Interface:Nicod axioms

The Nicod axioms, named for Jean Nicod, are an especially spartan axiomatic system for propositional logic, yet strong enough to derive all true statements. Nicod introduced it in his article A Reduction in the number of the Primitive Propositions of Logic.

Nicod makes use of a discovery by Henry Sheffer that all the logical truth functions can be expressed through a single one, the Sheffer stroke $$\vert$$, which classically means either Logical NAND or Logical NOR. Nicod defines the four classical truth functions negation, implication, disjunction and conjunction in terms of the version of the Sheffer stroke with the NAND meaning.

Let us first define well-formed formulas just as in Principia Mathematica, with $$p,q,r,s,t$$ as placeholders.  kind (wff) var (wff p q r s t) 

Truth functions
The basic truth function of the Nicod system is the Sheffer stroke $$\vert$$:  term (wff (| wff wff))  Through the Sheffer stroke, negation $$\neg$$, implication $$\rightarrow$$, disjunction $$\vee$$, and conjunction $$\wedge$$ can be defined:  def ((¬ p) (p | p)) def ((→ p q) (p | (q | q))) def ((∨ p q) ((p | p) | (q | q))) def ((∧ p q) ((p | q) | (p | q))) 

Nicod's axioms
Just as Principia, Nicod's system requires some rule of detachment. It is as follows:  stmt (applyRule (p (p | (q | r))) r)  Note that if $$q$$ were replaced by $$r$$ in this rule, it would just be the classical modus ponens. Note that if $$p$$ is true, then $$p\vert(q\vert r)$$ can only be true, if both $$q$$ and $$r$$ are true, so the rule is sound as it stands, despite $$q$$ not appearing in the consequent.

The only further axiom, called the Prop for proposition, is as follows.  stmt (Prop  ((p | (q | r)) | ((t | (t | t)) | ((s | q) | (¬ (p | s))))))  This is basically a generalised form of a combination of the Principia Identity $$p\rightarrow p$$ and Syllogism $$(p\rightarrow q)\rightarrow((q\rightarrow s)\rightarrow(p\rightarrow s))$$.

These two axioms are sufficient to derive all true formulas of classical propositional logic. However, one should mention that proving the Principia axioms from the Prop and the above rule detachment is not as easy a matter as Nicod imagined. For in both the Principia, implication are derived concepts. In Principia, implication is derived from negation and disjunction, while Nicod naturally derives it from the Sheffer stroke. Unfortunately, these two derivations are not compatible. This gap was first noticed by B. A. Bernstein.