Counterfactuals and dependencies on causal teams: expressive power and deduction systems
aa r X i v : . [ m a t h . L O ] A p r Counterfactuals and dependencies on causalteams: expressive power and deduction systems
Fausto Barbero Department of Philosophy, University of HelsinkiPL 24 (Unioninkatu 40), 00014 Helsinki, Finland
Fan Yang Department of Mathematics and Statistics, University of HelsinkiPL 68 (Pietari Kalmin katu 5), 00014 Helsinki, Finland
Abstract
We analyze the causal-observational languages that were introduced in Barbero andSandu (2018), which allow discussing interventionist counterfactuals and functionaldependencies in a unified framework. In particular, we systematically investigatethe expressive power of these languages in causal team semantics, and we providecomplete natural deduction calculi for each language. As an intermediate step towardsthe completeness, we axiomatize the languages over a generalized version of causalteam semantics, which turns out to be interesting also in its own right.
Keywords:
Interventionist counterfactuals, causal teams, dependence logic, teamsemantics.
Counterfactual conditionals express the modality of irreality : they describewhat would or might be the case in circumstances which diverge from the actualstate of affairs. Pinning down the exact meaning and logic of counterfactualstatements has been the subject of a large literature (see e.g. [11]). We areinterested here in a special case: the interventionist counterfactuals, whichemerged from the literature on causal inference ([10,9,7]). Under this reading,a conditional X = x € ψ says that ψ would hold if we were to intervene on thegiven system, by subtracting the variables X to their current causal mechanismsand forcing them to take the values x .The logic of interventionist counterfactuals has been mainly studied in thesemantical context of deterministic causal models ([5,6,3,15]), which consist The author was supported by grant 316460 of the Academy of Finland. The author was supported by Research Funds of the University of Helsinki and grant308712 of the Academy of Finland. Counterfactuals and dependencies on causal teams: expressive power and deduction systems of an assignment of values to variables together with a system of structuralequations that describe the causal connections. In [1], causal models weregeneralized to causal teams , in the spirit of team semantics ([8,12]), by allowinga set of assignments (a “ team ”) instead of a single assignment. This opens thepossibility of describing e.g. uncertainty , observations , and dependencies .One of the main reasons for introducing causal teams was the possibility ofcomparing the logic of dependencies of causal nature (those definable in terms ofinterventionist counterfactuals) against that of contingent dependencies (suchas have been studied in the literature on team semantics, or in database the-ory) in a unified semantic framework. [1] and [2] give anecdotal evidence ofthe interactions between the two kinds of dependence, but offer no generalaxiomatizations for languages that also involve contingent dependencies. Inthis paper we fill this gap in the literature by providing complete deductionsystems (in natural deduction style) for the languages COD and CO \\ / (from[1]), which enrich the basic counterfactual language, respectively, with atomsof functional dependence = ( X ; Y ) (“ Y is functionally determined by X ”), orwith the intuitionistic disjunction \\ / , in terms of which functional dependenceis definable. We also give semantical characterizations, for COD , CO \\ / and thebasic counterfactual CO , in terms of definability of classes of causal teams.The strategy of the completeness proofs is the following. We introducea generalized causal team semantics, which encodes uncertainty over causalmodels, not only over assignments. (This semantics is used as a tool towardscompleteness, but also has independent interest.) We then give completenessresults for this semantics, using techniques developed in [13,4]. Finally, weextend the calculi to completeness over causal teams by adding axioms whichcapture the property of being a causal team (i.e. encoding certainty about thecausal connections).The paper is organized as follows. Section 2 introduces the formal lan-guages and two kinds of semantics. Section 3 deepens the discussion of thefunctions which describe causal mechanisms, addressing issues of definabilityand the treatment of dummy arguments. Section 4 characterizes semanticallythe language CO and reformulates in natural deduction form the CO calculithat come from [2]. Section 5 gives semantical characterizations for COD and CO \\ / , and complete natural deduction calculi for both kinds of semantics. Let us start by fixing the syntax. Each of the languages considered in this paperis parametrized by a (finite) signature σ , i.e. a pair (Dom , Ran) , where
Dom is a nonempty finite set of variables , and
Ran is a function that associates toeach variable X ∈ Dom a nonempty finite set
Ran( X ) (called the range of X ) of constant symbols or values . We reserve the Greek letter σ for signatures. Note that we do not encode a distinction between exogenous and endogenous variablesinto the signatures, as done in [6]. Instead, we follow the style of Briggs [3]. Doing so willarbero, Yang 3
We use a boldface capital letter X to stand for a sequence h X , . . . , X n i of vari-ables; similarly a boldface lower case letter x stands for a sequence h x , . . . , x n i of values. We will sometimes abuse notation and treat X and x as sets.An atomic σ -formula is an equation X = x , where X ∈ Dom and x ∈ Ran( X ) .The conjunction X = x ∧ · · · ∧ X n = x n of equations is abbreviated as X = x ,and also called an equation. Compound formulas of the basic language CO [ σ ] are formed by the grammar: α :: = X = x | ¬ α | α ∧ α | α ∨ α | X = x € α The connective € is used to form interventionist counterfactuals . We abbre-viate ¬ ( X = x ) as X , x , and X = x ∧ X , x as ⊥ . Throughout the paper, wereserve the first letters of the Greek alphabet, α, β, . . . for CO [ σ ] formulas.We consider also two extensions of CO [ σ ] , obtained by adding the intuition-istic disjunction \\ / , or dependence atoms = ( X ; Y ) : • CO \\ / [ σ ] : ϕ :: = X = x | ¬ α | ϕ ∧ ϕ | ϕ ∨ ϕ | ϕ \\ / ϕ | X = x € ϕ • COD [ σ ] : ϕ :: = X = x | = ( X ; Y ) | ¬ α | ϕ ∧ ϕ | ϕ ∨ ϕ | X = x € ϕ We now define the team semantics of our logics over causal teams. We firstrecall the definition of causal teams adapted from [2].Fix a signature σ = (Dom , Ran) . An assignment over σ is a mapping s : Dom → S X ∈ Dom
Ran( X ) such that s ( X ) ∈ Ran( X ) for each X ∈ Dom . Denoteby A σ the set of all assignments over σ . A team T over σ is a set of assignmentsover σ , i.e., T ⊆ A σ .A system of functions F over σ is a function that assigns to each variable V in a domain En( F ) ⊆ Dom a set PA F V ⊆ Dom \ { V } of parents of V , and afunction F V : Ran( PA F V ) → Ran( V ) . Variables in the set
En( F ) are called endogenous variables of F , and variables in Ex( F ) = Dom \ En( F ) are called exogenous variables of F .Denote by F σ the set of all systems of functions over σ , which is clearlyfinite. We say that an assignment s ∈ A σ is compatible with a system offunctions F ∈ F σ if for all endogenous variables V ∈ En( F ) , s ( V ) = F V ( s ( PA F V )) . Definition 2.1 A causal team over a signature σ is a pair T = ( T − , F ) consisting of • a team T − over σ , called the team component of T , • and a system of functions F over σ , called the function component of T ,where all assignments s ∈ T − are compatible with the function component F . Any system
F ∈ F σ of functions can be naturally associated with a (di- result in more general completeness results. We identify syntactical variables and values with their semantical counterpart, followingthe conventions in most literature on interventionist counterfactuals, e.g. [5,6,3,15]. In thisconvention distinct symbols (e.g., x , x ′ ) denote distinct objects. We identify the set PA F V with a sequence, in a fixed lexicographical ordering. Counterfactuals and dependencies on causal teams: expressive power and deduction systems rected) graph G F = (Dom , E F ) , defined as ( X , Y ) ∈ E F iff X ∈ PA F Y . We say that F is recursive if G F is acyclic, i.e., for all n ≥ , E F has no subset of theform { ( X , X ) , ( X , X ) , . . . , ( X n − , X n ) , ( X n , X ) } . The graph of a causal team T ,denoted as G T , is the associated graph of its function component. We call T recursive if G T is acyclic. Throughout this paper, for simplicity we assumethat all causal teams that we consider are recursive.Intuitively, a causal team T may be seen as representing an assumptionconcerning the causal relationships among the variables in Dom (as encodedin F ) together with a range of hypotheses concerning the actual state of thesystem (as encoded in T − ). We now illustrate this idea in the following example. Example 2.2
