Some Instances of the Hexagon of Opposition in Mathematics, Aesthetics, and Politics


One of the things that logicians study is how opposition is expressed in formal or informal languages. The most obvious type of opposition appears when we negate a simple sentence: (1) “the cat is black” becomes (2) “the cat is not black”. In this case, (1) and (2) are strongly opposed to each other: they are contradictories. This sense of binarity is even stronger in mathematics, where natural numbers can be either even or uneven, with no other options. But often, we find more sophisticated ways to express opposition. A good example is that if we have eight people in front of us, we can refer to all, some, or none of them. We could, for example, say that all the people at the party have grey hair. What oppositions exist when we have not two, but three basic terms? It turns out that there are three types.

First of all, we have contradiction (strong opposition):

Diagram 1: Contradictories

We notice that there are not three but six corners here. This is because the negation of one of the original three options gives us either the first other option or the second. (“Not all” is the same as “some or none”, etc.) Three exclusive options (all-none-some) give rise to a total of six partially overlapping options. In the above diagram, a red line means that the corners are contradictory to each other. Contradiction here coincides with negation.

We can add another weaker notion of opposition to the diagram: contrariety. In this case, the corners connected to each other by a blue line are opposed but a third option is missing from either side of the opposition. For example, “all” is contrary to “none”, but “some” is a third option not covered by either. So contrariery is a weaker type of opposition than contradictoriness. There are three relations of contrariety in the following diagram:

Diagram 2: Contraries

The blue triangle touches the corners which are probably linguistically most familiar to us.

We can add a third notion of opposition, which is usually called subcontrariety. Two corners are subcontrary, signified by a green line, if taken together, they cover all the options with overlap. For example, “all or none” is subcontrary to “some or all”, since together, they cover all three options, overlapping on “all”.

Diagram 3: Subcontraries

To really give all the relevant structural relations that exist in such a hexagon, we should add six arrows. These are not instances of opposition, but of subalternation. To give a simple example, if it is true that all people in the room have grey hair, then it is a fortiori true that all or some people in the room have grey hair. So, this warrants an arrow from “all” to “some or all”, where the arrow means that the source corner is subaltern to the target corner. This gives us six arrows in total:

Diagram 4: Completed quantifier hexagon

These arrows point from the corners of the blue contrariety triangle to the corners of the green subcontrariety triangle.

This is one variant of the completed Hexagon as attributed to Augustin Sesmat and Robert Blanché, who discovered it independently in the 1950s.

It is interesting to note that not all corners of the hexagon have a single word from the English language corresponding to them (similarly for other languages). Nonetheless, all six options are easily expressed, and their meaning seems intuitive.

The vocabulary of all-some-none can be used to communicate important facts about domains of objects regardless of their size – it functions equally well for a domain of five objects as for an infinite domain. Only in the case of zero or exactly one object in the domain do things become a bit more complicated. (If we have a domain of n = 1, “some” will not describe any possible subset of the domain, i.e. there will be no option truly between “all” and “none”.)

If we assume the number of objects in our domain fixed (n > 1), we note that both “all” and “none” are true in exactly one scenario each, whereas “some” can be true in many distinct cases. Without additional vocabulary, no distinctions with regards to “some” can be made. (Think of “just a few”, “more than five”, “about half”, “nearly all”, etc.) This should not be regarded as a limitation but as a strength of these quantificational concepts. The active indifference to the middle region (“some”) limits us to three ranges and hence affords us an invariant conceptual schema applicable to domains of objects large or small. (Compare the sizes of the domains implied by “some of my siblings are female” and by “all natural numbers have a successor”.)

