Charles Sanders Peirce, George Spencer Brown, and Me • 6

Re: Laws Of Form DiscussionPeirce’s LawAMAM

The formal system of logical graphs is defined by a foursome of formal equations, called initials when regarded purely formally, in abstraction from potential interpretations, called axioms when interpreted as logical equivalences.  There are two arithmetic initials and two algebraic initials, as follows:

Arithmetic Initials

Logical Graph Initial I1

Logical Graph Initial I2

Algebraic Initials

Logical Graph Initial J1

Logical Graph Initial J2

Spencer Brown uses a different formal equation for his first algebraic initial — where I use  a \texttt{(} a \texttt{)} = \texttt{(~)}  he uses  \texttt{(} a \texttt{(} a \texttt{))} = ~~.  For the moment, let’s refer to my \mathrm{J_1} as \mathrm{J_{1a}} and his \mathrm{J_1} as \mathrm{J_{1b}} and use that notation to examine the relationship between the two systems.

It is easy to see that the two systems are equivalent, since we have the following proof of \mathrm{J_{1b}} by way of \mathrm{J_{1a}} and \mathrm{I_2}.


 a   a
 o---o
 |
 @

 =======J1a {delete}

 o---o
 |
 @

 =======I2  {cancel}

 @

 =======QED J1b

In choosing between systems I am less concerned with small differences in the lengths of proofs than I am with other factors.  It is difficult for me to remember all the reasons for decisions I made forty or fifty years ago — as a general rule, Peirce’s way of looking at the relation between mathematics and logic has long been a big influence on my thinking and the other main impact is accountable to the nuts and bolts requirements of computational representation.

But looking at the choice with present eyes, I think I continue to prefer the \mathrm{I_1, I_2, J_{1a}, J_2} system over the alternative simply for the fact that it treats two different types of operation separately, namely, changes in graphical structure versus changes in the placement of variables.

This entry was posted in Abstraction, Amphecks, Analogy, Animata, Automated Research Tools, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Cybernetics, Deduction, Diagrammatic Reasoning, Differential Logic, Duality, Form, Graph Theory, Iconicity, Information Theory, Inquiry, Laws of Form, Logic, Logical Graphs, Mathematics, Model Theory, Painted Cacti, Peirce, Peirce's Law, Pragmatic Maxim, Proof Theory, Propositional Calculus, Semiotics, Sign Relational Manifolds, Spencer Brown, Symbolism, Systems Theory, Theorem Proving, Topology, Visualization, Zeroth Order Logic and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , . Bookmark the permalink.

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