The following diagram illustrates a causal team T = ( T − , F ) . T − : U X Y Z F X ( U ) = U F Y ( X ) = X + F Z ( X , Y , U ) = ∗ Y + X + U The table on the left represents a team T − consisting of two assignments, eachof which is tabulated in the obvious way as a row in the table. For instance, theassignment s of the first row is defined as s ( U ) = s ( X ) = , s ( Y ) = and s ( Z ) = .The arrows in the upper part of the table represent the graph G T of the causalteam T . For instance, the arrow from U to Z represents the edge ( U , Z ) in G T .The graph contains no cycles, thus the causal team T is recursive. The variable U with no incoming arrows is an exogenous variable. The other variables areendogenous variables, namely, En( F ) = { X , Y , Z } . The function component isdetermined by the system of functions on the right of the above diagram. Eachequation defines the “law” that generates the values of an endogenous variable. Let S = ( S − , F ) and T = ( T − , G ) be causal teams over the same signature.We call S a causal subteam of T , denoted as S ⊆ T , if S − ⊆ T − and F = G .An equation X = x is said to be inconsistent if it contains two conjuncts X = x and X = x ′ with distinct values x , x ′ ; otherwise it is said to be consistent . Definition 2.3 (Intervention)
Let T = ( T − , F ) be a causal team over somesignature σ = (Dom , Ran) . Let X = x ( = X = x ∧ · · · ∧ X n = x n ) be a consistentequation over σ . The intervention do ( X = x ) on T is the procedure thatgenerates a new causal team T X = x = ( T − X = x , F X = x ) over σ defined as follows: • F X = x is the restriction of F to En( F ) \ X , • T − X = x = { s X = x | s ∈ T − } , where each s X = x is an assignment compatible with F X = x defined (recursively) as s X = x ( V ) = x i if V = X i , s ( V ) if V < En( T ) ∪ X , F V ( s X = x ( PA F V )) if V ∈ En( T ) \ X Example 2.4
Recall the recursive causal team T in Example 2.2. By applyingthe intervention do ( X = to T , we obtain a new causal team T X = = ( T − X = , F X = ) as follows. The function component F X = is determined by the equations: arbero, Yang 5 ( ( F X = ) Y ( X ) = X + F X = ) Z ( X , Y ) = ∗ Y + X + U The endogenous variable X of the original team T becomes exogenous in thenew team T X = , and the equation F X ( U ) = U for X is now removed. The newteam component T − X = is obtained by the rewriting procedure illustrated below:U X Y Z ... ... ... ... U X Y Z ... ... U X Y Z In the first step, rewrite the X -column with value . Then, update (recursively)the other columns using the functions from F X = . In this step, only the columnsthat correspond to “descendants” of X will be modified, and the order in whichthese columns should be updated is completely determined by the (acyclic) graph G T X = of T X = . Since the variable X becomes exogenous after the intervention, allarrows pointing to X have to be removed, e.g., the arrow from U to X . We referthe reader to [2] for more details and justification for this rewriting procedure. Definition 2.5
Let ϕ be a formula of the language CO \\ / [ σ ] or COD [ σ ] , and T = ( T − , F ) a causal team over σ . We define the satisfaction relation T | = c ϕ (or simply T | = ϕ ) over causal teams inductively as follows: • T | = X = x ⇐⇒ for all s ∈ T − , s ( X ) = x . • T | = = ( X ; Y ) ⇐⇒ for all s , s ′ ∈ T − , s ( X ) = s ′ ( X ) implies s ( Y ) = s ′ ( Y ) . • T | = ¬ α ⇐⇒ for all s ∈ T − , ( { s } , F ) = α . • T | = ϕ ∧ ψ ⇐⇒ T | = ϕ and T | = ψ . • T | = ϕ ∨ ψ ⇐⇒ there are two causal subteams T , T of T such that T − ∪ T − = T − , T | = ϕ and T | = ψ . • T | = ϕ \\ / ψ ⇐⇒ T | = ϕ or T | = ψ . • T | = X = x € ϕ ⇐⇒ X = x is inconsistent or T X = x | = ϕ . We write a dependence atom = (; X ) with an empty first component as = ( X ) .The semantic clause for = ( X ) reduces to: • T | = = ( X ) iff for all s , s ′ ∈ T − , s ( X ) = s ′ ( X ) .Intuitively, the atom = ( X ) states that X has a constant value in the team. It iseasy to verify that dependence atoms are definable in CO \\ / [ σ ] : = ( Y ) ≡ \\ / y ∈ Ran( Y ) Y = y and = ( X ; Y ) ≡ _ x ∈ Ran( X ) ( X = x ∧ = ( Y )) . (1)The selective implication α ⊃ ϕ from [1] is now definable as ¬ α ∨ ϕ . Its semanticclause reduces to: • T | = α ⊃ ψ ⇐⇒ T α | = ψ , where T α is the (unique) causal subteam of T withteam component { s ∈ T − | { s } | = α } . Note once more that the symbol x is used as both a syntactical and a semantical object. Counterfactuals and dependencies on causal teams: expressive power and deduction systems Example 2.6
Consider the causal team T and the intervention do ( X = fromExamples 2.2 and 2.4. Clearly, T X = | = Y = , and thus T | = X = € Y = .We also have that T | == ( Y ; Z ) , while T X = == ( Y ; Z ) (contingent dependenciesare not in general preserved by interventions). Observe that T | = Y , ∨ Y = ,while T = Y , \\ / Y = . Given a signature σ , write S em σ : = { ( s , F ) ∈ A σ × F σ | s is compatible with F } . The pairs ( s , F ) ∈ S em σ can be easily identified with the deterministic causalmodels (also known as deterministic structural equation models ) that are con-sidered in the literature on causal inference ([10],[9], etc.). One can identify acausal team T = ( T − , F ) with the set T g = { ( s , F ) ∈ S em σ | s ∈ T − } of deterministic causal models with a uniform function component F through-out the team. In this section, we introduce a more general notion of causalteam, called generalized causal team , where the function component F doesnot have to be constant thoroughout the team. Definition 2.7 A generalized causal team T over a signature σ is a set ofpairs ( s , F ) ∈ S em σ , that is, T ⊆ S em σ . Intuitively, a generalized causal team encodes uncertainty about whichcausal model governs the variables in
Dom - i.e., uncertainty both on the val-ues of the variables and on the laws that determine them. Distinct elements ( s , F ) , ( t , G ) of the same generalized causal team may also disagree on what isthe set of endogenous variables, or on whether the system is recursive or not.A generalized causal team is said to be recursive if, for each pair ( s , F ) inthe team, the associated graph G F is recursive. In this paper we only considerrecursive generalized causal teams.For any generalized causal team T , define the team component of T to bethe set T − : = { s | ( s , F ) ∈ T for some F } . A causal subteam of T is a subset S of T , denoted as S ⊆ T . The union S ∪ T of two generalized causal teams S , T is their set-theoretic union.A causal team T can be identified with the generalized causal team T g ,which has a constant function component in all its elements. Conversely, if T is a nonempty generalized causal team in which all elements have the samefunction component F , i.e., T = { ( s , F ) | s ∈ T − } , we can naturally identify T with the causal team T c = ( T − , F ) . In particular, a singleton generalized causal team { ( s , F ) } corresponds to asingleton causal team ( { s } , F ) . Applying a (consistent) intervention do ( X = x ) on ( { s } , F ) generates a causal team ( { s X = x } , F X = x ) as defined in Definition 2.3.We can then define the result of the intervention do ( X = x ) on { ( s , F ) } to bethe generalized causal team ( { s X = x } , F X = x ) g = { ( s X = x , F X = x ) } . Interventions onarbitrary generalized causal teams are defined as follows. arbero, Yang 7 Definition 2.8 (Intervention over generalized causal teams)
Let T bea (recursive) generalized causal team, and X = x a consistent equation over σ . The intervention do ( X = x ) on T generates the generalized causal team T X = x : = { ( s X = x , F X = x ) | ( s , F ) ∈ T } . Definition 2.9
Let ϕ be a formula of the language CO \\ / [ σ ] or COD [ σ ] , and T a generalized causal team over σ . The satisfaction relation T | = g ϕ (or simply T | = ϕ ) over generalized causal teams is defined in the same way as in Definition2.5, except for slight differences in the following clauses: • T | = g ¬ α iff for all ( s , F ) ∈ T , { ( s , F ) } 6| = α . • T | = g ϕ ∨ ψ iff there are two generalized causal subteams T , T of T suchthat T ∪ T = T , T | = ϕ and T | = ψ . We list some closure properties for our logics over both causal teams andgeneralized causal teams in the next theorem, whose proof is left to the reader,or see [2] for the causal team case.
