I was recently introduced to categorical arguments and, since the square of opposition is merely a mnemonic, it doesn't seem completely unreasonable to me that there could be a 3-dimensional analogue (also a mnemonic) that allows the inferences for three categorical propositions to be carried out by geometric means, like the modern/traditional square of opposition. But the more I think about it, the less attainable it seems.

The square of opposition works because every categorical proposition is of the form "something A something B" so we have two classes A,B involved for every one categorical proposition, but for three sets we would need at least two categorical propositions so that relationships between three classes could be introduced. So now we have to assign two propositions (a "couple") to every vertex in our "cube of opposition" and using elementary combinatorics we can conclude that the allowable combinations of A, E, I, and O propositions (excluding "couples" that do not introduce a third variable i.e. all x are y, no y are x is a "couple" that is not allowed) is greater than the 8 vertices of a cube. My question is this: is what I've described here even possible? I'm sure other people have had similar ideas since the idea is very intuitive, but is it even feasible? Maybe it is possible but its use as a mnemonic is ludicrous since it's so complicated?


The cube exists, is known (shockingly recently, though!), and is one of an infinite series of such constructs. As you say, it's not obviously of much use; just a mathematical curiosity (and a pretty obvious one at that IMO, but maybe the intersection of philosophers and algebraists/topologists is not well inhabited).

  • I agree; it is shocking to think that its conception is less than a decade old. Thanks for the link. – Hautdesert Nov 20 '12 at 0:13

Generalizations of the square of opposition have been known since at least the middle ages, although the usual context in which they were investigated was modal. The journal logica universalis often publishes historical and contemporary work on geometrical logical relations like hexagons of opposition, etc.

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