The underlying image is a fixed domain of objects (however large or small), with a coarse-grained but entirely standard vocabulary to refer a) specifically to the two limit cases (“all” or “none”) and b) to the range in the middle, ignoring the quantitative nuances that may exist in the middle. This vocabulary thus helps simplify any quantitative scenario according to the following picture:

Diagram 5: Outer limits

If we were so inclined, we could now start to understand the three types of opposition as relations on (the power-set of) a simple three-element set. The power-set of a three-element set has eight elements. If we ignore the empty set and the full set, we are left with six subsets. Three are the singleton sets of the original three options, three more are sets of two of the original options. Contradictory relations exist when two sets are each other’s complement in the Boolean algebra of subsets. One can easily describe the other two types of opposition in this fashion. The result is illustrated below:

Diagram 6: Hexagon of subsets

But for the moment, a mathematization of the underlying idea takes us away from the question why and when such conceptual hexagons emerge in ordinary language. This must have something to do with how this simple scheme actively suppresses some superfluous information in a general but useful way. We have seen how this works with concepts related to quantity. The question is: are there concepts that can be arranged hexagon-wise which cannot be explained as a special instance of the quantifier hexagon?

It has been pointed out by Robert Blanché in 1966 and by Jean-Yves Béziau more recently that many further (sets of) concepts can be presented quite nicely in the structure of a hexagon of opposition. The most famous example thereof is probably the modal hexagon of opposition, concerning the concepts of possibility, contingency, necessity, and impossibility:

Diagram 7: Modal hexagon

However, upon reflection, this hexagon is quite analogous to our earlier example. A ‘possible worlds’-interpretation of modal vocabulary allows us to regard this modal hexagon as a particular instance of the quantifier hexagon. For we can interpret “it is impossible that p” as “p holds in no world accessible from our own”. “It is necessary that p” becomes “p holds in every world accessible from our own”. The reader can easily complete this analogy by juxtaposing the two hexagons.

One may thus have the feeling that the two examples are not really all that different. Diagram 5 seems to describe the underlying simplification at work in both instances. With the modal vocabulary, the limit cases are of course “impossible” (in no accessible world) and “necessary” (in every accessible world); “contingent” (in some accessible world) then covers the wider middle ground.

If ternary (and senary) opposition is so fundamental to concept formation, we should be able to find examples of a different type. As far as I can tell, most of the current literature on the topic does not really investigate the relation between the stratification of the target domain to the hexagonal conceptualization thereof. Most of the time, it seems to be the structure of the hexagon itself which is foregrounded.[1] Let me introduce one class of examples which is different on this level. This will tease out what I mean by the stratification of the target domain.

In the quantification hexagon, the middle term was “some”, which we could call ‘quantitatively underdetermined’. (It is the middle term because “some” lies between “none” and “all”.) The outer terms were limit cases, only true in one scenario. The question can thus be posed whether other conceptual schemata can be found where the limit cases and the underdetermined cases are arranged differently. We may explore the following interesting option: what if the outer terms where quantitatively underdetermined, and the middle term a limit? This would yield the following diagram, which should be compared to diagram 5:

Diagram 8: Inner limit

Luckily, there is a beautiful example from set theory (the mathematical study of the higher infinite) which fits this schema. Mathematicians such as Georg Cantor have created all the concepts we need to fill in the corners of the following hexagon:

Diagram 9: Hexagon of cardinality (finite-infinite)

We can see that the semantics of these concepts cannot be mapped onto the earlier quantificational hexagon so easily. So, unlike the modal hexagon, this really is a different type of example altogether, organized around a limit term in the middle.

We can now ask whether the finite-infinite hexagon is an instance of another more general hexagon. We have already observed the importance of the middle term being a limit. A more general conceptual triple which gives rise to a hexagon would start from the following more abstract terms: inside, on the border, outside.[2] It seems that the more precise the border case is defined, the more precise the conceptual hexagon arising from such a scheme will be. “Countably infinite” is indeed the mathematically exact limit between the finite and the uncountably infinite. It seems that both here and before, the limit terms give the hexagon its proper ‘orientation’.

One can speculate what would happen if there were three limit cases and no underdetermined case at the base of a hexagon. In that scenario, we would merely be differentiating between three discrete options. But then there is no variance inside one of the regions, since there is no indeterminate range referred to by a single term. In that case, it seems that there is no real difference to common-sense discrete combinatorics. It seems to me that the salient examples of hexagons of opposition all seem to suppress some internal variance into a simpler structure.

