Basics: NOT

It is often said that one of the most significant discoveries in mathematics was the concept of zero, in the Indus valley sometime in the pre-Christian era. An equally important concept in logic is the operator NOT. While Aristotle, the founder of western logic, had discussed groupings of things in terms of what they are not in the Categories, chapter 10, the importance of NOT seems to have been realised first by George Boole in the nineteenth century. In this post I want to discuss it in the context of classification.

Aristotle wrote of four kinds of "contrarieties":

We must next explain the various senses in which the term ‘opposite’ is used. Things are said to be opposed in four senses: (i) as correlatives to one another, (ii) as contraries to one another, (iii) as privatives to positives, (iv) as affirmatives to negatives. Let me sketch my meaning in outline. An instance of the use of the word ‘opposite’ with reference to correlatives is afforded by the expressions ‘double’ and ‘half’; with reference to contraries by ‘bad’ and ‘good’. Opposites in the sense of ‘privatives’ and ‘positives’ are’ blindness’ and ‘sight’; in the sense of affirmatives and negatives, the propositions ‘he sits’, ‘he does not sit’.

We are concerned here with what the medieval logicians called, and we still translate as, privation, the opposite formed by adding "Not-" to the predicate or term. Aristotle's notion of a privative is roughly what in modern set theory we would call the complement. That is, in a Venn diagram (or actually an Euler diagram, since all Venn did was add the crosshatching):


Now the problem with something that is not some other thing is that all you know about it is that it lacks that other thing. The class of things it is the complement of can be indefinitely divided like this:


This leads to a regress, aptly discussed by Thomas Reid (from here, section I):

Another end commonly proposed by such divisions, but very rarely attained, is to exhaust the subject divided, so that nothing that belongs to it shall be omitted. It is one of the general rules of division, in all systems of logic, That the division should be adequate to the subject divided: a good rule without doubt, but very often beyond the reach of human power. To make a perfect division, a man must have a perfect comprehension of the whole subject at one view. When our knowledge of the subject is imperfect, any division we can make must be like the first sketch of a painter, to be extended, contracted, or mended, as the subject shall be found to require. Yet nothing is more common, not only among the ancient, but even among modern philosophers, than to draw, from their incomplete divisions, conclusions which suppose them to be perfect.

A division is a repository which the philosopher frames for holding his ware in convenient order. The philosopher maintains, that such or such a thing is not good ware, because there is no place in his wareroom that fits it. We are apt to yield to this argument in philosophy, but it would appear ridiculous in any other traffic.

Peter Ramus, who had the spirit of a reformer in philosophy, and who had a force of genius sufficient to shake the Aristotelian fabric in many parts, but insufficient to erect anything more solid than in its place, tried to remedy the imperfection of philosophical divisions by introducing a new manner of dividing. His divisions always consisted of two members, one of which was the contradictory of the other, as if one should divide England into Middlesex and what is not Middlesex. It is evident that these two members comprehend all England; for the Logicians observe, that a term along with its contradictory comprehend all things. In the same manner, we may divide what is not Middlesex into Kent and what is not Kent. Thus one may go on by divisions and subdivisions that are absolutely complete. This example may serve to give an idea of the spirit of Ramean divisions, which were in no small reputation about two hundred years ago.

Aristotle was not ignorant of this kind of division. But he used it only as a touchstone to prove by induction the perfection of some other division, which indeed is the best use that can be made of it. When applied to the common purpose of division, it is both inelegant and burdensome to the memory; and, after it has put one out of breath by endless subdivisions, there is still a negative term left behind, which shews you that you are no nearer the end of your journey than when you began.

Until some more effectual remedy be found for the imperfection of divisions, I beg leave to propose one more simple than that of Ramus. It is this - When you meet with a division of any subject imperfectly comprehended, add to the last member an et caetera. That this et caetera makes the division complete, is undeniable; and therefore it ought to hold its place as a member, and to be always understood, whether expressed or not, until clear and positive proof be brought that the division is complete without it. And this same et caetera shall be the repository of all members that may in future time shew a good and valid right to a property in the subject.

It is interesting that the one place Aristotle does discuss privative classifications by division, he rejects them as incoherent, in biology:

..., privative terms inevitably form one branch of dichotomous division, as we see in the proposed dichotomies. But privative terms in their character of privatives admit of no subdivision. For there can be no specific forms of a negation, of Featherless for instance or of Footless, as there are of Feathered and of Footed. Yet a generic differentia must be subdivisible; for otherwise what is there that makes it generic rather than specific? There are to be found generic, that is specifically subdivisible, differentiae; Feathered for instance and Footed. For feathers are divisible into Barbed and Unbarbed, and feet into Manycleft, and Twocleft, like those of animals with bifid hoofs, and Uncleft or Undivided, like those of animals with solid hoofs. Now even with differentiae capable of this specific subdivision it is difficult enough so to make the classification, as that each animal shall be comprehended in some one subdivision and in not more than one; but far more difficult, nay impossible, is it to do this, if we start with a dichotomy into two contradictories. (Suppose for instance we start with the two contradictories, Feathered and Unfeathered; we shall find that the ant, the glow-worm, and some other animals fall under both divisions.) For each differentia must be presented by some species. There must be some species, therefore, under the privative heading. Now specifically distinct animals cannot present in their essence a common undifferentiated element, but any apparently common element must really be differentiated. (Bird and Man for instance are both Two-footed, but their two-footedness is diverse and differentiated. So any two sanguineous groups must have some difference in their blood, if their blood is part of their essence.) From this it follows that a privative term, being insusceptible of differentiation, cannot be a generic differentia; for, if it were, there would be a common undifferentiated element in two different groups.

