Build an axiomatic system that is effective, complete but not valid.
I thought as the only deduction rule $\psi \rightarrow \neg \psi $ and only axiom n $\approx$ n where theorem is $\neg[$ n $\approx$ n] but this is not true. Is this correct or how I can make the construction. Thanks
If we assume a language with the $\lnot$ connective, we can use your proposed rule :
from $\psi$, infer $\lnot \psi$
to derive $\lnot (n ≈ n )$ from the axiom.
In this case the system is not sound, because we can prove a false theorem.
But from a theorem $\varphi$ wathever, applying the rule again, we can derive : $\lnot \varphi$, i.e. its negation.
Thus, the system is complete in a trivial way.
I use Polish notation.
Suppose our only rule of inferences are detachment from {$\vdash$C$\alpha$$\beta$, $\vdash$$\alpha$} we may infer $\beta$, and uniform substitution.
Suppose our only only axiom is Cpq.
Now what can we do with Cpq? Well, let's say we substitute p with a and q with b obtaining Cab. Thus, we have Cpq and Cab as theorems. Can we place either of these theorems into the antecedent of the other theorem? Sure, suppose we substitute p with Cab. Then we have CCabq as a theorem. Then since we have CCabq, and Cab as theorems, we can detach q as a theorem. But then we can substitute q by "falsum" or "0". Consequently, if we have two truth values this system is unsound (but if we only had one truth value it turns out that our system would qualify as sound, and Cpq could serve as the sole axiom for "one-valued" logic).