Theorem 2.10
Let T , S be (generalized) causal teams over some signature σ . Empty team property: If T − = ∅ , then T | = ϕ . Downward closure: If T | = ϕ and S ⊆ T , then S | = ϕ . Flatness of CO -formulas: If α is a CO [ σ ] -formula, then T | = α ⇐⇒ ( { s } , F ) | = c α for all s ∈ T − ( resp. { ( s , F ) } | = g α for all ( s , F ) ∈ T ) . The team semantics over causal teams and that over generalized causalteams with a constant function component are essentially equivalent, in thesense of the next lemma, whose proof is left to the reader.
Lemma 2.11 (i) For any causal team T , we have that T | = c ϕ ⇐⇒ T g | = g ϕ. (ii) For any nonempty generalized causal team T with a unique function com-ponent, we have that T | = g ϕ ⇐⇒ T c | = c ϕ. Corollary 2.12
For any set ∆ ∪ { α } of CO [ σ ] -formulas, ∆ | = g α iff ∆ | = c α . Proof.
By Lemma 2.11, { ( s , F ) } | = g β iff ( { s } , F ) | = c β for any β ∈ ∆ ∪ { α } . Thus,the claim follows from the flatness of CO [ σ ] -formulas. ✷ Consider a binary function f and an ( n + -ary function g defined as f ( X , Y ) = X + Y and g ( X , Y , Z , . . . , Z n ) = X + Y . Essentially f and g are the same function: Z , . . . , Z n are dummy arguments of g . We now characterize this idea in thenotion of two function components being equivalent up to dummy arguments. Definition 3.1
Let F , G be two function components over σ = (Dom , Ran) . • Let V ∈ Dom . Two functions F V and G V are said to be equivalent up todummy arguments, denoted as F V ∼ G V , if for any x ∈ Ran( PA F V ∩ PA G V ) , y ∈ Ran( PA F V \ PA G V ) and z ∈ Ran( PA G V \ PA F V ) , we have that F V ( xy ) = G V ( xz ) (where Counterfactuals and dependencies on causal teams: expressive power and deduction systems we assume w.l.o.g. the shown orderings of the arguments of the functions). • Let
Cn( F ) denote the set of endogenous variables V of F for which F V is aconstant function, i.e., for some fixed c ∈ Ran( V ) , F V ( p ) = c for all p ∈ PA F V .We say that F and G are equivalent up to dummy arguments , denotedas F ∼ G , if
En( F ) \ Cn( F ) = En( G ) \ Cn( G ) , and F V ∼ G V holds for all V ∈ En( F ) \ Cn( F ) . It is easy to see that ∼ is an equivalence relation. The next lemma showsthat the relation ∼ is preserved under interventions. Lemma 3.2
For any function components F , G ∈ F σ and consistent equation X = x over σ , we have that F ∼ G implies F X = x ∼ G X = x . Proof.
Suppose
F ∼ G . Then
En( F ) \ Cn( F ) = En( G ) \ Cn( G ) . Observe that En( F X = x ) = En( F ) \ X and Cn( F X = x ) = Cn( F ) \ X ; and similarly for G . It followsthat En( F X = x ) \ Cn( F X = x ) = (cid:0) En( F ) \ Cn( F ) (cid:1) \ X = (cid:0) En( G ) \ Cn( G ) (cid:1) \ X = En( G X = x ) \ Cn( G X = x ) . On the other hand, for any V ∈ En( F X = x ) \ Cn( F X = x ) = (cid:0) En( F ) \ Cn( F ) (cid:1) \ X , by the assumption, ( F X = x ) V = F V ∼ G V = ( G X = x ) V . ✷ We now generalize the equivalence relation ∼ to the team level. Let us firstconsider causal teams. Two causal teams T = ( T − , F ) and S = ( S − , G ) of thesame signature σ are said to be similar , denoted as T ∼ S , if F ∼ G . We saythat T and S are equivalent , denoted as T ≈ S , if T ∼ S and T − = S − .Next, we turn to generalized causal teams. We call a generalized causalteam T a uniform team if F ∼ G for all ( s , F ) , ( t , G ) ∈ T . By Lemma 3.2, weknow that if T is uniform, so is T X = x , for any consistent equation X = x . Forany generalized causal team T with ( t , F ) ∈ T , write T F : = { ( s , G ) ∈ T | G ∼ F } . Two generalized causal teams S and T are said to be equivalent , denoted as S ≈ T , if ( S F ) − = ( T F ) − for all F ∈ F σ . Theorem 3.3 (Closure under causal equivalence)
Let T , S be two (gen-eralized) causal teams over σ such that T ≈ S . We have that T | = ϕ ⇐⇒ S | = ϕ. Proof.
