Regularity Theories Reassessed

Abstract

For a long time, regularity accounts of causation have virtually vanished from the scene. Problems encountered within other theoretical frameworks have recently induced authors working on causation, laws of nature, or methodologies of causal reasoning – as e.g. May (Kausales Schliessen. Eine Untersuchung über kausale Erklärungen und Theorienbildung. Ph.D. thesis, Universität Hamburg, Hamburg, 1999), Ragin (Fuzzy-set social science. Chicago: University of Chicago Press, 2000), Graßhoff and May (Causal regularities. In W. Spohn, M. Ledwig, & M. Esfeld (Eds.), Current issues in causation (pp. 85–114). Paderborn: Mentis, 2001), Swartz (The concept of physical law (2nd ed.). http://www.sfu.ca/philosophy/physical-law/, 2003), Halpin (Erkenntnis, 58, 137–168, 2003) – to direct their attention back to regularity theoretic analyses. In light of the latest proposals of regularity theories, the paper at hand therefore reassesses the criticism raised against regularity accounts since the INUS theory of causation of Mackie (The cement of the universe. A study of causation. Oxford: Clarendon Press, 1974). It is shown that most of these objections target strikingly over-simplified regularity theoretic sketches. By outlining ways to refute these objections it is argued that the prevalent conviction as to the overall failure of regularity theories has been hasty.

Appendices

Appendix

First-Order Formalization

This appendix introduces the first-order formalizations of the core notions involved in a regularity theory of type (V). As indicated in “Hume’s Legacy,” the reason for a first-order representation of causal regularities lies in the relational constraints that distinguish causally interesting regularities from causally meaningless regularities as “Whenever there is a table, there is a table”. Causes and effects are instantiated by spatiotemporally proximate events, causal regularities are instantiated by different events, and factors constituting a complex cause are instantiated coincidently. These three relational constraints characterizing causal regularities must be accounted for by first-order means. I take them in turn.

Spatiotemporal Proximity of Causes and Effects

In accordance with Broad (1930), φ shall be defined to be a sufficient condition of ψ iff Eq. 18 holds and a necessary condition of ψ iff Eq. 19 holds:

$$ \forall \mu(\phi \mu\rightarrow\psi \mu)$$

(18)

$$ \forall \mu(\psi \mu\rightarrow\phi \mu)$$

(19)

μ is to be read as a metavariable running over variables and φμ and ψμ stand for any well-formed formulae with at least one free occurrence of μ.

Instances of causes and instances of their effects do not occur anywhere and anytime, but close by, i.e. within a certain spatiotemporal frame or within the same situation. The interpretation of the relation “...occurs in the same spatiotemporal frame as ...” cannot be fixed to a certain spatiotemporal interval independently of a causal process under investigation. However, in order to make sure that no causal element is covertly introduced into the notion of spatiotemporal proximity the latter shall be determined to be a symmetric relation, which – as mentioned above – causal relevance clearly is not. Certain instances of causes and effects can be said to be properly related only if they are in direct spatiotemporal contact, while in other cases instances of causes may well occur far away from the instances of their effects. The theory of special relativity only provides an upper bound for this interval: Instances of causes and effects must occur within each other’s light cones. Notwithstanding this lacking specificity, given a concrete causal process it is normally uncontroversial which events can be said to be properly related in order to be amenable to a causal interpretation.³¹ Moreover, whenever spatiotemporal proximity is unsuitably interpreted for a given causal process, no dependencies appear in corresponding empirical data.³²

If minimally sufficient and minimally necessary conditions should, at least in principle, be open for causal interpretations, the syntax of acceptable substitutions in Eqs. 18 and 19 must be restricted such that an instance x of a cause factor and an instance y of an effect factor are required to occur in the same spatiotemporal frame. In order to formally represent this spatiotemporal association, we introduce the symmetric relation Rxy. This induces a first approximation to a first-order representation of a causal regularity. Whoever claims that A is a (sufficient) cause of B, claims that for all events x of type A there is an event y of type B such that x and y occur in the same spatiotemporal frame. This is captured by Eq. 20.

$$\label{caus1} \forall x(Ax\rightarrow \exists y(By\wedge \mbox{R}xy))$$

(20)

