Show that: A is consistent iff Ded(A) does not equal Fml(L)

Question

For a language L and set of L-formulas A we must show that

A is consistent iff $Ded(A) \ne Fml(L)$ where

$Ded(A)$ = set of all formulas deduced from A

$Fml(L)$ = set of all L formulas

The => direction I am happy with, however I'm not sure where to start with the <= direction. My solutions just say it is an application of Modus Ponens however I still cant spot what I need to do.

edit: formatting

Answer 1

We assume that $A$ is inconsistent iff for all $\varphi, A \vdash \varphi$ and $A \vdash \lnot \varphi$.

For the $\Leftarrow$ direction : if $Ded(A) \ne Fml(L)$ [i.e. there is at least one $\psi \in Fml(L)$ such that $\psi \notin Ded(A)$], then $A$ is consistent, we can prove it by contraposition, showing that :

if $A$ is inconsistent, then for all $\psi \in Fml(L), A \vdash \psi$, i.e. $\psi \in Ded(A)$,

we have to use the proof system.

With natural Deduction, we have to use the $\lnot$-Elimination rule :

from $\varphi$ and $\lnot \varphi$, infer $\bot$

and then apply the $\bot$-rule : $\bot \vdash \psi$ to conclude that $A \vdash \psi$, for any $\psi$.

For Hilbert-style proof system, we need the tautology :

$B \rightarrow (\lnot B \rightarrow A).$