ | χ ) ∧ ( ψ | ψ ∨ χ )] ∨ [( ϕ | ψ ) ∧ ( ϕ | χ ) ∧ ( χ | ψ ∨ χ )], and this implies [( ϕ ∧ ψ | ψ ) ∧ ( ψ | ψ ∨ χ )] ∨ [( ϕ ∧ χ | χ ) ∧ ( χ | ψ ∨ χ )],that is equivalent to ( ϕ ∧ ψ | ψ ∨ χ ) ∨ ( ϕ ∧ χ | ψ ∨ χ ), which finally implies ( ϕ | ψ ∨ χ ).Now, let us fix a set of (atomic) conditional statements K , and let us define theconsequence relation associated to K : ϕ |∼ K ψ if K ⊢ LBC ( ψ | ϕ ). Our last claim is easilyderived from the previous lemma. Theorem 7.9. |∼ K is a preferential consequence relation. However, the following also well-known rule
Rational Monotonicity : if ψ |∼ ϕ and ψ |6∼ ¬ χ then ψ ∧ χ |∼ ϕ |∼ = |∼ K as the following example shows, and thus |∼ K is nota rational consequence relation in the sense of Lehmann and Magidor [38]. Example 7.10.
Fix a propositional language L with 2 propositional variables, say p, q ,so that it has four minterms ( α = p ∧ q, α = p ∧ ¬ q, α = ¬ p ∧ q, α = ¬ p ∧ ¬ q ), thatcorrespond to the 4 atoms of its Lindenbaum algebra L . The Lindenbaum algebra CL ofthe language of conditionals CL is isomorphic to the algebra C ( L ) with 24 atoms and 2 elements.Consider the following propositions: ϕ = α ∨ α , ψ = ⊤ , χ = α ∨ α and the followingset of only one conditional statement K = { ( ϕ | ψ ) } = { ( α ∨ α | ⊤ ) } . Note that ( ¬ χ | ψ ) = ( ¬ ( α ∨ α ) | ⊤ ) ≡ ( α ∨ α | ⊤ ) and ( ϕ | χ ∧ ψ ) = ( α ∨ α | α ∨ α ) ≡ ( α | α ∨ α ).Then we have:(1) K ⊢ ( ϕ | ψ )(2) K LBC ( ¬ χ | ψ ), i.e. ( α ∨ α | ⊤ ) LBC ( α ∨ α | ⊤ ), but(3) K LBC ( ϕ | χ ∧ ψ ), i.e. ( α ∨ α | ⊤ ) LBC ( α | α ∨ α )(1) and (2) are clear. Hence, let us prove (3). Via Theorem 7.6, it is enough to showthat there is an atom from C ( L ) below ( α ∨ α | ⊤ ) but not below ( α | α ∨ α ) (recallto this end Proposition 4.7). Consider the atom from C ( L ) ω β , with β = h α , α , α i , thatis, ω β = ( α | ⊤ ) ⊓ ( α | ¬ α ) ⊓ ( α | ¬ α ∧ ¬ α ). Then:- Clearly ω β ≤ ( α ∨ α | ⊤ ).- However, ω β ( α | α ∨ α ). Indeed, the least index i for which β i ≤ α ∨ α is i = 3: β = α ≤ α ∨ α , but α α . Then C ( L ) = ( α ∨ α | ⊤ ) → ( α | α ∨ α ) andhence, by Theorem 7.3 and Lemma 7.2, ( α ∨ α | ⊤ ) LBC ( α | α ∨ α ).Let us notice that Rational Monotonicity does hold whenever we further require therelation |∼ K to be defined by a complete theory K . Indeed, if K is complete, for each ϕ and ψ , one has ϕ |∼ K ψ iff ϕ |6∼ K ¬ ψ . Then, if we replace ϕ |6∼ K ¬ ψ by ϕ |∼ K ψ in theprevious expression of the Rational Monotony property, what we get is:If ψ |∼ K ϕ and ψ |∼ K χ then ψ ∧ χ |∼ K ϕ ,that is, we are back to the Cautious Monotony property (see [38, § ⊢ LBC ( ϕ | ψ ) ∨ ( ¬ ϕ | ψ ) . In general, this does not imply that |∼ K satisfies the so-called Conditional Excluded Middle property [5], namely either ψ |∼ K ϕ or ψ |∼ K ¬ ϕ. However if K is complete, then ψ |∼ K ¬ ϕ is logically equivalent to ψ K ϕ , yielding theproperty of Conditional Excluded Middle. In this section we discuss related algebraic approaches to conditional events, which wegroup in (three-valued) Measure-free conditionals and Conditional Event Algebras. The35ormer lead necessarily to non-Boolean structures whereas the latter to Boolean structures.Our approach combines elements of both. Boolean Algebras of Conditionals have a similarlanguage as Measure-free conditionals and share the underlying Boolean structure withConditional Event Algebras.
