Dear James, John, et al.
The questions arising in the present discussion take us back to the question of what we are using logical values like and for, which takes us back to the question of what we are using our logical systems for.
One of the things we use logical values like and for is to mark the sides of a distinction we have drawn, or noticed, or maybe just think we see in a logical universe of discourse or space
This leads us to speak of logical functions where is the so-called boolean domain But we are really using only “up to isomorphism”, as they say in the trade, meaning we are using it as a generic 2-point set and any other 1-bit set will do as well, like or my favorite colors for painting the areas of a venn diagram.
A function like is called a “characteristic function” in set theory since it characterizes a subset of where the value of is But I like the language they use in statistics, where is called an “indicator function” since it indicates a subset of where evaluates to
The indicator function of a subset of is notated as and defined as the function where if and only if I like this because it links up nicely with the sense of indication in the calculus of indications.
The indication in question is the subset of indicated by the function Other names for it are the “fiber” or “pre-image“ of It is computed by way of the “inverse function” in the rather ugly but pre-eminently useful way as