Effective completeness theorems for modal logic

MISCELLANEA LOGICA

Lecture Notes in Computer Science, 2013

A new semantics with the finite model property is provided and used to establish decidability for Gödel modal logics based on (crisp or fuzzy) Kripke frames combined locally with Gödel logic. A similar methodology is also used to establish decidability, and indeed co-NPcompleteness for a Gödel S5 logic that coincides with the one-variable fragment of first-order Gödel logic.

Lecture Notes in Computer Science, 2021

This paper presents a recent formalization of a Henkin-style completeness proof for the propositional modal logic S5 using the Lean theorem prover [5]. The proof formalized is close to that of Hughes and Cresswell [11], but the system, based on a different choice of axioms, is better described as a Mendelson system augmented with axiom schemes for K, T, S4, and B, and the necessitation rule as a rule of inference. The language has the false and implication as the only primitive logical connectives and necessity as the only primitive modal operator. The full source code is available online at https://github.com/bbentzen/mpl/ and has been typechecked with Lean 3.4.2. * The author wishes to thank Jeremy Avigad, Mario Carneiro, Rajeev Goré, and Minchao Wu for helpful suggestions and invaluable conversations on the topic covered herein.

Annals of Pure and Applied Logic, 1990

Russian Mathematical Surveys, 2016

Journal of Computer and System Sciences, 2017

Decidability of the validity problem is established for a family of many-valued modal logics, notably Gödel modal logics, where propositional connectives are evaluated according to the order of values in a complete sublattice of the real unit interval [0, 1], and box and diamond modalities are evaluated as infima and suprema over (many-valued) Kripke frames. If the sublattice is infinite and the language is sufficiently expressive, then the standard semantics for such a logic lacks the finite model property. It is shown here, however, that, given certain regularity conditions, the finite model property holds for a new semantics for the logic, providing a basis for establishing decidability and PSPACE-completeness. Similar results are also established for S5 logics that coincide with one-variable fragments of first-order many-valued logics. In particular, a first proof is given of $ Preliminary results from this work were reported in the proceedings of TACL 2013 (as an extended abstract) and WoLLIC 2013 [8].

Studia Logica - An International Journal for Symbolic Logic, 2006

The paper focuses on extending to the first order case the semantical program for modalities first introduced by Dana Scott and Richard Montague. We focus on the study of neighborhood frames with constant domains and we offer a series of new completeness results for salient classical systems of first order modal logic. Among other results we show that it is possible to prove strong completeness results for normal systems without the Barcan Formula (like FOL + K) in terms of neighborhood frames with constant domains. The first order models we present permit the study of many epistemic modalities recently proposed in computer science as well as the development of adequate models for monadic operators of high probability. Models of this type are either difficult of impossible to build in terms of relational Kripkean semantics.

A set F of formulas is complete relative to a given class of logics, if every logic from this class can be axiomatized by formulas from F . A set of formulas F is L-complete relative to a given class of logics, if every logic of this class can be L-axiomatized by formulas from F , that is, every of these logics can be defined by an L-deductive system with axioms and anti-axioms from F and inference rules modus ponens, modus tollens, substitution and reverse substitution. We prove that every complete relative to ExtInt (or ExtK4) set of formulas is L-complete. In particular, every logic from ExtInt (or ExtK4) can be L-axiomatized by Zakharyaschev's canonical formulas.

arXiv (Cornell University), 2018

Given a class C of models, a binary relation R between models, and a model-theoretic language L, we consider the modal logic and the modal algebra of the theory of C in L where the modal operator is interpreted via R. We discuss how modal theories of C and R depend on the model-theoretic language, their Kripke completeness, and expressibility of the modality inside L. We calculate such theories for the submodel and the quotient relations. We prove a downward Löwenheim-Skolem theorem for first-order language expanded with the modal operator for the extension relation between models.

Bulletin of the Section of Logic, 2021

This paper investigates a first-order extension of GL called \(\textup{ML}^3\). We outline briefly the history that led to \(\textup{ML}^3\), its key properties and some of its toolbox: the \emph{conservation theorem}, its cut-free Gentzenisation, the ``formulators'' tool. Its semantic completeness (with respect to finite reverse well-founded Kripke models) is fully stated in the current paper and the proof is retold here. Applying the Solovay technique to those models the present paper establishes its main result, namely, that \(\textup{ML}^3\) is arithmetically complete. As expanded below, \(\textup{ML}^3\) is a first-order modal logic that along with its built-in ability to simulate general classical first-order provability―"\(\Box\)" simulating the the informal classical "\(\vdash\)"―is also arithmetically complete in the Solovay sense.