Causal entailment

Schulz (2011) makes use of a notion of situation that slightly deviates from the one that we introduced in Ch. 3.1. Instead of treating situations as concrete objects in the ontology, as Kratzer (1989) does, Schulz uses a slightly more abstract notion in the tradition of Veltman (2005). Using a three-valued logic that adds u (for “undefined”) to the truth values, assignments of truth values to the complete set of atomic propositions fall into two sets: Those that assignuto some proposition are situations, while those that do not assignu to any proposition are possible worlds. In this sense, a situation is an incomplete description of a possible world. Some ways of talking about situations that are possible in the Kratzer semantics are not possible under this definition – e.g., an individual cannot directly be part of a Schulz situation – so that these two notions of situation are not straightforwardly interchangeable. In the remainder of this section, I will be using Schulz’s notion of situation unless indicated otherwise.

The second technical term that Schulz introduces is that of adynamics, used to represent causal dependencies. A dynamics is a partition on the set of propositional letters that divides them into those that are causally in-dependent, the setB(for “background variable”), and those that causally depend on some other variable, the set I (for “inner variable”). Addi-tionally, a dynamics provides a (two-valued) function that determines the truth values of inner variables based on the truth values of the background variables they depend on⁴.

(225) A B

C

In (225), A and Bare background variables, together determining the value of the inner variableC. The figure shows only the causal dependen-cies, but we need to additionally describe the truth function for C, based onAandB. For example, ifCis 1 if and only ifAandBare both 1, then the following information also needs to be encoded in the causal dynamics:

(226)

A B C

1 1 1

1 0 0

0 1 0

0 0 0

Given these ingredients, Schulz can formulate a notion of causal en-tailment based on the recursive application of an operator τ relative to a dynamicsDto a situations. τtakes a situations, which – by definition – has some undefined values. It then attempts to fill in these values based on the

4 I will limit myself to a somewhat informal introduction of the concepts necessary for our purposes here. For the formal definition of the underlying framework and the necessary constraints on dynamics, e.g. rootedness, see Schulz (2011).

values of other variables, following the causal dependencies specified in D. Once application ofτyields no further change, the resulting situation is returned, and all propositions verified by this resulting situation are taken to be causally implied by the original situations, relative toD.

Applyingτto a situationsand a dynamicsDyields the following: The resulting situation τD(s) agrees with sin all the values of the background variables, as well as in all values that are not undefined, or for which there is no truth function defined. For undefined values inswith a defined truth function,τD(s) obtains the value as defined by the truth function.

Take, for example, our dynamics above, specified in (225) and (226).

Relative to this dynamics, a situation which sets the values of A and B but leaves C undefined will always causally entail either C or ¬C: if A and B are both set to 1, then τ can compute the value of C as 1, and as 0 otherwise. This generalizes to larger dynamics (and situations) as well.

Take the example in (227) and (228) – (229):

(227) A B

C D

E

(228)

A B C

1 1 1

1 0 0

0 1 0

0 0 0

(229)

C D E

1 1 1

1 0 0

0 1 0

0 0 0

Here, the dynamics from (225) is extended: There is another back-ground variable D, and another inner variable E, which in turn depends on both an inner variable (C) and a background variable (D) for its value.

The truth conditions are simple conjunction again: E is 1 if and only ifC andDare both 1, as specified in (229).

A situation where the background variables (A,B,D) are all defined and the inner variables (C,E) undefined, will causally predict values for both inner variables, given the dynamics above. Here,τwill have to be applied in two steps though: First, the value ofCis computed by the truth table in (228). However, the first application ofτstill yields an undefined value for E, because althoughEis undefined in s, there is no defined truth function for it (becauseCis also undefined in S). Applyingτto the result of the first application now allows us to compute the value ofE, since we obtained a defined value forCin the first step and (229) is no longer undefined.

For concreteness, let’s take an initial situationsthat looks as follows:

(230) A=1, B=1, C=u, D=0, E=u

Applyingτto (230) yields a situations⁰ that looks as follows:

(231) A=1, B=1, C=1, D=0, E=u

How do we obtain (231)? First, s⁰ is required to agree with s in all defined values, that is, in the values of A, B and D. Then, we check the undefined values C and E: ForC, we have a defined truth table in (228), which for A = 1,B = 1 assigns C the value 1 in s⁰. However, for E, the truth table in (229) is not defined, becauseC insis still u. Since we have undefined values left, we applyτagain and yield the situations⁰⁰:

(232) A=1, B=1, C=1, D=0, E=0

Again,s⁰⁰agrees withs⁰, as described in (231), in all defined values, i.e.

in the values ofA, B,CandD. Then we check the undefined valueE: We now have a defined truth table in (229), because the values of bothCand Dare defined ins⁰. Together, they setEto 0.

Applyingτagain tos⁰⁰does not yield any changes, so thats⁰⁰is the final result – anything ins⁰⁰is causally entailed bysrelative toD.