The syntax of these logics extends that of previously developed modal languages for polynomial coalgebras by adding formulas that express the “termination” of certain functions induced by transition paths. A completeness theorem is proven for the logic of functors which have the Lindenbaum property that every consistent set of formulas has a maximal extension. This property is shown to hold if the deducibility relation is generated by countably many inference rules.
A counter-example to completeness is also given. This is a polynomial functor that is not Lindenbaum: it has an uncountable set of formulas that is deductively consistent but has no maximal extension and is unsatisfiable, even though all of its countable subsets are satisfiable. 相似文献