Dichotomies and the nature of antonyms

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).


3 Comments on “Dichotomies and the nature of antonyms”

  1. Dusan says:

    Are there any exact antonyms in English language that do not contain the word not? What do you think the implications are of using antonyms imprecisely the way most people are accustomed to using them?

    • David Cooke says:

      Good question, but I still can’t think of any counter-examples, the only completely unambiguous way to specify ‘everything that isn’t X’ seems to be ‘not X’. Or some other negation in place of ‘not’.

      In everyday speech, the imprecision probably doesn’t matter much. But in reasoning it might lead to the false assumption that there are only two alternatives in a situation where there are actually many – like saying “anyone who isn’t a capitalist must be a communist”. I had this in mind with the example about accept/reject, after discovering on Level 4 of TROM my habit of not accepting various effects, this had gone unnoticed because I didn’t overtly reject them either.

      • Dusan says:

        Thanks for your response, that was a great example. Can you address the “paradox” in trom where according to insanity lectures you are insane if you believe a thing can both exist and not exist at the same time, but according trom manual you can occupy any class such as being for and against someone at the same time and through RI we are bringing things into existence and knowing we created in this universe.

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