One of the most relevant approaches to study conditionals independently from conditionalprobability and outside modal logic has been to consider measure-free conditionals asthree-valued objects: given two propositions a, b of a classical propositional language L ,a conditional ( a | b ) is true when a and b are true, is false when a is false and b is true,and inapplicable when b is false. This three-valued approach actually goes back to deFinetti in the Thirties [13] and Schay in the Sixties [47], and was later further developedamong others by Calabrese [8], Goodman, Nguyen and Walker [26, 27, 28, 52], Gilio andSanfilippo [24], and Dubois and Prade [16, 17]. In particular the last authors have alsoformally related measure-free conditionals and non-monotonic reasoning. The rest of thissection is mainly from [17].The idea is to extend any Boolean interpretation v : L → { , } for the language L toa three-valued interpretation v : ( L | L ) → { , , ? } on the set of conditionals ( L | L ) builtfrom L and defined as follows: v ( a | b ) = (cid:26) v ( a ) , if v ( b ) = 1? , otherwise.Although the third truth-value “?” denotes inapplicable, it is usually be understood as“any truth-value”, i.e. one can take ? as the set { , } . Indeed, this interpretation iscompatible with taking v ( a | b ) as the set of solutions x ∈ { , } of the equation v ( a ∧ b ) = x ∧ ∗ v ( b ) , where ∧ ∗ is the Boolean truth-function for conjunction, once values for v ( a ) and v ( b ) havebeen fixed, and hence v ( a ∧ b ) as well. In particular, when b = ⊤ , there is a unique solution v ( a | ⊤ ) = { v ( a ) } , for every v and a .From an algebraic point of view, considering plain events belonging to a Boolean al-gebra A (f.i. the Lindenbaum algebra for the above propositional language L ), the abovethree-valued semantics led Goodman and Nguyen’s to the following definition of a condi-tional event [27], not as another event, but as a set of events in A :( a | b ) = { x ∈ A | x ∧ b = a ∧ b } , that turns out to actually be an interval in the algebra A , indeed, it can be checked that( a | b ) = { x ∈ A | a ∧ b ≤ x ≤ b → a } = [ a ∧ b, b → a ] . It is worth noticing that conditionals of the form ( a | ⊤ ), where ⊤ is the top element of A ,can be safely identified with the plain event a ∈ A , since [ a ∧ ⊤ , ⊤ → a ] = { a } . Moreover,this definition leads, in turn, to usually accepted equalities among conditionals like( a | b ) = ( a ∧ b | b ) = ( a ↔ b | b ) = ( b → a | b ) , a < ⊤ , then ( a | a ) = ( ⊤ | ⊤ ).From the above definition, it directly follows that the condition for the equality betweentwo conditionals is the same as we determined in Theorem 4.12 (except for the case whereboth are ⊤ C or ⊥ C ): ( a | b ) = ( c | d ) iff a ∧ b = c ∧ d and b = d, while the commonly adopted and compatible notion of ordering is the one defined by theinterval order: ( a | b ) ≤ ( c | d ) iff a ∧ b ≤ c ∧ d and b → a ≤ d → c, (9)that is of course a partial order, with T = ( ⊤ | ⊤ ) and F = ( ⊥ | ⊤ ) being the topand bottom conditionals respectively. It is interesting to notice that, if ⊥ < a < b and ⊥ < c < d , the left-to-right direction of (9) holds in every C ( A ). Indeed, under thosefurther conditions, ( a | b ) ≤ ( c | d ) gives, by Lemma 4.14 (ii), that a = a ∧ b ≤ c = c ∧ d .Moreover, a > ⊥ implies ⊥ < b → a , and b → a ≤ a < b . Similarly, ⊥ < d → c < d .Therefore, since ( a | b ) = ( b → a | b ) and ( c | d ) = ( d → c | d ), one has b → a ≤ d → c aswell.It is clear that, in the above setting of three-valued conditionals, any conditional canalways be equated to another one of the form ( a | b ) where a ≤ b . Thus the set ofconditionals built from events from a Boolean algebra A can be identified with the set A | A = { ( a | b ) : a, b ∈ A, a ≤ b } , that is in 1-1 correspondence with the set Int ( A ) ofintervals in A . Therefore, in this setting, conditioning can be viewed in fact as an externaloperation A × A → Int ( A ).Several attempts have been made to define reasonable operations on measure-free con-ditionals as internal operations on Int ( A ), in particular operations of “conjunction” ∧ ,“disjunction” ∨ and “negation” ¬ . There is a widespread consensus on the definition ofthe negation ¬ by stipulating ¬ ( a | b ) = ( ¬ a | b ), which coincides with the one on Booleanconditionals. As to conjunction, there have been some reasonable proposals in the liter-ature, corresponding to different possibilities of defining the truth-table of a three-valuedconjunction on { , , ? } (see [15, 17]), but two of them emerge as the major competingproposals, as they are the only ones satisfying the following qualitative counterpart ofBayes rule: ( a | b ) AND ( b | ⊤ ) = ( a ∧ b | ⊤ ). Goodman and Nguyen (cf. [27]): ( a | b ) ∧ I ( c | d ) = ( a ∧ c | ( ¬ a ∧ b ) ∨ ( ¬ c ∧ d ) ∨ ( b ∧ d )) Schay, Calabrese (cf. [47, 8]): ( a | b ) ∧ Q ( c | d ) = (( b → a ) ∧ ( d → c ) | b ∨ d )The operation ∧ I is in fact a genuine interval-based conjunction , in the sense that theinterval interpreting ( a | b ) ∧ I ( c | d ) is the result of computing all the conjunctions ofthe elements of the intervals interpreting the conditionals ( a | b ) and ( c | d ), that is, if( a | b ) ∧ I ( c | d ) = ( u | v ) then [ u ∧ v, v → u ] = { x ∧ y | x ∈ [ a ∧ b, b → a ] , y ∈ [ c ∧ d, d → c ] } .On the other hand, the operation ∧ Q , also known as quasi-conjunction [2], is tightlyrelated to the interval order on conditionals defined above, in the sense that ( a | b ) ≤ ( c | d ) Note that this a particular case of our Requirement R4 for Boolean algebras of conditionals, see (v)of Proposition 3.3 and (ii) of Corollary 3.4.