Hence, concludes Aristotle, privative classes are not genera in their own right, which is to say, they are not general classes. We would say they are not proper sets or proper subsets. They are what is left over when something that is a class is removed.

Now this is a matter of logic, and the NOT operator is one of the basic operators of most logics, and in fact of all computer languages. Using it and AND in sentential calculus you can derive all other operations. Which is all well and good, but what does it mean for science?

Aristotle makes the point: A classification founded in what something is not is not a classification proper. And in phylogenetics systematics, the classificatory techniques devised to deal with the evolutionary tree, privative classifications are regarded as delivering non-natural groupings. This is, to say the least, somewhat controversial, although to someone from a background of logic and philosophy it seems quite obvious. A group, say Mammalia, with some part excised, say, Primates, leaves no real thing: Mammalia-minus-Primates. What, to begin, would we call it? Non-Monkeys? Rocks and trees also comprise non-Monkeys, so that isn't a great name. Non-monkey-mammals just restates the group.

But the naming issue is trivial compared to the group itself. If we look to the defining characters of Primates, we get a number of dental, skeletal, physiological and other characters, nearly all of which are universal amongst the group. There are no such characters unique to all and only the complement of Mammalia, however, so inferences based on that grouping are doomed to fail at crucial points. And the inferences are what make a grouping natural in biology. If you have two members of the primates, any characters one has is likely to occur in the other, no matter how it is modified. We humans have all the characters of primates, some in highly modified ways such as our sacral tail, which we share with the African great apes but not other primates. We have thumbs, as they do, but ours are modified (as are the digits of many other primates also).

In phylogenetic systematics, also called cladistics, a privative group is called a paraphyletic group, which is to say it stands alongside a phyletic objects or branch of the tree. Here are some paraphyletic groupings:


Invertebrata (pale blue) is named for not having backbones, but it includes several quite disparate groups. Pisces (fishes, green) likewise includes groups that lack, in this case, a lifestyle (living on land), but includes air breathers, lobe-limbed fishes, and so on. Reptilia (orange) is all vertebrate land animals or secondarily seagoing animals (like the Galapagos iguana), which are neither birds nor mammals. And the blue group is not a paraphyletic group but another kind of non-natural group formed by having precisely one character, in this case warm blood (or endothermy), but that's another problem - of polyphyly (a group formed by convergent similarity rather than by lacking an evolutionary identity).

Classification by negation is sometimes held to be the same thing as classification by similarity, though. The systematists called - misleadingly - evolutionary systematists, felt that there were key adaptive radiations that marked otherwise paraphyletic groups out from their larger taxonomy and justified making them a distinct group. Reptilia is the famous case - birds are supposed to have unique characters (which they do - such as beaks, feathers, and hollow bones) that make the rest of the group unlike them, and hence needing a separate name.

Where this is most critical is where the group matters to a particular African Great Ape - us. And in particular where it matters to certain of these apes who study particular groups, especially ornithologists and primatologists. While most have happily accepted the phylogenetic turn in taxonomy if there are critics, they will often be of these disciplines, at least amongst zoologists. Botanists have their own concerns, not least the long standing taxonomy of some groups that goes directly back to Linnaeus. Here's an example of a paraphyletic classification based on not being something: apes. In ordinary language, being an ape and being a human are two distinct things, but there is no group that has defining properties if you remove humans from the ape group, so either we are Apes (I capitalise it to indicate it's an "official" name, which it isn't, not the English version at any rate), or there is no real group that answers to "ape", just a group that it suits humans to group together for personal reasons (i.e., to not be included with the monkeys and other apes. Something Linnaeus had absolutely no compunction doing, by the way).

Evolution produces a tree shape, after a fashion. Conceptually this is what mathematicians call a Directed Acyclic Graph, formed by the parent-child relations of organisms, and the parent-child relations of populations. Not all trees are well formed, because sometimes branches connect more than one together, as in hybrids, but for the purposes of classification, a single group, a real group is formed by a single cut of the tree diagram that represents the relations between taxa. Groups that are formed by excising parts of the branch are not natural groups, because the choice of inclusion or exclusion depends more on the person, the "Authority", than on facts about the groups involved. And this is precisely why a good number of scientists prefer paraphyletic groups, formed by not having some property the Authority thinks is important: because it makes the Authority important.

