Counterfactual Spaces

TL;DR

This paper introduces a new mathematical framework called counterfactual spaces that axiomatizes the stochastic nature of counterfactuals, separating them from interventions and broadening their theoretical scope.

Contribution

It develops two related frameworks, counterfactual probability and causal spaces, and combines them into counterfactual causal spaces, offering a novel formal approach to counterfactual reasoning.

Findings

Defines counterfactual probability and causal spaces as product measurable spaces.

Characterizes shared information between worlds via probability measures and causal kernels.

Enables mathematical treatment of a broader spectrum of counterfactuals.

Abstract

We mathematically axiomatise the stochastics of counterfactuals, by introducing two related frameworks, called counterfactual probability spaces and counterfactual causal spaces, which we collectively term counterfactual spaces. They are, respectively, probability and causal spaces whose underlying measurable spaces are products of world-specific measurable spaces. In contrast to more familiar accounts of counterfactuals founded on causal models, we do not view interventions as a necessary component of a theory of counterfactuals. As an alternative to Pearl's celebrated ladder of causation, we view counterfactuals and interventions are orthogonal concepts, respectively mathematised in counterfactual probability spaces and causal spaces. The two concepts are then combined to form counterfactual causal spaces. At the heart of our theory is the notion of shared information between the worlds, encoded completely within the probability measure and causal kernels, and whose extremes are characterised by independence and synchronisation of worlds. Compared to existing frameworks, counterfactual spaces enable the mathematical treatment of a strictly broader spectrum of counterfactuals.