37ff ( a | b ) ∧ Q ( c | d ) = ( a | b ). Dubois and Prade [17] consider that only the use of thequasi-conjunction ∧ Q is appropriate in the context of non-monotonic reasoning, whereconditionals ( a | b ) are viewed as non-monotonic conditional assertions or default rules“if b then generally a ”, usually denoted as b | ∼ a . In fact, consider an entailment re-lation between finite sets of conditionals and conditionals as follows: a finite set of con-ditionals K = { ( a i | b i ) } i ∈ I entails another conditional ( c | d ), written K | = ( c | d ), ifeither ( c | d ) ∈ K or there is a non-empty subset S ⊆ K s.t. C ( S ) ≤ ( c | d ), where C ( S ) = ∧ Q { ( a | b ) | ( a | b ) ∈ S } is the quasi-conjunction of all the conditionals in S .Then they show that this notion of entailment basically coincides with the nonmonotonicconsequence relation of System P. More explicitly, they show the following characterisationresult: K | = ( c | d ) iff ( c | d ) can be derived from K using the rules of system Pand the axiom schema ( x | x ).On the other hand, corresponding disjunction operations ∨ I and ∨ Q on conditionalsare defined by De Morgan’s laws from ∧ I , ∧ Q respectively.All these conjunctions and disjunctions are commutative, associative and idempotent,and moreover they coincide over conditionals with a common antedecent. More precisely,( a ∧ c | b ) = ( a | b ) ∧ I ( c | b ) = ( a | b ) ∧ Q ( c | b ). However, none of the algebras( A | A, ∧ i , ∨ i , ¬ , T , F ), for i ∈ { I, Q } , is in fact a Boolean algebra. For instance, Schay [47]and Calabrese [8] show that ∧ Q and ∨ Q do not distribute with respect to each other.It is worth noticing that the above conjunctions are defined in order to make theclass A | A of conditional objects closed under ∧ i , and hence an algebra. Therefore,for every a , b , a , b ∈ A , and for every i ∈ { I, Q } , there exists c, d ∈ A such that,( a | b ) ∧ i ( a | b ) = ( c | d ). On the other hand, our construction of conditional algebradefines a structure whose domain strictly contains all the elements ( a | b ) for a in A , and b ∈ A ′ . This allows us to relax this condition of closure as stated above. Indeed, for everypair of conditionals of the form ( a | b ) and ( a | b ) belonging to the conditional algebra,their conjunction is always an element of the algebra (i.e. the conjunction is a total, andnot a partial, operation), but in general it will be not in the form ( c | d ). Moreover ourdefinition of conjunction behaves as ∧ I and ∧ Q whenever restricted to those conditionalswith a common antedecent or context. In his paper [51], van Fraassen devises a minimal logic CE of conditionals whose languageis obtained from classical propositional language by adding a “conditional symbol” ⇒ .The algebraic counterpart of CE are the so-called proposition algebras . A propositionalgebra is a pair h F , ⇒i where F is a Boolean algebra (of events) and ⇒ is a partial binaryoperation on F such that (where defined) satisfying the following requirements:(I) ( A ⇒ B ) ∧ ( A ⇒ C ) = A ⇒ ( B ∧ C )(II) ( A ⇒ B ) ∨ ( A ⇒ C ) = A ⇒ ( B ∨ C )(III) A ∧ ( A ⇒ B ) = A ∧ B (IV) A ⇒ A = ⊤ P to the algebras and requiring that the following is satisfied(m2) If P ( A ) = 0 then P ( A ⇒ B ) = P ( A ∧ B ) /P ( A ).In the same paper van Fraassen uses the usual product space construction to providemodels of the above. In particular, starting from a Boolean algebra of ordinary events A and a probability measure P on it, he finds a way of representing conditional events suchthat: (1) conditional events ( b | a ) live in a bigger Boolean algebra A ∗ ; (2) ordinary events a are a special kind of conditional events, i.e. a should be identified with ( a | ⊤ ); (3) therule of modus ponens a ∧ ( b | a ) ≤ b holds true in A ∗ ; and (4) the probability measure P on A can be extended to probability measure P ∗ on A ∗ satisfying (m2) above.Goodman and Nguyen [26] build on this to define a Conditional Event Algebras (knownas Goodman-Nguyen-van Fraassen algebra). To compare this to our setting, let us recallthis construction in the case where the initial algebra A is finite. Consider the countableCartesian product Ω ∞ = at ( A ) × at ( A ) × . . . , i.e. the set of countably infinite sequencesof atoms of A . Then A ∗ is defined as the σ -algebra of subsets of Ω ∞ generated by all thecylinder sets of the form h a , . . . , a n , ⊤ , ⊤ , . . . i = { ( w , w , . . . , w n , w n +1 , . . . ) ∈ Ω ∞ | w i ≤ a i , for i = 1 , . . . , n } , for n ∈ N and a , . . . , a n ∈ A . Then, for every pair of events a, b ∈ A , Goodman andNguyen define the conditional event ( b | a ) as the element of A ∗ that is the followingcountable union of pairwise disjoint cylinder sets:( b | a ) := [ k ≥ h¬ a, k . . ., ¬ a, a ∧ b, ⊤ , ⊤ . . . i . (10)Note that the initial algebra of events A is isomorphic to the subalgebra of A ∗ of con-ditionals of the form ( a | ⊤ ) = h a, ⊤ , ⊤ , . . . i , with a ∈ A . Now, any probability measure P on A extends to a suitable probability P ∗ on A ∗ . Namely, one can define P ∗ on thecylinder sets of A ∗ as P ∗ ( h a , . . . , a n , ⊤ , ⊤ , . . . i ) = P ( a ) · . . . · P ( a n ) , and using Kolmogorov extension theorem (cf. [50, § P ∗ can be extended to the whole σ -algebra A ∗ .Finally, they show that the probability P ∗ on conditionals actually coincides with theconditional probability: if P ( a ) >
0, then one has: P ∗ (( b | a )) = P ∗ [ k ≥ h¬ a, k . . ., ¬ a, a ∧ b, ⊤ , ⊤ . . . i = X k ≥ P ∗ ( h¬ a, k . . ., ¬ a, a ∧ b, ⊤ , ⊤ . . . i )= P ( a ∧ b ) · X k ≥ (1 − P ( a )) k = P ( a ∧ b ) P ( a ) , (11)39here the last equality is obtained by using the well-known formula for the sum of ageometric series, in this case the ratio being 1 − P ( a ), and getting P k ≥ (1 − P ( a )) k =1 /P ( a ).Summing up, both our approach and the Goodman-Nguyen-van Frassen one aim atthe introduction of a formal structure to study conditional probability as measures ofconditionals. To this end we both require the underlying algebraic structure to be Boolean.On the other hand, and this marks a first technical difference, whenever the startingBoolean algebra A is finite, C ( A ) is finite as well while A ∗ is always infinite, and moreovereach conditional in A ∗ is defined itself as an infinitary joint, see (10) above. Note that, forequating P ∗ (( b | a )) with P ( a ∧ b ) /P ( a ), the derivation (11) necessarily requires P ∗ (( b | a ))to decompose in infinitely-many summands.A further important difference concerns the definition of conditional events. In fact,although in our approach the algebra C ( A ) itself defines the conditionals and character-izes their properties at a formal algebraic level, in A ∗ conditionals are identified with aparticular subset of its elements, namely those elements that can be expressed via (10)above. This difference owes mainly to the fact that the two approaches really capturedistinct intuitions about conditional probability. In this paper we have introduced a construction which defines, for every Boolean algebra ofevents A , its corresponding Boolean algebra of conditional events C ( A ). Our constructionpreserves the finiteness of A , in the sense that C ( A ) is finite whenever so is A . Actually, inthe case of finite algebras, we have provided a full characterization of the atomic structureof C ( A ) and proved that any positive probability measure P on A canonically extends toa (plain) positive probability µ P on C ( A ) that coincides with the conditional probabilityinduced by P , that is, it