The “Extended XOR Operator” as a Consistent interpretation of G. Spencer-Brown’s distinction

(Links updated: 13/10/2016) Richard Shoup has commented about Multiple Form Logic, and about my use of the XOR operator (as an interpretation of «Distinction»), as follows:

-XOR is at minimum a binary operator. NOR (if you must cast Distinction into a traditional operator) can be unary (= NOT). XOR is not an appropriate interpretation of the Distinction in logic, and leads to various mathematical difficulties, as Bricken has commented before. («Laws of Form forum», June 11 2007)

Well, forgetting -for a moment- the objection that «XOR is not a unary operator» (which is not a serious drawback, as I will try to show later), the remaining part of this comment seems to be the exact opposite of what is actually the case, i.e.

• XOR is an appropriate interpretation of the Distinction in logic,
• XOR leads to freedom from various mathematical difficulties (see ).

Furthermore,

William Bricken’s Logic (i.e. his «Boundary Algebra») is provably- a special instance of Multiple Form Logic (Theorem T12 in MF Logic).

UPDATE (20-8-2007): In response to certain points raised in the ‘Laws of Form forum‘ there is now a new web-page with complete algebraic proofs of interesting theorems in Multiple Form Logic, most notably theorem «ART-1«, expressing the XOR relation in M.F. logic:

• So, is XOR an appropriate interpretation of Distinction? And if so, in what way is it a «better» interpretation than NOR?

Well, first of all, we can regard the XOR operator as a programmable gate: If, in addition, A spreads out into multiple OR-ed parallel inputs, what we get is a programmable NOR/OR – gate. This programmable gate behaves as a NOR-gate when the control input X is set to 1, while it behaves like an OR- gate when the control input X is set to 0. It is not difficult to see now, that the control input X is effectively a programmable dictinction (in Brownian terminology) over the space of all the distinctions fed into A (as OR-ed parallel inputs): Evidently, what we get here is a programmable Brownian space of OR-ed distinctions, in which X determines if the whole space is to be placed inside another distinction, or if it is to be kept unchanged.

I.e. the input X determines if a distinction is to be drawn or not drawn (on the whole space). The new possibilities for Self-modifying Programmable Logic Circuits are evident, as are the connections with classic temporal circuits, with XOR feedback.

Now, what are the semantics of the XOR operator, in an expression like «A xor B»?

«Either A, or B, but not both».

Philosophically (or intuitively) a particular (local) observer can NOT be located both inside and outside a boundary. However, if one insists to be located both inside and outside a given boundary, then this boundary becomes effectively non-existent. (Since the very meaning of the term «boundary» entails restriction to be on one side).

In fact, the XOR operator as a depiction of «containment» or of «distinction», exhibits all the expected properties of such an interpretation, in all (four) cases:

1) If A and B are both empty spaces, there is no distinction between them (0).

2) If A expresses a distinction 1 and B is empty, the result is a distinction (1).

3) If A is empty and B expresses a distinction 1, the result is a distinction (1).

4) If both A and B are (the same) distinction 1, they cancel each other (0).

In addition, the XOR operator can also be regarded as unary, but if this is done then there are two possibilities: The second input (of XOR) is either a marked or an unmarked state. If it is a marked state, then the operator reduces to NOT (or NOR, if you prefer, with only one input). If it is an unmarked state, then the operator reduces to «self-sameness» or no distinction, leaving everything as it was before. I.e. the second input’s state is no more and no less than the distinction itself.

Now, there is only one «problem» with the XOR operator: It is symmetric, i.e.

A xor B = B xor A.

This commutative property (in Brownian terminology) means that we can turn the inside into the outside, and vice versa, i.e. However, this is partly illusory. For example, if there exist distinctions A, B in the spaces inside and outside a distinction X, then it is not possible to swap them: I.e. it is certainly possible to swap A and X in an expression “A xor X”, but it is not possible to swap A and B in an expression like “(A xor X) or B”. Nevertheless, if we insist, we may regard “containment” as an “extended XOR” (say ‘#’) where symmetry does not hold, i.e. “X # Y” is not “Y # X”. However, if we do this, then expressions like “A # B # A” are not reducible (by “cancellation”, Axiom 2 of Multiple Form Logic) to B. Numerous interesting new possibilities arise. E.g. We may define relations such as:

A # B # A = B # A

(and so on…)

In Conclusion:

“Laws of Form” is based on two implicitly defined relations between forms, which correspond to two “arithmetic axioms”. These two relations are then regarded (by George Spencer Brown and many of his disciples) as mathematically equivalent to the operators OR and NOR. If we adopt this interpretation, then using NOR (instead of XOR) has serious drawbacks, e.g. when modeling large composite expressions containing XOR. The number of (OR- and NOR-) terms required to model composite XOR-relations in this system rises steeply and prohibitively (as shown in  below). If we insist that XOR is “inappropriate”, just because George Spencer-Brown and William Bricken say so(!) then we achieve nothing but unnecessary mathematical (and expressive) difficulties.The myth that XOR “causes mathematical difficulties” is evidently the exact opposite of the truth.

Further references:

1) http://omadeon.atwebpages.com/logic/multiforms_1.html#or_and_xor (All we need is OR and XOR)

2) http://omadeon.atwebpages.com/logic/mf_efficiency.html (especially the complexity table in the end)

3) http://omadeon.atwebpages.com/logic/mflogic_simplified.html (The advantages of MF logic)

4) https://omadeon.wordpress.com/2007/05/31/equalitylogics (on “Equality Logic”).

5)

6) http://omadeon.atwebpages.com/logic/ (new home page of the Multiple Form Logic site, updated Oct. 2016)

7)http://omadeon.atwebpages.com/logic/proofs_2007.html (theorem «ART-1», expressing the XOR relation in terms of OR and NOR operations, and other theorem proofs)