Consider the double bonding A ⇒ B and B ⇒ A, or A ⇔ B.
In a double bonding in logic, the two fields A and B are co-extensive. If they are just two different names for the same thing, this is an innocent synonymy, as in the instances of nomenclatural synonymy in plant names. But if we consider them to be different (and by using the two names A and B we seem to be making that consideration), then it’s not at all innocent.
Then the statements A ⇒ B and B ⇒A together create a paradox where A and B are both identical and different; this can only be represented by an imaginary Boolean value as defined by Spencer-Brown (1969). The double bonding contains the seed of a feedback loop to an imaginary value.
This imaginary value can be approached more stealthily by making a series of bondings such as A ⇒ B, B ⇒ C, C ⇒ D and then adding D ⇒ A to create what Hofstadter (1979) called a strange loop. In other words, a function that re-enters itself, in this case at the fourth level.
The posssibility of double bondings as paradoxes or fallacies was noted by Lewis Carroll at the Mad Hatter’s tea party in Alice in Wonderland. Grammatically, “I see what I eat” could be equivalent to “I eat what I see.” But in English language syntax the order of antecedent and consequent expresses a convention that the first sentence means that Eat ⇒ See, but not that See ⇒ Eat.
Hofstadter, D. (1979) Gödel, Escher, Bach: An Eternal Golden Braid. (Basic Books: New York ).
Spencer-Brown, G. (1969) The Laws of Form. (Allen & Unwin: London).
A dichotomy is the division of a whole into two parts that are mutually exclusive and jointly exhaustive.
For example, the unity of 1 may be divided into two classes of entities, X and 1 – X, defined by an elective function x. George Boole showed that expanding the function f(x) gives the two constituents x and 1 – x only.
The two classes are mutually exclusive; this means that X (1 – X) = 0, or nothing can simultaneously be both X and not-X. They are also jointly exhaustive, X + (1 – X) = 1, as everything must be either X or not-X.
If the universe of discourse is limited to “birds” for the sake of a simple example, then x might be the definition of the group of bird species known as penguins. Then the class X contains all penguins, and the class 1 – X contains all the other birds that are not penguins. Note that one side of a true dichotomy always has a privative definition, it is defined only by what it is not.
A language such as English has many spurious dichotomies, pairs of words that are “antonyms” in the broad sense as offered by a thesaurus but do not satisfy the two equations above.
To define a dichotomy or decide whether it is a true or spurious one, it is necessary to first define the universe of discourse. For practical purposes, the universe of discourse is the real physical universe or a subset of it.
The two ends of a gradient such as pure black to pure white are not a true dichotomy in the real world even though black and white are mutually exclusive. But they are not jointly exhaustive because most of that gradient is neither black nor white, but grey. Not to mention all the other colours. Of course, in some hypothetical universe that consisted only of pure white and pure black things, black/white would be a true dichotomy.
Opposed pairs of concepts such as ‘freedom’ and ‘slavery’ are not true dichotomies either, except in a hypothetical universe where they are the only two things that exist so that x = (1 – w) and w = (1 – x). In the real world, their mathematical relationship can be expressed by relational statements of the form x ⇒ (1 – w); a slave is not free, but it does not necessarily follow that everyone who is not a slave is free.
On the other hand, direct opposites such as ‘known’ and ‘not known’ or ‘penguin’ and ‘not penguin’ are true dichotomies, as they are mutually exclusive and jointly exhaustive.
If antonyms are pairs of words with exactly opposite meanings, then the precise antonym of any word is formed by simply adding the prefix ”not”. This may seem trivial, but any attempt to invent more sophisticated antonyms leads to imprecision and confusion.
For example, the antonym of ‘accept’ is ‘not accept’. A thesaurus may suggest ‘reject’ as an antonym, but that is not an exact antonym, just an approximation. The pair accept/reject are more like the gradient of black to white because they are separated by a grey zone of neither accepting nor rejecting. This is because the concept of rejection is within not-acceptance, it is a narrower concept because there are many ways of not accepting something without actually rejecting it. Again it is a mathematical relation of the form x ⇒ (1 – w).
Sets of entities, of any kind, can be linked in logic by bonding postulates of the form A ⇒ B (meaning that A is a subset of B, implies B, is within B). The same statement can be written in reverse as B ⇐ A (meaning that B is a superset of A, is implied by A, includes A). A is called the antecedent, and B is the consequent. This does not imply either a causative or a temporal sequence between antecedent and consequent, but simply a logical relationship.
In each case, the set of entities classed as A is completely included within the set of B. A is never found without B although B may occur without A. This situation is described in Boolean algebra as a (1 – b) = 0, or a = ab
Uppercase letters here refer to sets of actual entities, whereas the postulates (in other words, the elective functions or decisions) that define those sets are indicated by the corresponding lowercase letters, following the usage of Boole (1847).
