Completeness and incompleteness for anodic modal logics

Journal of Applied Non-Classical Logics 19 (3):291-310 (2009)
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Abstract

We propose a new approach to positive modal logics, hereby called anodic modal logics. Our treatment is completely positive since the language has neither negation nor any falsum or minimal particle. The elimination of the minimal particle of the language requires introducing the new concept of factual sets and factual deductions which permit us to talk about deductions in the actual world. We start from a positive fragment of the standard system K, denoted by K⊃, ∧, ◊, which is a bimodal system with □ and ◊ as primitive. This system is then extended to a class of fragments of the Lemmon and Scott systems (cf. (Lemmon et al., 1977)), denoted by K⊃, ∧, ◊ + Gk;l;m;n + Gm;n;k;l. It is shown that such classes of systems are characterized 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 as introduced in the modal context by J. M. Dunn in (Dunn, 1995). We also obtain a surprising limiting result showing that the incompleteness phenomenon in modal logic is independent of negation.

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References found in this work

Positive modal logic.J. Michael Dunn - 1995 - Studia Logica 55 (2):301 - 317.
An Introduction to Modal Logic.E. J. Lemmon, Dana Scott & Krister Segerberg - 1979 - Journal of Symbolic Logic 44 (4):653-654.
A New Semantics for Positive Modal Logic.S. Celani & R. Jansana - 1997 - Notre Dame Journal of Formal Logic 38 (1):1-18.
Fragments of the propositional calculus.Leon Henkin - 1949 - Journal of Symbolic Logic 14 (1):42-48.

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