Nicod's axiom

Nicod's axiom

Nicod's axiom 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))))[1]

To utilize this axiom, Nicod made a rule of inference, called Nicod's Modus Ponens.

1. φ

2. (φ ⊼ (χ ⊼ ψ))

∴ ψ[2]

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))))[3]
  1. ^ http://us.metamath.org/mpegif/nic-ax.html
  2. ^ http://us.metamath.org/mpegif/nic-mp.html
  3. ^ http://www.wolframscience.com/nksonline/page-1151a-text

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