Abstract
We give a set of postulates for the minimal normal modal logicK + without negation or any kind of implication. The connectives are simply ∧, ∨, □, ◊. The postulates (and theorems) are all deducibility statements ϕ ⊢ ψ. The only postulates that might not be obvious are
.
It is shown thatK + 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.
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I wish to thank David McCarty for a most helpful conversation, in which he suggested the key idea that I try a construction in which certain sentences are “kept out”. I also want to acknowledge the helpful remarks of three anonymous referees and wish to thank Steve Crowley for a careful reading.
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Dunn, J.M. Positive modal logic. Stud Logica 55, 301–317 (1995). https://doi.org/10.1007/BF01061239
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DOI: https://doi.org/10.1007/BF01061239