What does it mean for a formula to be a truth functional component/compound of another formula? I'm reading Fine and Schurz's "Transfer Theorems for Multimodal Logics" in Logic and Reality: Essays on the Legacy of Arthur Prior edited by J. B. Copeland (Clarendon Press, 1996, pp. 169-213) and I've met the following definitions in page 177:

The set of truth-functional constituents$^1$ TC($\phi$) of a given formula $\phi$ is the set of all constituents of $\phi$ which are truth-functional components of $\phi$.


Finally, B($\Delta$) is the set of all Boolean, or truth-functional, compounds of formulae in $\Delta$.

$^1$ Consider this to mean formulas which are atomic.
Neither of which I can make sense of. I understand that truth-functionality relates to the  connectives of propositional logic, that the modal operators are not truth-functional and that component formulae usually mean formulae which are contained in another, bigger, formula, but I can't understand what the words "component" and "compound" may mean in this context, nor how they are different from one another.
While researching for this, I've found a paper by D. H. Sanford titled "What is a Truth-Functional Component" (free to read on site) which, although the title may sound convenient, was not of great help. Any help would be appreciated, and references to books/papers which explain these concepts would be great.
 A: Let us dwell on the question a bit longer.
Modal logic is an intensional logical system. Some people equate intension in this technical sense to meaning. However, this is a far too simplistic view of such a comprehensive concept as meaning. It is more appropriate to regard intension as a mode of meaning that stipulates how a statement relates to an extension (also a mode of meaning).
Hence in intensional systems of logic, there is at least one operator that puts statements into an intensional context which cancels truth-functionality, that is, we cannot set down a table with columns of statements as input and a column of truth-values as output. Thus, two propositions $P$ and $Q$ may have the same truth value, but $\Box P$ and $\Diamond Q$ diverges radically in their consequences - contrast them, for example, with the behaviour of negation $\neg$. Notice that being truth-functional or not is actually an attribution of logical operators.
The standard modal propositional logic is unimodal (a.k.a., monomodal); it has one modal (non-truth-functional) operator as primitive, either $\Box$ or $\Diamond$, the other one is defined by the one taken as primitive. The rest of the operators are truth-functional.
In the cited article, Fine and Schurz investigate stratified multimodal systems. For the sake of simplicity, they consider a bimodal system with modal operators $\Box_{1}$ and $\Box_{2}$ which do not interact logically.
By components, we take a synthetic point of view; we put together components to form an object. The logical systems $L_{l}$ and $L_{2}$ and formal languages $\mathscr{L}_{l}$ and $\mathscr{L}_{2}$ induced by $\Box_{1}$ and $\Box_{2}$, respectively, are the components of the logical system $L$ , the minimal normal bimodal logic containing both of them and and the associated formal language $\mathscr{L}$.
Constituents of an object give us an analytic view of it; we set apart constituents of an object to see the way it is built up. There is a distinction between the constituents of a formula $\phi$ and the subformulas of it. Keeping with the cited article, $SF_{1}(\phi)$ is the set of all $1$-subformulas of $\phi$ viewed as an $\mathscr{L}_{1}$-formula which are all formulas having an occurrence in $\phi$ which does not "properly lie in the scope of" a $\Box_{2}$-operator, because $\mathscr{L}_{1}$ does not have a syntactic rule to prefix $\Box_{2}$ to a formula. Thus, $\Box_{2} q$ cannot be a subformula in $\mathscr{L}_{1}$, but $\Box_{1}\Box_{2} q$ can. $1$-constituents form a set of formulas $\Sigma_{1}$ defined by $\mathbb{P}\,\cup\,\{\Box_{2}\phi\mid\phi\in\mathscr{L}\}$, where $\mathbb{P}$ is the set of propositional variables.
Notice that  $\Sigma_{1}$ is virtually the set of the atomic variables (i.e., not compound formulas) of $\mathscr{L}$ as seen relative to $\mathscr{L}_{1}$. $C_{1}(\phi)$ is the set of all $1$-constituents of $\phi$ which are the $1$-subformulas of $\phi$ which are $1$-constituents. Therefore, $C_{1}(\phi) = SF_{1}(\phi)\cap\Sigma_{1}$. Mutatis mutandis, we can define the same notions for $\Box_{2}$, and for $\mathscr{L}$ combine them, as $\Sigma = \Sigma_{1}\,\cup\,\Sigma_{2}$ and so on, in particular, $SC(\phi)$, the set of all subconstituents of $\phi$ in $\mathscr{L}$, defined by $SC(\phi) = SF(\phi)\,\cap\,\Sigma$.
It may be helpful to remark that $\Sigma=\mathbb{P}\,\cup\,\{\Box_{1}\phi\mid\phi\in\mathscr{L}\}\,\cup\,\{\Box_{2}\phi\mid\phi\in\mathscr{L}\}$, hence, the set of all constituents in $\mathscr{L}$ consists of truth-functional, non-compound formulas and non-truth-functional, compound (viz., modal) formulas. So, we see that $\Sigma$ is the set of all $\mathscr{L}$-formulas which are not truth-functionally compound.
Let us look into identification of the set of truth-functional constituents of a formula $\phi$, $TC(\phi)$ going over the authors' example on pp. 176-177:
$$\phi\equiv\Box_{2}q\wedge\Box_{1}\neg\Box_{1}\Box_{2}p$$
$SF_{1}(\phi) =\{\Box_{2}q, \Box_{1}\neg\Box_{1}\Box_{2}p, \neg\Box_{1}\Box_{2}p, \Box_{1}\Box_{2}p, \Box_{2}p\}$ since $\mathscr{L}_{1}$ has a syntactic rule to prefix $\Box_{1}$. Then, $C_{1}(\phi) =\{\Box_{2}q, \Box_{2}p\}$, the atomic formulas occurring in $\phi$.
$SF(\phi) =\{\Box_{2}q, \Box_{1}\neg\Box_{1}\Box_{2}p, \neg\Box_{1}\Box_{2}p, \Box_{1}\Box_{2}p, \Box_{2}p, p, q\}$ since $\mathscr{L}$ has a syntactic rules to prefix both $\Box_{1}$ and $\Box_{2}$.
$SC(\phi) =\{\Box_{2}q, \Box_{1}\neg\Box_{1}\Box_{2}p, \Box_{1}\Box_{2}p, \Box_{2}p, p, q\}$. Notice that we have omitted $\neg\Box_{1}\Box_{2}p$ from $SF(\phi)$ to obtain $SC(\phi)$, since a negated formula is a truth-functionally compound formula with respect to both $\mathscr{L}_{1}$ and $\mathscr{L}_{2}$. Consequently, $TC(\phi)=\{p, q\}$.
Suppose $\theta\equiv\Box_{2}q$ and $\psi\equiv\Box_{1}\neg\Box_{1}\Box_{2}p$, so that their conjunction gives the formula $\phi$ in the authors' example.
Let $\Delta=\{\theta, \psi\}$. Then $B(\Delta)=\{\neg\theta, \neg\psi, \theta\wedge\psi, \theta\vee\psi\ldots\}$. So, $\phi\in B(\Delta)$.
