Hypothetical Interventions and Belief Changes

Holger Andreas; Lorenzo Casini

doi:10.1007/s10699-019-09605-8

Foundations of Science

December 2019, Volume 24, Issue 4, pp 681–704 | Cite as

Hypothetical Interventions and Belief Changes

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Holger Andreas
Lorenzo Casini

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First Online: 11 May 2019

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Abstract

According to Woodward’s (Making things happen. A theory of causal explanation, Oxford University Press, New York, 2003) influential account of explanation, explanations have a counterfactual structure, and explanatory counterfactuals are analysed in terms of causal relations and interventions. In this paper, we provide a formal semantics of explanatory counterfactuals based on a Ramsey Test semantics of conditionals. Like Woodward’s account, our account is guided by causal considerations. Unlike Woodward’s account, it makes no reference to causal graphs and it also covers cases of explanation where interventions are impossible.

Keywords

Belief revision theory Causal models Causation Conditionals Explanation Interventionist account

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Notes

Acknowledgements

Lorenzo Casini acknowledges the generous support of the Swiss National Science Foundation, Grants Nos. CRSII 1 147685/1 and 100012E 160866/1.

Appendix: Revising a Prioritised Belief Base

To define full meet revisions for a prioritised belief base, we merely need to extend the definition of a remainder set $H \bot \alpha$ to the prioritised case. Let

$$\begin{aligned} {\mathbf {H}}=\langle H_1, \ldots , H_n\rangle \end{aligned}$$

be a prioritised belief base. That is, $H_1, \ldots , H_n$ are sets of formulas that represent explicit beliefs, and the indices represent an epistemic ranking of the beliefs. $H_1$ is the set of the most firmly established beliefs, the beliefs in $H_2$ have secondary priority, etc. We define the remainder set of such a prioritised belief base as follows:

Definition 4

(${\mathbf {H}} \bot \alpha$)

Let ${\mathbf {H}}=\langle H_1, \ldots , H_n\rangle$ be a prioritised belief base and $\alpha$ a formula. ${\mathbf {H}}'\in {\mathbf {H}} \bot \alpha$ iff (1) ${\mathbf {H}}'=\langle H'_1, \ldots , H_n'\rangle$ and (2) for all $i \;(1 \le i \le n)$, $H_1' \cup \ldots \cup H_i' \in (H_1 \cup \ldots \cup H_i ) \bot \alpha$.

Any member ${\mathbf {H}}'$ of ${\mathbf {H}} \bot \alpha$ is thus itself a prioritised belief base preserving the epistemic prioritisation of ${\mathbf {H}}$, i. e. any ${\mathbf {H}}'$ has the same structure of epistemic levels (1 to n) than ${\mathbf {H}}$ . The union of the epistemic levels of such a base ${\mathbf {H}}'$ – the set $H_1' \cup \ldots \cup H_n'$—does not entail $\alpha$ by classical logic.¹³

The next step is to define the contraction of ${\mathbf {H}}$ by $\alpha$ in terms of the componentwise intersection of the members of the prioritised remainder set:

$$\begin{aligned} {\mathbf {H}}\div \alpha =_{df} {\bigcap }\, {\mathbf {H}} \bot \alpha .\quad (\hbox {Def }\div ) \end{aligned}$$

The componentwise intersection of ${\mathbf {H}} \bot \alpha$ is defined in the obvious way: the set of the i-th level of ${\bigcap }\, {\mathbf {H}} \bot \alpha$$(1\le i \le n)$ is given by the intersection of the i-th levels of the members of ${\mathbf {H}} \bot \alpha$. Note that we use $\bigcap$ in a context-dependent manner; it means different things when applied to a set and to a sequence of sets.

It remains to define expansions for prioritised belief bases ${\mathbf {H}}=\langle H_1, \ldots , H_n\rangle$. Two types of definitions suggest themselves:

$$\begin{aligned}&{\mathbf {H}}+ \alpha =_{df}\langle \{\alpha \}, H_1, \ldots , H_n\rangle \quad \hbox {(Def +)}\\&{\mathbf {H}}+_i \alpha =_{df}\langle H_1,\ldots , H_i \cup \{\alpha \}, \ldots , H_n\rangle .\quad (\hbox {Def }+_i) \end{aligned}$$

For non-iterated revisions (where further revisions that follow upon a primary revision are not considered), the two definitions are equivalent as regards the resulting belief set. That is, $K(H+\alpha ) =K(H+_i \alpha )$ for all levels i. As we do not use iterated revisions, we chose the first definition, for the sake of simplicity.

Once expansions and contractions have been explained for prioritised belief bases, we can define revisions using the Levi identity in a straightforward manner:

$$\begin{aligned} {\mathbf {H}} *\alpha =_{df} ({\mathbf {H}} \div \lnot \alpha ) + \alpha = \left( {\bigcap } {\mathbf {H}} \bot \lnot \alpha \right) + \alpha .\quad (\hbox {Def }*) \end{aligned}$$

The last step is to define the belief set K of a prioritised belief base ${\mathbf {H}}$ by the inferential closure of some suitable logic. We simply take the consequence operation of classical logic:

$$\begin{aligned} K({\mathbf {H}})=_{df}Cn(H_1 \cup \ldots \cup H_n).\quad (K({\mathbf {H}})) \end{aligned}$$

Once $K({\mathbf {H}})$, ${\mathbf {H}}+\alpha$, and ${\mathbf {H}}*\alpha$ are defined, we can readily define the revision and the contraction of the belief set $K({\mathbf {H}})$ of the belief base ${\mathbf {H}}$:

$$\begin{aligned} K({\mathbf {H}})+\alpha =_{df} K({\mathbf {H}} +\alpha ) \\ K({\mathbf {H}})*\alpha =_{df} K({\mathbf {H}} *\alpha ). \end{aligned}$$

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Holger Andreas
- 1
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Lorenzo Casini
- 2

1.Department of Economics, Philosophy, and Political ScienceUniversity of British Columbia (Okanagan Campus)KelownaCanada
2.Département de PhilosophieUniversité de GenèveGenevaSwitzerland

Hypothetical Interventions and Belief Changes

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