satisfies the condition: µ P ( a | b ) = P ( a ∧ b ) P ( b ) , for any events a, b ∈ A . Finally we have introduced the logic LBC to reason aboutconditionals which is sound and complete with respect to a semantics defined in accordancewith the notion of Boolean algebra of conditionals. Moreover, we have pointed out thetight connections of this logic with preferential nonmonotonic consequence relations.Several interesting questions remain open and we aim to address them in future work.In this work we have not considered algebras with iterated conditionals. It would beinteresting to study whether it is possible to introduce in the algebras of conditionals C ( A ) a proper binary operator capturing the notion of iterated conditioning, for instancein the sense of whether an object of the form (( a | b ) | ( c | d )) can be defined as anotherelement of C ( A ) in a meaningful way, i.e. without having to resort to a meta-structure ofthe kind C ( C ( A )).As we have shown, given a positive probability P on an algebra of events A , we canextend it to a probability µ P on the whole algebra of conditionals C ( A ), and hence it isin principle possible to compute the probability of any compound conditional. However itwould be interesting to investigate whether there are more operational rules for computingthe probability of conjunctions and disjunctions of basic conditionals. In particular, onecould check whether rules appearing in e.g. [26, 33] apply also in our framework.40n the measure-theoretical side we leave open the problem of generalizing Theorem6.13 to the case when P is a not necessarily positive probability on A . Preliminaryinvestigations in this direction shows that a result of this kind needs a deeper algebraicanalysis of the Boolean algebras of conditionals C ( A ), in particular on the relation betweenthe congruences of A and those of C ( A ).At the foundational level, a pressing question will be to investigate Boolean algebrasof conditionals in light of de Finetti’s coherence criterion for conditional assignments. Inparticular we will investigate if, or up to which extent, the coherence of a “book” onconditional events ( a | b ) , . . . , ( a k | b k ) can be characterized in terms of the coherenceof an “unconditional book” on the ( a i | b i )’s viewed as elements of a Boolean algebra ofconditionals. A satisfactory solution to this problem would then motivate the extensionof this coherence-based analysis to non-probabilisitic measures of uncertainty, along thelines of [19].More generally, a natural question to be addressed in future research has to do withdefining non-probabilistic analogues of the relation established in the present paper be-tween Boolean algebras of conditionals and conditional probabilities. Most interestingtargets include possibility and necessity measures, ranking functions, belief and plausibil-ity functions, and imprecise probabilities.As to those latter recall that, as a consequence of Corollary 6.6, convexity fails, ingeneral, for sets of separable probabilities. This raises the question as to how separableprobabilities relate to the long standing problem (see e.g. [11]) of reconciling various formsof qualified stochastic independence with convex sets of probabilities. Acknowledgments
The authors are thankful to the anonymous reviewers for their helpful remarks and sug-gestions. They would also like to thank Didier Dubois, Marcelo Finger, Henri Pradeand Giuseppe Sanfilippo for fruitful discussions on the arguments of the present paper.Flaminio and Godo acknowledge partial support by the Spanish FEDER/MINECO projectTIN2015-71799-C2-1-P. Flaminio also acknowledges partial support by the Spanish Ram´ony Cajal research program RYC-2016-19799. Hosni’s research was funded by the Depart-ment of Philosophy “Piero Martinetti of the University of Milan under the Project “De-partments of Excellence 2018-2022 awarded by the Ministry of Education, University andResearch (MIUR). He also acknowledges funding from the Deutsche Forschungsgemein-schaft (DFG, grant LA 4093/3-1).
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Proofs
Proofs from Section 3
Proposition 3.8.
In every algebra C ( A ) the following properties hold for every a, c ∈ A and b ∈ A ′ :(i) ( a | b ) ≥ ( b | b ) iff a ≥ b ;(ii) if a ≤ c , then ( a | b ) ≤ ( c | b ) ; in particular a ≤ c iff ( a | ⊤ ) ≤ ( c | ⊤ ) ;(iii) if a ≤ b ≤ d , then ( a | b ) ≥ ( a | d ) ; in particular ( a | b ) ≥ ( a | a ∨ b ) ;(iv) if ( a | b ) = ( c | b ) , then a ∧ b = c ∧ b ;(v) ( a ∧ b | ⊤ ) ≤ ( a | b ) ≤ ( b → a | ⊤ ) ;(vi) if a ∧ d = ⊥ and ⊥ < a ≤ b , then ( a | ⊤ ) ⊓ ( d | b ) = ⊥ C ;(vii) ( b | ⊤ ) ⊓ ( a | b ) ≤ ( a | ⊤ ) .Proof. (i). By definition, ( a | b ) ≥ ( b | b ) iff ( a | b ) ⊓ ( b | b ) = ( b | b ) iff, by Proposition3.3 (iv), ( b | b ) = ( a ∧ b | b ) = ( a | b ). Finally, let us prove that if a b , then( a | b ) < ⊤ C . If a b , then a ∧ b < b and hence there exists c ∈ A , different from ⊥ , such that b = ( a ∧ b ) ∨ c . Therefore, from Proposition 3.3 (i) and Proposition 3.6(iii), ⊤ C = ( b | b ) = (( a ∧ b ) ∨ c | b ) = ( a ∧ b | b ) ∨ ( c | b ) and ( c | b ) = ⊥ C . Thus,( a | b ) = ( a ∧ b | b ) < ⊤ C .(ii). If a ≤ c , then a ∧ c = a , and hence ( a ∧ c | b ) = ( a | b ). Therefore by Proposition 3.3(ii), ( a | c ) ⊓ ( c | b ) = ( a | b ), and ( a | b ) ≤ ( c | b ). Moreover, if ( a | ⊤ ) ≤ ( c | ⊤ ), theneasily ( a → c | ⊤ ) = ⊤ C = ( ⊤ | ⊤ ). From Proposition 3.6 (i), a → c = ⊤ , and hence a ≤ c and the first part of (ii) holds.(iii). If a ≤ b ≤ d , then Proposition 3.3 (v) implies that ( a | d ) = ( a | b ) ⊓ ( b | d ), andhence ( a | d ) ≤ ( a | b ).(iv). It directly follows from Proposition 3.3 (iv).(v). First of all, ( a | b ) = ( a ∧ b | b ) ≥ ( a ∧ b | ⊤ ) from Proposition 3.3 (iv) and the lastclaim of (ii). In order to show the second inequality, assume first a ≤ b . Since b ∨ ¬ b = ⊤ we have ( a | b ) = ( a | b ) ⊓ ( b ∨ ¬ b | b ) = (( a | b ) ⊓ ( b | ⊤ )) ⊔ (( a | b ) ⊓ ( ¬ b | ⊤ )). Now, fromProposition 3.3 (v), ( a | b ) ⊓ ( b | ⊤ ) = ( a | ⊤ ), and clearly ( a | b ) ⊓ ( ¬ b | ⊤ ) ≤ ( ¬ b | ⊤ ),therefore ( a | b ) ≤ ( a | ⊤ ) ⊔ ( ¬ b | ⊤ ) = ( ¬ b ∨ a | ⊤ ). Now, in the general case, ( a | b ) =( a ∧ b | b ) ≤ ( ¬ b ∨ ( a ∧ b ) | ⊤ ) = ( ¬ b ∨ a | ⊤ ) = ( b → a | ⊤ ).(vi). We have ( a | ⊤ ) ≤ ( a | b ) from (ii). Then, ( a | ⊤ ) ⊓ ( d | b ) ≤ ( a | b ) ⊓ ( d | b ) = ( a ∧ d | b ) = ( ⊥ | b ) = ⊥ C .(vii). From Proposition 3.3 (iv) and (ii), it follows that ( b | ⊤ ) ⊓ ( a | b ) = ( b | ⊤ ) ⊓ ( a ∧ b | b ) ≤ ( b | ⊤ ) ⊓ ( a ∧ b | ⊤ ) = ( a ∧ b | ⊤ ) ≤ ( a | ⊤ ). Proposition 3.9.
