Quaternary Thinking

Fours are the most abundant structure in intellectual history and the least examined. Four elements, four causes, four humours, four temperaments, four gospels, four noble truths, four seasons, four directions, four quadrants of every consulting framework ever drawn. The abundance should be the first clue: a shape that turns up everywhere is either very deep or very cheap to produce.

Most of it is cheap, and there is a single question that separates the two cases. Is the four irreducible, or is it a product? Two independent binary distinctions, crossed, generate exactly four cells. If a fourfold decomposes that way, its real content is two yes-or-no questions, and the four is bookkeeping. If it does not, the four is telling you something two questions could not.

The honest fourfolds say which they are. The rest get cited as though four were a discovery.

The Honest Four

The best-behaved four in the literature is honest to the point of deriving itself.

Belnap’s four-valued logic answers the question of how a computer should reason about a database that has been told things by several sources, some of them wrong and some of them silent. The four values are true, false, both and neither, and they “correspond to the elements of the power set based on {T, F}.” That is the whole derivation. A source may have told you it is true, told you it is false, told you both, or told you nothing, and the powerset of a two-element set has four members. Not three, not five.

The values arrange into a lattice with “Both at the supremum and None at the infimum, and T and F on the wings,” and the ordering is information, not truth: neither is the state of knowing nothing, both the state of having been told everything, including the contradiction.

The credit needs to be laid out carefully, because it usually is not. The relational semantics “are due to Dunn (1976), from his 1966 PhD dissertation, with precedents in 1957 by Białynicki-Birula and Rasiowa.” Belnap’s A Useful Four-Valued Logic appeared in 1977, in a volume edited by Dunn and George Epstein. And the resulting system, First Degree Entailment, is not a freestanding curiosity: it “is a fragment of relevant logics.”

Notice what follows. This four is explicitly $2^2$. Belnap does not claim to have found four irreducible modes of being; he built a lattice out of a binary powerset because the application demanded exactly that. It is a genuine four and a product, and saying both is what makes it honest. Compare Ternary Thinking, where K3 and LP each take one bite out of this same structure: K3 keeps the gap, LP keeps the glut, FDE keeps both.

The 2x2 Test

Applied to the famous fours, the test is brutal.

Fourfold	The two axes	Verdict
Aristotle’s elements	hot/cold, wet/dry	Product, and he said so
Johari window	known to self, known to others	Product
BCG growth-share matrix	relative market share, market growth	Product
SWOT	internal/external, helpful/harmful	Product
Square of opposition	universal/particular, affirmative/negative	Product, but see below
Jung’s functions	rational/irrational, and the pair within each	Product
MBTI types	four binaries, giving 16	Product of products
Aristotle’s four causes	none	Irreducible

The elements are the cleanest confession in the set. Empedocles (c. 450 BC) proposed fire, earth, air and water as the four “roots” (rhizōmata). Aristotle then derived them, in On Generation and Corruption, by crossing two pairs of sensible qualities: fire is hot and dry, air hot and wet, water cold and wet, earth cold and dry. The four are not four primitives. They are the cells of a two-by-two whose axes are hot/cold and wet/dry, and the ancient world was perfectly clear about this. It is the modern citation of “the four elements” as an irreducible fourfold that loses the structure.

The management frameworks are the same move with worse provenance. The Johari window (Joseph Luft and Harrington Ingham, 1955) crosses known-to-self with known-to-others to yield Open, Blind, Hidden and Unknown. The growth-share matrix, sketched by Alan Zakon and popularised by Bruce Henderson in a 1970 BCG essay, crosses relative market share with market growth to yield Stars, Cash Cows, Question Marks and Dogs. Both are perfectly respectable as two-question instruments and neither needs the word “four” at all.

