## All Liar, No Paradox • Discussion 2

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 $\textsc{true}$ and $\textsc{false}$ 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 $\textsc{true}$ and $\textsc{false}$ 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 $X.$

This leads us to speak of logical functions $f : X \to \mathbb{B},$ where $\mathbb{B}$ is the so-called boolean domain $\mathbb{B} = \{ \textsc{false}, \textsc{true} \}.$  But we are really using $\mathbb{B}$ 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 $\mathbb{B} = \{ 0, 1 \}$ or $\mathbb{B} = \{ \textsc{white}, \textsc{blue} \},$ my favorite colors for painting the areas of a venn diagram.

A function like $f : X \to \mathbb{B} = \{ 0, 1 \}$ is called a “characteristic function” in set theory since it characterizes a subset $S$ of $X$ where the value of $f$ is $1.$  But I like the language they use in statistics, where $f : X \to \mathbb{B}$ is called an “indicator function” since it indicates a subset of $X$ where $f$ evaluates to $1.$

The indicator function of a subset $S$ of $X$ is notated as $f_S : X \to \mathbb{B}$ and defined as the function $f_S : X \to \mathbb{B}$ where $f_S (x) = 1$ if and only if $x \in S.$  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 $S$ of $X$ indicated by the function $f_S : X \to \mathbb{B}.$  Other names for it are the “fiber” or “pre-image“ of $1.$  It is computed by way of the “inverse function” $f_S^{-1}$ in the rather ugly but pre-eminently useful way as $S = f_S^{-1}(1).$

Regards,

Jon

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