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 studies on Gazania by Seranne Howis are a reminder that biodiversity can’t always be divided into discrete species. Speciation may form the kind of clearly articulated branching pattern that cladists like when evolution is driven by new niches becoming available one by one. But the sudden landscape-wide diversification of a clade and its sorting by natural selection into stable entities are separate, and almost contradictory, processes – rather like an explosion and the subsequent settling of the debris. Such explosions of diversity have occurred in the Mediterranean climate zones of south-western Australia and in South Africa, as part of wholesale vegetation changes caused by the cyclical climate changes of the last few million years. Many genera were reduced to small populations in refugia during the dry periods, and rapidly diversified again in the wet periods.
The South African biodiversity hotspot has given the world several genera of ornamental plants that have been evolving in this way. Some of them are real gifts to the plant breeder because in cultivation they function as coenospecies with all their genetic diversity available for use in hybridisation. For example, Watsonia (Iridaceae) has been divided into 52 morphological species that rarely hybridise in the wild as the flowers of sympatric populations have diversified to utilise different pollinators. But they all have the same chromosome number, and can all be interbred in cultivation with the F1 generation often showing hybrid vigour and high fertility. My ‘gut feeling’ from working with Watsonia is that it’s a genus with little evolutionary depth, all the species having similar genetic architecture and closely homologous genes. Gazania is another example, in which the process of speciation may not have proceeded even as far as it has in Watsonia.
Howis showed that Gazania includes seven valid, monophyletic species – each reproductively isolated, with a distinct morphology, habitat and genetic identity – but it has the majority of its diversity in a broad complex where morphological, ecological and genetic variation are only partially correlated. The complex may be called an ochlospecies, defined as a polymorphic species with chaotic infraspecific variation intractable to formal taxonomic treatment. In her 2007 thesis, Howis tentatively called this ochlospecies by the earliest published name, G. rigens (L.)Gaertn. But Howis et al. (2009) take a more conservative course by calling it the K-R complex – possibly because G. rigens is usually applied to the stoloniferous sand-binding forms that are distinct from the others morphologically and ecologically, but not genetically.
Another model for understanding genera like Gazania might be found in Vavilov’s theory of homologous variation, which is closely related to Nabokov’s concept of homopsis. A complex that has diversified only since the Pleistocene is likely to consist of populations with similar functioning genes that determine the morphology and physiology of individual plants. This is quite apart from the fine variations in the four non-coding chloroplast sequences and two nuclear spacers used in Howis’ study. The model predicts that similar traits of morphology and physiology would appear repeatedly in response to the appropriate environmental conditions. Homopsis is a type of homoplasy in which the phenotypic similarities are due to underlying genetic similarity, instead of being due to convergence from more diverse ancestors; this would be the case with the stoloniferous ‘rigens’ forms of Gazania that do not form a genetically coherent entity since natural selection has shaped them from the same gene pool as the rest of the complex.
The many forms of Gazania introduced to Australia, and now feral here, are all within the K-R complex. For practical purposes such as gardening books and legal declaration as weeds, a Latin binomial may be demanded. There are three possibilities:
- refer simply to the genus Gazania. As none of the seven distinct species are naturalised here, in practice this would mean the K-R complex.
- use the name Gazania rigens (L.)Gaertn. to signify the whole K-R complex. This species epithet has priority under the Code, having been published as Gorteria rigens L. in 1763.
- separate G. rigens out as the name of the stoloniferous forms that have been planted for sand stabilisation on our coasts. The earliest valid name for the residue of the K-R complex would then be Gazania rigida (Burm.f.)Roessler. This would be similar to the treatment in the 1986 Flora of South Australia, but with G. rigida replacing the later synonym G. linearis.
Howis, S. (2007) A taxonomic revision of the southern African endemic genus Gazania (Asteraceae) based on morphometric, genetic and phylogeographic data. Ph.D thesis, Rhodes University Botany Department. 293 pp.
Howis, S., Barker N.P. & Mucina, L. (2009) Globally grown, but poorly known: species limits and biogeography of Gazania Gaertn. (Asteraceae) inferred from chloroplast and nuclear DNA sequence data. Taxon 58(3): 871–882.
There are four ways of defining a term or a datum (Hubbard, 1951). The most obvious might be by description, or listing its essential characteristics. Then something can also be defined by its differences from similar things with which it might be confused, or conversely by classifying or associating it with things that it resembles. But the most useful is an action definition that specifies the cause and function of the thing.
This is very relevant to biological taxonomy, which is concerned with defining taxa as relatively stable entities within the diversity of living organisms.
Traditional taxonomic literature uses descriptive definitions (full description with reference to a type specimen) to establish taxa. It also uses differentiative definitions (diagnoses and keys) as tools for identification. In practice, differentiation becomes definition by elimination, as in the dichotomous keys used in floras and monographs to narrow down the possible identity of a specimen to a single name by gradient steps.
