Given the spot rates for a zero coupon security maturing at time 1 and a zero coupon security maturing at time 2, (where time 1 < time 2), it is possible through bootstrapping, to calculate the forward rate for the period between time 1 and time 2.
Similarly, given the spot volatilities for the period from $t_0$ to $t_1$ ($σ_{t_0,t_1}$) and from $t_0$ to $t_2$ ($σ_{t_0,t_2}$) respectively, it is possible to infer the expected volatility between $t_1$ and $t_2$ ($σ_{ t_1,t_2}$). This is the forward implied volatility (also known as the forward-forward volatility) for the period [$t_1$, $t_2$].
In ‘Dynamic Hedging’ Nicholas Nassim Taleb, assuming equal time intervals, presents a formula for computing the annualized forward implied volatility for the period between [$t_0, t_2$], , as follows:
$\sigma^2_{t_0,t_2}=\sigma^2_{t_0,t_1}+\sigma^2_{t_1,t_2}$
If the implied volatility were the volatility of independent log returns the use of the additive property of variance would be clear. Suppose, on the contrary, that the implied volatility were the volatility of the price of a share. How can you justify the additive form?