More loops – but not necessarily strange

A postulate, what it allows within a given universe, and what it forbids there, are three logically equivalent statements. Knowing one of them, it is possible to deduce the other two as if they were connected in a loop. For example, the material implication ‘if avocado then green’ would allow green avocados, green things that are not avocados, or things that are neither avocados nor green; but it excludes avocados that are not green.

Stephens (1994a) defined such a loop. It is like the ‘trinity’ mentioned by Spencer-Brown (1972), where the inside (what is included), the outside (what is excluded), and the boundary between them (the postulate) are all one form. He compared a sanity loop with an insanity loop.

Sanity loop

What Stephens called the sanity loop is the default condition in the real universe we inhabit, a universe defined by the law that a thing cannot both exist and not exist simultaneously.  In logic, ‘simultaneously’ does not imply time, but means ‘in the same universe of discourse’. Logical propositions may refer to time as subject matter, but are themselves outside of time, and each instant may be considered as a complete and self-consistent universe.

This is not a strange loop because, being logically equivalent, the three do not represent different levels of a hierarchy.

X = X                              a thing is itself                                                                          1
X + (1 – X) = 1               it either exists or it doesn’t                                                        2
X (1 – X) = 0                   it cannot both exist and not exist simultaneously                    3

These three equations can be proved equivalent by using Boole’s algebra:

X = X                                                                                                                               1
X – X = X – X                           subtract X from each side
X – X = 0                                 simplify RHS
X – X + 1 = 1                           add 1 to each side
X + (1 – X) = 1                         reorder                                                                            2
X² + X (1 – X) = X                     multiply by X
X + X (1 – X) = X                       X² = X
X (1 – X) = 0                              subtract X from each side                                             3
X – X² = 0                                 multiply out LHS
X – X = 0                                   X² = X
X = X                                        add X to each side                                                         1

The same proof can be written in a reverse sequence, just like going around a loop in a reverse direction:

X = X                                                                                                                               1
X – X = X – X                           subtract X from each side
X – X = 0                                 simplify RHS
X – X² = 0                                X = X²
X (1 – X) = 0                            factorise LHS                                                                   3
X + X (1 – X) = X                      add X to each side
X² + X (1 – X) = X                    X = X²
X + (1 – X) = 1                        divide by X                                                                       2
X – X + 1 = 1                           reorder
X – X = 0                                 subtract 1 from each side
X = X                                       add X to each side                                                           1

In the calculus of indications (Spencer-Brown, 1969), equations 1 and 2 can both be written as:

[X] X = []

and equation 3 as:

[[X][[X]]] = [[X] X] =

Since HTML cannot display Spencer-Brown’s mark as nested single brackets, I have used the convention of replacing it with pairs of square brackets.

Insanity loop

On the other hand, the insanity loop (Stephens, 1994b) is the condition at an IP (impossibility point), which by definition is not part of the real universe.

The three equations making up the insanity loop are the negations of equations 1 to 3:

X = (1 – X)                  a thing is not itself                                                                     4
X + (1 – X) = 0            it can neither exist nor not exist                                                5
X (1 – X) = 1               it both exists and does not exist simultaneously                       6

From 5 we see that 1 = 0 in this imaginary universe of discourse which contains the single class X , 1 – X.  If drawn as a Venn diagram it would be one shaded space without any division. Everything is nothing; and nothing is everything.

Substituting 4 into 4,
X = (1 – X)
(1 – X) = (1 – (1 – X))                                                                                  simplifies to (1 – X) = X
(1 – (1 – X)) = (1 – (1 – (1 – X)))                                                                  simplifies to X = (1 – X)
(1 – (1 – (1 – X))) = (1 – (1 – (1 – (1 – X))))                                                simplifies to (1 – X) = X
(1 – (1 – (1 – (1 – X)))) = (1 – (1 – (1 – (1 – (1 – X)))))                               simplifies to X = (1 – X)
This is a re-entering expression that oscillates between the two equivalents however far it is extended.

Substituting 4 into 5,
X + (1 – X) = 0
(1 – X) + (1 – (1 – X)) = 0
(1 – (1 – X)) + (1 – (1 – (1 – X))) = 0
(1 – (1 – (1 – X))) + (1 – (1 – (1 – (1 – X)))) = 0
This is a re-entering expression that always simplifies to 0 (since 1 = 0) however far it is extended.

Substituting 4 into 6,
X (1 – X) = 1
(1 – X) (1 – (1 – X)) = 1
(1 – (1 – X)) (1 – (1 – (1 – X))) = 1
(1 – (1 – (1 – X))) (1 – (1 – (1 – (1 – X)))) = 1
This is a re-entering expression that always simplifies to 1 however far it is extended.

If the same substitutions are done with equations 1 to 3 the results are trivial since we are just substituting X with X. The re-entry of equation 4 at an odd depth introduces an imaginary Boolean value into the insanity loop. And because the insanity loop is based on an imaginary Boolean, it contradicts the rules of Boole’s algebra which then cannot be used to prove the equivalence of its component equations.

However, the RHS of equations 4 to 6 are the negation of the RHS of equations 1 to 3 respectively: 1 – X, 1 – 1 and 1 – 0. And the LHS of equations 4 to 6 are identical to the LHS of equations 1 to 3. Since 1 to 3 have been proved equivalent, 4 to 6 must necessarily be equivalent also.

X and (1 – X) are coextensive (equation 4), therefore their conjunction X (1 – X) must equal 1 (equation 5). Equation 6 is the dual of equation 5, just as 2 and 3 are duals.

The relationship between the two loops can be demonstrated by writing them in the calculus of indications. We see that this simply means placing an additional mark over each expression in the sanity loop to get the insanity loop. Equations 4 and 5 both become:

[[X] X] =

and equation 6 (sharing the form of equation 3) is:

[[[X][[ X]]]] = [X][[ X]] = []

As with the NAND, this is another instance of a high-level abstraction having the same mathematical form as a subjective human experience. The concept of impossibility matches the phenomenon of insanity because an insane individual believes the impossible statement that X = (1 – X).

Impossibility and insanity seem to have a mathematical form that is as rigorous as the form of possibility and sanity. Such a form cannot be understood by considering it only as a negative, a departure from reality. It is also a return to a condition more ‘primitive’ or fundamental than the default state of this universe represented by the sanity loop. This may be the real reason insanity is so unconfrontable to those following the universe’s postulates. It’s a look behind the scenes, a pointer to how we create this universe; a condition where one instant is not separated from another and nonexistence equals existence as envisioned by Nāgārjuna and the other Mahāyana masters.

-oOo-

References

Boole, G. (1854) An Investigation of the Laws of Thought. (Walton & Maberley: London).

Spencer-Brown, G. (1969) The Laws of Form. (Allen & Unwin: London).

Spencer-Brown, G. writing as James Keys (1972) Only Two Can Play This Game. (Julian Press: New York).

Stephens, D.H. (1992) The Unstacking Procedure. Tape of 3 November 1992.

Stephens, D.H. (1994a) The Loop. Tape of 6 April 1994

Stephens, D.H. (1994b) Insanity. Tape of 30 June 1994.

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