SWOT is worth a paragraph for what its history shows about how fours propagate. Its origin is disputed between a 1965 SRI report by Robert Stewart, Otis Benepe and Arnold Mitchell using the framework “SOFT”, and a 1965 Harvard text by Learned, Christensen, Andrews and Guth that has the four components split into internal and external appraisal but no acronym. A 2023 study by Puyt and colleagues called the Harvard attribution “an academic urban legend.” Meanwhile Terry Hill and Roy Westbrook found in 1997 that “no-one subsequently used the outputs” of the analysis in later strategy stages. A four-cell frame with contested parentage and no demonstrated downstream use is a good picture of what a decorative quaternary looks like.

Where a product still earns its keep

The traditional square of opposition is the important exception, and it shows what the test is actually for.

Its four forms are generated by two binaries, quantity and quality: A “Every S is P”, E “No S is P”, I “Some S is P”, O “Some S is not P”. So far, a product. But the square’s content is not the four cells. It is the four relations between them: contradiction (they “cannot both be true and cannot both be false”), contrariety (“cannot both be true but can both be false”), subcontrariety (“cannot both be false but can both be true”) and subalternation, where the particular must be true if its universal is.

Those relations are not derivable from the axes. They are logical facts about the cells that only become visible once you lay them out as four, and the whole existential-import problem, where an empty subject term breaks the inferences, lives in the relations rather than in the forms. That is the refinement the crude test misses: a 2x2 earns its place when the relations between the cells carry content the axes do not. The square does. A consulting matrix whose quadrants are just four labels does not.

The test for any fourfold, then, is not “is it a product” alone. It is: strip the four back to its two axes and ask what is lost. Usually nothing. Occasionally the relations.

The Four Causes

Aristotle’s is the fourfold that survives the test cleanly, and it survives because it is not a partition of anything.

The four are four “explanatory roles that a thing can play,” each the answer to a different question: the material cause answers “What is it made out of?”; the formal cause answers “What is it?”, singling out “the essence or the what-it-is-to-be something”; the efficient cause answers “Where does change (or motion) come from?”; and the final cause answers “What is its good?“. The bronze statue takes all four at once: bronze, the shape, the caster’s art, and the statue itself as the end.

There are no two axes here. You cannot get “what is it made of” and “what is it for” out of crossing anything, because they are not values of a shared variable. Four independent questions happen to number four, and if a fifth good question about causation existed, Aristotle’s scheme would have accommodated it. That is the signature of a real enumeration as against a real partition: the count is contingent, and it does not pretend otherwise. See Teleology for what happened to the fourth of them.

The Four Corners

The catuṣkoṭi, literally “four corners,” has the same shape as Belnap’s four and does the opposite work with it.

The four positions are P, not-P, both P and not-P, and neither P nor not-P. Nāgārjuna deployed it, in the Mūlamadhyamakakārikā, not to assign one of the four but to deny all four. On causation: “things could be produced 1. from themselves, or 2. from other things, or 3. from both themselves and from other things, or 4. from neither,” and the answer is none of these. The Internet Encyclopedia of Philosophy describes the transformation he worked on it: he refined “the ‘four errors’ method from the strictly illocutionary and pragmatic tool it had been in early Buddhism into a logic machine that dissolved Buddhist metaphysical positions.”

Whether this is a four-valued logic is genuinely contested, and the honest reading is that it is not: “this denial is more a principled refusal to answer than a counter-thesis, it is more a decision than a proposition.” Assigning a truth value and refusing to assign any are different acts, whatever the surface symmetry.

The structural coincidence is still remarkable and worth stating plainly. The powerset of {true, false} has four members, and two traditions eighteen centuries apart both enumerated them exhaustively. One used the enumeration as a semantics and one used it as a ladder to be kicked away. The four is the same four, which is itself evidence that it is derived rather than discovered: anyone who starts from a binary and asks what an exhaustive enumeration over it looks like ends up here.

The Four That Are Chemistry

The genetic code is the fourfold most often held up as brute natural fact, and it is more interesting than that.

