Animated Logical Graphs • 37

Re: Richard J. LiptonLogical Complexity Of Proofs

Another dimension of proof style has to do with how much information is kept or lost as the argument develops.  For the moment let’s focus on classical deductive reasoning at the propositional level.  Then we can distinguish between equational inferences, which keep all the information represented by the input propositions, and implicational inferences, which permit information to be lost as the proof proceeds.

Information-Preserving vs. Information-Reducing Inferences
Implicit in Peirce’s systems of logical graphs is the ability to use equational inferences.  Spencer Brown drew this out and turned it to great advantage in his revival of Peirce’s graphical forms.  As it affects “logical flow” this allows for bi-directional or reversible flows, you might even say a “logical equilibrium” between two states of information.

It is probably obvious when we stop to think about it, but seldom remarked, that all the more familiar inference rules, like modus ponens and resolution or transitivity, entail in general a loss of information as we traverse their arrows or turnstiles.

For example, the usual form of modus ponens takes us from knowing p and p \Rightarrow q to knowing q but in fact we know more, we actually know p \land q.  With that in mind we can formulate two variants of modus ponens, one reducing and one preserving the actual state of information, as shown in the following figure.

1-Way and 2-Way

There’s more discussion of this topic at the following location.

To be continued …

Resources

Applications

cc: CyberneticsOntolog • Peirce (1) (2) (3) (4) (5) (6)Structural ModelingSystems

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.

6 Responses to Animated Logical Graphs • 37

  1. Pingback: Survey of Animated Logical Graphs • 3 | Inquiry Into Inquiry

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  6. Pingback: Sign Relations, Triadic Relations, Relation Theory • Discussion 2 | Inquiry Into Inquiry

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