Animated Logical Graphs • 29

Re: Animated Logical Graphs • 21
Re: Ontolog ForumJoseph Simpson

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).


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This entry was posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Constraint Satisfaction Problems, Deduction, Diagrammatic Reasoning, Duality, Equational Inference, Graph Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Painted Cacti, Peirce, Proof Theory, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Theorem Proving, Visualization and tagged , , , , , , , , , , , , , , , , , , , , , , , , , . Bookmark the permalink.

5 Responses to Animated Logical Graphs • 29

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