Deciding Modal Logics through Relational Translations into GF2 (original) (raw)

First-order classical modal logic

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.

Alternative Translation Techniques for Propositional and First-Order Modal Logics

2002

We describe and analyze techniques, other than the standard relational/functional methods, for translating validity problems of modal logics into first-order languages. For propositional modal logics we summarize the ✷-as-Pow method, a complete and automatic translation into a weak set theory, and then describe an alternative method, which we call algebraic, that achieves the same full generality of ✷-as-Pow but is simpler and computationally more attractive. We also discuss the relationships between the two methods, showing that ✷-as-Pow generalizes to the first-order case. For first-order modal logics, we describe two extensions, of different degrees of generality, of ✷-as-Pow to logics of rigid designators and constant domains.

Modal Deduction in Second-Order Logic and Set Theory - II

Studia Logica - An International Journal for Symbolic Logic, 1998

In this paper, we generalize the set-theoretic translation method for poly-modal logic introduced in [11] to extended modal logics. Instead of devising an ad-hoc translation for each logic, we develop a general framework within which a number of extended modal logics can be dealt with. We first extend the basic set-theoretic translation method to weak monadic second-order logic through a suitable change in the underlying set theory that connects up in interesting ways with constructibility; then, we show how to tailor such a translation to work with specific cases of extended modal logics.

Products of modal logics. Part 2: relativised quantifiers in classical logic

Logic Journal of IGPL, 2000

In the first part of this paper we introduced products of modal logics and proved basic results on their axiomatisability and the f.m.p. In this continuation paper we prove a stronger result-the product f.m.p. holds for products of modal logics in which some of the modalities are reflexive or serial. This theorem is applied in classical first-order logic; we identify a new Square Fragment (SF) of the classical logic, where the basic predicates are binary and all quantifiers are relativised, and for which we show the f.m.p. in the classical sense. Also we prove that SF not included in Guarded Fragment (and in Packed Fragment) and that it can be embedded into the equational theory of relation algebras. 1

Bundled fragments of first-order modal logic: (un)decidability

2018

Quantified modal logic provides a natural logical language for reasoning about modal attitudes even while retaining the richness of quantification for referring to predicates over domains. But then most fragments of the logic are undecidable, over many model classes. Over the years, only a few fragments (such as the monodic) have been shown to be decidable. In this paper, we study fragments that bundle quantifiers and modalities together, inspired by earlier work on epistemic logics of know-how/why/what. As always with quantified modal logics, it makes a significant difference whether the domain stays the same across worlds, or not. In particular, we show that the bundle forallBox\forall \BoxforallBox is undecidable over constant domain interpretations, even with only monadic predicates, whereas existsBox\exists \BoxexistsBox bundle is decidable. On the other hand, over increasing domain interpretations, we get decidability with both forallBox\forall \BoxforallBox and existsBox\exists \BoxexistsBox bundles with unrestricted predicates. In these ...

Modal Deduction in Second-Order Logic and Set Theory - I

Journal of Logic and Computation, 1997

We investigate modal deduction through translation into standard logic and set theory. Derivability in the minimal modal logic is captured precisely by translation into a weak, computationally attractive set theory . This approach is shown equivalent to working with standard rst-order translations of modal formulas in a theory of general frames. Next, deduction in a more powerful second-order logic of general frames is shown equivalent with set-theoretic derivability in an`admissible variant' of . Our methods are mainly model-theoretic and set-theoretic, and they admit extension to richer languages than that of basic modal logic.

Canonicity and Completeness Results for Many-Valued Modal Logics

Journal of Applied Non-classical Logics, 2002

We prove frame determination results for the family of many-valued modal logics introduced by M. Fitting in the early '90s. Each modal language of this family is based on a Heyting algebra, which serves as the space of truth values, and is interpreted on an interesting version of possible-worlds semantics: the modal frames are directed graphs whose edges are labelled with an element of the underlying Heyting algebra. We introduce interesting generalized forms of the classical axioms D, T, B, 4, and 5 and prove that they are canonical for certain algebraic frame properties, which generalize seriality, reflexivity, symmetry, transitivity and euclideanness. Our results are quite general as they hold for any modal language built on a complete Heyting algebra.