Causal vs. Semantic Regularities

Equation 20 not only describes causal regularities, but also what might be referred to as semantic regularities or regularities of set inclusion as “Whenever there is a soccer game, there is a sport event”. A semantic regularity is given in case of two predicates one of which has an extension that is included in the extension of the other. Accordingly, by satisfying the first of these predicates an object or event eo ipso satisfies the other predicate. As single objects or events are moreover spatiotemporally proximate to themselves, one and the same object or event satisfies antecedent and consequent of a semantic regularity. The soccer games and the sport events are identical, thus, they certainly are spatiotemporally proximate according to any spatiotemporal interval chosen as interpretation of R. In order to exclude semantic regularities from consideration when it comes to causal analyses, it must be stipulated that instances of causes differ from the instances of their effects. No token event ever causes itself. Hence, Eq. 20 must be specified such that all models of the specified formula feature different events as instances of the factors contained in antecedent and consequent, respectively:

$$ \forall x(Ax\rightarrow \exists y(By\wedge\mbox{R}xy \wedge x\neq y)) $$

(21)

Equation 21 states that for all events x of type A there is a different event y of type B in the same spatiotemporal frame as x. In order to conveniently abbreviate our notation, we introduce “↦”:

$$\label{mapsto1} Z_1\mapsto Z_2 =_{df} \forall x(Z_1x\rightarrow \exists y(Z_2y\wedge x\neq y\wedge \mbox{R}xy)) $$

(22)

Coincidence

The factors of a complex cause only become causally effective if coincidently instantiated. Hence, the factors contained in a minimally sufficient condition must be required to be coincidently instantiated. In order to symbolically represent coincident instantiations, I introduce the n-ary relation K with n being the number of conjuncts in a minimally sufficient condition apart from K itself. K subsists among the instances of the factors in a minimally sufficient condition iff these factors are coincidently instantiated. K can be seen on a par with any ordinary non-redundant factor within a minimally sufficient condition. If K does not hold among the instances of a conjunction of factors, the instantiations of these factors are not sufficient for the effect to occur. Thus, in contrast to R, K may remain uninterpreted.

Only the subset of minimally sufficient conditions that include K can possibly be causally interpreted. A possibly causally interpretable minimally sufficient condition is of the form:

$$ \forall x_1\forall x_2\ldots\forall x_n(A_1x_1\!\wedge\! A_2x_2\!\wedge\!\ldots\!\wedge\! A_nx_n\!\wedge\! Kx_1x_2\ldots x_n \rightarrow \exists y(By\!\wedge\! x_1\!\neq\! y\!\wedge\! x_2\!\neq\! y\!\wedge\!\ldots\!\wedge\! x_n\!\neq\! y\!\wedge\! \mbox{R}x_1x_2\ldots x_ny))$$

(23)

As Eq. 23 demonstrates, the factors of a possibly causally interpretable minimally sufficient condition are not required to be instantiated by the same event. As long as they occur coincidently, factors in a complex cause may well be instantiated by different events. The complexity of Eq. 23, which describes a minimally sufficient condition by explicitly mentioning three factors only, apparently calls for further abbreviations. To this end we adopt the convention that a conjunction of factors whose instances are related in terms of K shall simply be concatenated without conjunctor and without explicit mention of K. A universally or existentially quantified conjunction of factors A ₁ x ₁ ∧ A ₂ x ₂ ∧ ... ∧ A _n x _n  ∧ Kx ₁ x ₂...x _n , accordingly, is represented by A ₁ A ₂...A _n .

$$ A_1 A_2\ldots A_n=_{df}A_1x_1\wedge A_2x_2\wedge\ldots\wedge A_nx_n\wedge Kx_1x_2\ldots x_n$$

(24)

The quantifiers that bind the variables on the right-hand side of Eq. 24 can be left unspecified, because this abbreviated notation is only used in connection with “↦”, whose antecedent is determined to be universally quantified and whose consequent is existentially quantified by definition. Therefore, the context in which expressions of type A ₁ A ₂...A _n appear always clarifies the nature of the quantifiers involved. Given this notational convention, Eq. 23 can be transparently stated thus:

$$ A_1 A_2\ldots A_n \mapsto B$$

Minimal Theory

Before minimal theories can be formally represented, the abbreviated notation initiated in the previous sections needs to be extended. Minimally sufficient and minimally necessary conditions have been defined as (finite) open conjunctions and disjunctions. In order to account for that openness, two types of variables running over factors shall be introduced. For conjunctions of unknown or unspecified factors within sufficient conditions we shall implement the variables X ₁, X ₂, etc. Thus, these X-variables are to be read as running over factors that are not explicitly integrated within sufficient conditions.