The theorem is proved by induction on ϕ . The case ϕ = X = x € ψ follows from the fact that T X = x ≈ S X = x (Lemma 3.2). The case ϕ = ψ ∨ χ for causal teams follows directly from the induction hypothesis. We now givethe proof for this case for generalized causal teams. We only prove the left toright direction (the other direction is symmetric). Suppose T | = g ψ ∨ χ . Thenthere are T , T ⊆ T such that T = T ∪ T , T | = g ψ and T | = g χ . Consider S i = { ( s , F ) ∈ S | { ( s , F ) } ≈ { ( s , G ) } for some ( s , G ) ∈ T i } ( i = , ). It is easy tosee that S i ≈ T i ( i = , ) and S = S ∪ S . By induction hypothesis we havethat S | = g ψ and S | = g χ . Hence S | = g ψ ∨ χ . ✷ Thus, none of our languages can tell apart causal teams which are equivalentup to dummy arguments. However, one might not be sure, a priori, that agiven argument behaves as dummy for a specific function, as this might e.g. beunfeasible to verify if the variables range over large sets. For this reason, weare keeping this distinction in the semantics instead of quotienting it out. arbero, Yang 9
For any function component F over some signature σ , define a CO [ σ ] -formula Φ F : = ^ V ∈ En( F ) η σ ( V ) ∧ ^ V ∈ (Dom \ En( F )) ∪ Cn( F ) ξ σ ( V ) . where η σ ( V ) : = V (cid:8) ( W = w ∧ PA F V = p ) € V = F V ( p ) | W = Dom \ ( PA F V ∪ { V } ) , w ∈ Ran( W ) , p ∈ Ran( PA F V ) (cid:9) and ξ σ ( V ) : = V (cid:8) V = v ⊃ ( W V = w € V = v ) | v ∈ Ran( V ) , W V = Dom \ { V } , w ∈ Ran( W V ) (cid:9) . Intuitively, for each non-constant endogenous variable V of F , the formula η σ ( V ) specifies that all assignments in the (generalized) causal team T in questionbehave exactly as required by the function F V . For each variable V which,according to F , is exogenous or generated by a constant function, the formula ξ σ ( V ) states that V is not affected by interventions on other variables. If V ∈ Cn( F ) , then V has both an η σ and a ξ σ clause. Overall, the formula Φ F is satisfied in a team T if and only if every assignment in T has a functioncomponent that is ∼ -equivalent to F . This result is the crucial element foradapting the standard methods of team semantics to the causal context. Theorem 3.4
Let σ be a signature, and F ∈ F σ .(i) For any generalized causal team T over σ , we have that T | = g Φ F ⇐⇒ for all ( s , G ) ∈ T : G ∼ F . (ii) For any nonempty causal team T = ( T − , G ) over σ , we have that T | = c Φ F ⇐⇒ G ∼ F . Proof. (i). = ⇒ : Suppose T | = g Φ F and ( s , G ) ∈ T . We show G ∼ F . En( F ) \ Cn( F ) ⊆ En( G ) \ Cn( G ) : For any V ∈ En( F ) \ Cn( F ) , there are distinct p , p ′ ∈ Ran( PA F V ) such that F V ( p ) , F V ( p ′ ) . Since T | = η σ ( V ) , for any w ∈ Ran ( W ) , wehave that { ( s , G ) } | = ( W = w ∧ PA F V = p ) € V = F V ( p ) , { ( s , G ) } | = ( W = w ∧ PA F V = p ′ ) € V = F V ( p ′ ) . Thus, s W = w ∧ PA F V = p ( V ) = F V ( p ) , F V ( p ′ ) = s W = w ∧ PA F V = p ′ ( V ) . So, V < Cn( G ) , andfurthermore V is not exogenous (since the value of an exogenous variable is notaffected by interventions on different variables). Thus, V ∈ En( G ) \ Cn( G ) . En( G ) \ Cn( G ) ⊆ En( F ) \ Cn( F ) : For any V ∈ En( G ) \ Cn( G ) , there are distinct p , p ′ ∈ Ran ( PA G V ) such that G V ( p ) , G V ( p ′ ) . Now, if V < En( F ) \ Cn( F ) , then T | = ξ σ ( V ) . Let v = s ( V ) and Z = W V \ PA G V . Since { ( s , G ) } | = V = v and V < PA G V ,for any z ∈ Ran( Z ) , we have that { ( s , G ) } | = ( Z = z ∧ PA G V = p ) € V = v , { ( s , G ) } | = ( Z = z ∧ PA G V = p ′ ) € V = v . By the definition of intervention, we must have that v = s Z = z ∧ PA G V = p ( V ) = G V ( p ) , G V ( p ′ ) = s Z = z ∧ PA G V = p ′ ( V ) = v , which is impossible. Hence, V ∈ En( F ) \ Cn( F ) . F V ∼ G V for any V ∈ En( F ) \ Cn( F ) : For any x ∈ Ran( PA F V ∩ PA G V ) , y ∈ Ran( PA F V \ PA G V ) and z ∈ Ran( PA G V \ PA F V ) , since T | = η σ ( V ) and V < PA G V , for any w ∈ Ran( W ) with w ↾ ( PA G V \ PA F V ) = z , we have that { ( s , G ) } | = ( W = w ∧ PA F V = xy ) € V = F V ( xy ) . Then F V ( xy ) = s W = w ∧ PA F V = xy ( V ) = G V ( s W = w ∧ PA F V = xy ( PA G V )) = G V ( xz ) , as required. ⇐ = : Suppose that G ∼ F for all ( s , G ) ∈ T . Since the formula Φ F is flat, itsuffices to show that { ( s , G ) } | = η σ ( V ) for all V ∈ En( F ) , and { ( s , G ) } | = ξ σ ( V ) forall V ∈ (Dom \ En( F )) ∪ Cn( F ) .For the former, take any w ∈ Ran( W ) and p ∈ Ran( PA F V ) , and let Z = z abbreviate W = w ∧ PA F V = p . We show that { ( s Z = z , G Z = z ) } | = V = F V ( p ) . Since G ∼ F , by Lemma 3.2 we have that G Z = z ∼ F Z = z . Thus, s Z = z ( V ) = ( G Z = z ) V ( s Z = z ( PA G Z = z V )) = ( F Z = z ) V ( s Z = z ( PA F Z = z V )) ( since G X = x ∼ F X = x ) = F V ( s Z = z ( PA F V )) ( since V < Z ) = F V ( p ) . For the latter, take any v ∈ Ran( V ) and w ∈ Ran( W V ) . Assume that { ( s , G ) } | = V = v , i.e., s ( V ) = v . Since V < En( F ) \ Cn( F ) and F ∼ G , we know that V < En( G ) or V ∈ Cn( G ) . In both cases we have that { ( s , G ) } | = W V = w € V = v .(ii). Let T be a nonempty causal team. Consider its associated generalizedcausal team T g . The claim then follows from Lemma 2.11 and item (i). ✷ Corollary 3.5
For any generalized causal team T over some signature σ , T | = \\ / F ∈ F σ Φ F ⇐⇒ T is uniform . The intuituionistic disjunction \\ / was shown to have the disjunction prop-erty , i.e., | = ϕ \\ / ψ implies | = ϕ or | = ψ , in propositional inquisitive logic ([4]) andpropositional dependence logic ([13]). It follows immediately from Theorem 3.4that the disjunction property of \\ / fails in the context of causal teams, because | = c \\ / F ∈ F σ Φ F , whereas = c Φ F for any F ∈ F σ . Nevertheless, the intuitionisticdisjunction does admit the disjunction property over generalized causal teams. Theorem 3.6 (Disjunction property)
Let ∆ be a set of CO [ σ ] -formulas,and ϕ, ψ be arbitrary formulas over σ . If ∆ | = g ϕ \\ / ψ , then ∆ | = g ϕ or ∆ | = g ψ .In particular, if | = g ϕ \\ / ψ , then | = g ϕ or | = g ψ . Proof.
