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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References:

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