Proof a rule to be admissible.

Question

If I have a rule of the form $\phi_{0\dots n}/\psi$, can I show that it is admissible, i.e. that if all premises are true then the conclusion is also true by showing that $\models\bigwedge_{k=0}^{n}\phi_k\rightarrow\psi$ provided that the concerned logic fulfills the deduction theorem?

Answer 1

In general, no.

You can do such if you have a completeness meta-theorem and invoke it.

Say you work with a system which just has a conditional introduction rule and a conditional elimination rule:

{(cond $\alpha$ $\beta$), $\alpha$} $\vdash$ $\beta$

You might show that model-theoretically (semantically/using truth tables) the following holds:

|= (cond (cond (cond $\alpha$ $\beta$) $\alpha$) $\alpha$)

But, you can't admit the rule of inference

(cond (cond $\alpha$ $\beta$) $\alpha$) $\vdash$ $\alpha$ for the above system.

If you did then

$\vdash$ (cond (cond (cond $\alpha$ $\beta$) $\alpha$) $\alpha$)

which doesn't work for the above system, and you could find a model which satisfies the conditional introduce rule, the conditional elimination rule, but doesn't satisfy the last formula.