$$ X_i=_{df}Z_1x_1\wedge Z_2x_2\wedge Z_3x_3\wedge\ldots\wedge Z_nx_n, n\geq 1, $$

with i = 1,2,3,..., the variables Z ₁...Z _n running over factors, and quantification depending on whether X _i appears right or left of “↦”.

Building on the definition of X _i , we define the variables Y _A , Y _B , etc. to represent disjunctions whose disjuncts are not explicitly integrated within necessary conditions. The subscripts in case of the Y-variables correspond to the factors whose necessary condition a respective Y-variable complements.

$$ Y_x=_{df} X_1 \vee X_2 \vee X_3\vee\ldots\vee X_n,n\geq 1, $$

with x = A,B,C,... and quantification equally depending on whether Y _x appears right or left of “↦”.

With these notational means at hand, a factor A being part of a minimally sufficient condition of B, such that this minimally sufficient condition, in turn, is part of a minimally necessary condition of B can be expressed thus:

$$ (AX_1\vee Y_{B}\mapsto B)\wedge(B\mapsto AX_1\vee Y_{B}) $$

(25)

Equation 25 states that whenever A is instantiated coincidently with other factors X ₁ or one of the disjuncts in the domain of Y _B is instantiated, the factor B is instantiated in the same spatiotemporal frame by an event that differs from the instances of AX ₁ ∨ Y _B ; and whenever B is instantiated, there is either a coincident instantiation of AX ₁ or one of the disjuncts in the domain of Y _B is instantiated in the same spatiotemporal frame, such that the instances of B and of AX ₁ ∨ Y _B differ. Equation 25 can thus be seen as an abbreviation of expressions of the form of Eq. 26, where the incompleteness is indicated by dots instead of X- and Y-variables and k and i stand for arbitrary natural numbers.³³

$$ \forall x_1\ldots \forall x_i((A_1x_1\wedge A_2x_2\wedge\ldots\wedge A_ix_i\wedge Kx_1x_2\ldots x_i) \rightarrow \exists y(By\wedge x_1\neq y\wedge\ldots\wedge x_i\neq y \wedge \mbox{\footnotesize R}x_1\ldots x_iy)) \wedge \ldots\wedge \forall x_1\ldots \forall x_k((Z_1x_1\wedge Z_2x_2\wedge\ldots\wedge Z_kx_k\wedge Kx_1x_2\ldots x_k) \rightarrow \exists y(By\wedge x_1\neq y\wedge\ldots\wedge x_k\neq y \wedge \mbox{\footnotesize R}x_1\ldots x_ky)) \wedge \forall y(By\rightarrow(\exists x_1\ldots \exists x_i(A_1x_1\wedge A_2x_2\wedge\ldots\wedge A_ix_i\wedge Kx_1x_2\ldots x_i\wedge x_1 \neq y\wedge\ldots\wedge x_i\neq y \wedge \mbox{\footnotesize R}x_1\ldots x_iy)\vee \ldots\vee \exists x_1\ldots \exists x_k(Z_1x_1\wedge Z_2x_2 \wedge\ldots\wedge Z_kx_k\wedge Kx_1x_2\ldots x_k\wedge x_1\neq y\wedge\ldots\wedge x_k\neq y \wedge \mbox{\footnotesize R}x_1\ldots x_ky))) $$

(26)

Equation 26 clearly illustrates that minimal theories are highly intricate and non-transparent first-order expressions. In order to abbreviate our notation further, we introduce “$\Rightarrow$”:

$$Z_1\vee Z_2\Rightarrow Z_3 =_{df} (Z_1\vee Z_2\mapsto Z_3)\wedge(Z_3\mapsto Z_1\vee Z_2) $$

This allows for transparently expressing Eqs. 25 and 26 as follows:

$$AX_1\vee Y_{B}\Rightarrow B$$