Prove (using induction) that no proposition is a proper initial segment of another proposition

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I feel very confuse on math logic. For the question above, I don't even know what is the base case. Can anyone give me some hint?

This is the definition of proposition:

  • If $P\in Props$, then $P$ is a proposition.
  • And if $\varphi$ and $\psi$ are propositions, then the following are all propositions:
    • $\neg\varphi$
    • $(\varphi\land\psi)$
    • $(\varphi\lor\psi)$
    • $(\varphi\to\psi)$
    • $(\varphi\leftrightarrow\psi)$
  • No other string of symbols is a proposition.