DNA uses four bases, adenine, thymine, guanine and cytosine. But they divide twice over: purines (adenine, guanine) have two rings, pyrimidines (cytosine, thymine) have one; and the pairing strengths differ, since the adenine to thymine pair is held by two hydrogen bonds and the cytosine to guanine pair by three. Two independent binary distinctions, four bases. Even here, the four is not primitive.

What the four then determines is a three. Two bases give only $4^2=16$ combinations, too few for twenty amino acids; three give $4^3=64$, which is enough. George Gamow proposed the triplet scheme; Marshall Nirenberg and Heinrich Matthaei confirmed it in 1961 by feeding a poly-uracil sequence into a cell-free system and getting back a polypeptide of nothing but phenylalanine, establishing that UUU codes for phenylalanine.

So the codon is a triplet because the alphabet is four, and the alphabet is four because two chemical binaries cross. Nothing in that chain chose a number for its own sake, which is exactly why it is the strongest four in this note and also the least mystical.

Fours That Got Rounded

Two failure modes are worth naming, and they run in opposite directions.

Rounding a count down to four. The Cynefin framework (Dave Snowden, 1999) is routinely drawn as a 2x2 and taught as four domains. It has five: clear, complicated, complex, chaotic and confusion. It is also not a matrix. The domains on the right are “ordered: cause and effect are known or can be discovered,” those on the left “unordered,” and confusion sits in the centre, which is precisely the state a matrix has no cell for. The fifth domain is the one that names not knowing which domain you are in, and dropping it to make a tidy four throws away the most useful part.

Multiplying binaries into a taxonomy. The MBTI assigns “a binary letter value to each of four dichotomous categories,” giving sixteen types, and the criticism lands exactly where the test predicts. The underlying traits are not bimodal: scores “were actually distributed in a centrally peaked manner, similar to a normal distribution, indicating that the majority of people were actually in the middle of the scale.” Cutting a normal distribution at the mean four times produces a type that is mostly an artefact of where the cuts fell, which is why “between 39% and 76% of respondents” get a different classification on retest after five weeks. A 1991 National Academy of Sciences committee called “the popularity of this instrument in the absence of proven scientific worth” troublesome.

The underlying scheme is Jung’s, from Psychological Types (1921), and Jung’s own version is more careful. His four functions, sensation, intuition, thinking and feeling, are already sorted by a binary: thinking and feeling are the rational or judging functions, sensation and intuition the irrational or perceiving ones. Crossed with the attitudes of introversion and extraversion, they give eight types rather than sixteen, and Jung’s claim for them is modest in the right way: psychological functions “remain the same in principle under different conditions and cannot be reduced to each other.” Irreducibility is asserted about the functions, not about the number.

The Argument That Four Is Never Primitive

There is a formal result in the neighbourhood, and anyone attracted to fourfolds should know it exists.

Peirce’s reduction thesis holds that “all relations, relations of arbitrary adicity, may be constructed from triadic relations alone,” while “monadic and dyadic relations alone are not sufficient to allow the construction of even a single ‘non-degenerate’ triadic relation.” Three is where the ladder stops going up and refuses to come down. Four-place relations, five-place, any number: all buildable from triads. The construction needs specific resources, including negation, a generalisation of De Morgan’s relative product, and the teridentity relation, and whether the thesis holds “entirely depends on exactly what constructive resources are to be allowed,” which was Quine’s objection and remains the live question.

If it holds, it settles this note’s question in the relational domain. There is no irreducibly quadratic relation, because any four-place relation decomposes into triads. Peirce’s own categories stop at Firstness, Secondness and Thirdness for exactly this reason, and his examples run “possibility, actuality, necessity” and “quality, fact, habit (or rule or law)” with nothing after.

This does not touch Aristotle’s causes, which are four questions rather than a four-place relation, nor Belnap’s values, which are four states rather than a relation at all. But it does mean that the intuition “four is richer than three” has a formal argument standing against it, and the burden is on the fourfold.

When Four Is the Right Number

Three questions, and a candidate should answer all three before it is allowed the word.

