Strange loops and alternative denial in logic

Douglas Hofstadter coined the term ‘strange loop’ for a feedback cycle consisting of a series of stages that shift from one level of abstraction to another in an apparent hierarchy, yet paradoxically return to their starting point. He included many intriguing examples from the drawings of Escher and the music of Bach in his first book on this subject; but perhaps the most blatant example of a strange loop is a self-contradicting statement such as “This sentence is false.”

We experience a strange loop subjectively as a mental knot analogous to a Möbius strip. More formally, it’s a mathematical function that re-enters its own space, in the sense of Spencer-Brown (1969) who showed that some functions re-entering at odd (rather than even) depths give rise to imaginary Boolean values that are not stable as either 1 or 0. These imaginary Boolean values are closely related to the self-contradictory logical classes that Stephens (1994) called impossibility points: classes violating the basic rule of logic that a thing must either be or not-be.

An implication or single bonding such as A ⇒ B is not a loop.  A biconditional or double bonding of A ⇔ B is a loop but not necessarily strange; it becomes strange if A and B are at different levels in a conceptual hierarchy.  A series of implications that return to the first term such as A ⇒ B, B ⇒ C and C ⇒ A could also be a strange loop.

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There’s another form of strange loop that can arise when A and B are mutually exclusive and therefore their conjunction A,B is a null class. In informal English, mutual exclusivity means that you can choose either of two alternatives on its own, or you can have neither, but you can’t have them both together.

Such a loop is formed using the logical relation known as alternative denial or NAND, which can be written as ¬ (A ∧ B), or simply as A↑­B using the Sheffer stroke. This is the negation of conjunction, the logical relation that only allows A and B together but not the absence of one or both.  If A and B are in alternative denial, their conjunction A,B is necessarily a null class, but since not-A and not-B are not mutually exclusive their conjunction is not a null class. In fact, ¬ (A ∧ B) says explicitly that all possible classes except A,B are allowed.

Notice that B resembles the negative of A, written as not-A, but they are not exactly the same thing. A and not-A are a true dichotomy; but A and B are not. Nor is A co-extensive with not-B, or B coextensive with not-A, because they are not exact synonyms. Yet they are not unrelated: they are mutually referential, and so can form a self-referential loop.

A language is any system used to represent thoughts, and a word is a particle of meaning in such a system. Verbs expressing postulates that something will be done or happen may be called the root words of those postulates. And at this juncture we’ve stepped from pure abstract logic into the field of human will, human experience and human values. When A and B are mutually exclusive verbs, their mismatch may be experienced as a problem, a knot that needs to be untangled.

To take a common example, A might be ‘Accept’, and B might be ‘Reject’.  Rejection is one way of not accepting, but not the whole of it, and vice versa.  There is always a middle road, however narrow, between accepting a thing and rejecting it. It’s not a true dichotomy because there’s a third alternative to A or B, the middle way represented by not-A, not-B.

Some other such pairs are Display and Hide, or Love and Hate. It’s very easy to fall into the misconception that display and hide are a dichotomy, so that if we do not display something we are hiding it, and if we do not hide it we must be displaying it. That is the two-valued logic of philosophers like Ayn Rand. But are you displaying or hiding the Eiffel Tower at this moment? Probably neither, unless you’re a Parisian tour guide or the owner of a very big tarpaulin.

And check out almost any politician’s rhetoric for examples of spurious dichotomies like these. “Anyone who is against capitalism must be a communist.” It’s almost as if they were trying to gaslight us, perish the thought.

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The four postulates A, B, not-A and not-B can be arranged in a 2 by 2 matrix:

1. A                                3. B

2. Not-A                        4. Not-B

paralleling the basic game matrix described by Stephens (1979):

1. Be known                   3. Know

2. Be not-known            4. Not-Know

but with an important difference since A and B are such that B implies not-A and A implies not-B. In formal logic this condition is written A↑­B : B ⇒ not-A, A ⇒ not-B

The root words in the basic game matrix are Be Known and Know. When they are replaced by other root words such as Accept and Reject, a junior goal is formed within the basic goal. Any junior goal can be reduced to a special case of the basic goal of Know/BeKnown, since everything we try to do in life is a means of either knowing or making something known. Seeing, owning, learning are ways of knowing. Creating an effect in the world is a way of being known. In this context, the words ‘postulate’ and ‘goal’ are equivalent.