Any relationship that exists between two entities or two postulates can be exposited as a nested hierarchy of A ⇒ B relations. It’s hardly an exaggeration to call this relation the basis of all logical thought.
If A ⇒ B, a pair of conditions holds:
B is necessary for A: A needs B in order to exist, although B can exist without A. eg, water is necessary for plant growth.
A is sufficient for B; the presence of A guarantees B, although B might also exist under alternative conditions not involving A. eg, seeing plants growing is sufficient evidence to assume the presence of water.
Stephens (1994) pointed out that the necessity of B for A and sufficiency of A for B together form a tautology that arises from the way we have circumscribed A and B such that A ⇒ B. For example, if we agree that all dogs are mammals, or dog ⇒ mammal, then being a mammal is one of the necessary qualifications for being a dog, but being a dog is by itself sufficient to qualify an animal as a mammal. This type of tautology is ubiquitous in the systematic classification and naming of plants and animals. Thus species A may be assigned to genus B as one of its members so that A ⇒ B, and that genus is in turn assigned to a family. Thus the classification system of the plant kingdom is a nested hierarchy of A ⇒ B relations with A sufficient for B, and B necessary for A, at each level.
By the same logical process, a taxonomist may assign species M to another species, N, as a synonym if he considers them too similar to merit separate names. A synonymy is an example of what Boole (1854) called an abstract proposition as it is a proposition about species concepts, which are in turn propositions about actual, tangible specimens. Every scientific name of a species refers ultimately to one specimen, known as the type specimen. It will be seen from the paragraphs above that if name ‘M’ is a taxonomic synonym of ‘N’ they cannot be at precisely the same level in the hierarchy: M must be within N as a name applying to a subset of the whole set of individual organisms comprising species N. Therefore two names cannot both be taxonomic synonyms of each other.
The same issue arises with synonyms in ordinary language. There is always an asymmetry in rank, a difference in level between one word and another that is considered to be its synonym. The meaning of the latter must always be a subset within the former. A thesaurus might glibly suggest ‘vehicle’ as a substitute word for ‘car’. But ‘vehicle’ is a more inclusive concept than ‘car’: all cars are vehicles but not all vehicles are cars. Therefore cars are a subset of all vehicles, and the word ‘car’ is within ‘vehicle’ as a synonym.
However, the codes of biological nomenclature were drafted without reference to Boolean algebra. They can add a little confusion since the principle of priority mandates that the earliest-published name be used for the merged species, although this may not be the name associated with the most inclusive set. This arbitrary rule may give the paradoxical impression that a larger M can reside within a smaller N. For instance, many garden plants from China such as the Banksian rose (Rosa banksiae) and the weeping willow (Salix babylonica) were given their botanical names based on the selected horticultural forms first introduced into Europe, but those names must now apply to all wild populations of these species as well.
All the examples above are single bondings where A ⇒ B but not B ⇒ A. This can be expressed in Boolean algebra as a (1 – b) = 0 and b (1 – a) ≠ 0.
However, if A is both necessary and sufficient for B, then B exists if, and only if, A exists. This is the state of equivalence A ⇔ B, meaning that A and B are co-extensive, and either can be called the antecedent or the consequent. This is quite distinct from the taxonomic tautology mentioned above (where the antecedent is necessary for the consequent to be true, and the consequent is sufficient to prove the truth of the antecedent). An example of equivalence would be the relation between the concepts “the 4th of July” and “USA’s Independence Day”; then a statement that “July 4 is Independence Day in the USA” is quite true but adds no new information. If it is agreed that A and B refer to exactly the same things, they may be called nomenclatural synonyms rather than taxonomic synonyms as they differ only in name, not in the sets of entities to which they refer.
On the other hand, a mutual bonding of two non-equivalent entities – that is, A ⇒ B and B ⇒ A where A ≠ B – represents a logical contradiction. They cannot each be contained wholly inside the other if they are different in any way. This is what Stephens (1994) called a double bonding, and may be expressed in Boolean algebra as a (1 – b) = 0 and b (1 – a) = 0.
Therefore every double bonding contains a fallacy. Either one of the bonding postulates is untrue, or they do not both belong to the same logical type in the sense that Whitehead & Russell (1910) used this term, or the same level in the sense of Polanyi (1968).
Boole, G. (1847) The Mathematical Analysis of Logic. (Macmillan: Cambridge).
Boole, G. (1854) An Investigation of the Laws of Thought, on which are Founded the Mathematical Theories of Logic and Probabilities. (Macmillan: London).
Polanyi, M. (1968) Life’s irreducible structure. Science 160: 1308-1312.
Stephens, D.H. (1994) Relationships – Bonding. audio recording of 21 February 1994.
