  We develop a model to treat the inclusive non-elastic break up reactions
involving weakly bound three-cluster nuclei. Borromean, two-nucleon, halo
nuclei are candidates of unstable three-fragments projectiles. The model is
based on the theory of inclusive breakup reactions commonly employed in the
treatment of incomplete fusion and surrogate method. The theory was developed
in the 80's by Ichimura, Autern and Vincent (IAV) [Phys. Rev. C 32, 431 (1985)]
\cite{IAV1985}, Udagawa and Tamura (UT)[Phys. Rev. C 24, 1348 (1981)],
\cite{UT1981} and Hussein and McVoy (HM)[Nucl. Phys. A 445, 124 (1985)],
\cite{HM1985}. We extend these three-body theories to derive an expression for
the fragment yield in the reaction $A\,(a,b)\,X$, where the projectile is $a =
x_1 + x_2 + b$. The inclusive breakup cross section is found to be the sum of a
generalized four-body form of the elastic breakup cross section plus the
inclusive non-elastic breakup cross section which involves the "reaction" cross
section of the participant fragments, $x_1$ and $x_2$. The final result is
similar to the three-body case reviewed in Austern, et al. (Phys. Rep.
\textbf{154}, 125 (1987)), \cite{Austern1987}, but with important genuine
four-body effects added, both in the elastic breakup cross section, which now
contains the full correlations between the participant fragments, and in the
inclusive non- elastic breakup. These developments should encourage
experimentalists to seek more information about the $x_{1} + x_{2}$ system in
the elastic breakup cross section, and to theorists to further develop and
extend the surrogate method, based on the inclusive non-elastic breakup part of
the $b$ spectrum.
