There is an alternative semantics for multi-modal (epistemic) logics based on (world, agent) pairs (see, e.g., Grove, Artificial Intelligence, 1995). Denote it by \(Sem_{\langle w,a\rangle }\) . It is effectively applied to multi-modal logics with quantification over agents of knowledge (or over their names). In this paper, we consider propositional modal logic with quantification over propositions ( \(SOPML\) ) and introduce an alternative semantics for this logic also based on pairs, but pairs of a slightly different kind, namely, on (world, proposition) pairs. Denote it by \(Sem_{\langle w,p\rangle }\) . Some key properties of \(Sem_{\langle w,a\rangle }\) , as well as the principles for constructing this semantics will be taken as a basis to define \(Sem_{\langle w,p\rangle }\) . Within the framework of the introduced semantics, we define general (or Henkin) frames and present a decidable modal two-variable fragment of \(SOPML\) interpreted on these frames ( \(SOPML^H_{tvf}\) ). \(SOPML^H_{tvf}\) allows us to express all challenges considered in (Shtakser 2023a, J.Log.Lang.Inf.,) and inexpressible in a decidable modal loosely guarded fragment of \(SOPML\) presented in that paper. The fragment \(SOPML^H_{tvf}\) partially satisfies the principle of non-Fregean logic: two different atomic propositions with the same truth value can have different contents. Also we define relating connectives in \(SOPML^H_{tvf}\) and prove that the weak Boethius’ Thesis built using these connectives is a valid formula of this fragment.
