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) .