I’m studying the textbook Causal Inference: What If by Miguel A. Hernán, James M. Robins. On page 4, I came across a passage that seems nonsensical. The authors claim that, for each individual, the counterfactual outcome corresponding to the treatment they actually received is equal to the observed outcome:
To make our causal intuition amenable to mathematical and statistical analysis we will introduce some notation. Consider a dichotomous treatment variable $A$ (1: treated, 0: untreated) and a dichotomous outcome variable $Y$ (1: death, 0: survival). In this book we refer to variables such as $A$ and $Y$ that may have different values for different individuals as random variables. Let $Y^{a=1}$ (read $Y$ under treatment $a = 1$) be the outcome variable that would have been observed under the treatment value $a = 1$, and $Y^{a=0}$ (read $Y$ under treatment $a = 0$) the outcome variable that would have been observed under the treatment value $a = 0$. $Y^{a=1}$ and $Y^{a=0}$ are also random variables. Zeus has $Y^{a=1} = 1$ and $Y^{a=0} = 0$ because he died when treated but would have survived if untreated, while Hera has $Y^{a=1} = 0$ and $Y^{a=0} = 0$ because she survived when treated and would also have survived if untreated.
We can now provide a formal definition of a causal effect for an individual: The treatment $A$ has a causal effect on an individual’s outcome $Y$ if $Y^{a=1} \neq Y^{a=0}$ for the individual. Thus, the treatment has a causal effect on Zeus’s outcome because $Y^{a=1} = 1 \neq 0 = Y^{a=0}$, but not on Hera’s outcome because $Y^{a=1} = 0 = Y^{a=0}$. The variables $Y^{a=1}$ and $Y^{a=0}$ are referred to as potential outcomes or as counterfactual outcomes. Some authors prefer the term “potential outcomes” to emphasize that, depending on the treatment that is received, either of these two outcomes can be potentially observed. Other authors prefer the term “counterfactual outcomes” to emphasize that these outcomes represent situations that may not actually occur (that is, counter-to-the-fact situations).
For each individual, one of the counterfactual outcomes—the one that corresponds to the treatment value that the individual did receive—is actually factual. For example, because Zeus was actually treated $(A = 1)$, his counterfactual outcome under treatment $Y^{a=1} = 1$ is equal to his observed (actual) outcome $Y = 1$. That is, an individual with observed treatment $A$ equal to $a$, has observed outcome $Y$ equal to his counterfactual outcome $Y^a$. This equality can be succinctly expressed as $Y = Y^a$ where $Y^a$ denotes the counterfactual $Y^a$ evaluated at the value $a$ corresponding to the individual’s observed treatment $A$. The equality $Y = Y^a$ is referred to as consistency.
This seems nonsensical to me because, by definition, a counterfactual outcome is what would have happened if the opposite treatment had been received. Are the authors correct here (am I misunderstanding)? Or should I be looking for another textbook?
