TITLE: Why modal logic needs modalities
QUESTION [5 upvotes]: Why does modal logic need modalities like provability or others like necessity and possibility? Could they be replaced with equivalent predicates?

REPLY [3 votes]: A notable example of what you describe (if I understand you correctly) is Lewis's Counterpart Theory (cf. "Counterpart Theory and Quantified Modal Logic")
Counterpart Theory
Counterpart theory seeks to provide an alternate formalization of QML (Quantified Modal Logic). The language of counterpart theory contains four primitive predicates: $Wx$ ($x$
is a possible world), $Ixy$ (an individual $x$ inhabits a possible
world $y$), $Ax$ ($x$ is actual) and $Cxy$ ($x$ is a counterpart
of $y$). The domain of quantification consists of possible worlds
and the individuals that make up everything in those worlds. The following
postulates help to constrain the use of these predicates:
P1: $\forall x\forall y\,(Ixy\rightarrow Wy)$ (Nothing
is in anything except a world)
P2: $\forall x\forall y\forall z\,(Ixy\wedge Ixz\rightarrow y=z)$
(Nothing is in two worlds)
P3: $\forall x\forall y\,(Cxy\rightarrow\exists z\, Ixz)$
(Whatever is a counterpart is in a world)
P4: $\forall x\forall y\,(Cxy\rightarrow\exists z\, Iyz)$(Whatever
has a counterpart is in a world)
P5: $\forall x\forall y\forall z\,(Ixy\wedge Izy\wedge Cxz\rightarrow x=z)$
(Nothing is a counterpart of anything else in its world)
P6: $\forall x\forall y\,(Ixy\rightarrow Cxx)$ (Anything
in a world is a counterpart of itself)
P7: $\exists x\,(Wx\wedge\forall y\,(Iyx\leftrightarrow Ay))$
(Some world contains all and only actual things)
P8: $\exists x\, Ax$ (Something is actual)
The Translation Procedure 
Where $\phi$ is a closed sentence (i.e. a sentence with no free variables),
the possibilist sentence $\exists\beta(W\beta\,\&\,\phi^{\beta})$
($\phi$ holds in some possible world $\beta$) coincides in truth
conditions with the sentence $\Diamond\phi$. The sentence $\phi^{\beta}$
($\phi$ holds in world $\beta$) coincides in truth-conditions with
$\phi$ (of QML) when the quantifiers in $\phi$ are restricted to
the inhabitants of $\beta$. This restriction is achieved by replacing
every existential quantifier in $\phi$ with $\exists\alpha(I\alpha\beta\,\&\,\cdots)$
and every universal with $\forall\alpha(I\alpha\beta\supset\cdots)$.
Let $\iota\, x\,\forall y\,(Iyx\,\equiv\, Ay)$ denote the actual
world, abbreviate this $@$ (Lewis 1968, p. 114). The sentence $\phi^{@}$
(i.e., $\exists\beta(\forall\alpha\,(I\alpha\beta\equiv A\alpha)\,\&\,\phi^{\beta})$)
is equivalent to the sentence $\phi$ of QML. We can then go on to
fix the truth-conditions of $\phi^{\beta}$ ($\phi$ holds at world
$\beta$) for an infinite range of contexts as follows:
$\phi^{\beta}$ iff $\phi$, for atomic $\phi$
$(\sim\phi)^{\beta}$ iff $\sim\phi^{\beta}$
$(\phi\,\&\,\psi)^{\beta}$ iff $\phi^{\beta}\,\&\,\psi^{\beta}$
$(\phi\,\vee\,\psi)^{\beta}$ iff $\phi^{\beta}\,\vee\,\psi^{\beta}$
$(\phi\supset\psi)^{\beta}$ iff $\phi^{\beta}\supset\psi^{\beta}$ 
$(\phi\equiv\psi)^{\beta}$ iff $\phi^{\beta}\equiv\psi^{\beta}$
$(\forall\alpha\phi)^{\beta}$ iff $\forall\alpha(I\alpha\beta\supset\phi^{\beta})$
$(\exists\alpha\phi)^{\beta}$ iff $\exists\alpha(I\alpha\beta\,\&\,\phi^{\beta})$
$(\Box\phi\alpha_{1}\ldots\alpha_{n})^{\beta}$ iff $\forall\beta_{1}\forall\gamma_{1}\ldots\forall\gamma_{n}(W\beta_{1}\,\&\, I\gamma_{1}\beta_{1}\,\&\, C\gamma_{1}\alpha_{1}\,\&\ldots\,\&\, I\gamma_{n}\beta_{1}\,\&\, C\gamma_{n}\alpha_{n}\supset\phi^{\beta_{1}}\gamma_{1}\ldots\gamma_{n})$ 
$(\Diamond\phi\alpha_{1}\ldots\alpha_{n})^{\beta}$ iff $\exists\beta_{1}\exists\gamma_{1}\ldots\exists\gamma_{n}(W\beta_{1}\,\&\, I\gamma_{1}\beta_{1}\,\&\, C\gamma_{1}\alpha_{1}\,\&\ldots\,\&\, I\gamma_{n}\beta_{1}\,\&\, C\gamma_{n}\alpha_{n}\,\&\,\phi^{\beta_{1}}\gamma_{1}\ldots\gamma_{n})$ 
This provides you with a means of ridding yourself of the modal operators by re-interpreting them quantificationally (as quantifiers over worlds and possible individuals). Strictly speaking it isn't equivalent. While everything sentence of QML has a counterpart theoretic equivalent, not every sentence of Counterpart Theory has an equivalent in QML.
What I mean to show here is eliminability of the modal operators. This is one (famous) example of the sort of thing I imagine Henning had in mind in his answer. It is non-modal in the sense that it doesn't have modal operators.