Suppose ∆ = g ϕ and ∆ = g ψ . Then there are two generalized causalteams T , T such that T | = ∆ , T | = ∆ , T = ϕ and T = ψ . Let T : = T ∪ T . Byflatness of ∆ , we have that T | = ∆ . On the other hand, by downwards closure,we have that T = ϕ and T = ψ , and thus T = ϕ \\ / ψ . ✷ CO In this section, we characterize the expressive power of CO over causal teamsand present a system of natural deduction for CO that is sound and completeover both causal teams and generalized causal teams. arbero, Yang 11 In this subsection, we show that CO -formulas capture the flat class of causalteams (up to ≈ -equivalence). Our result is analogous to known characteriza-tions of flat languages in propositional team semantics ([14]), with a twist,given by the fact that only the unions of similar causal teams are reasonablydefined. We define such unions as follows. Definition 4.1
Let S = ( S − , F ) , T = ( T − , G ) be two causal teams over the samesignature σ with S ∼ T . The union of S and T is defined as the causal team S ∪ T = ( S − ∪ T − , H ) over σ , where • En( H ) = (En( F ) \ Cn( F )) ∩ (En( G ) \ Cn( G )) , • and for each V ∈ En( H ) , PA H V = PA F V ∩ PA G V , and H V ( p ) = F V ( px ) for any p ∈ PA F V ∩ PA G V and x ∈ PA F V \ PA G V . Clearly,
H ∼ F ∼ G and thus S ∪ T ∼ S ∼ T .A formula ϕ over σ determines a class K ϕ of causal teams defined as K ϕ = { T | T | = ϕ } . We say that a formula ϕ defines a class K of causal teams if K = K ϕ . Definition 4.2
We say that a class K of causal teams over σ is • causally downward closed if T ∈ K and S ⊆ T imply S ∈ K ; • closed under causal unions if, whenever T , T ∈ K and T ∪ T is defined, T ∪ T ∈ K ; • flat if ( T − , F ) ∈ K iff ( { s } , F ) ∈ K for all s ∈ T − ; • closed under equivalence if T ∈ K and T ≈ T ′ imply T ′ ∈ K . It is easy to verify that K is flat iff K is causally downward closed and closedunder causal unions. Any nonempty downward closed class K of causal teamsover σ contains all causal teams over σ with empty team component. The class K ϕ is always nonempty as the teams with empty team component are alwaysin K ϕ (by Theorem 2.10). By Theorems 2.10 and 3.3, if α is a CO -formula,then K α is flat and closed under equivalence. The main result of this section isthe following characterization theorem which gives also the converse direction. Theorem 4.3
Let K be a nonempty (finite) class of causal teams over somesignature σ . Then K is definable by a CO [ σ ] -formula if and only if K is flatand closed under equivalence. In order to prove the above theorem, we introduce a CO -formula Θ T , inspiredby a similar one in [13], that defines the property “having as team componenta subset of T − ”. For each causal team T over σ = (Dom , Ran) , define Θ T : = _ s ∈ T − ^ V ∈ Dom V = s ( V ) . Lemma 4.4 S | = Θ T iff S − ⊆ T − , for any causal teams S , T over σ . Proof. “ = ⇒ ”: Suppose S | = Θ T and S = ( S − , F ) . For any s ∈ S − , by downward closure, we have that ( { s } , F ) | = Θ T , which means that for some t ∈ T − , ( { s } , F ) | = V = t ( V ) for all V ∈ Dom . Since { s } and { t } have the same signature, this impliesthat s = t , thereby s ∈ T − .“ ⇐ = ”: Suppose S − ⊆ T − . Observe that S | = Θ S and Θ T = Θ S ∨ Θ T \ S . Thus,we conclude S | = Θ T by the empty team property. ✷ Lemma 4.5
Let S = ( S − , G ) and T = ( T − , F ) be causal teams over σ with S − , T − , ∅ . Then S | = Θ T ∧ Φ F ⇐⇒ S ≈ R ⊆ T for some R over σ. Proof.
By Lemma 4.4 and Theorem 3.4, we have that S | = Θ T ∧ Φ F iff S − ⊆ T − and G ∼ F . It then suffices to show that the latter is equivalent to S ≈ R ⊆ T for some R . The right to left direction is clear; conversely, if S − ⊆ T − and G ∼ F , then we can take R = ( S − , F ) . ✷ Consider the quotient set F σ / ≈ . For each equivalence class [ F ] ∈ F σ / ≈ choosea unique representative F . Denote by F σ the set of all such representatives. Proof of Theorem 4.3.
It suffices to prove the direction “ ⇐ = ”. For each F ∈ F σ , let K F : = { ( T − , G ) ∈ K | G ∼ F } . Clearly K = S F ∈ F σ K F . Let T F = S K F , which is well-defined as in Definition 4.1. Since K is closed undercausal unions, T F ∈ K . We may assume w.l.o.g. that T F = ( T −F , F ) . Let ϕ = _ F ∈ F σ ( Θ T F ∧ Φ F ) . It suffices to show that K ϕ = K . For any S = ( S − , G ) ∈ K , there exists F ∈ F σ such that S ∈ K F . Let R = ( S − , F ) . Clearly, S ≈ R ⊆ S K F = T F ,which by Lemma 4.5 implies that S | = Θ T F ∧ Φ F . Hence, S | = ϕ , namely S ∈ K ϕ .Conversely, suppose S = ( S − , G ) ∈ K ϕ , i.e., S | = ϕ . Then for every F ∈ F σ ,there is S F ⊆ S such that S = S F ∈ F σ S F and S F | = Θ T F ∧ Φ F . Thus, by Lemma4.5, we obtain that S F ≈ R F ⊆ T F for some R F . In particular, we have that S F = ( S −F , G ) ∼ ( T −F , F ) = T F , which gives G ∼ F . But since no two distinctelements in F σ are ∼ -similar to each other, and S F ∼ S for each F ∈ F σ , thiscan only happen if S −F = ∅ for all F ∈ F σ except one. Denote this uniqueelement of F σ by H . Now, S = S H ≈ R H ⊆ T F ∈ K . Hence we conclude that S ∈ K , as K is causally closed downward and closed under equivalence. (cid:3) The logic CO [ σ ] over (recursive) causal teams was axiomatized in [2] by meansof a sound and complete Hilbert style deduction system. In this section, wepresent an equivalent system of natural deduction and show it to be sound andcomplete also over (recursive) generalized causal teams. Definition 4.6
The system of natural deduction for CO [ σ ] consists of the fol-lowing rules: • (Parameterized) rules for value range assumptions: ValDef W x ∈ Ran( X ) X = x X = x ValUnq X , x ′ arbero, Yang 13 • Rules for ∧ , ∨ , ¬ : ϕ ψ ∧ I ϕ ∧ ψ ϕ ∧ ψ ∧ E ϕ ϕ ∧ ψ ∧ E ψϕ ∨ I ϕ ∨ ψ ϕ ∨ I ψ ∨ ϕ ϕ ∨ ψ [ ϕ ] ... α [ ψ ] ... α ∨ E α [ α ] ... ⊥ ¬ I ¬ α α ¬ α ¬ E ϕ [ ¬ α ] ... ⊥ RAA α • Rules for € : € Eff ( X = x ∧ Y = y ) € Y = y X = x € W = w X = x € γ € Cmp (1) ( X = x ∧ W = w ) € γ X = x € ⊥ € ⊥ E ϕ ⊥ € E (2) ( Y = y ∧ X = x ∧ X = x ′ ) € ϕ X = x ∧ X = x ∧ Y = y € ϕ € Ctr X = x ∧ Y = y € ϕ X = x € ϕ [ ϕ ] ... ψ € Sub X = x € ψ X = x ∧ Y = y € ϕ € Wk X = x ∧ X = x ∧ Y = y € ϕ X = x € ϕ X = x € ψ € ∧ I X = x € ϕ ∧ ψ X = x € ϕ ∨ ψ € ∨ Dst ( X = x € ϕ ) ∨ ( X = x € ψ ) X = x € ( Y = y € ϕ ) € Extr (3) ( X ′ = x ′ ∧ Y = y ) € ϕ ( X = x ∧ Y = y ) € ϕ € Exp (4) X = x € ( Y = y € ϕ ) ¬ ( X = x € α ) ¬ € E X = x € ¬ α X { X . . . . . . X k − { X k Recur (5) ¬ ( X k { X ) (1) γ is € -free. (2) x , x ′ . (3) X = x is consistent, X ′ = X \ Y , x ′ = x \ y . (4) X ∩ Y = ∅ .(5) X i , X j ( i , j ), and X { Y (meaning “ X causally affects Y ”) is defined as: X { Y : = _ n Z = z € (cid:0) ( X = x € Y = y ) ∧ ( X = x ′ € Y = y ′ ) (cid:1) | Z ⊆ Dom \ { X , Y } , z ∈ Ran( Z ) , x , x ′ ∈ Ran( X ) , y , y ′ ∈ Ran( Y ) , x , x ′ , y , y ′ o . Note that the above system is parametrized with the signature σ , and therules with double horizontal lines are invertible. We write Γ ⊢ σ ϕ (or simply Γ ⊢ ϕ when σ is clear from the context) if the formula ϕ can be derived from Γ by applying the rules in the above system.It is easy to verify that all rules in our system are sound for recursive(generalized) causal teams. The axioms and rules in the Hilbert system of [2]are either included or derivable in our natural deduction system, as shown inthe next proposition Proposition 4.7
The following are derivable in the system for CO [ σ ] : (i) α, ¬ α ∨ ϕ ⊢ ϕ (weak modus ponens)(ii) X = x € Y = y ⊢ X = x € Y , y ′ (Uniqueness)(iii) X = x € ϕ ∧ ψ ⊢ X = x € ϕ (Extraction)(iv) ¬ ( X = x € α ) ⊣⊢ X = x € ¬ α (v) W y ∈ Ran( Y ) ( X = x € Y = y ) (Definiteness) Proof.
