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Interpolation and preservation in \({\mathcal{ML}}_{\omega_1}\). (English) Zbl 0968.03026
Summary: We deal with the logic \({\mathcal M}{\mathcal L}_{\omega_1}\) which is the infinitary extension of propositional modal logic that has conjunctions and disjunctions only for countable sets of formulas. After introducing some basic concepts and tools from modal logic, we modify Makkai’s generalization of the notion of consistency property to make it fit for modal purposes. Using this construction as a universal instrument, we prove, among other things, interpolation for \({\mathcal M}{\mathcal L}_{\omega_1}\) as well as preservation results for universal, existential, and positive \({\mathcal M}{\mathcal L}_{\omega_1}\)-formulas.
MSC:
03B45 Modal logic (including the logic of norms)
03C75 Other infinitary logic
03C40 Interpolation, preservation, definability
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