In every algebra C ( A ) the following properties hold for all a, a ′ ∈ A and b, b ′ ∈ A ′ :(i) ( a | b ) ⊓ ( a | b ′ ) ≤ ( a | b ∨ b ′ ) ; in particular, ( a | b ) ⊓ ( a | ¬ b ) ≤ ( a | ⊤ ) ; ii) if a ≤ b ∧ b ′ , then ( a | b ) ⊓ ( a | b ′ ) = ( a | b ∨ b ′ ) ;(iii) ( a | b ) ≤ ( b → a | b ∨ b ′ ) ;(iv) ( a | b ) ⊓ ( a ′ | b ′ ) ≤ (( b → a ) ∧ ( b ′ → a ′ ) | b ∨ b ′ ) .Proof. (i). For all a ∈ A and b and c in A ′ , since ( b ∨ c | b ∨ c ) = ⊤ C , ( a | b ) ⊓ ( a | c ) = ( a | b ) ⊓ ( a | c ) ⊓ ( b ∨ c | b ∨ c ) = ( a | b ) ⊓ ( a | c ) ⊓ (( b | b ∨ c ) ⊔ ( c | b ∨ c ))) == (( a | b ) ⊓ ( a | c ) ⊓ ( b | b ∨ c )) ⊔ (( a | b ) ⊓ ( a | c ) ⊓ ( c | b ∨ c )) . (12)Now, ( a | b ) ⊓ ( a | c ) ⊓ ( b | b ∨ c ) ≤ ( a | b ) ⊓ ( b | b ∨ c ) = ( a ∧ b | b ) ⊓ ( b | b ∨ c ) = ( a ∧ b | b ∨ c ),where the first equality is due to Proposition 3.3 (iv) and the second one to Proposition3.3 (v). Analogously, we have ( a | b ) ⊓ ( a | c ) ⊓ ( c | b ∨ c ) ≤ ( a | c ) ⊓ ( c | b ∨ c ) = ( a ∧ c | b ∨ c ).Thus, by (12), we get that ( a | b ) ⊓ ( a | c ) ≤ ( a ∧ b | b ∨ c ) ⊔ ( a ∧ c | b ∨ c ) = ( a ∧ ( b ∨ c ) | b ∨ c ) = ( a | b ∨ c ), from Proposition 3.3 (iv).Obviously the particular case follows by taking b ′ = ¬ b .(ii). Now, assume a ≤ b ∧ b ′ . Then, from (12) the claim follows provided that, under thisfurther hypothesis, ( a | b ) ⊓ ( a | b ′ ) ⊓ ( b | b ∨ b ′ ) = ( a | b ) ⊓ ( b | b ∨ b ′ ) and ( a | b ) ⊓ ( a | b ′ ) ⊓ ( b ′ | b ∨ b ′ ) = ( a | b ′ ) ⊓ ( b ′ | b ∨ b ′ ). Let us prove the first equality, the second beingcompletely analogous.First of all ( a | b ) = ( a ∧ b | b ) (recall Proposition 3.3 (iv)), whence ( a | b ) ⊓ ( a | b ′ ) ⊓ ( b | b ∨ b ′ )) = ( a ∧ b | b ) ⊓ ( a | b ′ ) ⊓ ( b | b ∨ b ′ ) = ( a ∧ b | b ∨ b ′ ) ⊓ ( a | b ′ ) (the last equality followsfrom Proposition 3.3 (v)). Now, ( a ∧ b | b ∨ b ′ ) ≤ ( a | b ∨ b ′ ) ≤ ( a | b ′ ) because by hypothesis a ≤ b ′ ≤ b ∨ b ′ . Thus, ( a ∧ b | b ∨ b ′ ) ⊓ ( a | b ′ ) = ( a ∧ b | b ∨ b ′ ) = ( a | b ∨ b ′ ) since a ≤ b . Finally,since again a ≤ b ≤ b ∨ b ′ , from Proposition 3.3 (v), one has ( a | b ∨ b ′ ) = ( a | b ) ∧ ( b | b ∨ b ′ )as desired.(iii). In every Boolean algebra it holds that b → a = ¬ b ∨ a ≥ ¬ b . Thus b → a ≥ ¬ b ≥ c ∧¬ b .Thus, from Proposition 3.8 (i), ( b → a | c ∧ ¬ b ) = ⊤ C . Hence, ( a | b ) ≤ ( b → a | b ) and( a | b ) ≤ ( b → a | c ∧ ¬ b ), whence ( a | b ) ≤ ( b → a | b ) ⊓ ( b → a | c ∧ ¬ b ). The latter, from(i), is less or equal to ( b → a | ( c ∧ ¬ b ) ∨ b ) = ( b → a | c ∨ b ). This settles the claim.(iv). From (iii), ( a | b ) ≤ ( b → a | b ∨ b ′ ) and ( a ′ | b ′ ) ≤ ( b ′ → a ′ | b ∨ b ′ ). Thus,( a | b ) ⊓ ( a ′ | b ′ ) ≤ ( b → a | b ∨ b ′ ) ⊓ ( b ′ → a ′ | b ∨ b ′ ) = (( b → a ) ∧ ( b ′ → a ′ ) | b ∨ b ′ ). Proofs from Section 4
Proposition 4.3.
