In this paper, we explore possibilities and difficulties to prove super-quadratic formula size lower bounds from the following aspects. First, we consider recursive Boolean functions and prove their general formula size upper bounds. We also discuss recursive Boolean functions based on exact 2-bit functions. We show that their formula complexity are at least Ω(n²). Hence they can be candidate Boolean functions to prove super-quadratic formula size lower bounds. Next, we consider the reason of the difficulty of resolving the formula complexity of the majority function in contrast with the parity function. In particular, we discuss the structure of an optimal protocol partition for the Karchmer-Wigderson communication game.