Item (i) follows from ¬ E and ∨ E . Items (ii),(iii) follow from ValUnq , ∧ E and € Sub . For item (iv), the left to right direction follows from ¬ € E .For the other direction, we first derive by applying € ∧ I , and € ⊥ E that X = x € ¬ α, X = x € α ⊢ X = x € ¬ α ∧ α ⊢ X = x € ⊥ ⊢ ⊥ Then, by ¬ I we conclude that X = x € ¬ α ⊢ ¬ ( X = x € α ) .For item (v), we first derive by € Eff that ⊢ X = x € X = x , where X = x isan arbitrary equation from X = x . By ValDef we also have that ⊢ W y ∈ Ran( Y ) Y = y .Thus, we conclude by applying € Sub that ⊢ X = x € W y ∈ Ran( Y ) Y = y , whichthen implies that ⊢ W y ∈ Ran( Y ) ( X = x € Y = y ) by € ∨ Dst . ✷ Theorem 4.8 (Completeness)
Let ∆ ∪{ α } be a set of CO [ σ ] -formulas. Then ∆ ⊢ α ⇐⇒ ∆ | = c / g α. Proof.
Since our system derives all axioms and rules of the Hilbert system of[2], the completeness of our system over causal teams follows from that of [2].The completeness of the system over generalized causal teams follows from thefact that ∆ | = c α iff ∆ | = g α , given by Corollary 2.12. ✷ CO CO \\ / and COD
In this section, we characterize the expressive power of CO \\ / and COD overcausal teams. We show that both logics characterize all nonempty causallydownward closed team properties up to causal equivalence, and the two log-ics are thus expressively equivalent. An analogous result can be obtained forgeneralized causal teams, but we omit it due to space limitations.
Theorem 5.1
Let K be a nonempty (finite) class of causal teams over somesignature σ . Then the following are equivalent:(i) K is causally downward closed and closed under equivalence.(ii) K is definable by a CO \\ / [ σ ] -formula.(iii) K is definable by a COD [ σ ] -formula. By Theorems 2.10 and 3.3, for every CO \\ / [ σ ] - or COD [ σ ] -formula ϕ , the set K ϕ is nonempty, causally downward closed and closed under causal equivalence.Thus items (ii) and (iii) of the above theorem imply item (i). Since dependenceatoms = ( X ; Y ) are definable in CO \\ / [ σ ] (see (1)), item (iii) implies item (ii). Itthen suffices to show that item (i) implies item (iii). In this proof, we makeessential use of a formula Ξ T that resembles, in the causal setting, a similarformula introduced in [13] in the pure team setting. arbero, Yang 15 Given any causal team T = ( T − , G ) over σ , let T = ( A σ \ T − , G ) and G ∈ F σ be such that [ G ] = [ G ] . If T − , ∅ and | T − | = k + , define a COD [ σ ] -formula Ξ T : = ( χ k ∨ Θ T ) ∨ _ F ∈ F σ \{G } Φ F , where the formula χ k is defined inductively as χ = ⊥ , χ = ^ V ∈ Dom = ( V ) , and χ k = χ ∨ · · · |{z} k times ∨ χ ( k > . Lemma 5.2
Let S , T be two causal teams over some signature σ with T − , ∅ .Then, S | = Ξ T ⇐⇒ for all R : T ≈ R implies R * S . Proof.
First, observe that the formula χ k characterize the cardinality of causalteams S , in the sense that S | = χ k iff | S − | ≤ k . (2)Indeed, clearly, S | = χ iff S − = ∅ , S | = χ iff | S − | ≤ , and for k > , S | = χ k iff S = S ∪ · · · ∪ S k with each S i | = χ iff | S − | ≤ k .Now we prove the lemma. Let S = ( S − , H ) . “ = ⇒ ”: Suppose S | = Ξ T . If H 6∼ G , then T = ( T − , G ) ≈ ( T − , G ′ ) = R implies G ′ , H , thereby R * S . Now,suppose H ∼ G ∼ G . If S − = ∅ , then since T − , ∅ , the statement holds.If S − , ∅ , then by Lemma 3.4(ii), we know that no nonempty subteam of S satisfies W F ∈ F σ \{G } Φ F . Thus there exist S , S ⊆ S such that S − ∪ S − = S − , S | = χ k and S | = Θ T . (3)By (2), the first clause of the above implies that | S − | ≤ k . Since | T − | = k + > k ,this means that T − \ S − , ∅ . By Lemma 4.5, it follows from the second clause of(3) and the fact that S | = Φ G (given again by Lemma 3.4(ii)) that S ≈ R ⊆ T for some R . Thus, T − ∩ S − = ∅ . Altogether, we conclude that T − * S − . Thus,for any R such that R ≈ T , we must have that R − = T − * S − , thereby R * S .“ ⇐ = ”: Suppose T ≈ R implies R S for all R . If H 6∼ G ∼ G , then byLemma 3.4(ii) we have that S | = W F ∈ F σ \{G } Φ F , thereby S | = Ξ T , as required.Now, suppose H ∼ G . The assumption then implies that T − * S − . Let S = ( S − ∩ T − , H ) and S = ( S − \ T − , H ) . Clearly, S − = S − ∪ S − , and it suffices to showthat (3) holds. By definition we have that S − ⊆ ( T ) − , which implies the secondclause of (3) by Lemma 4.4. To prove the first clause of (3), by (2) it sufficesto verify that | S − | ≤ k . Indeed, since T − * S − , we have that T − ) S − ∩ T − = S − .Hence, | S − | < | T − | = k + , namely, | S − | ≤ k . ✷ Now we are in a position to give the proof of our main theorem of thesection.
