Hypercubes of Duality

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.

Appendix

Here we list all the orbits for the action of Example 2 when m=3. These are classified according to the number b of black dots in the cube-shape representation of a propositional expression, and their possible arrangements up to permutations of the variables p ₁, p ₂, p ₃. For each configuration $\mathcal{C}^{b}_{k}$, we give one generic propositional expression as example, with dimension $|\mathcal{C}^{b}_{k}|$. The index k just serves here as a numbering for different configurations with the same b. We use multiplicative notation (namely a dot) for conjunction and write E _ij for (p _i ⇔p _j ), with i≠j. Note that the actions of ι(1 ₁), ι(1 ₂), ι(1 ₃) preserve the numbers of white dots and black dots, whereas the action of ι(1 ₄) must exchange these numbers; so the values of b are restricted to 0, 1, 2, 3, 4.

The generic propositional expression for $\mathcal{C}^{4}_{6}$ is in fact logically equivalent to p ₁⇔p ₂⇔p ₃, which is not to be confused with E ₁₂⋅E ₁₃⋅E ₂₃, that is, $\mathcal{C}^{2}_{3}$. Note also that the ternary connective ‘If… , then… ; else…’, i.e. the propositional expression (p ₁⇒p ₂)∧(¬p ₁⇒p ₃), coincides with $(\mathcal{C}^{4}_{3})^{\iota(\boldsymbol {1}_{1})}$.

Now, in the next table, we count the number $n^{b}_{k}$ of orbits one obtains by permuting the variables p ₁, p ₂, p ₃ from each configuration $\mathcal{C}^{b}_{k}$, and then the number $m^{b}=\sum_{k} 2^{|\mathcal{C}^{b}_{k}|}\cdot n^{b}_{k}$ of propositional expressions whose orbits are generated by a cube-shaped domino with b black dots.

So the number of orbits is $\sum_{b,k} n^{b}_{k}=30$, and we have ∑ _b m ^b=256, which is all right equal to the number of propositional expressions in (at most) three variables—up to logical equivalence. For those familiar with finite group theory, we note that the total number of orbits under the action Example 2 can easily be obtained, for any m, by Burnside’s counting theorem.