Chapter

Around and Beyond the Square of Opposition

Part of the series Studies in Universal Logic pp 293-301

Hypercubes of Duality

  • Thierry LibertAffiliated withDépartement de mathématiques, Université Libre de Bruxelles (U.L.B.) Email author 

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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