In the case that there are two or three adjacent underdetermined terms, it may seem that the vocabulary also fails to simplify the field, in this case remaining too indiscrete. But a lot of examples are effectively like that. We now turn to two ‘impure’ examples of this type.

We all know that saying that something is “not ugly” is very different from saying that it is beautiful. In other words, beautiful and ugly are not contradictories, although they are at the opposite sides of a polarized spectrum. Saying that something is “not ugly” leaves open the option in which it is somewhere in between beautiful and ugly (“neutral”?). But to be in that in-between zone also means that it is not beautiful. – Is it not weird that we sometimes deny a negative with the true intention to deny a positive? Logic shows us that this behavior is not as ‘contradictory’ as it may seem. Rather, this option comes with the way such concepts are caught up in various forms of opposition:

Diagram 10: Hexagon of beauty (?)

“Not ugly” can become a euphemism for “neutral”, which is also “not beautiful” and hence, from the perspective of someone interested in the beautiful alone, on the side of the “ugly”.

One can be very critical of such double-distancing uses of language. But from a purely descriptive point of view, it seems that they are part and parcel of how we use language, and it is quite interesting to understand the underlying structural dimension to them. With regards to our earlier insistence on limits, it seems that they are more or less absent here. Is the neutral the limit between the beautiful and the ugly, or are the beautiful and the ugly the limits of a vast neutral range? Neither answer seems quite correct. Nonetheless, the three terms are ordered: neutral stands in between ugly and beautiful.

Here is another problematic example of a set of concepts which behaves in a hexagon-like fashion, this time relating to the chilly vocabulary of nationality and foreignness.

Diagram 11: Hexagon of belonging (?)

In the Dutch language, the word “allochtoon” refers to someone with a Dutch passport but who ‘came from elsewhere’ (from ἄλλος). Although it is fundamentally a distinction inside the group of people who have a Dutch passport (Dutch nationals), one grasps immediately how it can start to function as a problematic middle term between the fictional group of true natives and true foreigners.

Knowing something about opposition, we can anticipate the behavior of the middle term, without sanctioning its use, of course.

We can see quite clearly that this vocabulary has the effect of excluding the allochtoon from the problematic category of nativity, even though she has a Dutch passport and is thus a non-foreigner. As we can read from the diagram, someone with the intention to exclude the allochtoon can always put her in the category of the non-native, and hence in one category with the foreigner. But as a non-native, the allochtoon shares with the foreigner her ‘foreign origin’. We see how someone whose legal status is that of a Dutch national slip into the semantic field of that which is permanently marked as foreign.

My point in this final section has been that we can apply the same schema (the hexagon of opposition) to conceptual structures which we desire to critique. The existence of a hexagon is no proof for the dignity of the concepts involved. We have seen how the introduction of a middle term (allochtoon) can have the effect of positioning someone ‘towards the outside’ who is actually inside from a legal perspective (having a passport). The middle term is thus in a sort of perpetual double bind, which we have learned should not be mistaken for an inconsistency strictu sensu. Obviously, we are approaching the critique of racism here, which is certainly not reducible to the critique of binary thinking but must be addressed against the background of a more sophisticated web of oppositions (strongly suggested by the title of Frantz Fanon’s Peau noire, masques blancs).

References

Fanon, Frantz. Peau Noire, Masques Blanc. Paris: Seuil, 1952.

Béziau, Jean-Yves, and Gillman Payette, eds. The Square of Opposition: A General Framework for Cognition. Peter Lang, 2012.

Sesmat, A. Logique – II. Les Raisonnements, La Logistique. Paris: Hermann, 1951.

Blanché, R. “Sur L’opposition Des Concepts.” Theoria, no. 19 (1953): 89–130.

Moretti, Alessio. “The Geometry of Logical Opposition.” University of Neuchâtel, 2009. https://doc.rero.ch/record/12712/files/Th_MorettiA.pdf.

Notes

[1] The publication containing by far the largest number of examples, from the mundane to the technical to the extravagant, is Alessio Moretti’s thesis.

[2] Alternatively: on side A, between sides A and B, on side B.

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