Counterfactual Spaces
Junhyung Park, Fanny Yang, Thomas Icard
TL;DR
This work introduces a rigorous, measure-theoretic foundation for counterfactual reasoning by defining counterfactual probability spaces and counterfactual causal spaces, both based on product-structured worlds that encode inter-world information via cross-world measures and causal kernels. It treats counterfactuals and interventions as orthogonal concepts and generalizes existing formalisms by enabling arbitrary information sharing between parallel worlds and non-interventional counterfactuals. The framework extends to N-way worlds and provides formal mechanisms for interventions and conditional causal effects, all while avoiding strong assumptions like acyclicity or discreteness. By showing how SCMs and PO frameworks can be embedded within counterfactual spaces, the paper argues for a broader, more flexible foundations for causality and counterfactual reasoning with potential applications to actual causality and AI-driven reasoning.
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.