Does it decompose? Name the two axes. If you can, the four is a product, and you should ask what the axes are called before you ask what the quadrants are called. The elements decompose. The causes do not.

If it decomposes, do the relations between cells carry content? The square of opposition passes here and the growth-share matrix does not. This is the only thing that rescues a 2x2 from being two questions with extra steps.

Is the count contingent on the domain or imposed on it? Belnap has four because $2^2=4$. DNA has four because of ring counts and hydrogen bonds. Cynefin has five and gets drawn with four. Where the count is derived, it can be checked; where it is chosen, it usually turns out to be chosen for looking complete.

That last point is the one worth carrying out of this note. Fours feel complete in a way threes do not: they close, they square, they fill a page, they map onto the compass and the seasons and the mandala without effort. That feeling is the reason to be suspicious, not the reason to trust the structure. The same aesthetic pull that made the four elements survive two thousand years of not being true is what makes a consulting matrix persuasive in a meeting.

The discipline is the same one Ternary Thinking asks for, pointed at a different number: say what the count predicts, and if the answer is nothing, drop to the axes underneath. Most of the time there are two of them, and Binary Thinking is where the real claim was hiding.

Related Topics

• Binary Thinking - The two axes most fourfolds turn out to be made of
• Ternary Thinking - The third term, and Peirce’s argument that three is where irreducibility stops
• Dialectic - What happens when an opposition is treated as unstable rather than laid out in cells
• Quadrivium - Arithmetic, geometry, music, astronomy: a four that is a curriculum rather than a partition
• Four Freedoms - Interior, vocational, social and material freedom, a fourfold worth putting to the test above
• Syllogism - Where the square of opposition’s four forms do their work
• Teleology - The final cause, and what became of it
• Systems Thinking - The habit of asking what a framework’s cells actually predict
• Metacognition - Noticing the aesthetic pull of a complete-looking scheme
• Culture and Education - Domain overview

References

Four-valued logic

• Four-Valued Logic - Belnap’s four values as the powerset of {T, F}, the lattice, and IEEE 1364
• Paraconsistent Logic - Stanford Encyclopedia of Philosophy, on FDE, Dunn 1976, the 1966 dissertation, and the 1957 precedents
• Truth Values - Stanford Encyclopedia of Philosophy, on designated values and the Suszko reduction

The four corners

• Nāgārjuna - Stanford Encyclopedia of Philosophy, on the four alternatives regarding production
• Nāgārjuna - Internet Encyclopedia of Philosophy, on the catuṣkoṭi as refusal rather than counter-thesis
• Catuṣkoṭi - The four positions and their appearance in the Pali canon

Aristotle and the square

• Aristotle on Causality - Stanford Encyclopedia of Philosophy, the four causes as four explanatory roles
• The Traditional Square of Opposition - Stanford Encyclopedia of Philosophy, the four forms, the four relations, existential import
• Classical Element - Empedocles’ four roots and Aristotle’s derivation from hot/cold and wet/dry

Peirce

• Charles Sanders Peirce - Stanford Encyclopedia of Philosophy, the reduction thesis and the three categories
• Is Peirce’s Reduction Thesis Gerrymandered? - Sergiy Koshkin, on what the thesis depends on

Fours in use

• Genetic Code - The triplet arithmetic, Gamow, and the Nirenberg-Matthaei experiment of 1961
• Nucleobase - Purines and pyrimidines, and the two-versus-three hydrogen bonds
• Cynefin Framework - Snowden 1999, the five domains, and why it is not a matrix
• Psychological Types - Jung 1921, the four functions and the rational/irrational split
• Myers-Briggs Type Indicator - The retest figures and the centrally peaked distributions
• Johari Window - Luft and Ingham 1955, and the two crossed binaries
• SWOT Analysis - The disputed 1965 origins, Puyt 2023, and Hill and Westbrook 1997
• Growth-Share Matrix - Zakon, Henderson 1970, and the Slater and Zwirlein finding