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Directed frames. (English) Zbl 0689.03009

A directed frame is a frame \(<W,\leq >\), where the relation \(\leq\) on W is reflexive and transitive and for any \(x,y\in W\) there is a z such that \(x\leq z\) and \(y\leq z\). Directed frames frequently arise in mathematical practice and are associated with the logics which are stronger than intuitionistic logic because the principles of the weak excluded middle are provable in them. In this paper some questions related to their axiomatizability by first order calculi are investigated; in particular, it is shown that the class of directed frames with nested domains, and the class of directed frames with maximum element (and either nested or constant domains) are axiomatized both by first order intermediate calculi and by modal calculi. Furthermore, the class of directed frames with maximum cluster and either constant or nested domains is axiomatized by first order modal calculi.
Reviewer: Li Xiang

MSC:

03B45 Modal logic (including the logic of norms)
03B55 Intermediate logics
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References:

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