  There are logics where necessity is defined by means of a given identity
connective: $\square\varphi := \varphi\equiv\top$ ($\top$ is a tautology). On
the other hand, in many standard modal logics the concept of propositional
identity (PI) $\varphi\equiv\psi$ can be defined by strict equivalence (SE)
$\square(\varphi\leftrightarrow\psi)$. All these approaches to modality involve
a principle that we call the Collapse Axiom (CA): "There is only one necessary
proposition." In this paper, we consider a notion of PI which relies on the
identity axioms of Suszko's non-Fregean logic $\mathit{SCI}$. Then $S3$ proves
to be the smallest Lewis modal system where PI can be defined as SE. We extend
$S3$ to a non-Fregean logic with propositional quantifiers such that necessity
and PI are integrated as non-interdefinable concepts. CA is not valid and PI
refines SE. Models are expansions of $\mathit{SCI}$-models. We show that
$\mathit{SCI}$-models are Boolean prealgebras, and vice-versa. This associates
Non-Fregean Logic with research on Hyperintensional Semantics. PI equals SE iff
models are Boolean algebras and CA holds. A representation result establishes a
connection to Fine's approach to propositional quantifiers and shows that our
theories are \textit{conservative} extensions of $S3$--$S5$, respectively. If
we exclude the Barcan formula and a related axiom, then the resulting systems
are still complete w.r.t. a simpler denotational semantics.