Proof of Theorem 5.1.
We prove that item (i) implies item (iii). Let K be a nonempty finite class of causal teams as described in item (i). Since K is nonempty and causally downward closed, all causal teams over σ withempty team component belong to K . Thus, every causal team T ∈ C σ \ K hasa nonempty team component, where C σ denotes the (finite) set of all causal teams over σ . Now, define ϕ = ^ T ∈ C σ \K Ξ T . We show that K = K ϕ .For any S < K , i.e., S ∈ C σ \ K , since S ⊆ S and S − , ∅ , by Lemma 5.2we have that S = Ξ S . Thus S = ϕ , i.e., S < K ϕ . Conversely, suppose S ∈ K .Take any T ∈ C σ \ K . If T ≈ R ⊆ S for some R , then since K is closed underequivalence and causally closed downward, we must conclude that T ∈ K , whichis a contradiction. Thus, by Lemma 5.2, S | = Ξ T . Hence S | = ϕ , i.e., S ∈ K ϕ . (cid:3) CO \\ / over generalized causal teams In this section, we introduce a sound and complete system of natural deductionfor CO \\ / [ σ ] , which extends of the system for CO [ σ ] , and can also be seen as avariant of the systems for propositional dependence logics introduced in [13]. Definition 5.3
The system of natural deduction for CO \\ / [ σ ] consists of allrules of the system of CO [ σ ] (see Definition 4.6) together with the followingrules, where note that in the rules ∨ E , ¬ I , ¬ E , RAA and ¬ € I from Definition4.6 the formula α ranges over CO [ σ ] -formulas only: • Additional rules for ∨ : ϕ ∨ ψ ∨ Com ψ ∨ ϕ ( ϕ ∨ ψ ) ∨ χ ∨ Ass ϕ ∨ ( ψ ∨ χ ) ϕ ∨ ψ [ ϕ ]... χ ∨ Sub χ ∨ ψ • Rules for \\ / : ϕ \\ / I ϕ \\ / ψ ϕ \\ / I ψ \\ / ϕ ϕ \\ / ψ [ ϕ ]... χ [ ψ ]... χ \\ / E χϕ ∨ ( ψ \\ / χ ) ∨ \\ / Dst ( ϕ ∨ ψ ) \\ / ( ϕ ∨ χ ) X = x € ψ \\ / χ € \\ / Dst ( X = x € ψ ) \\ / ( X = x € χ ) The rules in our system are clearly sound. We now proceed to prove thecompleteness theorem. An important lemma for the theorem states that every CO \\ / [ σ ] -formula ϕ is provably equivalent to the \\ / -disjunction of a (finite) setof CO [ σ ] -formulas. Formulas of this type are called resolutions of ϕ in [4]. Definition 5.4
Let ϕ be a CO \\ / [ σ ] -formula. Define the set R ( ϕ ) of its resolu-tions inductively as follows: • R ( X = x ) = { X = x } , • R ( ¬ α ) = {¬ α } , • R ( ψ ∧ χ ) = { α ∧ β | α ∈ R ( ψ ) , β ∈ R ( χ ) } , • R ( ψ ∨ χ ) = { α ∨ β | α ∈ R ( ψ ) , β ∈ R ( χ ) } , • R ( ψ \\ / χ ) = R ( ψ ) ∪ R ( χ ) , arbero, Yang 17 • R ( X = x € ϕ ) = { X = x € α | α ∈ R ( ϕ ) } . The set R ( ϕ ) is clearly a finite set of CO [ σ ] -formulas. Lemma 5.5
For any formula ϕ ∈ CO \\ / [ σ ] , we have that ϕ ⊣⊢ \\ / R ( ϕ ) . Proof.
We prove the lemma by induction on ϕ . If ϕ is X = x or ¬ α for some CO [ σ ] -formula α , then R ( ϕ ) = { ϕ } , and ϕ ⊣⊢ \\ / R ( ϕ ) holds trivially.Now, suppose ψ ⊣⊢ \\ / R ( ψ ) and χ ⊣⊢ \\ / R ( χ ) . If ϕ = ψ ∧ χ , observing that θ ∧ ( θ \\ / θ ) ⊣⊢ ( θ ∧ θ ) \\ / ( θ ∧ θ ) (by \\ / E , \\ / I , ∧ I , ∧ E ), we derive by \\ / I , \\ / E that ψ ∧ χ ⊣⊢ (cid:0) \\ / R ( ψ ) (cid:1) ∧ (cid:0) \\ / R ( χ ) (cid:1) ⊣⊢ \\ / { α ∧ β | α ∈ R ( ψ ) , β ∈ R ( χ ) } ⊣⊢ \\ / R ( ψ ∧ χ ) . If ϕ = ψ ∨ χ , we have analogous derivations using the fact that θ ∨ ( θ \\ / θ ) ⊣⊢ ( θ ∨ θ ) \\ / ( θ ∨ θ ) (by ∨ \\ / Dst , \\ / I , \\ / E and ∨ Sub ) and \\ / I , \\ / E .If ϕ = ψ \\ / χ , then by applying \\ / I and \\ / E , we have that ψ \\ / χ ⊣⊢ (cid:0) \\ / R ( ψ ) (cid:1) \\ / (cid:0) \\ / R ( χ ) (cid:1) ⊣⊢ \\ / (cid:0) R ( ψ ) ∪ R ( χ ) (cid:1) ⊣⊢ \\ / R ( ψ \\ / χ ) . If ϕ = X = x € ψ , then X = x € ψ ⊣⊢ X = x € \\ / R ( ψ ) ( € Sub ) ⊣⊢ \\ / { X = x € α | α ∈ R ( ψ ) } ( € \\ / Dst , and \\ / I , € Sub , \\ / E ) ⊣⊢ \\ / R ( X = x € ψ ) . ( € \\ / Dst , and \\ / I , € Sub , \\ / E ) ✷ Theorem 5.6 (Completeness)
Let Γ ∪{ ψ } be a set of CO \\ / [ σ ] -formulas. Then Γ ⊢ ψ ⇐⇒ Γ | = g ψ. Proof.
We prove the “ ⇐ = ” direction. Observe that there are only finitelymany classes of causal teams of signature σ . Thus, any set of CO \\ / [ σ ] -formulasis equivalent to a single CO \\ / [ σ ] -formula, and it then suffices to prove the state-ment for Γ = { ϕ } .Now suppose ϕ | = ψ . Then by Lemma 5.5 and soundness we have that \\ / R ( ϕ ) | = \\ / R ( ψ ) . Thus, for every γ ∈ R ( ϕ ) , γ | = g \\ / R ( ψ ) , which further implies,by Lemma 3.6, that there is an α γ ∈ R ( ψ ) such that γ | = α γ . Since γ, α γ are CO [ σ ] -formulas, and the system for CO \\ / [ σ ] extends that for CO [ σ ] , we obtainby the completeness theorem of CO [ σ ] (Theorem 4.8) that γ ⊢ α γ . Applying \\ / I and Lemma 5.5, we obtain γ ⊢ \\ / R ( ψ ) ⊢ ψ for each γ ∈ R ( ψ ) . Thus, by Lemma5.5 and repeated applications of \\ / E , we conclude that ϕ ⊢ \\ / R ( ϕ ) ⊢ ψ. ✷ CO \\ / over causal teams The method for the completeness proof of the previous subsection cannot beused for causal team semantics, as it makes essential use of the disjunctionproperty of \\ / , which fails over causal teams. However, since causal teamscan be regarded as a special case of generalized causal teams, all the rules inthe system for CO \\ / over generalized causal teams are also sound over causalteams. We can then axiomatize CO \\ / over causal teams by extending the systemof CO \\ / for generalized causal teams with an axiom characterizing the propertyof being uniform, i.e. “indistinguishable” from a causal team. Definition 5.7
The system for CO \\ / [ σ ] over causal teams consists of all rules of CO \\ / [ σ ] over generalized causal teams (Def. 5.3) plus the following axiom: Unf \\ / F ∈ F σ Φ F By Theorem 3.4(ii), the axiom
Unf is clearly sound over causal teams.
