Registered: Oct 2003
Posts: 1

Wolfram says in Chapter 12 of the NKS Book "Logic and Language"

The Laws of Form presented by George Spencer Brown in 1969 introduce a compact symbolic notation for Nand with any number of arguments and in effect try to develop a way of discussing Nand and reasoning directly in terms of it. (The axioms normally used are essentially the Sheffer ones from page 773.)

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Registered: Sep 2003
Posts: 17

Yes the Brown form can be implemented via Nor. It is mentioned on the poster I left in Boston... The version for the Journal of Complex Systems includes a reference to the NOR approach but I prefer Discrete Delta to streamline binary arithmetic evaluation of algebraic terms. (Nor[0] returns !0 in Mathematica and Nor[1] returns !1, DiscreteDelta[True] simply returns unevaluated of course).

Concerning Nand versus Nor:

In the NKS book there are references about more efficient implementations of Nand on some machines this might justify a preference for Nand as the form can be interpreted as false and the empty state as true. This reversal is enough to turn Nand into a model for the form. Spencer-Brown presented both options (on page 113 in the appendix 2 īthe calculus interpreted for logicī of "The Laws of Form", Bookmasters Ashland Ohio for Cognizer Portland Oregon 1994 ).

A minor problem with the suggested identification of Nor and the form is that Nor should and does evaluate without input which happens to agree with the behavior of the form yet this might seem a bit counterintuitive for an implementation as the common language grammatical usage for this name suggests a choice. The same objection holds against Nand only among specialists as Nand is not a common word.

Concerning the formulas:

The interpretation of the Peirce notation may be true or not. The four interpreted form formulas given are also true in terms of my interpretation if square brackets indicate the form. Let me give some detail for implication.

Arithmetic evaluation of all possible values for A and B in statement four:

If A and B are empty unmarked space then only the form over A remains giving true.

If A is empty and B is the form then the mark over A consolidates with the form called B (axiom 1) thus giving true.

If A is the form and B is empty then the mark over A is cancelled(axiom 2) and no form remains thus false.

If A and B are the form then the mark over A is cancelled (axiom 2) but B remains a form thus true.

This matches the standard definition of implication in Mathematica: The logical implication P => Q is false only when P is true and Q is false.

Concerning the overall evaluation:

I think that the oscillatory alias imaginary values of variables (introduced in chapter 11 of Laws of Form for equations of the form:"[A]=A") offer even more potential than a powerful simplified notation. An answer might also mention that Spencer-Brown prefers to look at his work in terms of an algebra based on a primary arithmetics versus basing mathematics on logics.

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Registered: Feb 2004
Posts: 551

I'm not used to working with these webmail
systems, so forgive the formatting until
I can get used to how they work.

I have done some work on graphical transformation
systems that stem from C.S. Peirce's Alpha Graphs
and Spencer Brown's Laws of Form.

This work is documented at my Inquiry List:

http://stderr.org/pipermail/inquiry/

Here is an introductory discussion:

http://stderr.org/pipermail/inquiry...thread.html#126

Jon Awbrey

Last edited by Jon Awbrey on 02-26-2004 at 06:24 PM

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