Associative definitions may also be used in the protologue to discuss the relationships of a new taxon. Association was implicit in Aristotle’s concepts of genus and species, which Linnaeus adopted in the binomial nomenclature of plants and animals. Thus placing a new kind of cat into the genus Felis is a starting point for defining it. But an associative definition depends on the user’s prior knowledge of the thing being associated. It’s little use describing a lynx as a cat with certain characteristics to someone who has never seen any cats.
An action definition of a species would include its origin (the genus within which it evolved), its niche (functional role in the ecosystems where it lives), and its limits. e.g. “Watsonia hysterantha is a local derivative of W. meriana flowering in autumn from stored reserves, reduced in size and ground cover to exploit soil pockets in granite cliffs.”
An action definition of a higher taxon would similarly include its origin from a taxon of similar rank or the next higher taxon to which it belongs, the functional adaptations that caused this line to diverge, and its limits. e.g. “Cactaceae are caryophyllids adapted to seasonal drought by crassulacean acid metabolism and water-storing stems, with consequent slow growth rates making them vulnerable to competition”.
A species is a solution to the problems posed by its niche and environment. It is the lowest level taxon that can be called stable in that it is not liable to vanish within a few generations due to gene flow. There is no guarantee that a morphological species, based on museum specimens of distinctive appearance, has such stability. For practical purposes a taxonomic species is often taken as equivalent to a biological species: a reproductively isolated population or group of populations. The biological species concept was adopted by zoologists who considered only sexually reproducing organisms and ignored such things as clones and apomicts. But sterile or apomictic clones are morphologically distinct and function as species in ecology even though they do not fulfil the biological definition of a population as a unit within which genes are exchanged.
Like a key that fits a lock it wasn’t meant for, a species may fortuitously fit another environment quite different to the one where it evolved, as when an introduced crop species unexpectedly becomes a weed. But a population moved to a new habitat immediately becomes a new biological species because it is reproductively isolated from the original population and subjected to different selection pressures. Morphological changes may occur later, but it has already started on a new evolutionary trajectory. Bulbil watsonia is an example, a vegetatively reproducing biotype of Watsonia meriana that was reproductively isolated by becoming triploid and found a niche in habitats disturbed by human activity, first in its native South Africa and later overseas.
Hubbard, L. Ron (1951) Advanced Procedure and Axioms.
In a little-read paper from 1977, the philosopher Moshe Kroy almost casually expressed an insight that I find breathtakingly daring. Apart from its political significance, it has implications for taxonomy – which is why I’m including it in this blog – and for many other things as well. Kroy was analysing a seemingly trivial disagreement between two schools of Libertarians represented by Ayn Rand and Murray Rothbard to reveal its deeper roots. He resolved it by recognising that the problem only arose from the assumption made by Rand and many 20th century philosophers that individual humans are deterministic entities within the deterministic system that we call Community or Society. He wrote:
“Actually, within a deterministic context, any concept of entity as an ultimately discrete existent loses all significance. All entities easily reduce to parts of larger systems”.
“Only the stress on the individual as ultimately free establishes his individuality – being a distinct first cause.”
Entities are anything that we postulate and identify as complete units, separate from other such units. They range from planets to atoms, with our own bodies and the objects we use in daily life somewhere in the middle of that spectrum. The mental life of humans is all about perceiving entities in the world around us, understanding how those entities interact and using them to achieve our purposes. But the entities that we perceive in the universe of matter, energy, space and time are little more than convenient fictions. They reduce to parts of larger systems because they all ultimately have a common origin and are part of one big ‘machine’.
Taxonomists are concerned with organisms that are all products of a single evolutionary process from a common origin: the evolution of Earth’s biota, however many billion years it has taken, is a single event. No plant or animal is an individual since it has organic continuity with its ancestors and its siblings. Nor do species, genera and other taxa exist objectively as irreducible entities. At best, these are working hypotheses, useful divisions of the diversity that exists within biota. It is not surprising that taxonomists continue to have differing opinions on the extent to which some genus should be divided into species. The boundaries drawn between taxa may legitimately depend on the purpose of the classification.
In this paper, Kroy is concerned with the political freedoms of human beings, and he rightly saw that these freedoms have no theoretical basis as long as we consider people merely as physical entities in a deterministic system. On the other hand, each person – as distinct from the material body, the organism they temporarily inhabit – is a true individual. We are not products of evolution, or creations of some occult power, but are each the first cause of our own existence.
Kroy’s second paragraph demonstrates the influence of L. Ron Hubbard on his thought. He had encountered Scientology in his native Israel, and during 1976 was studying at Melbourne’s Church of Scientology while also lecturing in the Department of Philosophy at La Trobe University, but later diverged to follow his own intellectual path.