Lemma 5.8
For any set Γ ∪{ ψ } of CO \\ / [ σ ] -formulas, Γ | = c ψ iff Γ , \\ / F ∈ F σ Φ F | = g ψ. Proof. ⇐ = : Suppose T | = c Γ for some causal team T . Consider the generalizedcausal team T g generated by T . By Lemma 2.11, T g | = g Γ . Since T g is uniform,Corollary 3.5 gives that T g | = g \\ / F ∈ F σ Φ F . Then, by assumption, we obtain that T g | = g ψ , which, by Lemma 2.11 again, implies that T | = c ψ . = ⇒ : Suppose T | = g Γ and T | = g \\ / F ∈ F σ Φ F for some generalized causal team T . By Corollary 3.5 we know that T is uniform. Pick ( t , F ) ∈ T . Consider thegeneralized causal team S = { ( s , F ) | s ∈ T − } . Observe that T ≈ S . Thus, byTheorem 3.3, we have that S | = g Γ , which further implies, by Lemma 2.11(ii),that S c | = c Γ . Hence, by the assumption we conclude that S c | = c ψ . Finally, byapplying Lemma 2.11(ii) and Theorem 3.3 again, we obtain T | = g ψ . ✷ Theorem 5.9 (Completeness)
Let Γ ∪{ ψ } be a set of CO \\ / [ σ ] -formulas. Then Γ | = c ψ ⇐⇒ Γ ⊢ c ψ. Proof.
Suppose Γ | = c ψ . By Lemma 5.8, we have that Γ , \\ / F ∈ F σ Φ F | = g ψ , whichimplies that Γ , \\ / F ∈ F σ Φ F ⊢ ψ , by the completeness theorem (5.6) of the systemfor CO \\ / [ σ ] over generalized causal teams. Thus, Γ ⊢ ψ by axiom Unf . ✷ COD
We briefly sketch the analogous axiomatization results for the language
COD over both semantics.Over generalized causal teams, the system for
COD [ σ ] consists of all therules of the system for CO [ σ ] (Definition 4.6) together with ∨ Com , ∨ Ass , ∨ Sub (the “additional rules for ∨ ” from Definition 5.3) and the new rules for depen-dence atoms defined below: X = x DepI = ( X ) [ = ( X ) ] . . . [ = ( X n ) ] = ( Y ) DepI = ( X , . . . , X n ; Y ) ϕ ∀ x ∈ Ran ( X ) [ ϕ [ X = x / = ( X )] ]... ψ Dep E ( ∗ ) ψ = ( X , . . . , X n ; Y ) = ( X ) . . . = ( X n ) DepE = ( Y ) ( ∗ ) ϕ [ X = x / = ( X )] stands for the formula obtained by replacing a specific occurrence of = ( X ) in ϕ with X = x .arbero, Yang 19 These rules for dependence atoms generalize the corresponding rules in the pureteam setting as introduced in [13]. The completeness theorem of the system canbe proved by generalizing the corresponding arguments in [13]. Analogouslyto the case for CO \\ / , in this proof we use the fact that every formula ϕ is(semantically) equivalent to a formula \\ / i ∈ I α i in disjunctive normal form, whereeach α i is a CO [ σ ] -formula obtained from ϕ by replacing every dependence atom = ( X ; Y ) by a formula W x ∈ Ran( X ) ( X = x ∧ Y = y ) with y ranging over all of Ran( Y ) .The disjunctive formula \\ / i ∈ I α i is not in the language of COD , but we can provein the system of
COD (by applying the additional rules in the table above) that α i ⊢ ϕ ( i ∈ I ), and that Γ , α i ⊢ ψ for all i ∈ I = ⇒ Γ , ϕ ⊢ ψ. These mean in effect that “ ϕ ⊣⊢ \\ / i ∈ I α i ”. The completeness theorem for COD is then proved using essentially the same strategy as that for CO \\ / (Theorem5.6).Over causal teams, using the same method as in the previous section, thecomplete system for COD [ σ ] can be defined as an extension of the above gener-alized causal team system with two additional axioms and NoMix , definedas follows: (1) ^ V ∈ Dom (cid:16) β En ( V ) ⊃ ( ^ w ∈ W V W V = w € = ( V )) (cid:17) NoMix (2) ^ V ∈ Dom ^ { Ξ { a , b }∗ | ( a , b ) ∈ S em σ , { a } | = β En ( V ) , { b } 6| = β En ( V ) } (1) W V = Dom \{ V } , and β En ( V ) : = _ X ∈ W V β DC ( X , V ) , where each β DC ( X , V ) is the CO [ σ ] -formulafrom [2] expressing the property “ X is a direct cause of V ”: β DC ( X , V ) : = _ (cid:8) ( Z = z ∧ X = x ) € V = v , ( Z = z ∧ X = x ′ ) € V = v ′ | x , x ′ ∈ Ran( X ) , v , v ′ ∈ Ran( V ) , Z = Dom \ { X , V } , z ∈ Ran( Z ) , x , x ′ , v , v ′ (cid:9) . (2) Ξ { a , b }∗ is defined otherwise the same as Ξ { a , b } except that χ is redefined as χ : = ^ V ∈ Dom (cid:0) = ( V ) ∧ ^ w ∈ Ran ( W V ) ( W V = w € = ( V )) (cid:1) . The axiom states that the endogenous variables are governed by a uniquefunction; the axiom
NoMix guarantees that all members of the generalizedcausal team agree on what is the set of endogenous variables. Together, thesetwo additional axioms characterize the uniformity of the generalized causalteam in question (or they are equivalent to the formula
Unf in CO \\ / [ σ ] ), thusallow for a completeness proof along the lines of Section 5.3. We have answered the main questions concerning the expressive power and theexistence of deduction calculi for the languages that were proposed in [1] and[2], and which involve both (interventionist) counterfactuals and (contingent) dependencies. In the process, we have introduced a generalized causal teamsemantics, for which we have also provided natural deduction calculi. Wepoint out that our calculi are sound only for recursive systems, i.e., whenthe causal graph is acyclic. The general case (and special cases such as the“Lewisian” systems considered in [15]) will require a separate study. We pointout, however, that each of our deduction systems can be adapted to the caseof unique-solution (possibly generalized) causal teams by replacing the
Recur rule with an inference rule that expresses the
Reversibility axiom from [5].Our work shows that many methodologies developed in the literature onteam semantics can be adapted to the generalized semantics and, to a lesserextent, to causal team semantics. On the other hand, a number of peculiaritiesemerged that set apart these semantic frameworks from the usual team seman-tics: for example, the failure of the disjunction property over causal teams. Webelieve the present work may provide guidelines for the investigation of furthernotions of dependence and causation in causal team semantics and its variants.
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