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Positive modal logic. (English) Zbl 0831.03007

Summary: We give a set of postulates for the minimal normal modal logic \({\mathbf K}_+\) without negation or any kind of implication. The connectives are simply \(\wedge, \vee\), \(\square\), \(\lozenge\). The postulates (and theorems) are all deducibility statements \(\varphi \vdash \psi\). The only postulates that might not be obvious are: \(\lozenge \varphi \wedge \square \psi \vdash \lozenge (\varphi \wedge \psi), \quad \square(\varphi \vee \psi) \vdash \square \varphi \vee \lozenge \psi\).
It is shown that \({\mathbf K}_+\) is complete with respect to the usual Kripke-style semantics. The proof is by way of a Henkin-style construction, with “possible worlds” being taken to be prime theories. The construction has the somewhat unusual feature of using at an intermediate stage disjoint pairs consisting of a theory and a “counter- theory”, the counter-theory replacing the role of negation in the standard construction. Extension to other modal logics is discussed, as well as a representation theorem for the corresponding modal algebras. We also discuss proof-theoretic arguments.

MSC:

03B45 Modal logic (including the logic of norms)
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