= Truth-matrix countermodel for a Hilbert axiom set
{title2=$N(0)=N(1)=0$}
Take ordinary material <logical implication> on $\{0,1\}$, designate $1$, and interpret <negation> by the constant-zero operation $N$. Then $A\to(B\to A)$ and $(NB\to A)\to(NA\to B)$ are always designated, and <modus ponens rule> and <logical implication> discharge preserve consequence. But $(NB\to NA)\to(A\to B)$ is false for $A=1,B=0$. Thus the first two schemes and the deduction rule do not derive the last scheme. This detects the loss of a genuine classical-negation principle even though all the displayed schemes are tautologies under the usual classical <negation>.
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