Conceivably you could get some proof-theoretic understanding of "induction without the base", i.e., the phenomenon of how strengthening property $P_{\rm weak}$ to $P_{\rm strong}$ can make the implication $P(n) \to P(n+1)$ easier to prove. However, $P_{\rm strong}(1)$ and $P_{\rm weak}(1)$ are different, and it is hard to imagine how a theory of the base case could possibly be set up. Maybe when considering families of $P$ something could be said, but otherwise one is up against the full strangeness of the finite and accidental.