Whitehead, A.N. & Russell, B. (1910) Principia Mathematica. Vol.1 (Cambridge University Press: Cambridge) .
The surest way to mess up someone’s mind is to force them to do something that they already want to do. Or to forcibly prevent them doing something that they don’t want to do. Either way, they have to make an intolerable choice between submitting to force, or acting against their own will.
This is the essence of insanity, this is a double bind. As L. Ron Hubbard said in one of his better moments, fundamental aberration is the enforcement of basic truth.
But the converse is also true. If your mind is doing things outside your control, for example bringing up unwanted images or emotions, the solution is to do deliberately and consciously whatever the mind has been doing automatically. This will bring that automaticity under control so it can be used, or not, as appropriate. If someone is haunted by an unpleasant memory, they need to recall that memory by their own free will until its apparent power over them vanishes.
Take a look at whatever you call your “self”. Is it an entity with hard, permanent boundaries?
In my own experience, awareness extends across conceptual spaces that could be called fields, divided by boundaries that might be called discontinuities in awareness.
Michael Polanyi described a particular type of boundary in a series of publications in the 1960s. He introduced the theory of tacit knowledge, where information tacitly known at one level of reality is the basis of explicit understanding at a higher level. For example, whenever we read a text we are tacitly perceiving all the letters but normally notice only of the words or sentences that they spell. Many such levels may exist in a hierarchy, such as letters forming words according to rules of spelling, that form sentences according to the rules of grammar, that in turn carry meanings according to semantic rules.
Each level is a field containing a consistent set of concepts that is incomplete in that it allows its boundary to be ruled by the next higher level. The lower or proximal field contains things known tacitly but the distal field consists of things that are known explicitly, or are still unknown. The proximal field is experienced as self, the distal field as not-self or in other words the external world. For the purpose of this discussion I’ll call these Polanyi boundaries.
The old truism that anything has both an inside and an outside aspect is rediscovered from time to time. For example, the botanist Agnes Arber wrote that “The fact that each organism is both a unity intrinsic to itself, and also an integral part of the nexus which is the Whole, informs it with a basic duality.”
The subjective experience of being a self and separate from an external world – that is, the rest of the universe – was analysed by Gerbode in terms of the theory of tacit knowing. We tacitly know such things as the movement of our voluntary muscles, ideas with which we have identified, skills that have been learned and experiences internalised. All these things are within the aggregate that we think of as self. The other things that we perceive are considered to be separate from the self and therefore parts of an external world.
Another type of boundary that exists between opposing postulates in the mind was described by Stephens as occurring where postulate pairs such as “must know” and “must not be known” meet head-on like opposite flows forming a ridge, a mass that we experience as sensation. Such ridges might be called Stephens boundaries. Moreover, since one self-consciousness cannot simultaneously hold contradictory postulates, the boundary may effectively divide the mind into two fields that function as if they were independent entities.
Please note that I’m using the term postulate here to mean a causative thought, following the usage of Stephens and Gerbode, and before them of Hubbard. This isn’t quite the usual meaning of the term in English. Unfortunately, English doesn’t have any word that captures this concept exactly, and the Buddha’s Pali term saṅkhāra would be more precise. In Buddhist philosophy, saṅkhāra does not depend on self-consciousness but is actually a precondition for that consciousness.
At first sight, a Stephens boundary appears to separate a pair of entities that are both on the same level. The pair of postulates that define their boundary are not immediately recognisable as a rule imposed from a higher level that defines the boundary of the lower one.
But a Stephens boundary can also be seen as an instance of a Polanyi boundary. Both types of boundary represent an inconsistency that marks the limit of an internally consistent field. In fact, the contradictions between postulates are the source of the incompleteness or inconsistency that marks the boundary.
A pair of exactly opposed postulates forms a unity, just like the two ends of the same stick. More importantly, any Stephens boundary actually has higher and lower sides like a Polanyi boundary. The stick has a proximal and a distal end relative to the observer’s viewpoint.
The proximal field is experientially a self, which is normally a lower level field than the corresponding not-self. Self (the field of what we tacitly know) is a small portion of the whole universe (the field of what we explicitly know + what we tacitly know + everything that exists beyond our knowledge). In our everyday experience, the universe of discourse is whatever we perceive as the whole world. Any thing that we can readily view, including ourselves, is much smaller than the universe. Brotherhood with the universe can be a heady feeling when meditating under the summer stars, but taking that feeling too literally is the road to megalomania.
Any thing that we call our tacitly known “self” is an instance of what Stephens called a junior universe – an object that is selected as one side of a dichotomy, leaving the rest of the universe on the other side. Compulsive game playing compartmentalises a person into progressively smaller junior universes by successive dichotomies.
Could it be that a subjective sense of self arises from opposed postulates? If one being cannot hold both postulates simultaneously, there would be a division into self and not-self. The field of not-self can then be subdivided into various objects and even other living beings known as “them” or “you”.