P art i ( C ( A )) is a partition of C ( A ) .Proof. We have to prove the two following conditions:(a) F P art i ( C ( A )) = ⊤ C ,(b) for distinct t , t ∈ P art i ( C ( A )), t ⊓ t = ⊥ C .(a) We shall prove this claim by induction on i . The case i = 1 is easy as Seq ( A ) = {h α i | α ∈ at ( A ) } and clearly F α ∈ at ( A ) ( α | ⊤ ) = ( W α α | ⊤ ) = ⊤ C .If 1 < i , let 1 < j ≤ i and suppose the claim is true for j −
1, that is, F P art j − ( C ( A )) = ⊤ C . We have to prove that F P art j ( C ( A )) = ⊤ C as well. For each sequence β =46 β , . . . , β j − i ∈ Seq j − ( A ), consider its corresponding compound conditional ω β = ( β |⊤ ) ⊓ . . . ⊓ ( β j − | ¬ β ∧ . . . ∧ ¬ β j − ). By inductive hypothesis, we have G β ∈ Seq j − ( A ) ω β = G P art j − ( C ( A )) = ⊤ C . Further, let D ( β ) = at ( A ) \ { β , . . . β j − } be the set of n − j + 1 atoms of A disjointfrom { β , . . . β j − } . Then it is clear that G α ∈ D ( β ) ( α | ¬ β ∧ . . . ∧ ¬ β j − ) = ⊤ C , and thus ω β = F α ∈ D ( β ) ω β ⊓ ( α | ¬ β ∧ . . . ∧ ¬ β j − ).Therefore, since this holds for every sequence β ∈ Seq j − , we finally get: ⊤ C = G β ∈ Seq j − ( A ) ω β = G β ∈ Seq j − ( A ) G β ∈ D ( β ) ω β ⊓ ( β | ¬ β ∧ . . . ∧ ¬ β j − ) = G δ ∈ Seq j ( A ) ω δ = G P art j ( C ( A )) . (b) Let ω α , ω β ∈ P art i ( C ( A )) with α = β . To be more precise, if α = h α , . . . , α i i and β = h β , . . . , β i i , let 1 ≤ k ≤ i the minimum index such that α k = β k . Then it holds that( α k | ¬ α ∧ . . . ¬ α k − ) ⊓ ( β k | ¬ β ∧ . . . ¬ β k − ) = ⊥ C since ¬ α ∧ . . . ¬ α k − = ¬ β ∧ . . . ¬ β k − and α k ∧ β k = ⊥ . Then, the claim follows from observing that α ⊓ β ≤ ( α k | ¬ α ∧ . . . ¬ α k − ) ⊓ ( β k | ¬ β ∧ . . . ¬ β k − ). Corollary 4.8.
Let A be a Boolean algebra with | at ( A ) | = n . For every basic conditional ( a | b ) ∈ C ( A ) with a ≤ b , | at ≤ ( a | b ) | = n ! · | at ≤ ( a ) || at ≤ ( b ) | .Proof. We start proving the following
Claim 1. If a = α is an atom of A , then | at ≤ ( α | b ) | = n ! | at ≤ ( b ) | .Proof. (of Claim 1) Obviously, if b is an atom, and since we are assuming a ≤ b , it mustbe a = b = α . Then | at ≤ ( a | b ) | = | at ≤ ( α | α ) | = | at ( ⊤ C ) | = n !. Conversely, if b is not anatom, let α , . . . , α k ∈ at ( A ) (with k >
1) such that b = W ki =1 α i . Further, since α ≤ b ,there is i ≤ k such that α i = α . For simplicity, and without any loss of generality,assume i = 1. Hence, n ! = | at ≤ ( α ∨ α ∨ . . . ∨ α k | b ) | = k X i =1 | at ≤ ( α i | b ) | and hence, by a symmetric argument, | at ≤ ( α | b ) | = n ! k = n ! | at ≤ ( b ) | and Claim 1 is settled.Coming back to the proof of Corollary 4.8, and remembering from Proposition 4.1 that a | b = W α ≤ x ( α | y ), we have that, thanks to Claim 1, | at ≤ ( a | b ) | = | at ≤ ( a ) | · n ! | at ≤ ( b ) | .47 roofs from Section 6 For the proof of next lemma recall the tree structure T introduced in Section 5.1. Lemma 6.8.
The map µ P is a probability distribution on at ( C ( A )) , that is, X α ∈ Seq ( A ) µ P ( ω α ) = 1 . Proof.
Let | at ( A ) | = n and let T be as described in Section 4. Attach to each node of T the following values: • Level 0: recall that the root node is ( ⊤ | ⊤ ). Thus attach P ( ⊤ | ⊤ ) = 1. • Level 1: attach to each node ( α i | ⊤ ) of level 1 the value P ( α i ). By definition ofatom and since P is a probability measure, we have that X α ∈ at ( A ) P ( α ) = 1 . • Level i : For each node β i − at level i −
1, let h β , . . . , β i − i the partial sequencecorresponding to the path from ( ⊤ | ⊤ ) to β i − . Then attach to the n − i childrennodes the value P ( β | ¬ β ∧ . . . ∧ ¬ β i − ). Note again that X β ∈ at ( A ) \{ β ,...,β i − } P ( β | ¬ β ∧ . . . ∧ ¬ β i − ) = 1 . Recall also that, by construction of T , at ( C ( A )) is in 1-1 correspondence with the pathsfrom the root to the leaf nodes of the tree. Furthermore, the above procedure attachesto each node x of T , the value P ( x ) = P ω α ≤ x µ P ( ω α ). Thus, in particular, for every leaf l , P ( l ) = µ P ( ω α l ) where α l is the sequence in Seq ( A ) which uniquely corresponds to thepath from ⊤ | ⊤ to l . Thus, by the above construction, letting L T the set of leafs of T ,one has 1 = X l ∈ L T P ( l ) = X l ∈ L T µ P ( ω α l ) = X α ∈ Seq ( A ) µ P ( ω α ) . Thus, the claim is settled.
Lemma 6.10.
