I am currently studying metatheories of PL logic, and I am not sure why/when we need to use induction. For example, in proving soundness/completeness of a system we use induction, but in other cases where an induction seems called for, it is actually not needed.
There doesn't seem to be any significant difference in terms of the structure of the theorem that allows me to distinguish whether induction is needed or not; in the case of completeness, it is a conditional within the confines of universal quantifier (which applies to all formulas in the system). I would have thought that a simple if-then proof that proves $Γ \vdashφ$ by assuming $Γ \vDash φ$ would suffice.
On the other hand, the example below seems to apply to the entire system (because a system contains $n$ formulas, where $n$ equals to the set of natural number) which would lead me to think that induction is needed, but again I was wrong. (This is just one of many where I thought induction is needed but not really)
So why and when do we need to use induction? Is there any cue in the theorem that may help me to distinguish when is the time to use induction?
Theorem 3.7
The following general statements about the system S are equivalent.
(A) For all formulas $φ$ and all sets of formulas $Γ$, if $Γ \vdash φ$ then $Γ \vDash φ$.
(B) For all sets of formulas $Δ$, if $Δ$ is satisfiable then $Δ$ is consistent.
As the soundness theorem holds for S, statement (B) also holds for S.
As a consequence of the soundness theorem in the form (B) of Theorem 3.7, the system S is consistent. Prove this is true.
