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).
Watsonia pillansii L.Bolus is widespread in the eastern (i.e. summer rainfall) part of South Africa at low and medium elevations. This wide geographic range is associated with variation in ecological requirements and plant size, but the flower colour is generally bright orange to orange-red.
Plants grown from seed recently imported from South Africa have unbranched stems to 1.2 m high bearing up to 22 flowers. They are evergreen, with new shoots appearing in late summer immediately after flowering and before the previous season’s leaves have died. Each flower has a cylindric tube 3.5 to 5 cm long and acute perianth lobes to 24 mm long that flare widely when fully open; the colour in this strain whose exact provenance is unknown is a rather weak orange-juice orange on the lobes and deeper on the outside of the tube. The anthers and pollen are cream.
Watsonia pillansii is related to W. schlechteri in the section Watsonia, subsection Grandibractea.
The species has been in cultivation in Australia since the 19th century. Cultivars that may be selections of W. pillansii include ‘Flame’ (marketed by Lawrence Ball in the 1940s) and ‘Watermelon Shades’ (Cheers, 1997). Watsonia ‘Beatrice’ or the Beatrice Hybrids is a group name for various natural hybrids of W. pillansii (Eliovson, 1968) that were exported to Britain, America and Australia in the early 20th century. The name comes from Watsonia beatricis J.Mathews & L.Bolus, which was a taxonomic synonym of W. pillansii.
Cheers, G. ed. (1997) Botanica. (Random House Australia).
Eliovson, S. (1968) Bulbs for the Gardener in the Southern Hemisphere. (Reed: Wellington).
Goldblatt, P. (1989) The Genus Watsonia. (National Botanic Gardens: Kirstenbosch)
Watsonia fourcadei J.Mathews & L.Bolus has been in cultivation in Australia since the 19th century but does not appear to have contributed to the pedigrees of any hybrid cultivars bred in this country. It is widespread in the mountains of the southern Cape but absent from the Cape Peninsula where it is replaced by the related W. tabularis J.Mathews & L.Bolus.
Plants grown from seed recently imported from South Africa have flowers in a range of pink shades from pale salmon with a darker tube to the medium pink shown in the photo. They are evergreen, making most growth in mid summer to autumn but flowering in October to December.
The flowers have an arched, narrow cylindric tube about 6 cm long and perianth lobes 26 to 32 mm long incurved to form a cup-shaped limb.
Goldblatt, P. (1989) The genus Watsonia. 148 pp. (National Botanic Gardens: Kirstenbosch) ISBN 062012517
Watsonia schlechteri L.Bolus grows in the montane veld of the winter-rainfall parts of the Cape, South Africa. Plants grown from seed imported via Silverhill Seeds first flowered this year and are a good match for the lectotype of W. schlechteri.
The flowers are orange-vermillion with perianth lobes to 23 mm long. The buds end in a slightly downcurved point. There are no staminodal ridges in the perianth tube, a character that distinguishes it from the closely related W. pillansii. However, the fresh leaves of these specimens lack the strongly thickened margins and midvein that are used as another distinguishing character; these are only evident in dried material.
W. schlechteri is one of the smaller watsonias, usually much less than 1 metre tall with leaves about 40 cm long. It flowers in late December to January, resuming growth from offsets soon after flowering while the previous season’s shoot may still be green. Thus it has some leaves all year, or non-flowering plants may be briefly leafless before the new growth starts in late summer. Goldblatt notes that flowering in the wild is conditional on the plants not being shaded out by surrounding vegetation.
Like other watsonias native to high altitudes, it is at risk of damage in the agonisingly hot, dry summers we get here at sea level in Adelaide. The problem is to give the plants sufficient light without excess heat, a tall order on days of 42°C with northerly winds.
Goldblatt, P. (1989) The genus Watsonia. 148 pp. (National Botanic Gardens: Kirstenbosch).
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.