All Liar, No Paradox • Discussion 2

Re: Laws of FormJames BoweryJohn Mingers

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).



cc: CyberneticsOntolog ForumStructural ModelingSystems Science
cc: FB | Logical GraphsLaws of FormPeirce List

This entry was posted in Bertrand Russell, C.S. Peirce, Epimenides, Laws of Form, Liar Paradox, Logic, Logical Graphs, Mathematics, Paradox, Pragmatics, Rhetoric, Semantics, Semiositis, Semiotics, Sign Relations, Spencer Brown, Syntax, Visualization and tagged , , , , , , , , , , , , , , , , , . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.