An Admissible Semantics for Propositionally Quantified Relevant Logics
Robert Goldblatt and Michael Kane
The Routley-Meyer relational semantics for relevant logics is extended to give a sound and complete
model theory for many propositionally quantified relevant logics (and some non-relevant ones).
This involves a restriction on which sets of worlds are admissible as propositions, and an
interpretation of propositional quantification that makes
&forallpA true when there is some true admissible proposition that entails
all p-instantiations of A.
It is also shown that without the admissibility qualification
many of the systems considered are semantically incomplete, including all those that are
sub-logics of the quantified version of Anderson and Belnap's system E of entailment,
extended by the mingle axiom and the Ackermann constant t.
The incompleteness proof involves an algebraic semantics based on
atomless complete Boolean algebras.