KD45 with propositional quantifiers (Q6559161)

This paper is an investigation into the principle of ``epistemic modesty'' formalized by Steinsvold as follows: \(\square(\exists p)(\square p\wedge \neg p)\), where \(\square\) is the modality for belief. This principle states that it is rational to believe that some of our beliefs are false. The author presents two semantic systems, one constructed in terms of Kripke models, and the other in terms of topological spaces, and which are also due to Steinsvold. He then formulates a deductive system EM, which is an extension of the modal propositional system KD45\(_\square\) by means of the following axioms for propositional quantifiers: \((\exists p) (p \wedge\square (p \leftrightarrow \phi))\) if \(p \notin FV(\phi)\); \((\exists p) (\neg p \wedge\square (p \leftrightarrow \phi))\) if \(p \notin FV(\phi)\); \(\lozenge \phi \rightarrow (\exists p) (\lozenge(\phi \wedge p) \wedge \lozenge(\phi \wedge\neg p))\) if \(p \notin FV(\phi)\); \((\forall p)(\phi\rightarrow \psi) \rightarrow (\forall p)\phi\rightarrow(\forall p)\psi\), and the rule: \(\phi / (\forall p)\phi\) (if \(p\) is not free in any open assumption). It is then proved that the system EM is sound and complete with respect to both semantics and thus the two semantics are equivalent.