The aim of classification in science can be for many reasons. It may be as a teaching tool, or for the convenience of plant merchants, lawyers or lexicographers. Or it may be that it seeks to set up a type or class that is united by a single criterion or property. These are often important. There may be no natural group of warm blooded organisms, but if you have to house a number of organisms and maintain an environment they can live well in, it may be important to know whether or not they are. The thing is, that is not a classification of any theoretical importance. If you know that lizards, bees and snails need radiant heat to get going, that is all you know about that class. But if you know something is a vertebrate, or indeed a member of the Osteichthyes (bony fishes, which includes us, by the way), you actually know a lot about any individual species, on the basis of what the rest do or have, even if you have never encountered it before. A taxonomy is inductively projectible while a typology, formed from a single or small number of properties, is not.

And this gets us back to Aristotle and the privative predicate: the whole point is that what science seeks in its classifications is inductive projectibility - the ability to make solid and well founded inferences. Classes formed by privation do not help us here, and they are in the end informative only about the person who set them up, which while a kind of knowledge is not the knowledge we usually seek in a science. I would, in fact, say that this is not a good idea.


More like this

some loosely connected addenda thoughts, NOT in the contemporary world:

1: George Spencer Brown, "The Laws of Form", specifically on the subject of division.

2: negation seems not to be understood by the 'subconscious'; the concept to be negated is simply asserted and the negation ignored. This has serious implications for those of us with bad habits we need to change. It may also be related to how come it took so long - longer than '0' - to become properly handled.

3: the Schmidt Orthogonality Principle, which asserts that all logical (i.e. Boolean) propositions can be constructed using only expressions of the form ~(A & B). This has considerable consequences in the field of logic design as used in VLSI chips. It is also a good interview subject...


~A ::= ~(A&A)
(A&B) ::= ~(~(A&B)&~(A&B))
(A|B) ::= ~(~(A&A)&~(B&B))
(A^B) ::= ~(~(A&~(A&B))&~(B&~(A&B)))

and in fact this last is how the first computer I ever worked on actually implemented XOR functions in the ALU's Adder unit. It is a pretty and symmetrical diamond-shaped image when viewed as a logic schematic.

What really launched me into computers in my late teens (1966) was the visceral realization/apperception of the one-to-one relationship between the formal abstract logic (of the philosophical texts used in the Philosophy department at my University) to the actual physical electronic circuits which implement those logical equations, which in so doing perform useful computations. Without any visible moving parts. And I have to say that was a surprisingly hard association to make. The electronics was pretty easy, very basic Ohms law and simple semi-conductor junction physics; the philosophy and Propositional Calculus was equally straight-forward, something I learned and did well at in 2nd Form (UK equivalent to 6th grade). But there is something very different about the mind-set, or context, or something. I remember it took about a year of struggling to internalize an equivalence between ~(A&B) and the simple circuit of two diodes, a resistor, and a transistor. One of the few Aha! moments I can remember when it struck.

"negation seems not to be understood by the 'subconscious'; the concept to be negated is simply asserted and the negation ignored. This has serious implications ..."

Thanks for highlighting that observation. As I've worked to understand its serious implications, my approach to writing, speaking, and persuasion has changed. I feel it's helped my persuasive technique to become more effective.


It is often said that one of the most significant discoveries in mathematics was the concept of zero, in the Indus valley sometime in the pre-Christian era.

Actually John, Babylonian mathematics first produced the concept of zero in the Seleucid era, 300 BCE to 0, but their zero is purely a placeholder and not a number. The Indian concept of zero developed along with the decimal place value number system between Aryabhata (born 476 CE) and Brahmagupta (fl. 628 CE). Brahmagupta wrote the earliest known text that gives the arithmetical rules of addition, subtraction, multiplication and division for positive and negative numbers and zero considered as a number, although he didn't realise that division by zero is not definable. His work introduced the decimal place value number system with zero into Arabic mathematics from whence in the work of Al'Khwarizmi (circa 790-840) it made its way into Europe in the translation of Robert from Chester in the 12th century.

NOT is interesting as a logical operator in that it is the only single valued operator all others being two valued i.e. requiring two variables.

Great post! One thing I would like to add is that 'Classes formed by privation' are very much like artificial classifications. They may help us identify a group using presence as absence (e.g., lacking a backbone), but they do not necessarily represent evolutionary (i.e., natural) relationships. Notice I used the term 'relationship' rather than 'group'. The term 'group' may refer to a set (i.e., Venn diagram). Take the examples above: (A(BC)) is a set, whereas A(BC) is a relationship. This is why relationship are very difficult to represent mathematically. Confusing sets (phenograms etc.) with relationships has caused much of the kafuffle in phylogenetics.

Really cool post.

To Thony's point: the identity function is another such operator (although admittedly one of little practical use).

To Thony's point: the identity function is another such operator (although admittedly one of little practical use).

The identity function is not a logical operator ;)