Nicod's axiom

Nicod's axiom (named after Jean Nicod) is an axiom in propositional calculus that can be used as a sole wff in a two-axiom formalization of zeroth-order logic.

The axiom states the following always has a true truth value.

((φ ⊼ (χ ⊼ ψ)) ⊼ ((τ ⊼ (τ ⊼ τ)) ⊼ ((θ ⊼ χ) ⊼ ((φ ⊼ θ) ⊼ (φ ⊼ θ))))

To utilize this axiom, Nicod made a rule of inference, called Nicod's modus ponens.

1. φ

2. (φ ⊼ (χ ⊼ ψ))

∴ ψ

In 1931, Mordechaj Wajsberg found an adequate, and easier-to-work-with alternative.

((φ ⊼ (ψ ⊼ χ)) ⊼ (((τ ⊼ χ) ⊼ ((φ ⊼ τ) ⊼ (φ ⊼ τ))) ⊼ (φ ⊼ (φ ⊼ ψ))))