  I generalize acyclic deterministic structural equation models to the
nondeterministic case and argue that it offers an improved semantics for
counterfactuals. The standard, deterministic, semantics developed by Halpern
(and based on the initial proposal of Galles & Pearl) assumes that for each
assignment of values to parent variables there is a unique assignment to their
child variable, and it assumes that the actual world (an assignment of values
to all variables of a model) specifies a unique counterfactual world for each
intervention. Both assumptions are unrealistic, and therefore I drop both of
them in my proposal. I do so by allowing multi-valued functions in the
structural equations. In addition, I adjust the semantics so that the solutions
to the equations that obtained in the actual world are preserved in any
counterfactual world. I motivate the resulting logic by comparing it to the
standard one by Halpern and to more recent proposals that are closer to mine.
Finally, I extend these models to the probabilistic case and show that they
open up the way to identifying counterfactuals even in Causal Bayesian
Networks.
