Animated Logical Graphs • 29

Re: Animated Logical Graphs • 21
Re: Ontolog Forum • Joseph Simpson

JS:: I tend to view equivalence and distinction as relationships as opposed to operations. I do not know if this makes any significant difference in this context.

Dear Joe,

I invoked the general concepts of equivalence and distinction at this point in order to keep the wider backdrop of ideas in mind but since we’ve been focusing on boolean functions to coordinate the semantics of propositional calculi we can get a sense of the links between operations and relations by looking at their relationship in a boolean frame of reference.

Let $\mathbb{B} = \{ 0, 1 \}$ and $k$ a positive integer. Then $\mathbb{B}^k$ is the set of $k$ -tuples of elements of $\mathbb{B}.$

A $k$ -variable boolean function is a mapping $\mathbb{B}^k \to \mathbb{B}.$
A $k$ -place boolean relation is a subset of $\mathbb{B}^k.$

The correspondence between boolean functions and boolean relations may be articulated as follows.

Any $k$ -place relation $L,$ as a subset of $\mathbb{B}^k,$ has a corresponding indicator function (or characteristic function) $f_L : \mathbb{B}^k \to \mathbb{B}$ defined by the rule that $f_L (x) = 1$ if $x$ is in $L$ and $f_L (x) = 0$ if $x$ is not in $L.$
Any $k$ -variable function $f : \mathbb{B}^k \to \mathbb{B}$ is the indicator function of a $k$ -place relation $L_f$ consisting of all the $x$ in $\mathbb{B}^k$ where $f(x) = 1.$ The set $L_f$ is called the fiber of $1$ or the pre-image of $1$ in $\mathbb{B}^k$ and is commonly notated as $f^{-1}(1).$

Resources

Logic Syllabus • Relation Theory
Logical Equality • Logical Inequality
Boolean Function • Boolean-Valued Function

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