On the Arithmetico-Logical Principle of Duality

The Logical Calculus contains two operations ⊕ and ⊗, having respectively the sense of 'or' and 'and'. They are connected by the following laws:

(α) Commutative x⊕y=y⊕x.

x⊗y=y⊗x.

(β) Distributive x⊗(y⊕z) = (x⊗y)⊕(x⊗z).

x⊕(y⊗z) = (x⊕y)⊗(x⊕z).

(γ) Associative x⊕(y⊕z) = (x⊕y)⊕z.

x⊗(y⊗z) = (x⊗y)⊗z.

We see that in these axioms, ⊕ and ⊗ occur symmetrically.