In logic, Nicod's axiom (named after the French logician and philosopher Jean Nicod) is a formula that can be used as the sole axiom of a semantically complete system of propositional calculus. The only connective used in the formulation of Nicod's axiom is the Sheffer's stroke.


The axiom has the following form:

((φ | (χ | ψ)) | ((τ | (τ | τ)) | ((θ | χ) | ((φ | θ) | (φ | θ)))))[1]


Nicod showed that the whole propositional logic of Principia Mathematica could be derived from this axiom alone by using one inference rule, called "Nicod's modus ponens":

1. φ

2. (φ | (χ | ψ))

∴ ψ[2]


In 1931, the Polish logician Mordechaj Wajsberg discovered an equally powerful and easier-to-work-with alternative:

((φ | (ψ | χ)) | (((τ | χ) | ((φ | τ) | (φ | τ))) | (φ | (φ | ψ))))[3]

References

  1. ↑ "Nic-ax - Metamath Proof Explorer".
  2. ↑ "Nic-mp - Metamath Proof Explorer".
  3. ↑ "Note (A) for Implications for Mathematics and Its Foundations: A New Kind of Science | Online by Stephen Wolfram [Page 1151]".
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