Let α i , . . . , α i t ∈ at ( A ) . Then(i) µ P ( J α i , . . . , α i t K ) = P ( α i ) · P ( α i ) P ( ¬ α i ) · . . . · P ( α it ) P ( ¬ α i ∧¬ α i ∧ ... ∧¬ α it − ) .(ii) µ P ( J α i , . . . , α i t K ) = µ P ( J α i , . . . , α i j − , α i j +1 , . . . , α i t K ) · P ( α ij ) P ( ¬ α i ∧¬ α i ∧ ... ∧¬ α it − ) . Proof. (i). Let γ , . . . , γ l be the atoms of A different from α i , . . . , α i t . Thus, ω δ ∈ J α i , . . . , α i t K iff δ = h α i , . . . , α i t , σ i for σ any string which is obtained by permuting l − { γ , . . . , γ l } . 48herefore, letting Ψ = ¬ α i ∧ . . . ∧¬ α i t , K = P ( α i ) · P ( α i ) P ( ¬ α i ) · . . . · P ( α it ) P ( ¬ α i ∧¬ α i ∧ ... ∧¬ α it − ) and H = P σ P ( σ ) P (Ψ) · P ( σ ) P (Ψ ∧¬ σ ) · . . . · P ( σ l − ) P (Ψ ∧¬ σ ∧ ... ∧¬ σ n − ) , one has µ P ( J α i , . . . , α i t K ) = K · H. We now prove by induction on l that H = 1.(Case 0) The basic case is for l = 2. In this case H = P ( γ ) P (Ψ) + P ( γ ) P (Ψ) = P ( γ ∨ γ ) P (Ψ) . Thus, theclaim trivially follows because Ψ = γ ∨ γ .(Case l ) For any j = 1 , . . . , l we focus on the strings σ whose first coordinate σ = γ j .Therefore, H = P ( γ ) P (Ψ) · P σ : σ = γ ( P ( σ ) P (Ψ ∧¬ γ ) · . . . · P ( σ l − ) P (Ψ ∧¬ σ ∧ ... ∧¬ σ n − ) ) + . . .. . . + P ( γ l ) P (Ψ) · P σ : σ = γ l ( P ( σ ) P (Ψ ∧¬ γ ) · . . . · P ( σ l − ) P (Ψ ∧¬ σ ∧ ... ∧¬ σ n − ) ) . By inductive hypothesis, each term P σ : σ = γ j ( P ( σ ) P (Ψ ∧¬ γ ) · . . . · P ( σ l − ) P (Ψ ∧¬ σ ∧ ... ∧¬ σ n − ) ) equals 1.Thus, H = P lj =1 P ( γ j ) P (Ψ) = 1 since Ψ = W lj =1 γ j .(ii). The claim follows from (i) and direct computation. Lemma 6.11.
Let α | b be a basic conditional and let ¬ b = β ∨ . . . , ∨ β k with β k = α n .Then the following hold. For all t ∈ { , . . . , k − } , µ P ( B J α n , β , . . . , β t , α K ) = µ P ( J α n , β , . . . , β t − , α K ) · P ( β t ) P ( b ) . Proof.
We prove the claim by reverse induction on t . The basic case is hence for t = k −
1. In that case, by construction of the set J α n , β π (1) , . . . , β π ( t ) , α K , one has that J α n , β , . . . , β k − , α K is a leaf of B (recall Definition 5.4). Thus, B J α n , β , . . . , β k − , α K = J α n , β , . . . , β k − α K . Therefore, µ P ( B J α n , β , . . . , β k − , α K ) = µ P ( J α n , β , . . . , β k − , α K ) and by Lemma 6.10(ii), the latter equals µ P ( J α n , β , . . . , β k − , α K ) · P ( β k − ) P ( ¬ α n ∧ ¬ β ∧ . . . ∧ ¬ β k − ) . Therefore, the claim follows by noticing that b = ¬ α n ∧ ¬ β ∧ . . . ∧ ¬ β k − as we put α n = β k .Now, let t < k −
1. Thus, by construction of B , µ P ( T J α n , β , . . . , β t , α K ) = µ P ( J α n , β , . . . , β t , α K ) + X β l β ,...,β t } µ P ( B J α n , β , . . . , β t , β l , α K )and the inductive hypothesis ensures that µ P ( B J α n , β , . . . , β t , β l , α K ) = µ P ( J α n , β , . . . , β t , α K ) · P ( β l ) P ( b ) . µ P ( J α n , β , . . . , β t , α K ) + X β l β ,...,β t } µ P ( B J α n , β , . . . , β t , β l , α K ) == µ P ( J α n , β , . . . , β t , α K ) · (cid:18) P ( d ) P ( b ) (cid:19) , where d = W β l β ,...,β t } β l . Again, Lemma 6.10 (2) gives, µ P ( J α n , β , . . . , β t , α K ) = µ P ( J α n , β , . . . , β t − , α K ) · P ( β t ) P ( ¬ α n ∧ ¬ β . . . ∧ ¬ β t ) . Notice that 1 + P ( d ) P ( b ) = P ( b ∨ d ) P ( b ) and b ∨ d = ¬ α n ∧ ¬ β . . . ∧ ¬ β t . Thus, P ( b ∨ d ) = P ( ¬ α n ∧ ¬ β . . . ∧ ¬ β t ) . (13)Therefore, µ P ( B J α n , β , . . . , β t , α K ) = µ P ( J α n , β , . . . , β t , α K ) + X β l β ,...,β t } µ P ( B J α n , β , . . . , β t , β l , α K ) . = µ P ( J α n , β , . . . , β t , α K ) · (cid:16) P ( d ) P ( b ) (cid:17) = µ P ( J α n , β , . . . , β t − , α K ) · P ( β t ) P ( ¬ α n ∧¬ β ... ∧¬ β t ) · (cid:16) P ( b ∨ d ) P ( b ) (cid:17) = µ P ( J α n , β , . . . , β t − , α K ) · P ( β t ) P ( b ) where the last equality follows from (13). Lemma 6.12.