In a long footnote Kroy attributed this insight to Spinoza, who argued from an assumption of determinism in the Ethics that only one self-existent substance (substantium in the old scholastic sense) could exist in a universe, since everything else in that universe would be contingent on that substance. But it took a 20th century Libertarian to see the implications of Spinoza’s statement.
Kroy, M. (1977) Political freedom and its roots in metaphysics. Journal of Libertarian Studies 1(3): 205-213.
Many thanks to Dieter Zimmer for making his fascinating work A Guide to Nabokov’s Butterflies and Moths available to all readers online. This Web Book is particularly valuable as the original publication is already out-of-print and rare.
This book identifies all the butterflies and moths mentioned in Vladimir Nabokov’s scientific papers and his fiction: not a simple cataloguing job, since their nomenclature has changed repeatedly both during and since Nabokov’s time. As well as being an illustrated taxonomic reference work, it includes a concordance of the many references to Lepidoptera in his novels, poems and stories. It also lists the species named by Nabokov, and those named after himself, his family and characters in his fiction. But in my opinion the most interesting section is the treatment of Nabokov’s concept of the species and his views on evolution.
Vladimir Nabokov had abilities that are often associated with autism. His synaesthesia was well documented and was evident in his literary style. So was his eidetic ability to recall and cross-reference large numbers of mental images. In The Real Life of Sebastian Knight he wrote convincingly of how it feels to live in ‘constant wakefulness’ with every impression provoking a multitude of associative ideas. If there were such a thing as an aspie approach to taxonomy, the works of Nabokov might contain some clues to its nature.
While most zoologists were following the Neodarwinist fashion of downplaying morphological evidence and defining biological species based on observed or supposed limits to gene flow, Nabokov defined species on morphology and regarded biological data as secondary. Since he emphasised the morphology of the insects’ genitalia, the boundaries of his morphological species tended to coincide well with the practical limits to interbreeding of sympatric species in the field. He agreed that natural selection was the cause of an organism’s adaptation to its niche and habitat. But he remained sceptical of natural selection as the sole cause of the evolution of morphology of organisms and in particular the very widespread phenomenon of homoplasy (the evolution of similar characters in different clades and species).
Convergence was the old term for one kind of homoplasy: the independent development of similar but possibly superficial characters in widely separate clades. Nabokov understood that this phenomenon must be rare due to the number of genes involved and the statistical improbability of enough mutations with phenotypic effects in the right direction becoming fixed in a taxon. He introduced the new word homopsis for a more usual form of homoplasy: the repetition of characters in related species, due to similar sets of variations appearing in species with a similar genetic basis. This concept can be compared with homologous variation as conceived by the botanist Nicolai Vavilov – a Russian contemporary of Nabokov. Ever the synaesthete, Nabokov described a pattern of variation among species that contained gaps as a syncopated or jerky variational rhythm. For example, if two closely related moths had melanic variants and a third did not, this was an anomaly that called for an explanation.
In the 18th century Linnaeus considered genera to be more real than the species (literally, ‘appearances’) into which they could be divided. But in our time the genus has become an even more slippery concept than the species in biological nomenclature. Nabokov noted the limitations of Linnaeus’ binomial system where, following Aristotle, every species must belong to a genus. A genus of several species is defined by a particular combination of morphological characters that are common to them all. But a single-species genus has no reality beyond the implication that a common character combination would be revealed if some hypothetical, related species were to be found. Genus, species, and all taxonomic categories are noumena rather than phenomena. They exist only as mental constructs by which humans try to impose order on the kaleidoscopic variety of the world.
Many of Nabokov’s novels, and above all Pale Fire, are concerned with questions of identity. I suggest this is a key aspie characteristic: we’re so aware of everything around us that we sometimes have to think twice to find the boundary between self and not-self. Taxonomy is also concerned with postulating discrete entities among the continuous variation of organisms and drawing boundaries that identify them. It’s a branch of science that might have a natural appeal to aspies, as it did to Nabokov who took his first Cambridge degree in zoology, and later while a Lecturer in comparative literature at Harvard would work for up to ten hours per day on the Lepidoptera collection.
Nabokov, V. (1944) Notes on the morphology of the genus Lycaeides (Lycaenidae, Lepidoptera). Psyche 51: 104-138.
Nabokov, V. (1945) Notes on the Neotropical Plebejinae (Lycaenidae, Lepidoptera). Psyche 52: 1-61
Vavilov, N. I. (1922) The law of homologous series in variation. Journal of Genetics 12: 47-89.
Zimmer, D.E. (2001-2003) A Guide to Nabokov’s Butterflies and Moths. 438 pp. ISBN 3-00-007609-3