Around and Beyond the Square of Opposition pp 293-301 | Cite as

# Hypercubes of Duality

Chapter

## Abstract

We define hypercubes of duality—of which the modern square of opposition is an emblematic example—in proper mathematical terms, as orbits under some action of the additive group \(\mathbb{Z}_{2}^{m}\), with *m*∈ℕ. We then introduce a notion of dimension for duality in classical logic and show, for example, how propositional expressions in at most three variables can be classified according to that notion. The paper ends with logical formulations of some representation theorem for Galois connections on powersets, where we see an underlying square of duality.

### Keywords

Square of opposition Duality Galois connections### Mathematics Subject Classification

03B05 03B10 03B80### References

- 1.Adámek, J., Herrlich, H., Strecker, G.E.: Abstract and Concrete Categories: The Joy of Cats. Dover, Mineola (2004) Google Scholar
- 2.Dubreil, P.: Algèbre, Tome 1, 2ième édn. Cahiers scientifiques, Fascicule XX. Gauthier-Villars, Paris (1954) Google Scholar
- 3.Erné, M., Koslowski, J., Melton, A., Strecker, G.E.: A primer on Galois connections. In: Papers on General Topology and Applications. Annals of the New York Academy of Sciences, vol. 704, pp. 103–125 (1993) Google Scholar
- 4.Gottschalk, W.H.: The theory of quaternality. J. Symb. Log.
**18**, 193–196 (1953) MathSciNetMATHCrossRefGoogle Scholar - 5.Ore, O.: Galois connexions. Trans. Am. Math. Soc.
**55**, 493–513 (1944) MathSciNetMATHGoogle Scholar - 6.Reichenbach, H.: The syllogism revised. Philos. Sci.
**19**, 1–16 (1952) CrossRefGoogle Scholar

## Copyright information

© Springer Basel 2012