There are logics where necessity is defined by means of a given identityconnective: $\square\varphi := \varphi\equiv\top$ ($\top$ is a tautology). Onthe other hand, in many standard modal logics the concept of propositionalidentity (PI) $\varphi\equiv\psi$ can be defined by strict equivalence (SE)$\square(\varphi\leftrightarrow\psi)$. All these approaches to modality involvea principle that we call the Collapse Axiom (CA): "There is only one necessaryproposition." In this paper, we consider a notion of PI which relies on theidentity axioms of Suszko's non-Fregean logic $\mathit{SCI}$. Then $S3$ provesto 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 necessityand PI are integrated as non-interdefinable concepts. CA is not valid and PIrefines SE. Models are expansions of $\mathit{SCI}$-models. We show that$\mathit{SCI}$-models are Boolean prealgebras, and vice-versa. This associatesNon-Fregean Logic with research on Hyperintensional Semantics. PI equals SE iffmodels are Boolean algebras and CA holds. A representation result establishes aconnection to Fine's approach to propositional quantifiers and shows that ourtheories are \textit{conservative} extensions of $S3$--$S5$, respectively. Ifwe exclude the Barcan formula and a related axiom, then the resulting systemsare still complete w.r.t. a simpler denotational semantics.
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