Let α | b be a basic conditional and let b ≥ α . Then,(i) µ P ( S ) = P ( α ) .(ii) For all ≤ j ≤ n , µ P ( S j ) = P ( α ) · P ( α j ) P ( b ) .Proof. (i) immediately follows from Lemma 6.11 (i) noticing that, indeed, S = J α K byLemma 5.1 (i). Let hence prove (ii). In particular, and without loss of generality, let usprove the claim for α j = α n . As in the statement, let us call β , . . . , β k those atoms of A such that ¬ b = W ki =1 β i . In particular, without loss of generality due to Lemma 5.1 (ii),let us put β k = α n .Notice that µ P ( S n ) = µ P ( B J α n , α K ) which, by construction and thanks to Lemma6.11, equals µ P ( J α n , α K ) + k − X j =1 µ P ( B J α n , β j , α K ) = P ( α n ) · P ( α ) P ( ¬ α n ) + k − X j =1 µ P ( J α n , α K ) · P ( β j ) P ( b ) , Therefore, since by Lemma 6.10 (i), µ P ( J α n , α K ) = P ( α n ) · P ( α ) P ( ¬ α n ) , we get µ P ( S n ) = P ( α n ) · P ( α ) P ( ¬ α n ) k − X j =1 P ( β j ) P ( b ) = P ( α n ) · P ( α ) P ( ¬ α n ) P (cid:16) b ∨ W k − j =1 β j (cid:17) P ( b ) . b ∨ W k − j =1 β j = ¬ α n . Thus, P (cid:16) b ∨ W k − j =1 β j (cid:17) = P ( ¬ α n ) and µ P ( S n ) = P ( α n ) · P ( α ) · (cid:18) P ( b ) (cid:19) which settles the claim. Proofs from Section 7
Theorem 7.3. CL / ≡ ∼ = C ( L ).The proof needs some previous elaboration. For each satisfiable proposition ϕ ∈ L , let ϕ ∗ be its expression in full DNF. We assume that there is a unique such full DNF expressionfor each formula, i.e. ϕ ∗ is the unique representative of the class [ ϕ ] of CPL formulasequivalent to ϕ in full DNF. Thus, let • L ∗ be the set of satisfiable propositional formulas from L in full DNF, • V ∗ = { ( ϕ ∗ | ψ ∗ ) : ϕ ∗ ∈ L ∗ ∪ {⊥} , ψ ∗ ∈ L ∗ } , • Free ( V ∗ ) be the freely generated Boolean algebra with generators V ∗ Let us define an equivalence relation on
Free ( V ∗ ) as follows: Φ ∗ ≡ Ψ ∗ iff LBC ⊢ Φ ∗ ↔ Ψ ∗ . (the latter due to rules (R1) and (R2) and CPL reasoning).Actually, ≡ is indeed a congruence relation on Free ( V ∗ ), that satisfies the five prop-erties:- ( ϕ ∗ | ϕ ∗ ) ≡ ⊤ - ( ϕ ∗ | ψ ∗ ) ∧ ( γ ∗ | ψ ∗ ) ≡ (( ϕ ∧ γ ) ∗ | ψ ∗ )- ¬ ( ϕ ∗ | ψ ∗ ) ≡ (( ¬ ϕ ) ∗ | ψ ∗ )- ( ϕ ∗ | ψ ∗ ) ≡ (( ϕ ∧ ψ ) ∗ | ψ ∗ )- ( ϕ ∗ | ψ ∗ ) ∧ ( ψ ∗ | γ ∗ ) ≡ ( ϕ ∗ | γ ∗ ), when ⊢ CP L ϕ → ψ → γ Therefore, ≡ and ≡ C as defined in Section 3 are the same congruence on Free ( V ∗ )and, consequently, Free ( V ∗ ) / ≡ ∼ = Free ( V ∗ ) / ≡ C . Now, let L ∗ be the Lindenbaum algebra of the language L ∗ . Notice that, as booleanalgebra, L ∗ contains a bottom element, although the symbol ⊥ does not belong to thelanguage L ∗ as we previously remarked. Thus, V ∗ = L ∗ × ( L ∗ ) ′ , whence they generateisomorphic free algebras, that is to say, Free ( V ∗ ) ∼ = Free ( L ∗ × ( L ∗ ) ′ ). Therefore, passingat the quotients, one has Free ( V ∗ ) / ≡ C ∼ = Free ( L ∗ × ( L ∗ ) ′ ) / ≡ C and the latter, by definition,equals C ( L ∗ ). Thus, we conclude that Free ( V ∗ ) / ≡ C ∼ = C ( L ∗ )Finally notice that, since L ∗ is actually isomorphic to the Lindenbaum algebra L of L , onefinally has the following Lemma A.1. C ( L ) ∼ = Free ( V ∗ ) / ≡ . It is well-known that contradictory formulas like ⊥ = ϕ ∧ ¬ ϕ cannot be expressed in full DNF. Forthis reason we need to add that symbol to L ∗ to be a possible consequent of a basic conditional. CL ∗ of CL built from basic conditionals ( ϕ ∗ | ψ ∗ ). If Φis an LBC-formula, we will denote by Φ ∗ the formula from CL ∗ obtained by replacing allpropositional formulas ϕ in basic conditionals in Φ by their DNF expressions.The relation ≡ which is a congruence of Free ( V ∗ ) clearly also is an equivalence relationon CL ∗ and hence, since ≡ is compatible with Boolean operations, the structure CL ∗ =( CL ∗ / ≡ , ∧ , ∨ , ¬ , ⊥ , ⊤ ) is a Boolean algebra. Lemma A.2. CL ∗ ∼ = Free ( V ∗ ) / ≡ .Proof. Since CL ∗ and Free ( V ∗ ) / ≡ are finite algebras we only need to exhibit a bijec-tion between their supports. Notice that the elements of CL ∗ and those of Free ( V ∗ ) / ≡ are indeed the same elements, i.e., equivalence classes of Boolean combinations of ba-sic conditionals whose antecedent and consequent are in DNF. Further the equivalenceclasses are determined by the same equivalence relation. Thus, clearly, the sets CL ∗ / ≡ and F ree ( V ∗ ) / ≡ are actually the same set and hence the claim follows.Now we can finally prove the desired isomorphism claimed in Theorem 7.3: CL / ≡ ∼ = C ( L ) Proof.
Due to Lemmas A.1 and A.2 it is enough to prove that CL ∼ = CL ∗ , that is, foreach formula Φ of CL , Φ is a theorem of LBC iff Φ ∗ can be proved from the DNF-instances( A ∗ − ( A ∗ of the axioms of LBC, axioms and rules of CPL, but without the use of(R1) and (R2).Every formula Φ ∗ is logically equivalent, in CPL, to Φ and also every ( Ai ) is clearlylogically equivalent to its DNF. Thus if Φ ∗ can be proved from ( A ∗ − ( A ∗ without (R1)and (R2), then Φ ∗ is a theorem of LBC and hence so is Φ.Conversely, assume that π = Ψ , . . . , Ψ k (where Ψ k = Φ) is a proof of Φ in LBC. Firstof all, we can safely replace each Ψ i by its DNF Ψ ∗ i without loss of generality so defining alist of formulas π ∗ all in DNF. Further notice that, in replacing Ψ i by Ψ ∗ , any occurrenceof the rule ( R
1) becomes(R1) ∗ from ⊢ CP L ϕ ∗ → ψ ∗ , derive ( ϕ ∗ | χ ∗ ) → ( ψ ∗ | χ ∗ ).Then, since in DNF we have ⊢ CP L ϕ ∗ ↔ ψ ∗ iff ϕ ∗ = ψ ∗ , (R1) ∗ yields the followingderived rule: if ϕ ∗ = ψ ∗ , derive ( ϕ ∗ | χ ∗ ) ↔ ( ψ ∗ | χ ∗ ), i.e., essentially, for all ϕ , derive( ϕ ∗ | χ ∗ ) ↔ ( ϕ ∗ | χ ∗ ), which is trivial and hence can be omitted in π ∗ . The same obviouslyapplies to (R2).Thus, summing up, π ∗ is a proof of Φ ∗ made of formulas in CL ∗∗