Ceteris Paribus and Ceteris Rectis Laws: Content and Causal Role

Abstract

This paper has three goals. The first goal is to work out the difference between literal ceteris paribus (cp) laws in the sense of “all others being equal” and ceteris rectis (cr) “laws” in the sense of “all others being right” (Sects. 2, 4). While cp laws involve a universal quantification, cr generalizations involve an existential quantification over the values of the remainder variables Z. As a result, the two differ crucially in their confirmability and lawlikeness. The second goal is to provide a classification of different kinds of cr generalizations (indefinite, definite and normic), including certain transition cases between cr generalizations and cp laws (Sect. 3). The third goal is to work out what cp laws and all kinds of cr assertions have in common: they figure as an information source for assertions of causal influence between variables (Sect. 5).

Appendix

Proof of the equivalence of “DEP” and “Cr” stated in (17)

DEP ⇒ Cr: If (for some given x, y, z) P(y|x, z) ≠ P(y|z), then by probability theory there must also exist a y*∈Val(Y) such that P(y*|x,z) > P(y*|z). By letting Δx = {x} and Δz = {z} this implies an cr assertion of the form (13)(i) (with empty C).

Cr ⇒ DEP: Assume “Cr”, i.e. for some Δy, Δx and Δz, P(Y ∈ Δy | X ∈ Δx, Z ∈ Δz) > P(Y ∈ Δy|Z ∈ Δz). Y ∈ Δy is equivalent with an exclusive disjunction Y ∈ y₁ $\dot{ \vee }$ ··· $\dot{ \vee }$Y ∈ y_n; and likewise for X ∈ Δx and Z ∈ Δz. By probability theory, P(A₁ $\dot{ \vee }$A₂|B₁ $\dot{ \vee }$B₂) > P(A₁ $\dot{ \vee }$A₂) implies that P(A₁|B₁) > P(A₁) or P(A₂|B₂) > P(A₂) or P(A₁|B₂) > P(A₁) or P(A₂|B₁) > P(A₂). By generalizing this result to n-fold disjunctions one derives from “Cr”: ∃x,y,z: P(Y = y|X = x, Z = z) > P(Y = y|Z = z); i.e. DEP(X,Y|Z).□

Proof of theorem (18)

By (17), the cr-claim in (18) is equivalent with “DEP(X,Y|Z)”. If (V,E,P) satisfies the causal Markov condition (cMC), then (V,E,P) satisfies the following principle (D) of “d-connection”:

(D) For all X,Y∈V and Z ⊆ V − {X,Y}: If DEP(X,Y|Z), then X and Y are connected by some path π such that no intermediate cause Z (Z’ → Z→ Z’’) or common cause Z (Z’ ← Z→ Z’’) on π is in Z, and every common effect Z on π (Z’ → Z ← Z’’) is in Z or has an effect in Z.

This is a consequence of the first part of theorem 1.2.4 in Pearl (2009, 18).

For (18.1): By (D), DEP(X,Y|Z) entails that

(*) there exists a path π = X − C₁ − ··· − C_n − Y (with n ≥ 0) such that no common or intermediate cause C_i on this path π is in Z, and every common effect on π is in Z.

By our assumption,

(**) Z contains all and only the non-effects of X in V − {X,Y}.

We have two cases:

Case (1): If X − Y holds, then X → Y and Y → X are compatible with (*) and (**).

Case (2): X − C₁ − ··· −C_n − Y holds for n ≥ 1.

Step (a): Then for no C_i (1 ≤ i ≤ n), X − ···←C_i −− Y can hold in E, for then C_i would be a non-effect of X and a common or intermediate cause on path π, which by (*) must not be in Z. But this contradicts (**).

Step (b): By step (a), X → C₁ →→ C_n − Y must hold.

Step (c): X →→ C_n ← Y cannot hold, since then C_n would be an effect of X and a common effect on π, which by (*) must be in Z, in contradiction of (**). Therefore in case (2), X →→ Y must hold.

For (18.2): This is an obvious consequence of (18.1), since t(X) < t(Y) and the assumption that causal arrows are forward-directed in time precludes the second possibility X ← Y.

For (18.3): Spirtes et al. (2000, 50) define an intervention variable (“policy variable”) I_X for X as a direct cause of X which not connected with any other variable in V. So if the second possibility of (18.1), X ← Y, would hold, the causal structure would be I_X → X ← Y; but then by (D) and since I_X is not connected with any other variable in V, DEP(I_X,Y) could not hold.□