Conversely, resolving the postulate opposition would resolve the perceived boundary of a self. An experimental test of this hypothesis would be to erase some contradictory postulates from one’s mind and observe what happens to the sense of self. Does it expand?
Arber, A. (1954) The Mind and the Eye: A study of the biologist’s standpoint. (Cambridge University Press).
Gerbode, F.A. (2013) Beyond Psychology: an Introduction to Metapsychology. 4th edn (Applied Metapsychology International Press: Ann Arbor).
Polanyi, M. (1968) Life’s irreducible structure. Science 160: 1308-1312.
Stephens, D.H. (1979) The Resolution of Mind: A Games Manual. (privately published: Sydney).
Further to the previous post, I’d been thinking about why the basic pair of complementary postulates in the mind is Know and Be Known, instead of anything else. Is this an arbitrary? Could it be something different in another universe?
But, of course!
Know and Be Known are the two essential properties of Life, the one thing that ultimately exists. They add up to BEING.
” … yet always a twoness in that many. And that twoness so near unite to oneness as sense to spirit, yet so as not to confound to unity the very heart and being of God, who is Two in One and One in Two.” – E.R. Eddison, The Mezentian Gate.
The duality of knowing and being known can explain the duality of self and not-self. Why am I sure that I exist? Because I can know (sense, see, feel, understand) things. Why am I sure that those things exist objectively outside me? Because they can be known (sensed, seen, felt, understood).
These are also are the two sides of the communication cycle: that is, receipt point and source point.
In Eddison’s novel The Mezentian Gate, Life has created a universe of experience by dividing itself into Love and Beauty, the knower and the known. Similarly, all our experience in this universe that we inhabit depends on a division into self and not-self. That division opens the door to the possibility of games, aberration and all the states of woe. But if the two were collapsed into one without division there would be no consciousness.
I would like to present another public domain edition of The Resolution of Mind by Dennis Stephens.
Dennis Stephens (1927 – 1994) was one of the first dianetic auditors in England, where he worked with L. Ron Hubbard in the 1950s and contributed to the development of scientology in that period. He later acknowledged Hubbard as “the man who took psychology out of the brain and gave it back to the people.” His other major influence was the mathematician George Boole, “the man who took logic out of the esoteric.”
Stephens developed Hubbard’s dianetic techniques for viewing the past under control of a therapist into a simpler procedure that he called timebreaking. Timebreaking is done solo because a person must take responsibility for their own mind if they are ever to become cause over it. Many of us who have found timebreaking valuable had the benefit of previous experience with dianetics and scientology, but Stephens intended this technique to also work for people with no knowledge of these sciences.
His second breakthrough was to extend Hubbard’s ideas on game theory and goals. In his opinion, research in scientology went astray in a search for the opposing goals and identities that form the core of the mind. Stephens realised that the basic goal package had been there all along in the two basic abilities of life stated in Hubbard’s first axioms of scientology: to create things to be known, and to know things that have been created. He then used Boolean algebra to analyse the structure of games that arise from opposing and complementary goals; I have found this theory to be workable and valid.
In several ways the life of Stephens was a contrast and complement to that of Hubbard. Stephens’ goal in life was to know, rather than to be known. He was passionate about learning and finding the truth, but didn’t give a shit about becoming rich or famous. His childhood in the semi-slum neighbourhood of Tottenham had taught him a disgust for capitalism and a desire to find “a better way”. He founded no group or movement, he never claimed to be the source of all wisdom, but quietly wrote up his discoveries for anyone who might find them useful. And he expected users of his system to think for themselves, take responsibility for their own progress, and make new discoveries.
Stephens’ system is known by the acronym TROM from the title of this, his only published book. It is available at several places on the web as noted below, as are his other research notes which he recorded as audio tapes in the last years of his life. I’m posting this edition as one more source of The Resolution of Mind, hoping that many others will pass it on – in any medium, in English and other languages. It began as my working copy, with typographical errors from earlier editions cleared up to make reading easier. I have also arranged the four addendum sections in chronological order, added a table of contents and used a font that I hope readers will find easy on the eyes.
Other editions of The Resolution of Mind are available as free downloads from:
Boole, G. (1854) An Investigation of the Laws of Thought, on which are Founded the Mathematical Theories of Logic and Probabilities. (Macmillan: London). This is the original Boolean algebra.
Hubbard, L.R. (1956) The Fundamentals of Thought. (HASI: London). This summarises Hubbard’s version of scientology at the time when he and Stephens were working together. The discussion of postulates and games is very relevant to TROM.
Hubbard, L.R. (1957) 18th Advanced Clinical Course. A year on, and Hubbard was considering a theory of how memory works very similar to the one Stephens later adopted.