Stephens demonstrated the value of the basic game matrix in making decisions in life, and as a therapy. This property of the basic game is a consequence of Know and Be Known not being mutually exclusive. They are complementary conditions that satisfy each other, represented in English by the active and passive forms of the same verb. Neither of the classes Be Known, Know or Be not-known, Not-know is a null class.

But in the NAND matrix, B is a subset of not-A and A is a subset of not-B. Therefore B becomes a negative of A and not-B becomes a double negative, effectively a positive but on a lower level to the starting point of A. This matrix functions as a descending spiral if the four postulates are followed in sequence.

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I first encountered NAND matrices 35 years ago when studying the theory of implant GPMs. Now, that term needs some explanation. Implants were believed to be mental control operations that had been done to us in long-vanished civilisations, and consisted of postulates enforced by painful energy. They would constrict our ability to think just as malware reduces the efficiency of a computer. GPM stands for Goals Problem Mass, the apparent mass generated by the problem of two opposing goals rather like a standing wave between two forces meeting head-on. GPM implants are particularly severe because they consist of balanced pairs of conflicting postulates, logical knots such as the NAND.

According to the chronology of Hubbard (1963) the earliest implant GPMs were the ones now called the Invisible Picture implants. These have the logical form described above, the form of A↑­B : B ⇒ not-A,  A ⇒ not-B, being a sequence of contradictory items in NAND relationships.

The basic game matrix in TROM Level Five undercuts any implanted GPM because it is closer to truth (Stephens, 1992). We now know that each item in these implants is reducible to one of the four legs of the basic game matrix. Must Be Known (MBK) is an postulate to put something there to be known and Must Not Be Known (MNBK) is the opposing postulate – to not do this. So a sequence can run MBK, MNBK, MNBK, MBK over and over, pairs of true dichotomies that stand in alternative denial to each other.  Or it may change to MK (Must Know), MNK (Must Not Know), MNK, MK. One word in each item, the root word, varies through the sequence.

Here’s a made-up example (I’m not plagiarising from Hubbard’s published materials):

Grow                        MBK
Don’t grow               MNBK
Shrink                       MNBK
Don’t shrink              MBK
Build                         MBK
Don’t build               MNBK
Demolish                  MNBK
Don’t demolish         MBK

Find                         MK
Don’t find                MNK
Lose                         MNK
Don’t lose                MK

Note that in each pair of verbs in a NAND relationship, the negative of one is similar (but not identical) to the positive of the other. Thus ‘don’t find’ and ‘lose’ both fit within the broader negative concept of ‘must not know’, while ‘find’ and ‘don’t lose’ are within the positive ‘must know.’ The structure has a regular + + – – + + – – pattern down the page.

An impossibility point (Stephens, 1994) would occur between each pair of opposing items, meaning either MK/MNK or MBK/MNBK, generating the perceived mass that holds the charge in the GPM. This generated mass would tend to persist because – unlike the basic game matrix – a GPM lacks complementary pairs of items that would enable the release of charge.

This may all sound like science fiction, but the structure of implant GPMs makes them resemble artefacts composed by someone who knew about the basic game matrix and was deliberately perverting it. Whether they are really artefacts, or just consequences of the basic game, or something else, I can attest that they have real power that feels like mass or charge in the mind when first contacted. They have been found on many cases around the world; they’re not something dreamed up one sunny afternoon at Saint Hill.

Going even further into speculative fiction, suppose the structure of the Invisible Picture implant was copied by later implanters who devised new GPMs as variations on that original sequence. Hubbard also discovered many other implant GPMs that are structured with different forms of contradictory postulates instead of the NAND. It’s as if there had been several lines of development from several original implant authors.

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The previous section is rather a digression. My main point is that Spencer-Brown’s imaginary Boolean values are closely related to Stephens’ impossibility points within the less well-defined popular concept of strange loops.

Stephens and Spencer-Brown were pioneers of that basic area of knowledge where mathematics and psychology converge. Perhaps someone with more mathematical insight than this blogger will be able to clarify the formal relationship between strange loops, impossibility points and imaginary Booleans.

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References

Hofstadter, D. (2007) I Am a Strange Loop. (Basic Books: New York).

Hubbard, L.R. (1963) Routine 3N line plots. HCO Bulletin of 14 July 1963, Saint Hill.

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

Stephens, D.H. (1979) The Resolution of Mind. Public domain text.

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

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

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