PA$^{-}\vdash I\Sigma_n \leftrightarrow I\Pi_n$.
Here $I\Sigma_n$ refers to the induction principle restricted to $\Sigma_n$ formulas. PA$^{-}$ is just PA without induction.
I was reading the paper by Paris and Kirby. In it they more or less give a one line proof for the above claim.
"Suppose $\theta(0) \wedge \forall x\ (\theta(x)\rightarrow\ \theta(x+1))$, but $\neg\theta(a)$, then induction also fails for $\exists w(a = y + w\ \wedge\ \neg\theta(w))$ and $\forall w(a = y + w\ \rightarrow\ \neg\theta(w))$"
I honestly have no idea what they are talking about here, hence if anyone could elaborate more on their reasoning I would deeply appreciate it.
Cheers